Statistical guide to data analysis of avian monitoring programs

              Statistical Guide to

Data Analysis of Avian

Monitoring Programs

Biological Technical Publication

BTP-R6001-1999

U.S. Fish & Wildlife Service

Statistical Guide to

Data Analysis of Avian

Monitoring Programs

Biological Technical Publication

BTP-R6001-1999

Nadav Nur

Point Reyes Bird Observatory, Stinson Beach, CA 94970

Stephanie L. Jones

U.S. Fish & Wildlife Service, Mountain-Prairie Region, Denver, CO 80225

Geoffrey R. Geupel

Point Reyes Bird Observatory, Stinson Beach, CA 94970

U.S. Fish & Wildlife Service

Authors

Nadav Nur

Point Reyes Bird Observatory

4990 Shoreline Hwy.

Stinson Beach, CA 94970-9701

415/868 1221

email: NadavNur@prbo.org

Stephanie L. Jones

Nongame Migratory Bird Coordinator

U.S. Fish & Wildlife Service, Mountain-Prairie Region

P.O. Box 25486 DFC

Denver, CO 80225

303/236 8145 ext. 608

email: Stephanie_Jones@fws.gov

Geoff Geupel

Point Reyes Bird Observatory

4990 Shoreline Hwy.

Stinson Beach, CA 94970-9701

415/868 1221

email: GGeupel@prbo.org

Suggested citation

Nur, N., S.L. Jones, and G.R. Geupel. 1999. A statistical

guide to data analysis of avian monitoring programs.

U.S. Department of the Interior, Fish and Wildlife

Service, BTP-R6001-1999, Washington, D.C.

ii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Chapter I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Recommended Monitoring Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Methods for Assessing Abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Demographic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Statistical Terminology and Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

General Considerations of Study Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Analysis of Vegetation and Habitat Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter II. Assessment of Abundance and Species Composition Using Point Counts . . . . . . . . . . . . . . . . . . . . . . 8

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Community Similarity Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Analyzing Vegetation Data in Relation to Point Count Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Power and Sample Size Analysis Using TRENDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Using MONITOR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Power and Sample-Size Analyses: Other Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter III. Demographic Monitoring: Mist-nets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Analysis of Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Analysis of Adult Survival. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chapter IV. Demographic Monitoring: Nest-monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Additional Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Alternatives to the Mayfield Method: Systematic Searching and Time-to-Failure Analysis . . . . . . . . . . . . . . 34

Vegetation Analysis in Relation to Nest-monitoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Chapter V. Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Chapter VI. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

iii

Table of Contents

iv

Tables

1. Monitoring methods used in landbird population monitoring and their characteristics. . . . . . . . . . . . . . . . . 2

2. Potential objectives of a monitoring program and typical number of years needed for a method

to achieve results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3. Example of data from point count observations conducted at three point count stations, three times

during the breeding season. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4. Calculation of diversity, similarity and evenness indices using total bird detections across sites in

burned and unburned aspen (Populus tremuloides) stands in Wyoming (from Dieni 1996). . . . . . . . . . . . . . 12

5. Linear regression analysis of number of Black-headed Grosbeaks during the breeding season. . . . . . . . . 14

6. Sample output for linear regression analyses using STATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7. Analysis of point count data on Sacramento River: relationship of bird species richness to

Damage Index, controlling for vegetation/habitat characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8. Analysis of mist-net captures, Sacramento River 1993: relationship to Damage Index for the

six species with adequate sample size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9. Analysis of mist-net captures, Sacramento River, 1993: relationship of HY, and proportion HY birds

caught in relation to Vegetation Damage Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10. Evaluation and summary of available computer program software used for the analysis of animal

marking and surveying studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11. Results of SURGE analysis of Wrentits, by territory status. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12. Summary of models in JOLLY and JOLLYAGE (Pollack et al. 1990). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

13. Power analysis for detecting differences in survivorship between two groups. . . . . . . . . . . . . . . . . . . . . . . 36

14. Logistic regression analyses of Grasshopper Sparrow presence/absence in relation to habitat

features (from Holmes and Geupel 1998). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Figures

1A. Trend, log-linear, P = 0.001, Black-headed Grosbeak, Palomarin 1980-1992. . . . . . . . . . . . . . . . . . . . . . . . 13

1B. Trend, linear-no transformation, P = 0.004, Black-headed Grosbeak, Palomarin 1980-1992. . . . . . . . . . . 13

2A. Normal probability plot, residuals of log-transformed data, Black-headed Grosbeak. . . . . . . . . . . . . . . . 16

2B. Normal probability plot, residuals of untransformed data, Black-headed Grosbeak. . . . . . . . . . . . . . . . . 16

3. Bird species richness in relation to Vegetation Damage Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4A. Distribution of residuals: species richness vs. Vegetation Damage Index. . . . . . . . . . . . . . . . . . . . . . . . . . 19

4B. Quantile-quantile plot of residuals of species richness vs. Vegetation Damage Index against

normal distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5. Probability of detecting Grasshopper Sparrows in relation to Index of Perennial Grass Cover. . . . . . . . . 40

List of Tables and Figures

v

This Statistical Guide is intended to aid field

biologists wishing to analyze data gathered in

standardized monitoring programs for landbirds. It

grew out of the needs expressed by the Western

Working Group of Partners in Flight, and we thank

the members of that group for providing the

incentive to develop this document. It is not

intended to replace good statistical texts, but to

supplement them. We encourage readers, and

especially users, of this Guide to forward their

comments, corrections, and other advice to the

senior author for incorporation into future versions

of this Guide.

This work has been a contract between Point Reyes

Bird Observatory and the U.S. Fish & Wildlife

Service. This is PRBO Contribution 679.

References to commercial products does not imply

endorsement.

Acknowledgments

We thank John R. Sauer, J. Scott Dieni, Ken Gerow,

Daniel R. Petit, and Jon Bart for multiple reviews of

earlier drafts; John Cornely, Barry Noon, Kathie

Purcell, C.J. Ralph, Len Thomas, and Jerry Verner

also provided helpful discussion and comments on an

earlier draft of this document. The authors, not the

above named reviewers, should be held responsible

for any errors or outlandish opinions expressed

here. We thank Jim Nichols for providing a helpful

preprint. We thank the USFWS Nongame

Coordinators: Tara Zimmerman, Bill Howe, Steve

Lewis, Diane Pence, Richard Coon, Kent Wohl,

together with Dan Petit and John Trapp, for support

and encouragement. Special thanks to all the field

biologists who took the time to assist us in doing this

document and are out there doing the work, facing

the challenges, and balancing the issues: Adrianna

Araya, Grant Ballard, Sharon Browder, Mike

Bryant, Claire Caldes, Lynn Clark, Paula Gouse,

Ron Garcia, Todd Grant, Bill Haglan, Jeanne

Hammond, Laura Hubers, Craig Hultberg, Beth

Madden, Steve Martin, Bob Murphy, Lark Osborne,

Fritz Prellwitz, Pam Rizor, Vickie Roy, Kelli Stone,

Julian Wood, Kodiak and McDougall Jones and

many more.

Preface

This Guide is intended to provide guidance to field

biologists wishing to analyze data collected on

terrestrial bird populations, as part of an avian

population monitoring program. A second objective

is to provide information that will help biologists

design such programs. The audience is similar to

that for the Handbook of Field Methods (Ralph et

al. 1993), the Monitoring Bird Populations by Point

Counts (Ralph et al. 1995), and in many ways this

Statistical Guide to Data Analysis of Avian

Monitoring Programs can be a useful complement

to the field methods handbook. At the same time,

we feel this Statistical Guide can be of use to field

biologists studying other organisms besides

terrestrial birds. In our view, all field biologists

will benefit from taking the equivalent of 2 or 3

semester courses in statistics and we assume that

readers of this guide have completed at least this

basic level in statistics.

This document is not intended to fill deficiencies in

basic knowledge of statistics, nor is it a substitute

for a good statistical text. Rather, this Guide is

intended as a supplement to these texts. Our aim is

to provide practical advice in the design and analysis

of field ecological data and to provide timely

information about current statistical computer

programs. Two good statistical texts are provided by

Neter et al. (1990) and Kleinbaum et al. (1988). Both

of these texts are “intermediate” in level; that is,

they assume the reader has had a basic,

introductory course in statistics. Other texts by

Snedecor & Cochran (1989), Sokal & Rohlf (1995)

and Zar (1996) all provide a good, general statistical

background. Intermediate level guides for

practicing ecologists are provided by Crawley (1993),

Bart and Notz (1996) and Bart et al. (1998).

Noteworthy specialized statistical ecological texts

include Ludwig & Reynolds (1988), Skalski &

Robson (1992), and Draper & Smith (1981). The last

two mentioned have many biological examples. Also

see the informative review by Lancia et al. (1996).

Computer Programs

Computer programs for summarizing and analyzing

data with general statistical packages are available,

for many different levels, prices and target

audiences. Ellison (1992) reviewed a number of

general statistical packages, but that review is

somewhat out of date. One versatile statistical and

graphical package, available for DOS, Windows,

and UNIX platforms, is Stata (StataCorp. 1999)

(obtained from Stata Corporation, 702 University

Drive East, College Station, TX 77840). Specialized

computer software programs have been created to

assist with analysis of capture/recapture data (used

for analyses of survivorship, also population size);

these are reviewed and summarized in this and

additional specialized computer programs are

mentioned in the respective sections of this Guide.

Recommended Monitoring Methods

A wide range of methods have been used to conduct

avian monitoring, each tailored to meet a different

set of objectives in the face of different constraints.

This Guide does not address all methods that are

available, especially those that are more widely used

for research or inventory. Below is a short review of

monitoring methods available, based on Butcher

(1992) and Ralph et al. (1993). The reader is referred

to these references (and others cited below) for

additional information. Table 1 describes the

variables measured and subjectively assesses the

relative strengths and weaknesses of each method.

“Strength” and “weakness” is assessed relative to

the quality of the data gathered to meet the

objective and we have not attempted to factor in cost

per datum. Table 2 provides a list of monitoring

objectives, monitoring methods and the typical time

required by the various methods to achieve those

objectives (from Geupel & Warkentin 1995).

Descriptions of monitoring methods, their

applications and comparisons, and their limitations

can be found in Ralph and Scott (1981), Verner

(1985), Butcher (1992), Ralph et al. (1993), Buckland

et al. (1993) and Geupel & Warkentin (1995).

Methods

Area search—A method in which observers are

allowed to roam for a fixed time in a specified area,

usually 20 minutes per 3 hectare area (Loyn 1986,

Slater 1994). This technique has a wide appeal to

volunteers but standardization of data collection is

difficult.

1

I. Introduction

2 Statistical Guide to Data Analysis of Avian Monitoring Programs

Table 1. Monitoring methods used in landbird population monitoring and their characteristics.

Methods are grouped under “survey” and “demographic.” Positive or high level is denoted by “+”,

negative or low level denoted by “–” and partial level denoted by “+/–“. Modified from Table 1 in Butcher

(1992). “Color banding” is assumed to include nest-searching. “Rare” species refers to species that are

locally (not just globally) rare.

Survey Demographic

Fixed Spot Area Variable Mist Nest Color

Variables Measured distance map Search distance net Search banding

Index to abundance + + + + +/– +/– +

Density – + – + – – +

Survivorship (adult) – – – – + – ++

Productivity – – – – + + +

Recruitment – – – – + – +

Habitat Relations + + + + +/– + +/–

Nest Site Characteristics – – – – – + +

Predation/Parasitism – – – – – + +

Individuals Identified – – – – + – +

Breeding Status Known – + – – +/– + +

General Characteristics

Habitat specificity + + + + +/– + +

Rare species measured + +/– + +/– – +/– +/–

Canopy species measured + + + + – +/– –

Area sampled known + + + + +/– + +

Large area sampled + – + + +/– – –

Use in non-breeding season + +/– + + + – +

Table 2. Potential objectives of a monitoring program and typical number of years needed for a method to

achieve results.

Actual number of years depends on study design and will vary depending on sample size (e.g., number of

census stations, detection or capture rates, number of nests found). We assume that the priorities of the

monitoring program reflect local or site-specific needs (adapted from Geupel & Warkentin 1995).

Method

Single Point Repeat Area Spot Mist Nest

Objective Countsa Pt. Countsb Searchc mapping nettingd monitoringd

Inventory, species presence/absence 1 1 1 1 1 na

Inventory locally rare species 2-3 1-3 1-3 1-3 1-3 na

Determine species richness 2-3 1-3 1-3 1-3 na na

Determine relative abundance 1-2 1-2 1-3 1-2 3-5 na

Determine species breeding status/seasonality na 1-3 1-3 1-3 1-3 1-3

Determine population trend 6-10 5-9 10+ 5-9 6-10 na

Determine productivity na na na na 1-3 1-2

Determine adult survivorship na na na 3-5e 3-5 na

Determine life history traits na na na 2-4 na 1-2

Habitat association or preference 1-2 1-2 1-2 1-3 na 1-2

Identify habitat features 4-6 3-5 3-5 2-4 na 1-2

Determine cause of pop. change na na na na 3+ 3+

a Each point count censused one time in a season.

b Each point count censused 3 or more times in a season.

c Each plot censused 3 or more times in a season.

d Most authors/programs recommend this method in conjunction with population surveys.

e Possible if birds have been uniquely color-banded.

na Not applicable or not possible.

Methods for Assessing Abundance

Point counts—Fixed radius point counts are the

basic method recommended for most monitoring

studies, and are most widely used (Hutto et al. 1986,

Ralph et al. 1993, Ralph et al. 1995). These can

provide a cost-effective method of estimating the

relative abundance of birds.

Line transects—Fixed-width transects can provide

coverage of a greater area than point counts, but

with fewer independent data points or replicates.

Variable distance methods—Estimating distance at

which birds are detected can be incorporated into

both point count and line transect surveys.

Standardization of distance estimation may be

difficult, as abilities to accurately estimate distances

may vary greatly between observers.

Spot-mapping—Can provide good density

information and information on many aspects of

avian life history. It is expensive per data point and

may be better applied to research projects or to high

priority areas or species.

Demographic Methods

In general, demographic monitoring methods can be

used to identify proximal causes of population

declines and provide insight into causes of habitat

associations. They can identify population problems

prior to the detection of declines based on

abundance surveys. Ultimately, these methods can

be used to identify “source” or “sink” populations.

However, these methods require much effort per

station.

Constant effort mist-netting—Provides information

on productivity and survivorship of populations,

but is limited by area covered (which is generally

unknown) and lack of habitat specificity. However,

many species can be monitored at the same time,

without expending extra effort.

Nest monitoring—Provides site-specific and

habitat-specific information on productivity and

reproductive status. Available personnel usually

limit the number of plots that can be studied, and

studying additional species normally requires

increased effort.

Color-banding—When combined with nest

monitoring, using unique color-band combinations to

follow the fates of individuals will provide the most

complete and unbiased measures of demographic

parameters. However, it is the most intensive

method of all. It is not a method recommended for

general monitoring, but like spot-mapping, best

suited for research projects or for high priority

areas and species.

Statistical Terminology and Principles

The following is a selective review of some statistical

terms relevant to a biologist conducting a

monitoring study. Our intention here is to re-acquaint

the reader with terms and principles that

may have rested dormant for many years.

Accuracy—An estimator is accurate if it produces

estimates that are, on average, close to the true

value, i.e., without bias or with a minimum of bias.

Accuracy is independent of precision (below). An

estimate can be accurate but not precise, precise but

not accurate, or both accurate and precise. The

difficulty is that often the “true” value is unknown

and therefore accuracy is difficult to judge, except

for simulated data where an investigator knows the

true values.

Bias—The difference between the average estimate

(more precisely, the expected value of the estimate)

and the true value. Bias is not the same as “error”,

rather it is one kind of error, systematic error. If an

estimate is as likely to be an overestimate as it is to

be an underestimate, the estimator in question is

unbiased, even though there will always be error

associated with an estimate. To minimize bias would,

by definition, maximize accuracy.

Precision—Precision refers to the variability of the

estimate: the smaller the variability (and thus the

smaller the standard error) of the estimate, the

greater the precision. As mentioned above, precision

is independent of accuracy. An estimate can be very

precise, but wildly inaccurate (i.e., strongly biased).

Type I and Type II errors—Rejecting the null

hypothesis when it is correct is committing a Type I

error. The probability of committing a Type I error

is symbolized α [alpha] and is the significance level

of a test of statistical inference. Accepting the null

hypothesis when it is incorrect is committing a Type

II error; the probability of making such an error is

symbolized ß [beta].

Power—The probability of detecting a biological

effect, if there is one. More precisely, power is the

probability of rejecting the null hypothesis when the

null hypothesis is incorrect. Normally, the null

hypothesis is an hypothesis of no effect (i.e., no

difference). Power is equal to 1–ß. Power cannot be

calculated unless one specifies the alternative

hypothesis: one must specify the magnitude of the

effect or difference. A given test will have greater

power the greater the magnitude of the effect, and

conversely, the smaller the true difference between

groups, the less the power to detect that difference

for a given sample size. Power is discussed in

greater depth in Chapter II of this Guide.

Introduction 3

Poisson distribution—Among several discrete

distributions (binomial, geometric, negative

binomial), this distribution is one of the most likely

to be encountered or utilized in ecological studies

(Ludwig and Reynolds 1988). Many random

processes, in which events occur independently of

each other, in space or time, conform to a Poisson

distribution. Suppose one set up a grid of 100 one-cm

squares (10 cm × 10 cm). The number of rain drops

falling per square in a short interval of time is likely

to be Poisson-distributed. Suppose that in one

minute, 100 rain drops fell on the 100 squares. If this

process was indeed a Poisson process, then we would

expect that in 1minute, on average, 37 squares

would receive 0 drops, 37 squares would receive 1

drop, 18 squares would receive 2 drops, and 8

squares would receive 3 or more drops. For a

Poisson process, the mean of occurrences (per unit

time or space, in the rain drop example = 1.0 drops

per square per minute) will equal the variance of

those occurrences. Thus, for a Poisson-distributed

variable, only one parameter is specified.

Another useful distribution is the binomial

distribution. If N independent trials were conducted

and the probability of a “hit” (representing success,

failure, death, etc.) on any one trial is p, then the

number of hits in total is binomially distributed, with

mean = Np, and Variance = Np(1–p). As a result,

variance is neither independent of the mean (as it is

in the normal distribution) nor is it equal to the

mean (as it is in the Poisson distribution); moreover,

variance is maximized when p =1–p = 0.5. As p

approaches 0 or 1, the variance will shrink to zero.

The binomial distribution is, for example, utilized in

logistic regression (Chapter V). Note that the

binomial distribution has two parameters, N and p.

When the number of trials is large and the

probability of a “hit”, p, is low, then the binomial

distribution can be approximated by the Poisson

distribution.

Replicates—Replicates are independent repetitions

or measurements within the experimental design. If

repetitions are not independent then these “repeats”

are sometimes referred to as pseudoreplicates

(Hurlbert 1984, Bart et al. 1998). Suppose 100 point

count stations in a given habitat type have been

surveyed three separate times during the breeding

season. The 300 data points obtained should not be

treated as 300 replicates or samples, because bird

data obtained on different days in the same season

are not independent. Whether or not the 100 point

count stations are independent or not is difficult to

say a priori, but if spaced far enough apart (Ralph

et al. 1995 recommend spacing of at least 250 m), so

that the same individuals are not being counted at

different stations, the 100 point count stations can be

treated as independent. Assuming independence

among adjacent point count stations, and if the 100

point count stations were divided evenly among 4

habitats, then there would be 25 replicates.

As far as the three repeats per point count station

are concerned, one can average the data, select the

repeat with the highest score for each individual

species, or sum the data from each of the three

visits. If one wished to compare results among the

three visits (e.g., asking whether there was a

seasonal, within-year trend), one can analyze the 300

observations, using “point count station” as a

categorical variable to be controlled for; this is an

example of a repeated-measures design, in which

“point count station” is a blocking variable.

Independence of observations—This is an important

issue in statistical analysis, and is often

misunderstood. To start, what is required are that

outcomes be independent from one observation to

another, after controlling for factors or variables

that might be influencing the outcome. Suppose the

point count stations have been spaced 100 m apart

on transects of 1 km length. An investigator might

not feel comfortable in treating observations from

different stations on the same transect as being

independent of each other. One solution would be to

classify the transect as the unit of observation, i.e.,

pooling data from all point count stations on the

same transect, and analyze data accordingly.

Another solution would be to include in the analysis

a “transect effect.” This would control for the fact

that stations on the same transect are more likely to

be similar in outcome than are stations on different

transects. In this way one can investigate

differences among and within transects.

A second point is that the independence refers to the

outcome, not the independent variables or factors.

Suppose one related bird species richness to

vegetation. As long as bird species richness varies

independently from station to station (after

controlling for various factors), it would not matter

that all stations on a transect shared some of the

same vegetation characteristics. In other words,

there is no requirement that vegetation

characteristics be independent from one observation

unit to another.

General Considerations of Study Design

General study design considerations will apply to

most monitoring techniques and studies. Neter et al.

(1990) provides a good discussion of experimental

design, also see Skalski & Robson (1992) and

Crawley (1993); those wishing more detail can

consult specialized texts such as Hicks (1982). A

helpful and interesting discussion of the issues and

the process for designing an avian monitoring study

on one site such as a National Wildlife Refuge is

4 Statistical Guide to Data Analysis of Avian Monitoring Programs

given in Johnson (In Press). In this section, we

discuss some general points concerning design of a

study. Later when discussing each methodology in

turn (point counts, mist-netting and nest-monitoring),

we return to questions of design.

Throughout this Guide, the use of “station” refers to

one independent monitoring site, e.g., one point

count station (if observations are deemed

independent of other stations), one line transect, one

mist-netting array, one nest-monitoring plot, etc. It

is important to correctly determine the unit of

analysis early in the study design.

Design—The first and most important

consideration in designing a study is its objectives.

Statistical inference (in particular, tests of

statistical significance) may be of little interest, in

which case statistical power need not be considered

in determining the sample size needed. A biologist

may instead wish to monitor a particular area

mainly as a descriptive tool. If data are gathered

in a standardized fashion (Ralph et al. 1993), the

data from one area can contribute to regional or

national monitoring programs, which likely have

statistical inference as an objective. In many cases

the number of stations will be limited by available

resources or by the physical areas of interest.

Some field biologists will be able to establish one,

or at most, a couple of demographic monitoring

stations (e.g., one mist-net array or one

nest-monitoring plot). In those cases placement

of the station will usually be constrained by the

location and size of the habitat of interest, by the

density of the species of special concern, or be

centered on the location of the habitat or species

of interest.

Data from just a single demographic monitoring

station may be valuable for several reasons:

1. the data provide a description of temporal

patterns, which data can be combined with other

sources of data, 2. the data can allow statistical tests

of trends over time, given sufficient number of years

of data collection (possibly 10 years or more for a

single station), and 3. the data can be combined with

data from other monitoring stations.

Not every monitoring program needs to have

hypothesis testing as its goal from the outset. A

monitoring program may be able to collect valuable

data that can later be analyzed (by itself or as part

of a larger study), and that analysis would surely

include hypothesis testing and tests of statistical

significance. But it is pointless to erect contrived

hypotheses before data collection has begun, simply

in order to justify the establishment of a monitoring

program. After data have been collected, the

investigator will have a much better idea of how to

formulate meaningful hypotheses. This point does

not apply to experimental studies, where explicit

hypothesis formulation is an essential ingredient to

a successful study.

Assuming statistical inference is an important

consideration, one needs to determine whether the

objective is to determine trends through time,

establish bird-habitat relationships, compare effects

of different treatments, or other possible objective.

Choice of objective will influence questions of sample

size and allocation of stations (see Randomization

below).

Assuming that statistical inference is a goal, the

question of necessary sample size needs to be

related to statistical power, i.e., the ability to detect

an effect if there is one. Statistical power is an

elusive concept in part because it is arbitrary.

Calculations of sample size in the past have used

power values ranging from 50% to 95%. Clearly, the

greater the desired power, the greater the sample

size necessary to achieve that power. Generally, this

Guide uses values of 50% and 80%. In designing a

study one would not ordinarily consider 50% power

to be adequate and we do not recommend a study be

designed to achieve 50% power. Nevertheless 50%

power presents a useful level for a posteriori

investigations, where someone has already collected

these data and the biologist wishes to consider the

statistical power of the data to detect effects of

interest. Conversely, in designing a study, 80%

power is a commonly used and often-recommended

benchmark, but it is nothing more than a

benchmark.

Power calculations and sample size calculations

both rely on the presumed magnitude of the effect

in question. Clearly, the greater the presumed

effect (e.g., the greater the difference between

the two groups), the greater the power will be to

detect that effect, and, conversely, the smaller the

necessary sample size to detect an effect at a

specified power. The difficulty here is that the true

difference between groups is unknown, and

furthermore one cannot necessarily use the

observed magnitude of an effect (e.g., observed

difference between two groups) as the criterion

for judging power.

It is easy to fall into the trap of estimating power,

retrospectively, using the observed magnitude of an

effect, and several general statistical packages

appear to encourage users to do so, without

appropriate warnings (discussed in Thomas and

Krebs 1997). The problem is that if a statistically

significant effect is found, one would not normally

calculate power retrospectively. If the investigator

looks for an effect and finds there is one, then there

is little need to determine the probability of having

Introduction 5

found that effect. Therefore, retrospective power

calculations are usually pursued only when no

significant effect is detected. But given that no effect

was detected (statistically), it could be because the

observed magnitude of an effect was substantial, but

power was weak, or because the observed

magnitude of an effect was small, even negligible.

However, power will always be low to detect a

negligible effect. It is not very informative to

calculate that, given the negligible effect observed,

yes, one’s power to detect a negligible effect is

negligible. Thus, to be useful, retrospective power

analysis requires that only effects of a priori

interest be examined. In other words, in conducting

power analysis, the magnitude of the effect of

interest needs to be fixed independently of the data

at hand. The biologist must decide what is the

magnitude of an effect worth considering; this is a

biological, not a statistical, issue that is sometimes

difficult to settle.

Randomization—Randomization is an important

part of experimental design, owing to the work of

Sir Ronald Fisher in the early 20th century.

Randomization is used to combat biases that can

undermine survey and experimental studies. The

most important bias concerns assignment to

treatments. By randomizing assignment to

treatment (e.g., grazed vs. ungrazed), extraneous

differences among experimental units can be

minimized. Even here one would likely use

randomization subject to constraint. Suppose one

had five land units, each one that can be divided into

four plots. Randomly choosing treatment for the 20

plots could result in an unbalanced design. Instead,

one can randomly choose treatment, subject to the

constraint of 10 plots for each treatment. An even

better design would use land unit as a blocking

variable. Within each block (here, land unit), one

randomly assigns treatment to plots, with the

constraint that there must be two plots for each

treatment. Of course, in many studies assignment to

treatment is not always under the investigator’s

control.

Randomization should also be applied to minimize

other types of bias, if feasible. If two treatments

are being compared using point counts, using two

observers, one should not assign one observer to

conduct point counts in treatment A and the other

observer to conduct point counts in treatment B.

In this case, observer identity and the effect of the

treatment would be confounded. Instead, the two

treatments should be divided between the two

observers, as randomly or equitably as possible.

Another bias concerns order of observation. If

several plots are to be visited each day, one should

not visit the plots in the same order each time, but

should vary the order. It is not usually feasible to

visit point count stations in a random order, but

one can usually randomize the starting point on

each visit.

The final source of bias concerns inclusion in a study.

The sample to be studied will likely be the most

representative of the population in question if it is

randomly selected; however, this is often not

feasible. Nevertheless, we recommend incorporating

some randomness into every study. For example, one

could lay out a grid of point count stations, centered

on a randomly selected starting point as suggested

by Sauer (1998). This approach can be adapted for

those setting up transects of point count stations:

the starting point for a transect can be randomly

selected among a subset of possible points. Another

approach is to set up a grid of possible stations and

then randomly determine whether or not to include

individual stations in the study. Hutto et al. (1996)

and Hutto and Paige (1995) provide other

suggestions for randomizing point count stations

across broad areas.

Analysis of Vegetation and Habitat Characteristics

Data on vegetation and habitat features can play an

important role in avian monitoring studies. These

data can be gathered at different scales and in many

different ways. Methods of vegetation data

collection are described in many publications,

including Ralph et al. (1993), the BBIRD program

protocol (Martin et al. 1997), and Hays et al. (1981).

One of the most influential vegetation assessment

protocols developed for use with bird studies is by

James and Shugart (1970), with modifications by

Noon (1981). The analyses of vegetation data

collected in conjunction with point counts and

nest-monitoring are discussed in the appropriate

sections.

Vegetation data can be collected and analyzed at

several different scales. The broadest is habitat

classification and is qualitative (categorical) rather

than quantitative. This level includes most

vegetation maps and can be used to select the

vegetation types for study. The next broadest scale

is the “stand” level. This scale is commonly used to

ground-proof aerial photographs and, depending on

methods, to construct bird-habitat (or bird-vegetation)

correlations, making use of point count

and line transect data. The third scale involves

vegetation used to characterize the study area at a

smaller scale than the first methods, often within a

radius of 11.28 m following James and Shugart

(1970). In some studies, plots are centered on nests

or other sites of bird use (“use sites”), while others

(“non-use sites”) are randomly placed for

comparison within the study area. This scale allows

data that are more quantitative in nature to be

collected, compared to other scales. Examples of

6 Statistical Guide to Data Analysis of Avian Monitoring Programs

studies using this scale are Knopf et al. (1988) and

Larson & Bock (1986). This scale provides a good

means to establish bird-habitat relationships; such

data can be gathered quickly, accurately and

efficiently. The finest scale of vegetation

measurement is around the nest, nest plant or other

micro-habitat features (Martin & Roper 1988;

Martin et al. 1997).

Currently there is little agreement among biologists

on the methods, and even the scale, of vegetation

data collection needed to correlate with bird

abundance, habitat needs, distribution and behavior.

Therefore, it is not possible at this time to

recommend a single approach for analysis of

vegetation data since the data analytic approach will

depend on how the data were collected.

Introduction 7

Several techniques have been used for estimating

abundance of birds (Verner 1985, Bibby et al. 1992,

Butcher 1992, Skalski & Robson 1992, Buckland et

al. 1993, Greenwood 1996, Lancia et al. 1996). In the

past, two widely used and promoted methods have

been point counts and line-transects (Ralph & Scott

1981, Buckland et al. 1993). Capture/recapture data

is a third method used to estimate populations

(Greenwood 1996, Lancia et al. 1996). Following the

recommendations of the National Monitoring

Working Group of Partners in Flight (Butcher 1992)

and Ralph et al. (1993) and Ralph et al. (1995), we

restrict our attention to point counts. Line-transects

can also yield valuable data regarding population

abundance and species composition; however, the

design and analysis of transect data is beyond the

scope of this Guide (Ralph & Scott 1981, Buckland et

al. 1993). We assume that data will be collected using

fixed radius point counts, as described in Ralph et al.

(1993), rather than unlimited distance point counts

or variable distance point counts (Ralph & Scott

1981).

Throughout this Guide we discuss how to analyze

data gathered in a typical monitoring program and

then discuss design of monitoring programs,

especially sample size. Ideally, one should first put

careful thought into designing a monitoring

program before data collection and analysis.

However, here we discuss data analysis first in order

to give the reader a better idea of what sorts of data

can be gathered and what are some inferences that

can be drawn from data collected in a monitoring

program.

Analysis

Point count data have commonly been analyzed

with respect to 1. relative abundance, 2. species

richness, 3. species diversity and 4. community

similarity. An alternative to the analysis of relative

abundance, has been 5. the analysis of species

presence/absence (i.e., a species is scored as 1 if one

or more individuals are detected, and 0 if

otherwise). (We recommend not using the term

“frequency of occurrence” to characterize such

analyses, because of ambiguity of this terminology.)

However, from the point of maximizing statistical

power, the analysis of relative abundance (i.e.,

number of individuals detected per station) is to be

preferred to an analysis of presence/absence. The

latter discards information, leading to a loss of

statistical power. On this point we are in agreement

with Dawson (1981),

“[E]ither frequency of occurrence or average

number [per station] is adequate measure for

species which occur usually as one or none in each

counting unit. On the other hand, frequency

becomes an increasingly insensitive measure for

species found in larger numbers.”

Presence/absence may be very helpful as a

descriptive tool. That is, it may be informative to

state that a species was present at 40% of stations in

habitat x and 60% of stations in habitat y. Another

advantage of presence/absence data is that some

analytic methods can be used for such data but not

for total detections. For example, logistic regression

can be used with presence/absence, but not with

total detections. Logistic regression is discussed in

more detail in Chapter V and an example is provided

below of the analysis of presence/absence data.

Nevertheless, more sophisticated variants on

logistic regression can use total detections (e.g,

“ordered logistic regression”, StataCorp. 1999). Also,

Poisson regression, an analytic method that has

much in common with logistic regression, can

analyze total detections (Kleinbaum et al. 1988). As

its name implies, Poisson regression assumes that

the number of detections per station is Poisson-distributed,

but some software (e.g., EGRET)

includes the capability of testing this assumption

(and modifying the analysis if data do not conform to

this assumption).

Relative abundance is analyzed as number of

detections per unit area. The number of individuals

are determined at each point count station and this

datum can be entered into regression analyses or

analysis of variance (ANOVA). Results from several

point count stations can be averaged to produce a

summary statistic (Example 1). If a point count

station is surveyed more than once per season, one

can either sum the number of detections over all

point count surveys or calculate an average number

per point-count survey. As long as each station is

surveyed the same number of times (e.g., three

times), the two measures (average vs. sum) will

8

II. Assessment of Abundance and Species

Composition Using Point Counts

differ only by a constant, in this case, three. A third

commonly used method is to use the maximum

number of detections over the course of the three

surveys. In analyzing relative abundance these three

methods can be expected to yield similar patterns.

The number of individuals detected at a point count

station is a function of the absolute abundance and

the probability of detecting an individual (given that

it is present). Analyses of relative abundance

assume that differences in detectability can be

ignored, for the purposes of the study. In contrast,

variable distance methods (often referred to as

distance sampling; Buckland et al. 1993) attempt to

estimate detectability. The assumption that

differences in detectability are unimportant should

be kept firmly in mind when considering surveys of

relative abundance. Recent studies confirm that

detectability is influenced by a number of different

factors (Buckland et al. 1993, McShea & Rappole

1997, Gutzwiller & Marcum 1997).

Absolute abundance. Point count data are often used

to determine relative abundance; however, absolute

abundance may be estimated using variable distance

methods (Buckland et al. 1993, Ramsey & Scott,

1981). An important assumption of variable distance

methods is that at the center point of the

observation, all individuals are detected (i.e.,

detectability = 100%). It is possible to relax this

assumption if, instead, the true absolute density can

be independently determined at the center point,

but this is often not feasible. A second important

assumption is that individuals do not move towards

or away from the observer before being detected.

Buckland et al. (1993) provide extensive discussion of

these and other assumptions. The same authors

have developed a program DISTANCE that carries

out such analyses (Laake et al. 1993, Web site:

).

Species richness is analyzed as total number of

species detected. A total can be calculated for each

point count station, or for each group of point count

stations (Example 1).

There are a plethora of indices for species diversity

(Magurran 1988, Ludwig & Reynolds 1988). The

utility of diversity indices has been strongly

questioned by some (Verner & Larson 1989), and

their use has limitations. It has been argued that

species richness, a component of species diversity, is

more easily and more accurately measured. Species

richness is highly correlated with species diversity

and can be interpreted more clearly (Verner&

Larson 1989). An example of the value of a diversity

index (but one that is admittedly extreme) is a

comparison of two communities, each containing five

species and each with a total of 100 individuals.

Community A contains 96 individuals of species 1

and 1 individual of each of the other 4 species;

community B contains 20 individuals of each of five

species. Which community is more diverse? If one

feels that both are equally diverse, then species

richness is all one needs to take into account.

However, if one’s view is that community B is more

diverse, because its bird community is more

heterogeneous, then one is justified in using a

diversity index. However, keep in mind that more

assumptions are required to estimate diversity than

species richness. In particular, calculations of

species diversity assume that relative abundance is

accurately estimated and ignores the differences in

detectability among species that can skew estimates

of relative abundance.

The most widely used diversity index is referred to

as Shannon’s index, or as the Shannon-Wiener index

or the Shannon-Weaver index (Krebs 1989).

Shannon’s index, which is derived from information

theory, reflects both species richness and evenness

of distribution among species present. An equation

for the Shannon index, using natural logarithms

(ln) is:

where S = number of species in the sample, and pi

is the proportion of all individuals belonging to the

ith species. The original Shannon index was

calculated in terms of logarithm base 2, and thus H'

was expressed in terms of bits; however, it is more

common and more convenient to use natural

logarithms, as we have done above. A useful

transformation of H' is given by eH', which has been

labeled N1 (MacArthur 1965).

N1 expresses diversity in terms of species instead of

bits and thus is easier to interpret. N1 provides the

number of species that would, if each were equally

common, yield the same H' value as the actual

sample. For example, suppose there are 3 species, 20

of species A, 20 of species B and 10 of species C.

Using the above equation, H' = 1.055 and N1 = 2.87.

These three species, in their uneven distribution,

yield the same diversity value as would 2.87 species

of equal abundance. A comparison of species

richness (= S = 3) with N1 (= 2.87) gives us a

measure of evenness of species distribution. That is

the species distribution is maximally even when

S=N1.

For a fixed S, the maximum diversity (Hmax) is equal

to –ln (1/S) = ln(S) and therefore the ratio of

observed diversity to maximum diversity is a

measure of evenness (E): E = H'/Hmax = H'/ln S

i=S

H′ = Σ(pi)(lnp), i=1, 2,…S

i=1

Assessment of Abundance and Species Composition Using Point Counts 9

(Examples 1 and 2). If some species are more

detectable than others this will bias one’s measure of

diversity, either upwards or downwards. If the

Shannon index is calculated for a number of

samples, the indices themselves will be normally

distributed, making it possible to use parametric

statistics to compare sets of samples using diversity

indices (Magurran 1988). Further techniques for the

analysis of diversity patterns are described in

Magurran (1988) and Pielou (1975).

Example 1:

Calculation of Summary Statistics

The following is a simple and hypothetical example

of data collected using point counts (Table 3).

Observations were made at 3 point count stations at

3 different times during the breeding season.

Species are uniquely identified by a single letter

(A, B, C, etc.).

From these data, summary statistics can be

calculated, first of all summing (or averaging) across

the three survey periods, and then summing (or

averaging) across the three point count stations

whose data have already been summed over the 3

survey periods. Such a summarization is shown in

Table 3B.

The results shown in Table 3A for each point count

station can be used in a statistical analysis (e.g.,

regression or ANOVA) (Example 3). The biologist

may also summarize results for a group of point

count stations characterized by an important

similarity, e.g., all stations at a specific site, or all

stations in a specific habitat on a refuge, or other

unit of interest.

The row titled “Average” in Table 3B (second from

the bottom), simply averages the results from point

count stations 1-3. The row titled “Cumulative”

(bottom) shows the total number of individuals seen

at the 3 point count stations (a measure of

abundance), the total species richness for the 3

stations, and the species diversity as measured for

all 3 stations taken together. Thus, the average

station had 5 species, but the three stations together

had 7 different species. For average number of

individuals seen per point count survey, the

“Cumulative” value is simply three times that of the

“Average” value (i.e., 12.33 = 4.11 × 3). Thus the only

difference between these two measures is that in

one case one sums the number of individuals and

divides by the number of point count stations and in

the other case one sums and does not divide. Any

statistical results will be identical whichever

measure of individuals detected is used, except for a

10 Statistical Guide to Data Analysis of Avian Monitoring Programs

Table 3. Example of data from point count observations conducted at three point count stations, three times

during the breeding season.

A. Results by species. “A, A” indicates two individuals of species A were seen, “A, A, A” indicates three individuals,

“A, B, C” indicates one individual of three species, etc.

Point Count Survey Species Number of Species

Station Number Observed Individuals Richness

1 1 A, A, B, C 4 3

1 2 B, B, C, D 4 3

1 3 A, C, D, E 4 4

2 1 B, B, B, C 4 2

2 2 B, B, D 3 2

2 3 B, B, F 3 2

3 1 B, C, C, D 5 3

3 2 B, C, E, F, F 5 4

3 3 B, C, E, F, F, G 6 5

B. Summarization of data from Table 4A.

Point Count Average Number Cumulative Ecological

Station Individuals Species Richness Species Diversity1 Eveness = E

1 4.0 5 4.69 0.960

2 3.33 4 2.56 0.678

3 5.0 6 5.24 0.924

Average 4.11 5.0 3.86 0.839

Cumulative 12.33 7 5.55 0.881

1 Shannon’s index expressed as N1

constant (in this case, 3, the number of point count

stations).

In contrast to measures of abundance, Average

species richness and Cumulative species richness,

will generally not be so simply related to each other.

At one extreme average species richness will equal

cumulative species richness where there is complete

overlap of species at each point count station. At the

other extreme, cumulative species richness will be

three times that of average species richness

(assuming one is summarizing data from three point

count stations) provided there is no species overlap

at any point count station. Reality will usually fall

somewhere in between. Either way of summarizing

species richness can be justified. The same holds for

species diversity; the average diversity (per point

count station) and the diversity of the group of point

count stations are both legitimate ways to

characterize diversity.

Community Similarity Indexes

Another method of comparing communities is to

measure the degree of association or similarity in

community composition between sites or samples.

For example, two sites may be identical in species

richness, but both have completely different species.

For this purpose, a wide range of similarity indices

have been developed (Magurran 1988). Two such

indices that are widely used and that rely only on

presence/absence data are the Jaccard index and

Sorensen index (Krebs 1989):

where j = the number of species found at both site A

and B, a = the number of species in site A and b =

the number of species found in site B. These indices

are designed to equal 1 where the species from the

two sites are the same and 0 if the sites have no

species in common. Example 2 and Table 4 provide

an example of a calculation of Jacard and Sorenson

similarity coefficients.

One of the advantages to these indices is their

simplicity, but the indices do not account for

differences in the abundance of species. All species

count equally in the equation whether they are

abundant or rare. For this reason, quantitative

indices of similarity have much appeal as an

alternative. Again, many such indices have been

developed (Magurran 1988, Krebs 1989). Here we

just mention one of the simplest, the Renkonen

2j

Sorenson Cs = _______

a+b

j

Jaccard Cj = _______

a+b–j

index, also called the Percentage Similarity index.

The formula for the Renkonen index (P) is:

where pA

i is the percentage of species i in sample A

and pB

i is the percentage of species i in sample B

and S is the number of species found in either

sample. With no overlap between samples the index

equals 0, with complete similarity the Renkonen

index equals 100%. Table 4 provides an example of

the Renkonen index.

Example 2:

Calculation of Community Similarity Indices

The following is an simplified example of data

collected using point counts (Table 4). Observations

were pooled using the highest number counted

during 3 different surveys in the breeding season,

and pooled across 5 paired treatment-control plots

(modified from Dieni 1996). Community similarity

and diversity indices can be calculated and

comparisons made using these data.

The row titled “number of individuals” in Table 4 is

the sum of the total number of individuals counted in

each site. The columns titled “pa” and “pb” are the

proportion of each species in the total; i.e., the

number of individuals divided by the total number of

individuals for that site. The calculation of Jaccard’s

index is the number of species in common (j) to both

sites divided by the difference between the sum of

the number of species in each site minus the number

in common. Sorenson’s index is 2 times j divided by

the summation of the number of species in both

sites.

Other indices may also be informative including the

Renkonen index which is calculated by taking the

summation of the minimum of either pa or pb. Other

examples of the calculations of indices that may be

useful are shown in Table 4.

Linear Regression

To introduce linear regression, and provide a simple

example of trend analysis we consider the following.

Example 3:

An Example of Simple Regression

Black-headed Grosbeaks (Pheucticus

melanocephalus) have been surveyed at the

Palomarin station of Point Reyes National Seashore

during the breeding season for many years. Here we

present data from 1980-1992 (13 years) and wish to

i=S

P = Σ minimum (pA

i , pB

i)

i=1

Assessment of Abundance and Species Composition Using Point Counts 11

determine if there has been a trend for numbers to

increase or decrease during this period.

Keep in mind four key assumptions of linear

regression analysis:

1. Normality of residuals

2. Homoscedasticity; that is, there are no systematic

differences in variance of residuals

3. Independence of the outcome variable (i.e.,

independence of residuals), and

4. That we are interested in testing the hypothesis

(HA) that there is some sort of linear relationship

between dependent and independent variable. In

this case, the hypothesis is that bird abundance is

decreasing or increasing with time, in a linear

fashion.

Note that assumptions 1-3 refer to residuals, i.e., the

difference between the observed value of the

dependent (i.e., outcome) variable and the predicted

value from a regression model, we have to fit a

regression model before we can evaluate the

residuals. Figure 1 shows observed data and fitted

regression lines for this example, for

log-transformed data (Figure 1A), and for

untransformed data (Figure 1B). The log

transformation is commonly used in analyses of

linear models (e.g., regression and ANOVA;

additional examples below). There are two reasons

for using a logarithmic transformation:

12 Statistical Guide to Data Analysis of Avian Monitoring Programs

Table 4. Calculation of diversity, similarity and evenness indices using total bird detections across sites in

burned and unburned aspen (Populus tremuloides) stands in Wyoming (modified from Dieni 1996).

Number of Individuals Statistical Transformations

Minimum

Species Burned Control pa or pb pa pa ln pa pb pb ln pb

Red-tailed Hawk 2 0 0.000 0.004 –0.021 0.000 0.000

American Kestrel 1 0 0.000 0.002 –0.012 0.000 0.000

Northern Flicker 36 20 0.036 0.068 –0.182 0.036 –0.119

Western Wood-Pewee 21 39 0.040 0.040 –0.128 0.070 –0.186

Dusky Flycatcher 13 9 0.016 0.024 –0.091 0.016 –0.066

Tree Swallow 47 29 0.052 0.089 –0.215 0.052 –0.153

Clark’s Nutcracker 3 0 0.000 0.006 –0.029 0.000 0.000

Black-capped Chickadee 13 18 0.024 0.024 –0.091 0.032 –0.110

White-breasted Nuthatch 0 3 0.000 0.000 0.000 0.005 –0.028

Red-breasted Nuthatch 1 6 0.002 0.002 –0.012 0.011 –0.049

House Wren 127 142 0.239 0.239 –0.342 0.254 –0.348

Hermit Thrush 0 1 0.000 0.000 0.000 0.002 –0.011

American Robin 38 47 0.072 0.072 –0.189 0.084 –0.208

Warbling Vireo 163 199 0.307 0.307 –0.363 0.355 –0.368

Orange-crowned Warbler 14 40 0.026 0.026 –0.096 0.071 –0.189

Brewer’s Blackbird 3 0 0.000 0.006 –0.029 0.000 0.000

Western Tanager 2 2 0.004 0.004 –0.021 0.004 –0.020

Pine Siskin 33 3 0.005 0.062 –0.173 0.005 –0.028

American Goldfinch 1 0 0.000 0.002 –0.012 0.000 0.000

Cassin’s Finch 13 2 0.004 0.024 –0.091 0.004 –0.020

Number of individuals 531 560

Number of species 18 15

Number of species in common (j) 13

Summations 0.827 1.0 –2.095 1.0 –1.903

Jaccard (Cj) 0.650

Sorenson qualitative (Cs) 0.788

Renkonen index (P) 0.827

Shannon diversity (H) 2.095 1.903

Shannon evenness (E) 0.725 0.703

Shannon maximum value (Hmax) 2.890 2.708

1. Linear models assume additivity, but the

relationship between the dependent variable and an

independent variable may be multiplicative, i.e., with

an increase of each unit in x, y increases by a

constant proportion. Exponential growth or decline

of a population is a good example of a multiplicative

model. In this case, we may wish to fit a model in

which Black-headed Grosbeak numbers increase or

decrease by d% per year; our objective is to estimate

the value d, and test whether it is significantly

different from zero.

By taking logarithms, one can convert a

multiplicative relationship,

y = abx,

into an additive relationship,

log(y) = log(a) + (log(b))(x).

What was once a multiplicative relationship can be

rewritten in an additive form,

y′ = a′ + b′x.

2. The logarithmic transformation can often

normalize residuals (as shown below), thus

conforming to an important assumption of

regression analysis, as well as of ANOVA, ANCOVA,

and similar analysis.

A regression analysis on the log-transformed data is

appropriate, but before doing so, we present typical

output from STATA (Table 5) from a regression

analysis with annotated comments (numbers below

correspond to numbers on the output). Table 5A

shows analysis of log-transformed data; Table 5B

shows analysis of untransformed data.

1. Sums of Squares (“SS” in Table 5), degrees of

freedom (“df ”), and Mean Squares (“MS”) are

provided for the model being examined. This output

is usually of greater interest in ANOVA than in

regression analyses. Sums of squares are included in

R2 and R2a (#3, below). “Model” refers to

independent variables (in this case, only one) and

does not include the “constant” term.

2. The F statistic (“F”) for the entire model

(excluding the constant) is shown, and the P-value

associated with that statistic (“Prob >F”). The

degrees of freedom of the numerator (the first term

within the parentheses) equals the number of

parameters in the model, excluding the constant. If

a model includes linear trends for two independent

variables, the numerator df is equal to 2. If the

model, instead, includes quadratic and linear terms

for a single independent variable then the

numerator df is also equal to 2. If the model includes

linear trends for two independent variables and

their interaction, then the numerator df is equal to 3,

and so on.

The overall P-value, while of some interest, should

be of less concern than P-values for individual terms.

A model which contains one very significant

independent variable and one insignificant

independent variable can generate a highly

significant overall P-value, though such a model

Assessment of Abundance and Species Composition Using Point Counts 13

Figure 1. A. Linear trend in log(number Black-headed Grosbeaks observed) in relation to year (1980 to

1992), (statistical analysis in Table 5A). Triangles indicate log(number observed) in each year; solid line

indicates best-fitting trend using linear regression analysis. The trend depicted is a log-linear trend.

B. As in A. but numbers observed are untransformed. Statistical analysis in Table 5B; trend depicted is a

linear trend. Note that trend line fits observations better for log-transformed data (Figure 1A) than for

untransformed data (Figure 1B); e.g., with a higher R2 0.637 vs. 0.545.

Figure 1A. Trend, log-linear, P=0.001

Black-headed Grosbeak, Palomarin 1980-1992

Figure 1B.Trend, linear—notransformation,P=0.004

Black-headed Grosbeak, Palomarin 1980-1992

would be undesirable. On the other hand, if two

independent variables are highly correlated, each

variable could be insignificant (when controlled for

the other), yet the overall model could be very

significant and provide a good predictive model.

3. R2 (“R-square”) and adjusted R2 (“Adj R-square”).

The first statistic is often referred to as the

coefficient of determination. While it should be

familiar to all field biologists, much confusion still

surrounds its use or abuse (Anderson-Sprecher

1994). The second statistic is probably unfamiliar to

many, yet should be more widely known and used

(Neter et al. 1990, Kleinbaum et al. 1988). R2 can be

interpreted as the proportion of variation in the

dependent variable that can be accounted for by the

model in question. Both statistics provide a measure

of the predictive ability of a model. If R2 and

adjusted R2 are low, this means that much variation

in the Y variable is not accounted for by the model,

but this does not reflect on the adequacy of the

model. In Table 5A, R2 = 0.637, meaning that 36% of

the variation in Black-headed Grosbeak numbers is

not accounted for by an exponential decline in

numbers with increasing year.

There are several drawbacks to R2. For one, any

regression model will have a positive R2 associated

with it, even a regression model that links two

variables that are completely unrelated. To provide

an example, we generated two random variables X,

Y, integers chosen from a uniform distribution (0,

100) and which were independent of each other.

Values for X were (3, 67, 98, 63, 25, 90, 34, 4, 31, 78)

and for Y were (44, 91, 30, 92, 26, 56, 57, 90, 81, 47).

Regressing Y on X we obtain R2 = 0.021 (P = 0.69).

We would feel uncomfortable in stating that “X

accounted for 2.1% of the variation in Y,” since in

reality we know that it accounts for no such

variation. The second drawback is that as one adds

additional terms (additional independent variables),

R2 will always increase (Neter et al. 1990). Adjusted

14 Statistical Guide to Data Analysis of Avian Monitoring Programs

Table 5A. Linear regression analysis of number of Black-headed Grosbeaks, breeding season,

log-transformed (=ltotbrs) vs. year.

Source SS df MS Number of obs = 13

Model 2.71854704 1 2.71854704 F (1, 11) = 19.31

Residual 1.54865857 11 .140787142 Prob > F = 0.0011

Total 4.26720561 12 .355600468 Rsquare = 0.6371

Adj Rsquare = 0.6041

Root MSE = .37522

ltotbrs Coef. Std. Err. t P>|t| [95% Conf. Interval]

year .1222173 .0278129 4.394 0.001 .183433 .0610016

_cons 245.1384 55.23646 4.438 0.001 123.5637 366.713

Table 5B. Linear regression analysis of number of Black-headed Grosbeaks, breeding season,

untransformed (=totalbrs) vs. year.

Source SS df MS Number of obs = 13

Model 298.291209 1 298.291209 F (1, 11) = 13.20

Residual 248.631868 11 22.6028971 Prob > F = 0.0039

Total 546.923077 12 45.5769231 Rsquare = 0.5454

Adj Rsquare = 0.5041

Root MSE = 4.7543

totalbrs Coef. Std. Err. t P>|t| [95% Conf. Interval]

year 1.28022 .3524085 3.633 0.004 2.055866 .504574

_cons 2554.44 699.8845 3.650 0.004 1014.004 4094.875



 



 





 

 

 

 

R2 (R2a) was developed to counteract these

drawbacks. Adjusted R2 is defined as

where n = number of observations and p = number

of parameters (including the constant), SSE equals

Sums of Squares of the Residual and SSTO equals

Total Sums of Squares. Note that

In other words, R2a is equal to R2 after multiplying

the proportion of unexplained variance by

(n–1)/(n–p). This ratio (the adjustment factor) is

always equal to or greater than one, and therefore

R2a will always be less than or equal to R2. As n gets

large this ratio diminishes, and as p gets large, the

ratio increases. The properties of R2a are that:

1. If there is no relationship between two variables,

R2a will, on average, be equal to zero. Thus, under

the null hypothesis, R2a provides an unbiased

measure of the true relationship between the two

variables. In other words, if Y and X are completely

unrelated, R2a but not R2 will, on average, equal

zero. In the example cited above (of random X and

Y), R2a = –0.101. Any R2a less than 0 makes it

unambiguously clear that one variable does not have

value in predicting the other.

2. R2a will not necessarily increase as one adds

parameters. If the gain in R2 is small, then R2a can

decrease because the gain in R2 does not offset the

decrement due to the increase in p. Thus, R2a can

provide a good means of selecting the best

predictive regression model. In fact, the model

which maximizes R2a is also the model that

minimizes Mean Square Error (equivalently, Root

MSE), which is a measure of residual variation

about the predicted regression line.

4. Root Mean Square Error (“Root MSE”). This

provides a measure of the variability about the

regression line. In other words, it is the residual

variation left after allowing for the effect of, in this

case, year on Black-headed Grosbeak numbers. It is,

literally, the square-root of the Mean Square

associated with the Residual term (i.e., “error”).

Root MSE would equal the standard deviation of the

outcome variable if there were no explanatory power

to the independent variable (i.e., R2 = 0.0);

otherwise, Root MSE is less than the standard

 SSE 

______

SSTO R2 =1–

 SSE 

______

SSTO

 n–1

____

 n–p R2a = 1 –

deviation. Note that Root Mean Square Error in this

example is the measure of variance which the

programs MONITOR and TRENDS ask for

(described in detail below).

5. The regression coefficients (“Coef.”), their

standard errors (“Std. Err.”), and results of t tests,

examining whether t is significantly different from

zero, are shown (“t” and “P>|t|,” respectively).

Shown first is the regression coefficient for the

independent variable, Year. From Table 5A, our best

estimate (assuming that linear regression

assumptions are met) is that the number of birds

observed declines at an instantaneous rate of 0.122

units, expressed in natural logarithms. This

translates to an 11.5 percent decline per year, i.e.,

each year the number of detected birds is 0.885 times

that of the previous year. When the untransformed

data are analyzed (Table 5B), the best estimate is a

decline of 1.28 birds per year.

Shown below the coefficient for year is the

coefficient for the intercept term (here termed

“constant”). The value of the intercept term

provides the predicted value when the independent

term (here Year) equals zero. Thus its value depends

on how the independent variable is coded. Year = 0

might refer to the year 0, to the year 1900, or to any

other year so designated. The designation is

arbitrary and won’t affect the regression coefficient

for the term, Year. Note that STATA evaluates the

regression coefficient for Year using a two-sided

test, which we consider appropriate.

6. The 95% confidence interval for the regression

coefficients are presented. We recommend that

biologists examine confidence intervals for

regression coefficients; a confidence interval can

provide clear evidence of the precision (or lack of

precision) of our analysis. For an example, where an

analysis indicates no significant effect, a confidence

interval may indicate that a very broad range of

values is consistent with the data.

Comparing Table 5A and 5B (corresponding to

Figure 1A and 1B), we see that log-transformation

(Table 5A, Figure 1A) produces a better fitting

model (higher R2, more significant P-value) than

does analysis of untransformed data. This implies

that Black-headed Grosbeaks are declining at a,

more or less, constant proportion rather than at, a

more or less a constant decrease, using the absolute

number of individuals. This result makes biological

sense. Evaluating residuals confirms that the log-transformed

model is preferable. For example, we

can evaluate whether skewness and kurtosis of

residuals deviates from normality for each model

using the Skewness/Kurtosis test in the program

STATA (StataCorp. 1999). For log-transformed data,

Assessment of Abundance and Species Composition Using Point Counts 15

we cannot reject the hypothesis of normality (P =

0.25), whereas for untransformed data we can reject

the assumption of normality (P = 0.0003) (results

obtained using “sktest” in the program STATA).

Results will not always be this clear-cut; we may

want to use graphical methods to examine normality

of residuals. Figure 2 shows a normal probability

plot for transformed (Figure 2A) and untransformed

data (Figure 2B). We won’t go into the details of

these plots (interested readers can refer to

Kleinbaum et al. 1988, Neter et al. 1990); the main

point is that if residuals are normally distributed,

the data points will fall on the straight line shown.

For the log-transformed data there is a reasonably

good match between data points and the line; for

untransformed data there is not. The graphical

method does not determine whether or not the

residuals are normally distributed. It does indicate

to what extent transformation is or is not improving

the normality of residuals.

Example 4:

Application of Simple and Multiple Regression

We now tackle a more complex example, taken from a

study by the Point Reyes Bird Observatory,

conducted for the California Department ofFish&

Game (Nur et al. 1994).We use this example as an

opportunity to provide guidance in carrying out

multiple regression analysis. In July1991an herbicide

was accidentally spilled in and near the Sacramento

River, close to Dunsmuir,CA, resulting in the death of

all aquatic forms of life for a 36-mile stretch of river.

In addition, terrestrial fauna and flora along the river

were thought to have been impacted. Nur et al. (1994)

report results of an avian monitoring project

designed to assess the impact of the spill on

terrestrial bird populations.Aquantitative measure

of presumed impact was developed by California

Department ofFish&Game biologists, relying on

defoliation, leaf death and other symptoms of stress

exhibited by the riparian vegetation, which we term

the Vegetation Damage Index. Sites along the river

varied in the degree of impact, depending on the

exposure to the herbicide. In general sites closer to

the spill site in the downstream direction received

greater damage, and therefore higher values of the

damage index. Point counts were laid out in transects

of 7 stations per transect, stations spaced 300m

apart, with each transect parallel to the river and1800

min length, with one transect per “site”. All transects

were in riparian habitat.

In general, there was a tendency for areas with high

damage to show low species richness (Figure 3). In

particular, there was an overall significant linear

trend for bird species richness to decline with

increasing damage, when analyzing all 55 point

count stations. Output for this analysis is shown in

Table 6A (using the program STATA). Note that in

Table 6A, R2 = 0.149, meaning that 85% of the

variation in species richness among point count

16 Statistical Guide to Data Analysis of Avian Monitoring Programs

Figure 2. Evaluating the assumption of normality using graphical techniques. Comparison of “Normal

probability plots” depicting residuals from analysis of log-transformed (Figure 2A) and untransformed

(Figure 2B) observations of Black-headed Grosbeaks. Figure 2A and 2B depict the empirical cumulative

distribution function expected if the variable were normally distributed (y-axis) vs. the observed cumulative

distribution function (x-axis). If the variable in question were normally distributed then the graphed points

would fall exactly on the solid line and the correlation between the two cumulative distribution functions

would be +1.0. Log-transformed data conform better to a normal distribution than do untransformed

observations.

Figure 2A. Normal probability plot,

residuals of log-transformed data

Black-headed Grosbeak

Figure 2B. Normal probability plot,

residuals of untransformed data

Black-headed Grosbeak

stations is not accounted for by differences in the

damage index. Our interpretation of this result is

that species richness data from individual point

count stations are very variable. The model is,

however, highly significant, and we have no reason to

think the model is inadequate.

One needs to keep in mind that there are two

different objectives for which one can use regression

models: (i) hypothesis testing, and (ii) prediction. In

this case, a model with only vegetation damage

would poorly predict species richness at a specific

point count station. However, such a model achieves

the objective of confirming the hypothesis that

biological damage resulting from the spill was

associated with diminished species richness.

Also keep in mind that the magnitude of R2 depends

on the unit of analysis. If one were to average data

from several point counts and then use the averaged

data in a regression analysis, this would have little

effect on the P-value, yet would increase R2

substantially. This is because some of the variation

in the dependent variable has been eliminated by

using mean species richness values in the regression

analysis, rather then species richness at individual

point count stations.

We confirmed that the linear regression analysis in

Table 6A is appropriate, first by examining

normality of residuals: P = 0.50, using the skewness/

kurtosis test (“sktest” of STATA). In other words,

residuals do not appear to deviate from normality.

We demonstrate this point graphically in Figure 4.

Figure 4A shows the frequency distribution of

residuals compared to a normal distribution; Figure

4B shows a quantile-normal plot for the residuals

from Table 6A. (Quartiles and percentiles are

examples of quantiles; a quantile-normal plot shows

quantiles for the distribution of interest vs. quantiles

from a normal distribution which matches the first

distribution in terms of mean and variance.) As with

a normal-probability plot (Figure 2), if the

distribution is indeed normal, then the data points

(quantiles in this case) would fall on the solid line

shown in the Figure 4B. In this case, there seems to

be a very good match, implying that residuals are

approximately, normally-distributed.

That bird species richness was correlated with the

Vegetation Damage Index is not by itself adequate

evidence for a causal link. In a similar fashion to the

analysis in Table 6A, a suite of vegetation

characteristics were examined, to determine

whether bird species richness, diversity and

abundance were related to habitat or vegetation

features. If so, such habitat variables could be

confounding any relationship of the bird fauna to the

impact of the spill. In one scenario, there could be no

true functional relationship between bird species

richness and vegetation damage, but a correlation

between the two can arise if both are correlated with

a vegetation feature. In another scenario, the true

causal relationship between bird species richness

and vegetation damage could be strong but it could

be masked, wholly or in part, because both are

correlated with a vegetation feature. For example, if

biological damage from the spill lowered species

richness, and the presence of willow (Salix spp.)

increased species richness, then if biological damage

was greatest in an area where willows were most

abundant, the correlation between biological

damage and bird species richness could be very

weak despite a strong causal relationship between

the latter two variables.

Nur et al. (1994) examined 25 habitat features to

determine whether they might be correlated with

abundance, species richness and/or species diversity.

They found that only two habitat features were

significantly correlated with abundance, species

richness and diversity. The latter variables were

positively correlated to the presence of willow

species and negatively with the presence of big-leaf

maple (Acer macrophyllum), i.e., the more big-leaf

maple, the fewer the bird species detected. The

independent variables were indices based on percent

cover that was willow (on a 0 to 10 scale,

corresponding to 0 to 100%), and percent cover of

big-leaf maple. Results of simple linear regression of

bird species richness in relation to willow cover and

Assessment of Abundance and Species Composition Using Point Counts 17

Figure 3. Bird species richness from 55 point count

stations along the Sacramento River, in relation to

Vegetation Damage Index. Higher values imply

greater damage from spill of metam sodium

(statistical results in Table 6A). Least squares line of

best fit is shown. Data at each point count station

have been “jittered” (Stata Corp. 1997) to reduce

overlap of points.

Figure 3. Bird species richness in relation to

Vegetation Damage Index.

in relation to big-leaf maple cover are shown in

Tables 6B and 6C, respectively.

The next step in the analysis was to conduct a

multiple regression analysis including the three

independent variables (Table 7). In this case, the

primary interest was the effect of damage index

while controlling for the two habitat variables. The

results indicate that damage was still inversely

correlated with species richness, even after

controlling for one or the other habitat variable, or

after controlling for both of the habitat variables

(Table 7). These results give support to the view that

biological damage due to the spill reduced species

richness along the river. The results do not support

the alternative view that the inverse association

between species richness and damage was

coincidental, reflecting habitat or vegetation

differences among sites along the river.

The degree and direction of differences among

independent variables (if it exists) can be assessed

by comparing regression coefficients in the

simple regression analysis (Table 6) and in the

corresponding multiple regression analysis

18 Statistical Guide to Data Analysis of Avian Monitoring Programs

Table 6. Sample output for linear regression analyses using STATA. See text, Example 4.

A) model: species richness [specrich] = Vegetation Damage Index [vegdindx]

Source SS df MS Number of obs = 55

Model 118.969398 1 118.969398 F (1, 53) = 9.28

Residual 679.466965 53 12.8201314 Prob > F = 0.0036

Total 798.436364 54 14.7858586 Rsquare = 0.1490

Adj Rsquare = 0.1329

Root MSE = 3.5805

specrich Coef. Std. Err. t P>|t| [95% Conf. Interval]

vegdindx 1.663229 .5459849 3.046 0.004 2.758336 .5681219

_cons 9.368597 .5243445 17.867 0.000 8.316895 10.4203

B) model: species richness [specrich] = willow cover [willotco]

Source SS df MS Number of obs = 55

Model 78.2553574 1 78.2553574 F (1, 53) = 5.76

Residual 720.181006 53 13.5883209 Prob > F = 0.0200

Total 798.436364 54 14.7858586 Rsquare = 0.0980

Adj Rsquare = 0.0810

Root MSE = 3.6862

specrich Coef. Std. Err. t P>|t| [95% Conf. Interval]

willotco .0838145 .0349257 2.400 0.020 .0137624 .1538667

_cons 8.184659 .549244 14.902 0.000 7.083015 9.286303

C) model: species richness [specrich] = big-leaf maple Cover [bigleaco]

Source SS df MS Number of obs = 55

Model 111.19329 1 111.19329 F (1, 53) = 8.58

Residual 687.243074 53 12.9668504 Prob > F = 0.0050

Total 798.436364 54 14.7858586 Rsquare = 0.1393

Adj Rsquare = 0.1230

Root MSE = 3.601

specrich Coef. Std. Err. t P>|t| [95% Conf. Interval]

bigleaco .0290063 .0099058 2.928 0.005 .0488740 .0091387

_cons 9.799174 .6043525 16.214 0.000 8.586997 11.01135

(Table 7). The effect of damage index was similar

when analyzed by itself or after controlling for

willow tree cover (ß = –1.66 ± 0.55 vs. ß = –1.58 ±

0.53). This indicates that willow cover did not

confound the relationship between vegetation

damage and species richness. On the other hand, the

apparent effect of damage index, was stronger when

analyzed by itself than after controlling for big-leaf

maple (ß = –1.66 ± 0.55 vs. ß = –1.26 ± 0.56). Big-leaf

maple tended to be more prevalent in areas

where biological damage was greater (in fact, there

was a significant correlation between the two,

P<0.01) and thus part of the apparent reduction in

species richness with increasing damage may be

attributed to the influence of big-leaf maple.

Analyzing Vegetation Data in Relation to

Point Count Data

In Example 4 (Tables 6-7), we provide an example of

vegetation data analysis coupled with analysis of

data on bird populations. In this case, the objective

was to determine whether the relationship between

vegetation damage and species richness was due to a

direct effect of spill-induced damage, or whether the

correlation was spurious and due to the fact that

both were correlated with additional habitat

variables. There was evidence that bird species

richness was related to habitat variables (specifically

the presence of willow and big-leaf maple), but these

relationships could not by themselves account for

the observation that bird species richness declined

as vegetation damage increased.

Collecting data on many habitat and vegetation

features doesn’t answer the question of which

habitat and vegetation features are causally related

to bird abundance or distribution. If data on

important variables are not collected then

interpretation of the data that were collected can be

compromised. There is still the problem of sifting

through the data to determine which features are

most closely related to the response variable in

question. Many techniques have been used by

investigators to evaluate multi-dimensional data,

including logistic regression, discriminant analysis,

principal component analysis, correspondence

analysis and MANOVA (Ludwig & Reynolds 1988,

Trexler & Travis 1993). It is beyond the scope of this

Guide to review these various techniques; however,

an example of the use of discriminant analysis is

presented in the section on vegetation analysis in

relation to nest-monitoring.

Design

Even if statistical inference is of no concern to the

investigator, one must decide on sample size. Ralph

et al. (1995) recommend at least 30 point count

stations per habitat and per area of interest. If one

wished to monitor two habitats in each of two areas

then this would necessitate 120 point count stations.

This is only a base number and the number of

stations should be increased if few individuals of a

species or group of species of interest are detected.

Where statistical inference is a goal, then sample size

will be dictated by considerations of statistical

Assessment of Abundance and Species Composition Using Point Counts 19

Figure 4. Evaluation of the assumption of normality of residuals using graphical techniques. Residuals of

linear regression analysis of species richness are depicted (statistical model in Table 6A). Figure 4A)

Frequency distribution of residuals (histogram), superimposing a frequency distribution for a normally-distributed

variable with the same mean and variance as the observed variable. Figure 4B) Same residuals

as in A) but graphed using a quantile-normal plot. The quantiles for the observed distribution (residuals as

in Figure 4A) are plotted against the quantiles from a normally-distributed variable with the same mean and

variance as the variable in question. Normality is demonstrated if the observations fall on the solid line.

Both A) and B) confirm that residuals are normally distributed.

Figure 4A. Distribution of residuals: species

richness vs. Vegetation Damage Index

Figure 4B. Quantile-quantile plot of residuals of

species richness vs. Vegetation Damage Index

against normal distribution

power; examples include comparisons among

treatments and monitoring programs that assess

temporal trends (Gerrodette 1987, 1991, Link &

Hatfield 1990; for a general review, see Thomas &

Krebs 1997).

Dawson (1981) has derived an equation for

determining the sample size, in this case the number

of point count stations necessary to detect an effect

of interest with 50% power. He assumed that each

station is surveyed once, and that the number of

detections (for each species or group of species) at

each station follows a Poisson distribution. If the

distribution of bird-detections deviates from Poisson

distribution, then the formula would need to be

revised. Under a Poisson distribution the mean

number of detections and the variance in the

number of detections per point count station would

be equal. If the variance substantially exceeds the

mean, or vice versa, one could either modify his

formula, or use other formulas (as modified below).

Dawson also assumed that there were two groups

(two habitats, two treatments, etc.) of interest. The

formula for sample size calculations is

[1]

(3.84)(20000)

n > ______________

(d2)(m)

where m = average number of detections per

sampling unit and d = percent difference between

group 1 and group 2, defined as

For example, if m1 = 2.5 and m2 =1.5, then m = 2.0

and d = 50.

Thus, where the average number of individuals

detected per station is equal to 1, the number of

stations per group necessary to achieve 50% power

to detect a 50% difference in mean abundance is

30.7. To detect a 25% difference would require 123

point count stations per group. That is, to detect

half the difference (all else being equal) requires 4

times the sample size! This exemplifies a general

rule—precision increases, and therefore standard

errors decrease, in proportion to the square root of

sample size. Note that as average number of

detections increases, sample size (number of

stations) decreases linearly—and proportionally.

This emphasizes that calculations of necessary

sample sizes reflect the average number of

detections. Thus, if the average number of

individuals detected per station is 0.5 rather than

m1–m2

_______

 m 

d = (100)

20 Statistical Guide to Data Analysis of Avian Monitoring Programs

Table 7. Analysis of point count data on Sacramento River. Relationship of bird species richness to Damage

Index, controlling for vegetation/habitat characteristics (n = sample size).

A. Multiple regression analysis of bird species richness per point count station in relation to vegetation

damage and willow cover

Model Statistics Independent variable1

R2a = 0.201, R2= 0.231, P = 0.0011, n =55 Vegetation Damage Index ß = –1.577 ± .525, t = –3.00, P = 0.004

Willow cover2 ß = +0.0770 ± .0326, t = +2.36, P = 0.022

B. Multiple regression analysis of bird species richness per point-count station, in relation to vegetation

damage and big-leaf maple cover.

Model Statistics Independent variable1

R2a = 0.186, R2= 0.216, P = 0.0018, n = 55 Vegetation Damage Index ß = –1.264 ± .562, t = –2.25, P = 0.029

Big-leaf maple cover2 ß = –0.0213 ± .0102, t = –2.10, P = 0.041

C. Multiple regression analysis of bird species richness per point-count station, in relation to vegetation

damage, willow cover, and big-leaf maple cover

Model Statistics Independent variable3

R2a = 0.229, R2= 0.272, P = 0.0010, n = 55 Vegetation Damage Index ß = –1.273 ± .547, t = –2.33, P = 0.024

Willow cover2 ß = +0.0651± .0329, t = +1.98, P = 0.053

Big-leaf maple cover2 ß = –0.0170 ± .0101, t = –1.68, P = 0.100

1 Both considered simultaneously

2 On 0 – 10 scale

3 All three considered simultaneously

1.0, then twice as many point count stations are

required, i.e., 246 and 61.4 point count stations

would be required per group to detect a 25% and

50% difference, respectively.

As stated earlier, it is common practice to use a

higher level of statistical power (e.g., 80%) in

designing studies. Dawson (1981) only considered

50% power, but his results can be extended to

consider more stringent levels of power as follows.

For other levels of power substitute the following

(approximate) values for the 3.84 in Equation 1: for

70% power, substitute 6.15; for 80% power substitute

7.84; for 90% power substitute 10.50.

These values were derived from a more general

formula for comparison of means using two-sample

t tests, i.e.,

where n is the required sample size for each sample,

σ1

2 refers to variance in sample 1, etc., μ1 refers to

mean value in sample 1, etc., and zsubscript is a

“normal deviate” (also called z-score). For example,

for α = 0.05, z(1–α/2) = z0.975 =1.96; for power = 0.8,

z0.8=0.84; for power=0.5, z0.5=0.0; for power=0.9,

z0.9=1.28; and so on (Snedecor & Cochran 1989).

The experimenter would set the difference between

means; σ1 and σ2 are fixed by the investigator

(i.e., determined independently). For a Poisson-distributed

variable, μ1 = σ1

2 and μ2 = σ2

2.

Thus, with 1 individual detected on average per

point count station, approximately 250 and 63 point

count stations would be required per group to

achieve 80% power to detect between-group

differences of 25% and 50%, respectively. With 0.5

individual detected per point count, at least double

these sample sizes (500 and 126) would be required

per group to achieve the same 80% power. Note that

the minimum recommendation of Ralph et al. (1995),

i.e., 30 point count stations per habitat or treatment,

provides 80% power to detect a 50% difference given

an average of 2.0 detections per station, but only

yields 50% power to detect a 50% difference given

1.0 detections per station.

Where there are three or more groups of interest,

we recommend the same number of point count

stations per group be maintained. Thus to detect a

50% between-group difference with 50% power and

0.5 individual detected, on average, per station, one

would need either 123 point count stations (total)

allocated among 2 groups or 184 point count stations

[2]

(σ1

2+σ2

2) (z1–α/2+z1–β)2

________________________

( μ1–μ2)2

n =

for 3 groups. This is admittedly a conservative

approach. It would maintain the same degree of

precision per group, whether or not there are two or

more groups.

Buckland et al. (1993) present a formula for

calculating sample size (number of point count

stations) necessary to achieve a specified precision

in estimating population size. They assume that one

has conducted a pilot study of k0 stations and

detected a total of n0 individuals (assuming no

aggregations or clustering of individuals, as is the

case in flocks). Their formula for number of

stations, K, is

where CV(D) is the coefficient of variation of

abundance, i.e., the standard error (not the

standard deviation) of abundance divided by mean

abundance. For example if one wished to estimate

abundance such that the standard error was 20% of

the mean value, then CV(D) = 0.2. b is a factor that

depends on several variables and can be estimated

from pilot data (Buckland et al 1993); however, they

state that it will usually be about 3. Thus if 15

individuals are detected at 10 point count stations

on the pilot study, CV(D) = 0.2 (by design) and b =

3, then K = 3.0⁄0.04 × 0.667 = 50 point count stations.

This will be sufficient to establish a confidence

interval for abundance that ranges ± 40% of the

true value.

Power and Sample Size Analysis Using TRENDS

Recently, there has been a veritable explosion of

software now available for determination of power

and/or necessary sample size to achieve specified

power (Thomas & Krebs 1997, available on the

world-wide web at http://www.interchg.ubc.ca/cacb/

power/review/). Two specialized, free programs are

available for monitoring programs that evaluate

trends, whether those trends are temporal or

spatial.

For analysis of trend data using linear regression, a

user-friendly program, TRENDS (Gerrodette 1987,

1991), has been written by Tim Gerrodette and is

available, free of charge, from the Internet

(ftp://ftp.im.nbs.gov/pub/software/CSE/wsb21515/

trends.zip; also available from T. Gerrodette,

Southwest Fisheries Science Center, P.O. Box 271,

La Jolla, CA 92038, in which case please provide him

a 3.5″ IBM-compatible floppy disk). A User’s Guide

is provided with the software. We offer the following

as guidance in using and interpreting results from

TRENDS.

 k0 

___

 n0 

 b 

 _________ 

 [CV(D)]2 

K =

Assessment of Abundance and Species Composition Using Point Counts 21

TRENDS can be used for either temporal trends or

spatial trends, with regard to changes in abundance.

The program TRENDS can compute any one of the

following parameters:

1. Number of samples (either number of occasions,

for temporal trends, or number of sites for spatial

trends) (n);

2. Rate of change (expressed as proportional decline

or increase of the total population, e.g., –0.10 refers

to 10% decline per unit of time or space; 0.05 refers

to 5% growth per unit of time or space, etc.) (r);

3. Measure of variation about the trend line, which

Gerrodette (1989) refers to as “initial coefficient of

variation” (CV1);

4. Significance level (α);

5. Power (1-ß, where ß is the probability of making a

Type II error).

If four of these parameters are specified, the fifth

parameter is strictly determined, and its value can

be calculated by TRENDS. For example, if one

specifies number of samples, magnitude of the rate

of change, the variation about the trend line, and the

a level, TRENDS calculates power. In the same way

one can calculate the necessary number of temporal

or spatial samples to achieve a specified power to

detect an effect of specified magnitude.

If one is evaluating temporal trends, then the

number of samples refers to the number of sampling

occasions—most commonly the number of years

studied. If several surveys or point counts

contribute to a single year’s data, these would be

averaged together to yield a single datum for that

year. In the program TRENDS, if 10 point counts

are conducted in each of 10 years, the number of

samples is 10, not 100.

The measure of variation about the trend line is

given as the coefficient of variation (= standard

deviation [or standard error] divided by the mean),

symbolized in the program as CV1. CV1 is inversely

related to precision. Gerrodette refers to CV1 as

“initial coefficient of variation”, which is misleading.

The best way to obtain an estimate of CV1 is to use

data from a trend analysis to determine root mean

square error (Table 5) and then divide this by the

mean (or expected value). Gerrodette provides an

example where CV1 was estimated from replicate

counts on the same population in the same year

which was the initial year of the study. We strongly

advise against this practice because doing so

estimates only the part of the variation due to

measurement error. An additional part of the

variation about the trend line is due to stochastic

variation in abundance, which also needs to be

incorporated into CV1.

The other three parameters are straight forward. In

addition, TRENDS requires one to make additional

specifications:

6. Whether to use a 1- or 2-tailed test,

7. Whether population change is linear or

exponential,

8. Whether one’s test statistic is the z or t statistic,

and

9. How CV1 is related to abundance.

Although the first three are straight-forward, we do

have some recommendations. First, a 2-tailed test is

almost always the appropriate test, because the

possibility of an increase in population cannot be

ruled out, and would be of interest just as much as a

decline in population. Secondly, for temporal trends,

exponential growth is to be preferred. Exponential

growth implies that the growth or decline is a

constant percentage, e.g., a population increases at

10% per year. Furthermore, this assumption is in

accord with the definition of r, the rate of change.

Thirdly, if one is estimating CV1 from data, then the

t statistic is appropriate (Link & Hatfield 1990).

The relationship of CV1 to abundance is complex.

TRENDS assumes that variance (VAR) in a

population estimate is (i) proportional to abundance

(A), (ii) proportional to A2, or (iii) proportional to A3.

TRENDS does not allow for VAR to be independent

of abundance, or for VAR to be inversely

proportional to abundance. Noting that mean

abundance is proportional to abundance and that the

CV is the square-root of VAR divided by the mean,

options (i) to (iii) imply that:

(i) CV is proportional to 1/√A,

(ii) CV is independent of A, or

(iii) CV is proportional to √A.

TRENDS allows one to choose among these three

options. Which option one chooses will affect

calculated power, necessary sample sizes, etc.

(Link & Hatfield 1990). As guidance for choosing

among the options, Gerrodette (1987) offers the

following: for quadrats, strip transects, line

transects, or catch per unit effort (CPUE), CV is

proportional to 1/√A. For distance sampling, CV is

independent of A. For the single mark-recapture

using the Peterson method, CV is proportional to

22 Statistical Guide to Data Analysis of Avian Monitoring Programs

√A. Presumably, option (i) applies to standard

(fixed-distance) point count data, since a point

count can be thought of as a line transect of length

zero (Buckland et al. 1993).

Example 5:

Power Calculation Using TRENDS

To provide an example of one of the uses of

TRENDS, we consider surveys of Black-headed

Grosbeaks (based on data given in Example 3 and

Figure 1). CV1 for a single annual count (of log-transformed

data) is estimated to be 0.284 (root

mean square error [see Table 5A] divided by mean

value). Assuming α=0.05, exponential population

change (i.e., constant percentage change each year),

a 2-tailed test, use of the t-statistic, and CV1

proportional to 1/√A, the probability (power) to

detect a 5% decline per year after 10 years is 34%.

If, instead, CV1 is independent of A, then the power

to detect a 5% decline per year after 10 years is 29%.

It is difficult to say whether option (i) or option (ii) is

more appropriate, but the difference between the

two estimates is small. Thus, the power to detect a

substantial decline (amounting to 40% decline after

10 years) is fairly weak. If we increase the time scale

to 15 years, however, power increases to 89% (under

option i). Under this scenario we would have

appreciable power to detect a decline, but after 15

years the population will have declined by 54%. An

alternative means of increasing power would be to

increase the precision of our annual estimate, which

implies lowering the variance about the trend line,

i.e., decreasing CV1. If CV1 could be lowered from

0.284 to 0.200, power would increase (assuming 5%

decline over a 10 year period) from 34% to 59%.

CV1 might be lowered if sources of error could be

reduced (e.g., conduct surveys at the same time of

year) or if replicate surveys were carried out and

the results for each year then averaged.

One can easily use TRENDS to determine, instead,

the minimum number of years required to attain

80% power to detect a 5% decline per year (14 years,

assuming option i).

Using MONITOR

Whereas TRENDS uses an analytic approach to

determine sample size, power, etc., MONITOR

(developed by James Gibbs) uses computer

simulation. Link & Hatfield (1990) argue that

computer simulation is to be preferred to analytic

solutions, because the latter can only provide

approximate results. The program MONITOR is

easy to use (ftp://ftp.im.nbs.gov/pub/software/

monitor), and a Manual is readily available as well.

Users of the program should take into account the

following points.

1. Similar to TRENDS, only one data point per plot

(or transect or route) is allowed per time unit (e.g.,

per year). Unlike TRENDS, MONITOR can allow

for analyses conducted on several plots at once.

Where several plots (transects, routes, etc.) are

analyzed at once, MONITOR calculates a weighted

trend (see below for further discussion of

weighting).

2. Whereas “plots”, can refer to “routes” or

“transects,” it can also refer to individual point count

stations if they will be analyzed in this way (and not

simply pooled across a transect or route). The

maximum number of “plots” is 250.

3. When several surveys are conducted for each plot

in the same year (or breeding season or other

interval of interest), MONITOR averages across

these data (i.e., collapsing the data into a single data

point per plot per year).

4. A critical variable is “variance of plot counts.”

This variance is used to simulate variation about the

specified trend line. The manual suggests that

within-year variation (determined from multiple

surveys) can be used to estimate between-year

variance, but this will generally not be valid. The

correct estimate is the variance about the trend line,

just as with TRENDS.

5. As with TRENDS, the trend can be linear or

exponential. We strongly recommend an exponential

trend for reasons discussed above, unless data at

hand indicate a linear trend is more appropriate.

6. Data from multiple plots can be weighted

according to mean abundance, but variance about

the plot-specific trends is not used in weighting. This

is less than satisfactory, because it means that a

poorly-estimated trend has as much weight as a

well-estimated trend.

7. When data are collected from several plots,

MONITOR de-means the values (subtracting off

the mean value for each plot) before calculating the

variance. Otherwise, variance due to habitat

differences among plots will be included in the

estimate of sampling variance (which we are

interested in). However, this de-meaning is

undesirable because it over-corrects. That is,

suppose we have n plots that are true replicates. In

this case, all between-plot differences are due to

sampling variation, which will have been completely

removed by de-meaning. A better approach would

be to use a covariate (or set of covariates) to

characterize habitat variation, and then use

residuals from a regression on the habitat

covariate, to provide an appropriate measure of

variance.

Assessment of Abundance and Species Composition Using Point Counts 23

Power and Sample-Size Analyses: Other Sources

Several stand-alone statistical packages are now

available that can calculate power for a variety of

statistical tests and situations (reviewed by Thomas

& Krebs 1997). In their review, Thomas and Krebs

mention five programs that they could recommend.

Of these we highlight two: the first is PASS (Power

And Sample Size; available from NCSS Statistical

Software, 329 North 1000 East, Kaysville, UT 84037;

http://www.ncss.com/pass.html). When a class of

ecology graduate students was asked to compare

PASS with three other recommendable power and

sample size programs, 17 out of 19 students

preferred PASS! It is flexible, accurate, easy to use,

and easy to learn. The cost is moderate ($249). The

other program we mention (also reviewed by

Thomas and Krebs) is GPOWER (Erdfelder et al.

1996); though this program did not score as highly as

PASS, it is free (http://www.psychologie.uni-trier.

de:8000/projects/gpower.html). Thomas and

Krebs (1997) examined several general-purpose

statistical programs with built-in power analyses,

but found none that they could recommend.

Two other valuable sources for power analysis are

on the web. The Patuxent Wildlife Research Center

of USGS has an excellent page, that includes a

power analysis program for calculating power

for monitoring programs, using the data in a

manner similar to TRENDS. This is available at

http://www.im.nbs.gov/powcase/powcase.html.

Also available is a web page dedicated to the

discussion and calculation of power analyses at

http://www.im.nbs.gov/powcase/powlinks.html.

This site has both MONITOR and TRENDS

available as freeware.

A number of statistical texts treat the problem of

determining power. Fleiss (1981) provides an

excellent practical treatment of the problem when

the outcome is binary (only one of two outcomes), or

can be expressed as a rate or proportion. Thus, his

text can be very useful for studies of survival or

studies in which the outcome is presence or absence.

Cohen (1988) gives an extensive non-technical

treatment of power analysis for ANOVA.

24 Statistical Guide to Data Analysis of Avian Monitoring Programs

Mist-nets can be used to provide estimates of many

parameters: 1. relative abundance, 2. species

composition (richness, diversity), 3. productivity, as

measured by production or abundance of HY

(Hatching Year) birds, and 4. annual adult survival.

In addition, one can, in theory, estimate 5. offspring

survivorship to breeding age using data from mist-nets

but this is an area that is only now being

investigated by researchers. Regarding abundance

and species composition, methods of analysis are

the same as described for analysis of point count

data. Recent examples of analyses of trends in

abundance include Johnson & Geupel (1996) and

Chase et al. (1997); Silkey et al. (1999) discuss the

validity of inferring population trends from mist-net

capture data. Nur et al. (1994) analyzed

patterns of abundance along the upper Sacramento

River (Example 4) using mist-net capture data and

using point-count data. Mist nets cannot, however,

provide an absolute measure of abundance. On the

other hand, they can provide an age-specific, and

sometimes sex-specific, measure of abundance,

with a resolution that cannot be matched by point-count

or line-transect data.

Analysis of Productivity

The number of HY birds caught in a standardized

mist-netting study can provide an index of

production of young (DeSante & Geupel 1987, Nur

& Geupel 1993b, DeSante et al. 1993). Such data

have been analyzed in three ways: (i) analysis of

total number of HY birds caught; (ii) analysis of

number of HY birds caught per AHY (After

Hatching Year, i.e., adult) caught; or (iii) analysis of

per cent of all birds who are HY. Among these

parameters, (iii) is just a transformation of (ii), and

vice versa, provided that all birds are classified as

HY or AHY (total = AHY + HY). This can be seen

as follows: let HY/AHY=R. Then proportion of all

birds that are HY = HY/(AHY + HY), can be

written as

Thus, (iii) only re-expresses (ii), but the

interpretation of (ii) is more direct: the number

of fledged young per adult.

proportion HY = ____1___

1 + (1–

R)

Nur & Geupel (1993a, 1993b) point out that there are

hazards with including the number of AHY birds

caught as a measure of productivity (as do indices ii

and iii above): notably, the catchment area of HY

and AHY can differ markedly (Nur & Geupel 1993a).

Secondly, many AHY birds are transient, i.e., not

breeding locally. As an alternative, one can use the

measure in (i), HY birds alone. If one finds

differences in the number of HY birds caught in two

sets of sites, or can establish a trend in HY numbers,

this provides information about the production of

young on a population level, but it may or may not

indicate differences or trends in productivity per

pair. Thus we recommend that, if variations in

breeding population size can safely be ruled out, in

comparing areas or comparing years, the number of

HY be analyzed by itself (see examples in Nur &

Geupel 1993b). Otherwise, the biologist should

analyze (ii) or (iii). Example 6 demonstrates

different ways of analyzing productivity.

Example 6:

Analyses of Productivity.

This example is taken from the study of the impact

of the herbicide metam sodium on landbird

populations of the Sacramento River, described in

Example 4. Table 8 shows a species-by-species

analysis of productivity as measured in two ways,

HY birds per 100 net-hours, and proportion of HY

birds in the catch.

To examine patterns of productivity we selected

those species with sufficient sample size. Our

criteria were: (1) at least 36 individuals caught (of all

age classes) from the 9 sites, and (2) at least 12 HY

individuals caught, total, from the 9 sites. Six species

met both criteria (Table 8). The “36 individual

criterion” implied that each site averaged 4 or more

individuals caught, which we considered a minimal

acceptable number. We would have preferred to

impose a minimum of 45 individuals (i.e., 5

individuals caught per site on average), but then we

would have had fewer than 6 species to analyze. The

second criterion, at least 12 HY individuals caught

among the 9 sites, may seem too low a threshold (an

average of 1.33 HY individuals caught per site).

Nevertheless, we wished to include possible

instances where reproductive success was poor or

25

III. Demographic Monitoring: Mist-nets

nil at a number of the 9 sites; such apparent

reproductive failure might be especially informative.

Thus, a hypothetical species with 5, 4, 2, 1, 0, 0, 0, 0, 0

captures at 9 sites would qualify with respect to the

“12 HY” criterion. On the other hand, a species with

12 HY captures at 1 site, and 8 sites without any HY

captures is not particularly informative. We thus set

a 3rd criterion: HY captures at a minimum of 3 sites.

All species that met criteria (1) and (2), also met the

3rd criterion.

To analyze the HY capture data, we log-transformed

capture rates (birds caught per 100 net-hours), for

each species at each site. To avoid taking the log of 0

(which is undefined), we added a constant—in this

case, 1—before log-transforming. Had there been no

zeroes in the data set, we would not have added any

constant; there would have been no need to. Whereas

adding a constant before log-transformation is

standard practice, it can lead to bias (Thomas 1996).

However, the direction of bias is conservative: adding

a constant makes it somewhat more difficult to detect

an effect (e.g., to detect a trend).We recommend that

investigators try two different constants (e.g., adding

0.5 and adding 1) and determine if results are similar.

If they are, then the investigator has some confidence

that his or her results are not unduly sensitive to the

chosen constant.

For analysis of the proportion of HY in the catch, we

used the logit-transformation. In the case where

total captures = HY + AHY,

logit(proportion of HY) = loge(HY/AHY).

26 Statistical Guide to Data Analysis of Avian Monitoring Programs

Table 8. Analysis of mist-net captures, Sacramento River 1993: Relationship to Damage Index for the

six species with adequate sample size (at least 12 HY individuals caught, and at least 36 individuals,

total, caught, among 9 sites).

A) Dependent Variable: Hatching Year birds per 100 net hoursa

Species Analysis Number HY Caught

Black-headed Grosbeak ß = –0.369 ± 0.192, P = 0.096, R2a= 0.251, R2= 0.345 12

McGillivray’s Warbler ß = –0.519 ± 0.405, P = 0.240, R2a= 0.074, R2= 0.190 39

Orange-crowned Warbler ß = –0.226 ± 0.078, P = 0.023, R2a= 0.479, R2= 0.544 15

Song Sparrow ß = –0.565 ± 0.357, P = 0.16, R2a= 0.159, R2= 0.264 55

Spotted Towhee ß = –0.649 ± 0.227, P = 0.024, R2a= 0.472, R2= 0.538 32

Yellow-breasted Chat ß = –0.462 ± 0.169, P = 0.029, R2a= 0.449, R2= 0.518 13

B) Dependent Variable: Proportion of Hatching Year birdsb

Species Analysis Number of Sites

Black headed Grosbeak ß = –0.901 ± 0.566, P = 0.15, R2a= 0.161, R2= 0.266 9

McGillivray’s Warbler ß = –1.054 ± 1.115, P > 0.3, R2a= 0.006, R2= 0.130 9

Orange-crowned Warbler ß = +1.623 ± 0.432, P = 0.007, R2a= 0.622, R2= 0.669 9

Song Sparrow ß = –0.819 ± 0.690, P > 0.2, R2a= 0.170, R2= 0.274 9

Spotted Towhee ß = –1.707 ± 0.789, P = 0.074, R2a= 0.344, R2= 0.438 8

Yellow-breasted Chat ß = –1.497 ± 1.117, P > 0.2, R2a= 0.117, R2= 0.264 7

a Hatching Year Birds caught per 100 net-hours, log-transformed, i.e. ln((HY caught + 1)/100 net-hours). Results of simple regression

analyses for effect of Vegetation Damage Index. Sample size = 9 sites for each analysis.

b Proportion of Hatching Year birds, logit-transformed Results of simple regression analyses for effect of Vegetation Damage Index.

Sample size (number of sites) for each analysis is shown.

Table 9. Analysis of mist-net captures, Sacramento River, 1993: Relationship of HY, and proportion HY birds

caught in relation to Vegetation Damage Index. Results of simple regression analyses; independent variable

in each model is Vegetation Damage Index. Number of sites (sample size) is 9. Capture rates have been

log-transformed, i.e. ln((number of birds + 1)/100 net-hours).

Dependent Variable Analysis

HY birds/100 net-hours ß = –0.795 ± 0.201, P = 0.006, R2a= 0.646, R2= 0.690

Proportion of HY birds caught

logit-transformed ß = –0.360 ± 0.120, P = 0.028, R2a= 0.453, R2= 0.521

The logit transformation is a commonly used

transformation in biological analysis and forms the

basis of logistic regression (Chapter V). Note that

the logit(proportion of HY) is undefined when the

denominator (in this case number AHY caught) is

zero.We could have added a constant to the

denominator to avoid this “problem” but did not;

we consider it biologically appropriate that our

measure of productivity is undefined when there

are (apparently) no adults present. For the analysis

in Table 8B, sites could not be included where

logit(proportion of HY) was undefined, i.e., where

no AHY were caught. This applied to two of the

six species.

As shown in Table 8, of the six species analyzed,

three showed a significant decline in capture rate

with increasing biological damage. Analyses of the

HY/AHY ratio indicated a consistently downward

trend with increasing damage (5 out of 6 species had

a negative slope), but no species had a significant

negative trend. These results suggest that sample

sizes of individual species were likely too small to

reveal significant patterns, and a pooled analysis

was carried out, shown in Table 9. Analyses of all

HY and AHY caught for all terrestrial bird species

were pooled and the results confirmed a significant

decrease in productivity with increase in damage

symptoms.

Analysis of Adult Survival

Survival can be analyzed in two ways: using

capture/recapture methods or analyzing “return

rate”. Return rate is the proportion of individuals

observed in one time period (we refer to this

period as t), which are observed again (resighted,

recaptured, etc.) in the following time period

(period t +1). Thus return rate is the product of two

processes: survival from period t to period t +1, and

resighting (or recapture) in period t +1. Resighting

probability is defined as the probability an individual

is resighted at time t +1, given that an individual

has survived until time t +1 (Clobert et al. 1987,

Nur & Clobert 1988). In short,

return rate = survival × recapture

probability .probability.

(We use “recapture” in a broad sense to refer to

both resighting and recapture.) The justification

for analyzing return rate as a means of studying

survival is the assumption that recapture probability

is 100% or, at least, that it can be treated as a

constant. This assumption is likely to be violated

when one is comparing the sexes, or comparing

different species or even different populations.

Capture/recapture methodology analyzes both

parameters, survival and recapture probability.

In this way, survival can be estimated independently

of recapture probability and one can test for

differences in survival as well as differences in

recapture probability (Lebreton et al. 1992). It

would seem that capture/recapture methodology

provides a superior means to analyze survival and,

in theory, it does. However, there are three

drawbacks to its usage:

1. Capture/recapture methods require at least three

field seasons to estimate survival for one year,

instead of two. 2. More data are required to carry

out these analyses than with return-rate analyses,

because two parameters are being estimated instead

of one. 3. The optimal software for survival analyses

is not yet available, one that combines flexibility,

statistical power, and ease of use, without requiring

specialized instruction. In the meantime, there are

several programs available which can fill the gap (for

more detailed discussion, see Lebreton et al. 1993).

Table 10 summarizes statistical programs that are

available for analyzing capture/recapture data

(based on Lebreton et al. 1992). Below we discuss six

programs that have been widely used (SURGE,

RELEASE, MARK, SURPH, JOLLY, and

JOLLYAGE).

General Comments. Capture/recapture models

such as SURGE require at least three field seasons

(usually years) in order to estimate survival between

the first season and the second, though it is possible,

making some assumptions, to derive survival

estimates for the period between the second and

third field seasons. Thus ten field seasons would

yield estimates of survival for each of eight years,

and so on. It is strongly recommended that the

capture occasions be equally spaced and generally

speaking the programs SURGE, RELEASE,

JOLLY, and JOLLYAGE assume this. If one is

seeking to estimate annual survival, then the

capture “occasion” is the year or breeding season.

In each year, an individual is either caught or

re-sighted (scored a “1”), or not observed (scored a

“0”). This allows one to construct a capture history

for each individual (a string of 1’s an