Power analysis for detecting trends in the presence of concomitant variables

Ecology, June, 1998 by David M. Nickerson, Arnold Brunell

INTRODUCTION

Wildlife management often requires the detection of trends in abundance over time, after adjusting survey counts for concomitant variables that are known to affect observability (e.g., rainfall, temperature, etc.). Here, as in Gerrodette (1987, 1991) and Link and Hatfield (1990), we wish to determine whether counts are increasing or decreasing over time using a straight line regression. In some applications (e.g., a power analysis) it is unnecessary to precisely model the pattern of change. In these cases, it is sufficient to determine whether there is, over time, a general propensity for the counts to become greater or less than their initial levels. This can be easily and sufficiently accomplished by modeling the counts after adjustments with a simple linear regression over time.

In this article we shall consider designing a sampling scheme for the detection of trends in abundance in the presence of concomitant variables. In this case, there are several questions that can be addressed. For instance, one question of interest is, How much data do we collect to be reasonably certain that we declare significantly different from 0, a trend of a given magnitude? Another question of interest would be, For a given sampling intensity what magnitude of trend would we be reasonably certain to declare significant? To answer these questions sufficiently, an a priori power analysis must be conducted.

The objectives of this article are similar to those of Gerrodette (1987, 1991) and Link and Hatfield (1990). Here, however, we extend the power analysis to the case of trends in counts adjusted for concomitant variables as in the analysis of covariance (see, e.g., Cochran 1957 for a general discussion of analysis of covariance). Specifically, we will exhibit the necessary computations of the probability of declaring significantly different from 0 trends of a given magnitude after adjustment for the effects of concomitant variables for a given amount of sampling.

METHODS

Let us consider the usual set-up in analysis of covariance for the one-way classification with k ([[greater than or equal to] 1) covariates or concomitant variables. Here, t ([greater than or equal to] 2) may represent the number of years in which data are collected on abundance and the k covariates. Within the ith year we have [n.sub.i] observations on abundance ([Y.sub.ij]) and the k covariates ([X.sub.ij1], [X.sub.ij2], ..., [X.sub.ijk]), j = 1, 2, ..., [n.sub.i] with i = 1, 2, ..., t. Here, the [n.sub.i]'s are not necessarily equal and we require [n.sub.i] [greater than or equal to] 2 for at least one year. We are assuming that we have included all the covariates that affect observed counts (but not population size), being careful to eliminate those that are unnecessary. Of the two actions, the greater danger is the exclusion of those covariates which are necessary.

As a matter of convenience (see, e.g., Cochran 1957), we will require for the mth covariate, m = 1, 2, ..., k,

[Mathematical Expression Omitted]

This can be easily achieved by redefining each covariate as

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted]

and

n = [summation of] [n.sub.i] where i = 2 to t

Consider now the usual analysis of covariance model

[Y.sub.ij] = [[Mu].sub.i] [[Beta].sub.1][X.sub.ij1] [[Beta].sub.2][X.sub.ij2] ... [[Beta].sub.k][X.sub.ijk] [[Epsilon].sub.ij] (1)

where the [[Epsilon].sub.ij]'s are independent and identically distributed (i.i.d.) normal random variables with mean 0 and variance [[Sigma].sup.2], i = 1, 2, ..., t and j = 1, 2, ..., [n.sub.i]. Here, we are assuming that the effects of the covariates on observed counts are additive and can be sufficiently accounted for by linear terms in Eq. 1. Also, the [[Mu].sub.i]'s are the mean counts and are generally only indices to abundance.

It is important to note that correct specification of the functional form of the effects of the covariates is critical. Incorrect specification can lead to serious bias in the estimates of [[Mu].sub.i], [[Mu].sub.2], ..., [[Mu].sub.t] and invalidate the analysis.

In a typical analysis of covariance, one of the major objectives is to estimate the mean count for each year, adjusting for the effects of the k covariates in the given year. Given Eq. 1, these adjusted means are simply the least squares estimates of [[Mu].sub.1], [[Mu].sub.2], ..., [[Mu].sub.t] obtained simultaneously with the estimates of [[Beta].sub.1], [[Beta].sub.2], ..., [[Beta].sub.k], the effects of the covariates.

Eq. 1 can be written as

[Y.sub.ij] = [[Mu].sub.1][Z.sub.ij1] [[Mu].sub.2][Z.sub.ij2] ... [[Mu].sub.t][Z.sub.ijt] [[Beta].sub.1][X.sub.ij1] [[Beta].sub.2][X.sub.ij2] ... [[Beta].sub.k][X.sub.ijk] [[Epsilon].sub.ij] (2)

where

[Z.sub.ijm] = 1, if i = m

= 0, if i [not equal to] m

i = 1, 2, ..., t, j = 1, 2, ..., [n.sub.i] and m = 1, 2, ..., t. Consequently, in matrix notation, Eq. 2 becomes

[Mathematical Expression Omitted]

where Y = [[Y.sub.1[prime]], [Y.sub.2[prime]], ..., [Y.sub.t[prime]]][prime] with [Y.sub.i] = [[Y.sub.i1], [Y.sub.i2], ..., [Y.sub.i[n.sub.i]]][prime], [Mu] = [[[Mu].sub.1], [[Mu].sub.2], ..., [[Mu].sub.t]][prime], [Beta] = [[[Beta].sub.1], [[Beta].sub.2], ..., [[Beta].sub.k]][prime], [Epsilon] = [[[Epsilon].sub.1][prime], [[Epsilon].sub.2][prime], ..., [[Epsilon].sub.t][prime]][prime] with [[Epsilon].sub.i] = [[[Epsilon].sub.i1], [[Epsilon].sub.i2], ..., [[Epsilon].sub.i[n.sub.i]]][prime], Z = [[Z.sub.i][prime], [Z.sub.2][prime], ..., [Z.sub.t][prime]][prime] with