Simple Transformation

The simplest application of the delta method is when we manipulate a single variable (Table 1). For example, we might have a known number of nesting female birds (N), with a subsample of nests from which we obtain an average clutch size (μ̂ _c) and its sampling variance, vâr(μ̂_c). From this sample of nests, a biologist might need to predict the number of eggs produced by the population, as well as the sampling variance for this prediction. To apply the delta method, we must start by describing the relationship between the demographic rates in question. The estimate of total egg production (𝑝̂) by the nesting population would be: 𝑝̂  =  n · μ̂_c. But, we also need to derive the variance for 𝑝̂.

Table 1

General rules for calculating sampling variances using the delta method. Examples are provided for each set of simple relationships. In the functions provided, c is a constant, and θ is a parameter (e.g., survival rate or density estimate).

To use the delta method to arrive at this approximation, using equation 1 with a single variable transformation, we simply have:

Table 2

Simple rules for calculating derivatives for use in the approximation of variance using the delta method. In the functions provided, c is a constant, and x is a parameter (e.g., survival rate or density estimate). Derivative rules after Larson and Hostetler (1982).

Changing Temporal Scale

Let's evaluate a second example with a single variable. Consider that we want to take a daily nest survival rate estimate (Ŝ_d), and extrapolate it to a weekly nest survival estimate (Ŝ_w). Following the process outlined in our first example, above, we first need to establish the relationship as: Ŝ_w  =  (Ŝ_d)⁷.

Next, substitute values in equation 1:

We have an estimate of var(Ŝ_d) from our daily nest survival analysis, so we only need the derivative of Ŝ_d⁷. In this case, our parameter of interest (x) for the derivative is Ŝ_d, and it is raised to the power of 7. We use the x^c rule in Table 2, to find:

More examples of changes in temporal scale for survival rate estimates of interest to avian ecologists are provided in Table 3 (an on-line variance calculator for these examples can be found at < http://snr.unl.edu/powell/research/research.htm).

Table 3

Approximations for sampling variance of survival estimates, by the delta method, when changing the scale of temporal units.

Annual Population Growth

Annual, discrete population growth (λ) can be defined for avian populations as: λ  =  S_A + B·S_J (Pulliam 1988). Avian biologists commonly use this relationship to analyze source-sink dynamics. If λ_i < 1 then populations are classified as sinks during year i (mortality exceeds reproduction); similarly, if λ_i > 1 then populations are classified as sources (reproduction exceeds mortality). Thus, it becomes necessary to derive vâr(λ ˆ) to rigorously determine if the population is increasing or decreasing; indeed, confidence intervals for each quantity could be used to reject H_o: λ  =  1. However, I am not aware of a published method for deriving var(λ) when λ  =  S_A + B·S_J.

Because λ is a function of more than one random variable, each of which has an associated sampling variance estimate, it follows that λ will have a variance that is a function of the sampling variance of the individual parameters. Further complicating matters, fecundity (B) is defined as the number of female fledglings per year, and B is a function of three random variables, B  =  π • φ • ψ, where π equals mean number of female fledglings per successful nest, φ equals nest survival probability, and ψ equals the average number of nests built per female per year. Thus, var(B) is a function of the variances of π, φ, and ψ.

To derive vâr(𝐵̂), where 𝐵̂  =  π̂ · φ̂ · ψ̂ (assuming independence of π, φ, and ψ), we use equation 1:

(7)

After finding the three partial derivatives (use the cx rule in Table 2) in equation 7, we arrive at:

(8)

We now have a derived variance for 𝐵̂, which could be useful on its own merits. But, to obtain vâr(λ ˆ), the next step is to use vâr(𝐵̂) with direct estimates (potentially from mark-recapture survival analyses) of vâr(𝑆̂_A) and vâr(𝑆̂_J) to approximate the variance of λ.

To derive vâr(λ̂), where λ̂  =  𝑆̂_A + 𝑆̂_J · 𝐵̂,

(9)

After taking the three partial derivatives (use rules in Table 2) in equation 9, we arrive at:

(10)

Mean Annual Density

Let's consider a second example of the transformation of multiple variables. Biologists often obtain annual estimates of density for a bird species over multiple years. Consider a situation in which a biologist is interested in the effects of prescribed burning on grassland bird densities. To compare densities in different treatment types (burned, b, and control, c) across years (i), it is necessary to calculate the mean density (𝐷̄^b_i, 𝐷̄^c_i) in each treatment. The sampling variance of the mean density for the burned treatment, vˆr(𝐷̄^b), is not simply the average sampling variance of the annual estimates of 𝐷̂^b_i used to calculate 𝐷̄^b. But, as expected, vâr 𝐷̄^b is certainly a function of the annual sampling variances of 𝐷̂^b_i.

For our example, let's consider the data from only the burned portion of the above experiment and assume we have five years of density estimates, 𝐷̄^b₁, 𝐷̄^b₂, 𝐷̄^b₃, 𝐷̄^b₄, and 𝐷̄^b₅. Again, our goal is to obtain 𝐷̄^b and vâr(𝐷̄^b). For simplicity, we'll assume that the densities are estimated from separate datasets, and the annual estimates are independent. The relationship of the parameters is:

(12)

To calculate the partial derivatives, each 𝐷̂^b_i becomes the parameter of interest (x) and all other terms in equation 11 are constants (c). We use the cx and c rules in Table 2. By the c rule, the derivative of the string of constants is 0 (Table 2). Thus, with respect to each 𝐷̂^b_i in equation 12, the partial derivative is:

(14)

Effect Size (correlated Variables)

The use of the delta method is most straightforward when dealing with single variable transformations. When working with multiple variables, it is possible (probable in many cases) that variables X₁, X₂, ..., X_n (equation 1) will not be independent. In this case, the covariance between variables must be considered when approximating the sampling variance (equation 2).

Doherty et al. (2002) incorporated covariance into an approximation of the sampling variance of effect size. Although a useful example of the use of the delta method for multiple variables that are not independent, Doherty et al. (2002) do not provide instructions to guide similar applications. To understand how Doherty et al. (2002) used the delta method to arrive at the formula in their manuscript, we begin with the function in question. Doherty et al. (2002) estimated effect size (θ̂) as the ratio of male and female fidelity rates (F^m and F^f, respectively), where . Sampling variance for the individual fidelity rates is available, but we now need vâr(θ̂).

To use the delta method (equation 2) to approximate vâr(θ̂), incorporating covariance between F^m and F^f, we have:

We use two rules to find the partial derivatives in equation 15. For the partial derivative when F^m is our parameter of interest (x), our constant (c) is 1/F^f, and we use the x/c rule (Table 2). When, F^f is our parameter of interest (x), our constant (c) is F^m, and we use the c/x rule (Table 2). Thus, the partial derivatives are: . Substituting the partial derivatives into equation 15 and simplifying, we have:

The estimate for covariance can be obtained from software packages that provide variance-covariance matrices. But, in other cases, covariance matrices must be derived. MacKenzie et al. (2006) and Cooch and White (2006) provide additional examples of how to incorporate covariance matrices into the delta method.