FW662 Lecture 14 – Estimating Variance Components 1Lecture 14. Estimating Variance Components. Reading: Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, and K. H. Pollock. 1987. Design and Analysis Experiments for Fish Survival Experiments Based onCapture-Recapture. Am. Fish. Monograph No. 5, Pages 260-278.We have discussed various sources of variance that impact the dynamics of a population:demographic, environmental and spatial (process), individual heterogeneity, and geneticvariances. In addition, the concept of sampling variance, or the uncertainty of our estimates ofpopulation parameters, has been frequently mentioned. This chapter covers the statisticalmethodology to estimate the different variance components from data.Consider the example situation of estimating survival rates each year for 10 years from a deerpopulation. Each year, the survival rate is different from the overall mean, because of snowdepth, cold weather, etc. Let the true, but unknown, overall mean be S. Then the survival ratefor each year can be considered to be S + some deviation:Environmental VariationMean Year Year i i i1 S S + e1S12 S S + e2S23 S S + e3S34 S S + e4S45 S S + e5S56 S S + e6S67 S S + e7S78 S S + e8S89 S S + e9S910 S S + e10S10Mean S¯S¯SThe estimator of S is :¯SFW662 Lecture 14 – Estimating Variance Components 2¯S 'j10i'1Si10ˆF2'j10i'1(Si&¯S)210with the variances of the :Siwhere the random variables are selected from a distribution with mean 0 and variance . IneiF2reality, we are never able to observe the annual rates because of sampling variation ordemographic variation. For example, even if we observed all the members of a population, wewould still not be able to say the observed survival rate was Si because of demographic variation. Consider flipping 10 pennies. We know that the true probability of a head is 0.5, but we will notalways observe that value exactly. The same process operates in a population as demographicvariation. Even though the true probability of survival is 0.5, we would not necessarily seeexactly ½ of the population survive on any given year.Hence, what we actually observe are the quantities:Environmental Variation + Sampling VariationMean Year Year i i i1 S S + e1 + f1ˆS12 S S + e2 + f2ˆS23 S S + e3 + f3ˆS34 S S + e4 + f4ˆS45 S S + e5 + f5ˆS56 S S + e6 + f6ˆS67 S S + e7 + f7ˆS78 S S + e8 + f8ˆS89 S S + e9 + f9ˆS910 S S + e10 + f10ˆS10Mean S¯SˆSwhere the are as before, but we also have additional variation from sampling (or demographiceiFW662 Lecture 14 – Estimating Variance Components 3Var(ˆSi*Si) 'ˆSi(1 &ˆSi)niˆS 'j10i'1ˆSi10variation), .fiThe standard approach to estimating the sampling variance separately from the environmentalvariance is to take replicate observations within each year in this example, so that the within cellreplicates can be used to estimate the sampling variance, whereas the between cell variance isused to estimate the environmental variation. Years are assumed to be a random effect, andmixed model analysis of variance procedures are used (e.g., Bennington and Thayne 1994). Thisapproach assumes that each cell has the same sampling variance. An example of the applicationof a random effects model is Koenig et al. (1994). They considered year effects, species effects,and individual tree effects. The assumption of constant variance within cells across a variety of treatment effects is often nottrue, i.e., the sampling variance of a binomial distribution is a function of the parameter estimate. Another common violation of this assumption is caused by the variable of interest beingdistributed lognormally, so that the coefficient of variation is constant across cells, so that the cellvariance is a function of the cell mean. Further, the empirical estimation of the variance fromreplicate measurements may not be the most efficient procedure. Again, the binomial estimatorof a survival rate is a good example of this. Therefore, the remainder of this chapter describesmethods which can be viewed as extensions of the usual variance component analysis based onreplicate measurements within cells. We will be examining estimators for the situation where thewithin cell variance is estimated by some other estimator than from the moment estimator basedon replicate observations.Assume that we can estimate the sampling variance for each year, given a value of for theSiyear. For example, the sampling variation for a binomial is where is the number of animals monitored to see if they survived. Then, can we estimate thenivariance term due to environmental variation, given that we have estimates of the samplingvariance for each year?If we assume all the sampling variances are equal, the estimate of the overall mean is still just themean of the 10 estimates: with the theoretical variance beingFW662 Lecture 14 – Estimating Variance Components 4var(ˆS) 'F2% E[var(ˆS*S)]10ˆvar(ˆS) 'j10i'1(ˆSi&ˆS)210(10 & 1)ˆE(var(ˆS|S)] 'j10i'1ˆvar(ˆSi*Si)10ˆF2'j10i'1(ˆSi&ˆS)2(10 & 1)&j10i'1ˆvar(ˆSi*Si)10wi'1F2% var(ˆSi*Si)ˆS 'j10i'1wiˆSij10i'1wivar(ˆS) '1j10i'1wii.e., the total variance is the sum of the environmental variance plus the expected samplingvariance. This total variance can be estimated asWe can estimate the expected sampling variance as the mean of the sampling variances so that the estimate of the environmental variance obtained by solving for F2However, normally, the sampling variances are not all equal, so that we have to weight them toobtain an unbiased estimate of . The general theory says to use a weight, F2wiso that the estimator of the weighted mean iswith theoretical variance (see Box 1 for a derivation of this result)FW662 Lecture 14 – Estimating Variance Components 5ˆvar(ˆS) 'j10i'1wi(ˆSi&ˆS)2j10i'1wi(10 & 1)1j10i'1wi'j10i'1wi(ˆSi&ˆS)2j10i'1wi(10 & 1)1 'j10i'1wi(ˆSi&ˆS)2(10 & 1)j10i'1wi(ˆSi&ˆS)2(10 & 1)'P210 & 1,"L10 & 1j10i'1wi(ˆSi&ˆS)2(10 & 1)'P210 & 1,"U10 & 1and empirical variance estimatorWhen the are the true (but unknown) weights, we have wigiving the followingHence, all we have to do is manipulate this equation with a value of (that is imbedded in theF2term) to obtain an estimator of .wiF2To obtain a confidence on the estimator of , we can substitute the appropriate chi-squareF2values in the above relationship. To find the upper confidence interval value, , solve theˆF2Uequationand for the lower confidence interval value, , solve the