variance and confidence interval of derived

[jlaake]

The following was sent to me as an email. I'll post the question here and then reply to it as others may have the same question.

I am using RMark under linux to analyze closed capture data on zebras in Kenya. I am using Huggins models to analyze my data, with differing capture probabilities for males and females (ie sex as an individual covariate). I have managed to successfully run various models (equivalent to Otis et al Mo, Mb, Mt, Mth etc). I find the design data specification to be easy and simpler than PIMS in Program Mark. Thank you very much for your package.

I would be extremely grateful if you could answer one question. I am a bit confused how to get total population estimates from Huggins model outputs. The output includes results of derived parameters (Model$results$derived$(estimate,se,lcl,ucl). But, it provides two estimates (one for males and one for females, my two "groups"). Can you please explain how I may combine these estimates to get one estimate (with standard error and confidence limits) for total population size?

Thank you for your help.

best,
Siva

[jlaake]

As part of the model results there should be an object derived.vcv

data(dipper)
model=mark(dipper)
model$results$derived.vcv

which is the variance-covariance matrix. It is NULL in the dipper example below because there are no derived parameters for CJS models. In general, variances of functions of parameters can be solved with the delta method -- See the appendix of the Gentle Intro to Mark that is online on phidot.org. In this case it is a simple sum. The variance of the sum is the sum of the variances and covariances. So your point estimate is the sum of the estimates for the groups and the variance is the sum of all the elements of derived.vcv. If you assume a log-normal distribution for the sampling distribution for your estimate then the ci is

C=exp(1.96*sqrt(log(1+cv(est)^2))
lowerlimit=est/C
upperlimit=est*C

where est is your estimated abundance and cv(est)^2 = var of the estimate/est^2 and log is the natural logarithm.

As a point of clarification here, you use the term individual covariate for sex but it was used for "groups". Factor (categorical) variables like sex are typically used as a grouping variable and "individual covariate" is reserved to refer to a variable that is not used as a grouping variable but is entered directly into the design matrix. An individual covariate in MARK cannot be a factor variable without creating dummy variables for each column of the design matrix. This may seem like a subtle point and regardless of how you did it you would get the same answer; however, if you did enter it as an individual covariate you would only get a single estimate of total abundance whereas by entering it as a grouping variable you get an estimate of abundance for each level of the factor variable (eg Male and Female for sex). So using sex as an individual covariate is an alternate way of deriving the total abundance.

regards --jeff