# Issues when data is unavailable when sampling a finite population without replacement

I am specifying a sample of weekly work records to be entered into a spreadsheet for an estimation of means on two variables. There are about 250 employees, I am only interested in weeks between a specific date and the present, the total universe of possible work weeks is 28,538. Because we wanted to assure that the sample was representative by employee and calendar quarter, the population was stratified by calendar quarter and employee, and random sampling with replacement was used within the strata to define a sample of 2190 employee weeks.

I recently learned that the data for the first year of the time period we're interested in is unavailable. The unavailable population data includes 50 weeks out of 341 weeks in the period of interest, and also includes 23 out of the 250 employees. Overall, of the 2190 lines specified in the sample as designed, about 175 lines are unavailable.

A biased sample representing one employee has already been drawn from the population, the obtained sample variance on the statistic of interest was 115.37, n=276.

I realize that the unavailability of data for a certain time period systematically biases the sample's representativeness across calendar quarters, and that bias is introduced by the fact that 23 employees are not represented at all. My question is, what other concerns are introduced by this limitation in the availability of data from which to sample?

My statistics experience comes from laboratory experimental design; I now find myself charged with questions regarding sampling and power in a naturalistic environment, any insight would be most helpful. Thanks.

I think you can reasonably employ a two-way random effects model $$y_{it} = \mu + u_i + v_t + \epsilon_{it}$$ with $i$ enumerating employees, and $t$, weeks. If all 250 employees have been with the firm for the whole duration of the study, and if you are sampling at the same rates in all strata, then you have an uninformative sample design that does not need any weighting. You will even get a (nearly) balanced data set in terms of the number of available employees and weeks (where the "nearly" qualifier is due to the missing data problem). So you can simply run ANOVA on this and pull the overall mean out of that. Model-based inference in finite population sampling is as respectable as randomization-based approach. See, e.g., Valliant et. al. (2000) (where you might also stumble upon a better approach for this problem).
A reservation regarding this model is, of course, that the data must be missing at random (if not completely at random). This would be violated if, for instance, the employees demonstrated such a low productivity in these weeks that they were put on probation, or sent off to an additional training, or something of that kind. An extension you might need to consider is to allow $v_t$ to be a more sophisticated time-series process, such as AR(p).