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## Stratified samples (5 replies)

This is an interesting question. The three forms of sampling amenable to useful calculations of estimates of sampling error are random sampling, systematic sampling and stratified random sampling. (See the esteemed Dr. W. G. Cochran's book "Sampling Techniques", Wiley, 1977.) Not including data from one strata of a stratified random sample results in what Dr. Deming called a "judgment sample". This does not necessarily mean that the results are not useful, only that no valid estimate can be made of the sampling error.

What's your relative level of knowledge of the missing strata and the other strata. Is there some degree of prior knowledge from earlier sampling or is this an entirely new area for you?

Are you assaying each stratified sample, or forming a composite from them?

The variance of the lot mean calculated from stratified sampling is

var(av)= [(Nstrat-Nsamp)/(Nstrat-1)]/var(bs)+var(ws)/(Nstrat x Nws).

Nstrat = Number of strata in the lot, Nsamp = Number strata from which samples are taken, Nws= number of samples taken from each stratum, var(bs) = variance between-strata and var(ws) = variance between samples taken within strata. If every stratum of the lot is sampled Nstrat-Nsamp = 0 and as the consequence of this the between-strata variance is eliminated from the variance of the lot mean. If the some of the strata are missing then the between-strata variance has to include in the calculations and the variance of the lot mean increases.

Assumptions: The stratum means are not correlated and the missing strata are not outliers (e.g., they belong to the same population with the other strata). You can read more, e.g. in Pentti Minkkinen, Practical Applications of Sampling Theory, Chemo metrics and Intelligent Laboratory Systems, 74 (2004) 85-94.

If your strata are auto correlated, e.g. samples taken from a process stream, then it safe to replace the missing value(s) by interpolation from the closest available values. This was studied in Maaret Paakkunainen, Jarmo Kilpeläinen, Satu-Pia Reinikainen, and Pentti Minkkinen, Effect of missing values in estimation of mean of auto-correlated measurement series, AnalyticaChimicaActa, 505 (2007) 209-215.

Following from earlier remarks, I would suggest that if you wish to find an optimal estimate of the missing sample assay, you should Krieg the value using the surrounding values. The process of kriging uses the data you have to estimate a variogram or covariance function for your data. If the covariance function can be found, this function describes the correlation between the consecutive values of your samples. There is an optimal manner of estimating the covariance function and the overall mean of your data set, given that the data set is not suffering a strong drift in the mean with time.

If you are interested in putting a confidence interval on the mean of your sample values over a specified time period, it is very important to investigate the correlation between sample values over time. This means that you should be estimating the covariance function on an on-going basis. There are very sound methods of finding the confidence interval for the mean if you have estimated the covariance for your data.

I will also add that keeping track of the variogram or covariance function for your data will also allow you to estimate your overall sampling and analysis error for your samples.

In stratified samples if we lost one strata how to deal with this situation