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The data comes from agricultural market research on farming. The sample was derived based on stratification of farming industries (sheep, beef, grains, etc.) and random sampling within each stratum.

We have population estimates (frequencies, percentages/proportions) of these industry strata. Likewise, we have frequencies and proportions of each stratum in the sample.

A weight was calculated for each farming stratum by dividing the population proportion by the sample proportion.

I'm not sure what that weight means. What I know is weight is an inverse probability of selection of a unit into the sample. Can you give a hint?

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Let $N$ be the population size and $n$ the sample size, let $N_h$ and $n_h$ be the population and sample sizes for stratum $h$.

Then, the weight you defined is given by

$ W_h = \frac{N_h/N}{n_h/n} = \frac{N_h}{n_h}\frac{n}{N}$

where $\frac{n}{N}$ is the sampling fraction $f$ for the whole sample and $\frac{N_h}{n_h}$ is the inverse of the sampling fraction, i.e., of the probability of selection, in the $h$-th stratum, $f_h$. Put it differently, $w_{hi}=N_h/n_h$ is the inverse probability sampling weight of a unit $i$ in stratum $h$ that you are familiar with.

Writing it as $W_h = \frac{f}{f_h}$, you can see that \begin{equation} \begin{cases} W_h < 1 ,\qquad f_h > f\\ W_h = 1 , \qquad f_h = f\\ W_h > 1 , \qquad f_h <f\\ \end{cases} \end{equation}

These are relative weights showing by how much a given stratum was under- or oversampled. These are OK weights to deal with the ratio-type statistics (means, proportions, regression estimates). For the totals, e.g. total acreage under a given crop, or a total harvest, you need the correct inverse probability weights rather than relative weights.

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  • $\begingroup$ Thanks for your mathematical representation. I think I might not be clear about the question. What I wanted to know was the interpretation of this weighting scheme. For example, as in Levy and Lemeshow's text Sampling of Populations, weights are the inverse selection probability, or the number of individuals in the target population represented by each sample unit. Following that, weights are generally greater than 1. But here we will expect to see a fair number of cases where weights are either less than or equal to 1. $\endgroup$
    – NonSleeper
    Apr 16, 2017 at 20:45
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    $\begingroup$ I think you're talking about the weights attributed to each sample unit during estimation. It is true that expansion estimators weight each unit by the reciprocal of their inclusion probability (not selection probability, althought they coincide in simple random sampling), see for example the well known Horvitz-Thompson estimator. However, your question is about stratification weights, the weights attributed to each strata in the computation of the final estimator. $\endgroup$
    – Roberto
    Apr 17, 2017 at 9:16
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    $\begingroup$ I edited @Roberto's answer to provide more context. These relative weights are unfortunate; the population weights should be used whenever possible. $\endgroup$
    – StasK
    Apr 18, 2017 at 13:47

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