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Has there been any research done on how to estimate the population median using multistage sampling? Simple random sampling is not possible in my case I would really like to use something like multistage sampling.

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Suppose you are interested in the median $m$. For the ordered values, $x_1, x_2, \dots x_n$, compute the estimated weighted proportions $\hat{F}(x_1), \hat{F}(x_2) \dots$, where $\hat{F}(x_i)$ is the estimated proportion of observations $ X $ that are $ \le x_i$.

\begin{equation} \hat{F}(x_i)= \frac{\sum_{j=1}^n w_j I(x_j\le x_i)}{\sum_{j=1}^n w_j} \end{equation}

Here $w_j$ is the sampling weight for observation $j$.

Imagine that 37 and 41 are successive values of x in the sample such that $\hat{F}(37) = 0.46$ and that $\hat{F}(41) = 0.53$. Since $F(m)=0.50$, it's obvious that the estimated median $\hat{m}$ must be between 37 and 41. The value of $\hat{m}$ can be calculated by linear interpolation.

$$ \begin{aligned} \hat{m} &= 37 + \frac{0.50 -0.46}{0.53 -0.46}\times(41-37) \\ & = 37 + \frac{.04}{.07}\times 4 = 39.28 \end{aligned} $$

Note that this result depends only on the sampling weights, not on any other aspects of the survey design. The first reference that I know of is Woodruff (1952). Other quantiles are estimated in a similar way.

What has been the subject of research is the estimation of confidence intervals for quantiles. Woodruff (1952) contained a method that is based on the asymptotic normality of the sample weighted proportions $\hat{F}$. The method is still widely used. See the quick reference list below.

Some survey packages that estimate quantiles with standard errors/confidence intervals

  • R: Tom Lumley's Survey package includes the svyquantile function.
  • Stata: Stas Kolenikov's epctile command (findit epctile in Stata)
  • SUDAAN
  • SAS SVYMEANS
  • WesVar

References

Binder DA (1991) Use of estimating functions for interval estimation from complex surveys. Proceedings of the ASA Survey Research Methods Section 1991: 34-42

Dorfman A, Valliant R (1993) Quantile variance estimators in complex surveys. Proceedings of the ASA Survey Research Methods Section. 1993: 866-871

Francisco, C. A. and Fuller, W. A. 1991. “Quantile Estimation With a Complex Survey Design,”. The Annals of Statistics, 19: 454–469.

Shah BV, Vaish AK (2006) Confidence Intervals for Quantile Estimation from Complex Survey Data. Proceedings of the Section on Survey Research Methods. http://www.amstat.org/sections/SRMS/Proceedings/y2012/files/304420_73075.pdf

Sitter, Randy R, and Changbao Wu. 2001. A note on Woodruff confidence intervals for quantiles. Statistics & probability letters 52, no. 4: 353-358.

Woodruff, Ralph S. 1952. Confidence intervals for medians and other position measures. Journal of the American Statistical Association 47, no. 260: 635-646.

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