Solved – Is it possible to manually calculate standard deviation for a multiply-imputed survey variable based on the standard error (SE)

multiple-imputationstandard deviationsurvey

I am analyzing a multiply-imputed complex sample survey data using Stata. For normally distributed numerical variables I want to report the mean and standard deviation. However, the Stata command for estimating mean of a multiply-imputed survey data mi estimate: svy: mean [varlist] does give the standard error of the mean, not the standard deviation. I tried to search for valuable help using Google but in vain. My question is this: Under such circumstances, is it possible to obtain an unbiased estimate of the standard deviation using the formula $\sigma$ $=$ $SE$ $\Huge.$ $\sqrt{n}$?

Best Answer

If you really want to get to the standard deviation of the population distribution, you should mi xeq : generate y2 = y*y the squares of the variables, and then

mi estimate (sd : sqrt( _b[y2] - _b[y]*_b[y] ) ) : svy : mean y y2
mi testtransform sd

Note that the interface of multiple imputation and inference with complex survey data is extremely poorly researched into given the ubiquity of the issue. I outlined the literature and the steps elsewhere on statalist.

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Solved – Negative imputed values

I think you just continue with your analysis. For one thing, negative income is not impossible. If you lose money (e.g. on investments) and don't make any, that would be negative income.

Solved – Cluster-robust SE in Stata when using a survey design

A standard error for regress with a cluster() option will always be larger than that from svy: regress, with the ratio of squared standard errors equal to $\dfrac{n-1}{n-k}$, where $n$ is the sample size and $k$ is the number of predictors, including the intercept.

Here's a modification of your example to demonstrate this. The change in definition of cluster should make no difference.

sysuse auto, clear
set seed 92122
gen mkr = substr(make,1,2)
codebook mkr // 23 clusters
*run the "manual" regression
reg price mpg weight [pw = gear], cluster(mkr)
scalar v1 = _se[mpg]^2
scalar k = e(rank) // number of predictors + 1
scalar n = e(N)

*using svy design
svyset mkr [pw = gear]
svy : reg price mpg weight
scalar v2 = _se[mpg]^2
di " v1 = " v1   "    v2 = " v2
di " n =  "n   "    k = " k
di " v1/v2 = "v1/v2
di " (n-1)/(n-k) = "(n-1)/(n-k)

The results of the display statements:

v1 = 5364.5846 v2 = 5217.6097
n = 74 k = 3
v1/v2 = 1.028169
(n-1)/(n-k) = 1.028169

The values in the last two lines are identical. Page 469 of the Stata 14 Manual entry for _robust, (http://www.stata.com/manuals14/p_robust.pdf) refers to a multiplier $\dfrac{n}{n-k}$ that should make the two calculations equivalent. I verified the ratio $\dfrac{n-1}{n-k}$ in other examples. I can't account for the difference.

Best Answer

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Solved – Negative imputed values

Solved – Cluster-robust SE in Stata when using a survey design

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