Re: st: How to test for equality of variance in data with sampling weights

[Steven Samuels <ssamuels@albany.edu>]

Yohannes wrote to me privately that he has a household indicator, say  
hh_id; that household is the only PSU he can identify in the data  
set; and that in a single urban setting, there is no stratum  
variable. In that case he would set up his analysis with:
 "svyset hh_id [pweight=finalwgt]"

As Stas indicated in HIS recent email, one can compute a SD as a  
function of expectations and use either -testnl- or (my choice) - 
nlcom-   because:

 SD(income)= square root of E(income^2)-(E(income))^2

However  -nlcom- will produce an erroneous standard error unless the  
variable is standardized by subtracting off the mean: inc= inc - mean 
(inc) and then squaring : inc2= inc*inc  Then E(inc2) = var(income)   
and sqrt(E(inc2)) estimates the SD of income

As the SD is apt to have an asymmetric distribution, I suggest that  
the Johannes estimate the SE for the log(SD) and then convert back to  
the SD scale.

Johannes actually wants to compare SD's in two groups, assumed to be  
Male & Female gender here for illustration.   In that case, I  
recommend that he compute a CI for the ratio, rather then for the  
difference, and that he do this on the log scale and then convert  
back to the ratio scale.



Below is code that should work.  This utilizes the linearization  
method.  Possibly Johannes might wish to try a jackknife estimate of  
the variance-covariance matrix.
Steve

/***************************CODE  
FOLLOWS*********************************************/

capture program drop _all

/* First a little program to back transform calculations done on the  
log scale after -nlcom- */
program antilog
local lparm  el(r(b),1,1)
local se     sqrt(el(r(V),1,1))
local bound  invttail(e(df_r),.025)*`se'  //For 95% CI's
local parm   exp(`lparm')
local ll     exp(`lparm'  - `bound')
local ul     exp( `lparm' + `bound')
di  "parm =" `parm'  "    ll = " `ll'  "   ul = " `ul'
end

/* Get Estimate of the Mean for each Group */

svy: mean income, over(gender)

/* If gender has value labels (e.g. 1=Male 2=Female) use the  
following syntax */
gen     inc=income-[income]Male   if gender==1
replace inc=income-[imcome]Female if gender==2

/* Use this syntax if gender has no value label, but values 1 & 2 as  
above */
gen     inc=income-[income]1 if gender==1
replace inc=income-[imcome]2 if gender==2

/* Now compute the square term */

gen inc2=inc*inc

svymean: inc2, over(gender)   //estimate for inc2 is the estimated  
Variance of income

/* Individual SD's. Log Scale */
nlcom  .5*log([inc2]Male)
antilog
nlcom  .5*log([inc2]Female)
antilog

/* CI for the ratio of SD's--No Log */
nlcom sqrt([inc2]Male/[inc2]Female)

/* CI for ratio of SD's after Log Transformation. The square root can  
be omitted, because log(A^.5)-log(B^.5) = log(A)-log(B)
The t-statistic is apt to be very different from that of the no-log  
version above*/

nlcom log([inc2]Male/[inc2]Female)
antilog




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