DATA DEMO INVOLVING STRATIFIED SAMPLES
======================================= 10/10/05

DATA from Example 4.3, Table 4.2

> ACLS.fr <- data.frame(matrix(c(9100, 1950, 5500, 
     10850, 2100, 5500, 9000, 915, 633, 658, 855, 667, 833, 824, 636,
     451, 481, 611, 493, 575, 588, 38, 27, 18, 19, 36, 13, 26),
     ncol=4, dimnames=list(c("LIT","CLAS","PHIL","HIST","LING",
     "POLT","SOCY"), c("Members","Mailed","Returned","Female.Pct"))))

## Stratified estimate of % Females, assuming SAME among responders 
##   as  non-responders
> attach(ACLS.fr)
  pstr <- sum(Members*Female.Pct)/sum(Members) ## ans 24.65%

## SE taking percentages as fractions, and numbers returned as
##   sample-sizes:
> SEpstr <- sqrt(sum((1-Returned/Members)*(Members/44000)^2*
    (Female.Pct/100)*(1-Female.Pct/100)/(Returned-1)))
> SEpstr                                                    
[1] 0.007118622

## We can ask: suppose the same total number of returns were
   distributed equally [resp. proportionately to members] among 
   the 7 Professional Societies, and the observed fractions 
   female were the same, then how would the SE change ?

> nalt1 <- round(sum(Returned)/7)
  nalt2 <- round(sum(Returned)*Members/44000)
  rbind(Equal=nalt1, PropSize=nalt2, Actual=Returned)
         [,1] [,2] [,3] [,4] [,5] [,6] [,7]
Equal     548  548  548  548  548  548  548
PropSize  793  170  479  946  183  479  784
Actual    636  451  481  611  493  575  588

## SE under Actual allocation was .0071.
## Under Equal Allocation:

> sqrt(sum((1-nalt1/Members)*(Members/44000)^2*
    (Female.Pct/100)*(1-Female.Pct/100)/(nalt1-1)))
[1] 0.00743646

## Under Proportional Allocation:

> sqrt(sum((1-nalt2/Members)*(Members/44000)^2*
    (Female.Pct/100)*(1-Female.Pct/100)/(nalt2-1)))
[1] 0.006522492


## Now let's find the optimal allocation and associated SE.

>  Shsq <- Female.Pct/100*(1-Female.Pct/100)
   nopt <- round(sum(Returned)*Members*sqrt(Shsq)/
       sum(Members*sqrt(Shsq)))
> nopt
[1] 918 180 439 884 209 384 820

> sqrt(sum((1-nopt/Members)*(Members/44000)^2*Shsq/(nopt-1)))
[1] 0.006474661       
### So proportional allocation was already close to optimal!

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NB. In this Example: all one can actually estimate is the
proportion of females among those who WOULD respond if asked.