1. Estimating population size and a ratio
Consider first estimating the population mean
An obvious choice:
Alternative: Estimate N as well, whether N is known or not

2. Illustration when the studyvariable is a constant c with no variation.
Second eq. Follow since y is fixt.
3rd eq. is obvious

3. If sample size varies then the “ratio” estimator performs better than the H-T estimator, the ratio is more stable than the numerator
Example:

4. H-T estimator varies because n varies, while the modified H-T is perfectly stable
Sum y_i/pi_i
Y_i=c and pi_i=pi for all i
t_ht = nc/pi
n is bin (N,pi)
N* y_tilde -> N* (sum(c/pi) / sum (1/pi)) = N*((c/pi)sum_s 1) / 1/pi*sum_s 1

5. Review of Advantages of Probability Sampling
Objective basis for inference
Permits unbiased or approximately unbiased estimation
Permits estimation of sampling errors of estimators
Use central limit theorem for confidence interval
Can choose n to reduce SE or CV for estimator

6. Remaining issues in design-based inference
Estimation for subpopulations, domains
Choice of Survey strategy –
discuss several different sampling designs
appropriate estimators
More on use of auxiliary information to improve estimates
More on variance estimation

7. Estimation in domains
-Let D be the number of domains in the population. We have D subsets, U1,…Ud,…,UD for which estimates are needed.
-Let Nd be the size of domain d and nd be the number of sampled units that falls into domain d.
Then we have the following partitions of the population and the sample.
Each unit in the population in the population belongs to exactly one domain.
Hence, the set of population units denoted by U is the union of the D non-overlapping subsets Ud. The set of the sampled elements units denoted by s is the union of the D non-overlapping subsets sd

8. Estimation in domains Ex: Population = all adults aged 16-64
Examples of domains:
Women
Adults aged 35-39
Men aged 25-29
Women of a certain ethnic group
Adults living in a certain city
The objective is to estimate the domain totals or domain means.

9. Estimating domain means Simple random sample from the population
e.g., proportion of divorced women with psychiatric problems.
Note: nd is a random variable

10. The estimator is a ratio estimator:

12. Can then treat sd as a SRS from Ud
Whatever size of n is, conditional on nd, sd is a SRS from Ud – conditional inference
Example: Psychiatric Morbidity Survey 1993
Proportions with psychiatric problems
Domain d nd SE
women
4933
0.18
Divorced women
314
0.29

13. Estimating domain totals
Nd is known: Use
Nd unknown, must be estimated

14. Stratified sampling
Basic idea: Partition the population U into H subpopulations, called strata.
Nh = size of stratum h, known
Draw a separate sample from each stratum, sh of size nh from stratum h, independently between the strata
In social surveys: Stratify by geographic regions, age groups, gender
Ex –business survey. Canadian survey of employment. Establishments stratified by
Standard Industrial Classification – 16 industry divisions
Size – number of employees, 4 groups, 0-19, 20-49, , 200+
Province – 12 provinces
Total number of strata: 16x4x12=768

15. Reasons for stratification
Strata form domains of interest for which separate estimates of given precision is required, e.g. strata = geographical regions
To “spread” the sample over the whole population. Easier to get a representative sample
To get more accurate estimates of population totals, reduce sampling variance
Can use different modes of data collection in different strata, e.g. telephone versus home interviews

16. Stratified simple random sampling
The most common stratified sampling design
SRS from each stratum
Notation:

17. th = y-total for stratum h:
Consider estimation of th:
Assuming no auxiliary information in addition to the “stratifying variables”
The stratified estimator of t:

18. A weighted average of the sample stratum means.
Properties of the stratified estimator follows from properties of SRS estimators.
Notation:

19. Estimated variance is obtained by estimating the stratum variance with the stratum sample variance
Approximate 95% confidence interval if n and N-n are large:

20. Estimating population proportion in stratified simple random sampling
ph : proportion in stratum h with a certain characteristic A
p is the population mean: p = t/N
Stratum mean estimator:
Stratified estimator of the total t = number of units in the
with characteristic A:

21. Estimated variance:
55)

22. Allocation of the sample units
Important to determine the sizes of the stratum samples, given the total sample size n and given the strata partitioning
how to allocate the sample units to the different strata
Proportional allocation
A representative sample should mirror the population
Strata proportions: Wh=Nh/N
Strata sample proportions should be the same:
nh/n = Wh
Proportional allocation:

23. The stratified estimator under proportional allocation
Inclusion probabilities:
the same for all units in the population, but it is not an SRS
The equally weighted sample mean ( sample is self-weighting: Every unit in the sample represents the same number of units in the population , N/n)

24. Variance and estimated variance under proportional allocation

25. The estimator in simple random sample:
Under proportional allocation:
but the variances are different:

26. Total variance = variance within strata + variance between strata
Implications:
No matter what the stratification scheme is:
Proportional allocation gives more accurate estimates of
population total than SRS
Choose strata with little variability, smaller strata variances. Then
the strata means will vary more and between variance becomes
larger and precision of estimates increases compared to SRS

27. Constructing stratification and drawing stratified sample in R
Use API in California schools as example with schooltype as stratifier.
3 strata: Elementary, middle and high schools.
Stratum1: Elementary schools, N1 =4421
Stratum 2: Middle schools, N2 = 1018
Stratum 3: High schools, N3 = 755
5% stratified sample with proportional allocation:
n1 = 221
n2 = 51
n3 = 38
n = 310

28. R-code: making strata >x=apipop$stype
# To make a stratified variable from schooltype:
>make123 = function(x) +{
+ x=as.factor(x)
+ levels_x = levels(x)
+x=as.numeric(x)
+attr(x,"levels") = levels_x
+ x +}
> strata=make123(x)
> y=apipop$api00
> tapply(y,strata,mean)
# 1=E, 2=H, 3 = M. Will change stratum 2 and 3

29. > x1=as.numeric(strata<1.5)
> x2=as.numeric(strata<2.5)-x1
> x3=as.numeric(strata>2.5)
> stratum=x1+2*x3+3*x2
> tapply(y,stratum,mean)
> # stratified random sample with proportional allocation
> N1=4421
> N2=1018
> N3=755
> n1=221
> n2=51
> n3=38
> s1=sample(N1,n1)
> s2=sample(N2,n2)
> s3=sample(N3,n3)

30. > y1=y[stratum==1]
> y2=y[stratum==2]
> y3=y[stratum==3]
> y1s=y1[s1]
> y2s=y2[s2]
> y3s=y3[s3]
> t_hat1=N1*mean(y1[s1])
> t_hat2=N2*mean(y2[s2])
> t_hat3=N3*mean(y3[s3])
> t_hat=t_hat1+t_hat2+t_hat3
> muhat=t_hat/6194
> muhat
[1]
> mean(y1s)
[1]
> mean(y2s)
[1]
> mean(y3s)
[1]

31. > varest1=N1^2*var(y1s)*(N1-n1)/(N1*n1)
> se=sqrt(varest1+varest2+varest3)
> se
[1]
> semean=se/6194
> semean
[1]
> CI=muhat+qnorm(c(0.025,0.975))*semean
> CI
[1]
#CI = (647.7, 676.1)

32. Suppose we regard the sample as a SRS
> z=c(y1s,y2s,y3s)
> mean(z)
[1]
> var(z)
[1]
> sesrs=sqrt(var(z)*( )/(6194*310))
> sesrs
[1]
Compared to 7.25 for the stratified SE.
Note: the estimate is the same, 661.9, since we have proportional allocation

33. Optimal allocation
If the only concern is to estimate the population total t:
Choose nh such that the variance of the stratified estimator is minimum
Solution depends on the unkown stratum variances
If the stratum variances are approximately equal, proportional allocation minimizes the variance of the stratified estimator

34. Result follows since the sample sizes must add up to n

35. Called Neyman allocation (Neyman, 1934)
Should sample heavily in strata if
The stratum accounts for a large part of the population
The stratum variance is large
If the stratum variances are equal, this is proportional allocation
Problem, of course: Stratum variances are unknown
Take a small preliminary sample (pilot)
The variance of the stratified estimator is not very sensitive to deviations from the optimal allocation. Need just rough approximations of the stratum variances

36. Optimal allocation when considering the cost of a survey
C represents the total cost of the survey, fixed – our budget
c0 : overhead cost, like maintaining an office
ch : cost of taking an observation in stratum h
Home interviews: traveling cost +interview
Telephone or postal surveys: ch is the same for all strata
In some strata: telephone, in others home interviews
Minimize the variance of the stratified estimator for a given total cost C

37. Solution:

38. In particular, if ch = c for all h:
We can express the optimal sample sizes in relation to n

39. Other issues with optimal allocation
Many survey variables
Each variable leads to a different optimal solution
Choose one or two key variables
Use proportional allocation as a compromise
If nh > Nh, let nh =Nh and use optimal allocation for the remaining strata
If nh=1, can not estimate variance. Force nh =2 or collapse strata for variance estimation
Number of strata: For a given n often best to increase number of strata as much as possible. Depends on available information

40. Sometimes the main interest is in precision of the estimates for stratum totals and less interest in the precision of the estimate for the population total
Need to decide nh to achieve desired accuracy for estimate of th, discussed earlier
If we decide to do proportional allocation, it can mean in small strata (small Nh) the sample size nh must be increased

41. Poststratification
Stratification reduces the uncertainty of the estimator compared to SRS
In many cases one wants to stratify according to variables that are not known or used in sampling
Can then stratify after the data have been collected
Hence, the term poststratification
The estimator is then the usual stratified estimator according to the poststratification
If we take a SRS and N-n and n are large, the estimator behaves like the stratified estimator with proportional allocation

42. Poststratification to reduce nonresponse bias
Poststratification is mostly used to correct for nonresponse
Choose strata with different response rates
Poststratification amounts to assuming that the response sample in poststratum h is representative for the nonresponse group in the sample from poststratum h