1. The European Statistical Training Programme (ESTP)

2. Chapter 8: Weighting adjustment
Handbook: chapter 8
What is weighting adjustment?
Post-stratification
Linear weighting
Multiplicative weighting
Calibration estimation
Other weighting issues
An example

3. Introduction What is weighting adjustment?
Assignment of weights to observed (responding) persons.
Use of weighted values to compute estimates.
Why weighting?
Reducing of the bias due to nonresponse.
Increasing the precision of estimates (decreasing the variance).
Required ingredients: auxiliary variables
Usually categorical variables.
Strongly correlated with target variables of the survey.
Individual values are measured in the survey.
Distribution in population (or full sample) must be available.

4. Introduction Principle
Make response representative with respect to auxiliary variables.
If auxiliary variables are correlated with target variable, then the sample will also be representative with respect to target variable.
Use as much as possible auxiliary variables.
Auxiliary variables
Usually only limited number available (statistical institute).
Examples: age, gender, marital status, region.
They are not the most effective ones.
Statistics Netherlands has many more auxiliary variables in the Social Statistical Database (SSD).

5. Introduction Adjustment weighting to correct for unit-nonresponse
Use of auxiliary information
A set of variables that have been measured in the survey and for which information on the population distribution is available
Calculate adjustment weights
Example: inclusion weight ci = 1 / πi
This can also be written as
Where wi is the inclusion weight ci times a correction weight di.

6. Post-stratification
Suppose auxiliary variable X has L categories. It divides the population U into L strata U1, U2, …, UL.
The number of elements in stratum Uh is denoted by Nh for h=1,2,...,L. So N = N1 + N NL
The sample consists of n elements that can also be divided into the same strata, then n = n1 + n nL
In case of simple random sampling without replacement
With inclusion probabilities ci = n / N the post-stratification estimator becomes
The estimator is equal to a weighted sum of sample stratum means

7. Post-stratification
In case of nonresponse the post-stratification estimator becomes
The bias of this estimator is equal to
This can also be written as

8. Post-stratification A closer look at the bias:
The bias is small if the biases within the strata are small
This is the case when within strata
there is little or no relationship between the target variable and the response behaviour.
all response probabilities are more or less equal.
all values of the target variable are more or less equal.
Therefore, it is important to construct homogeneous strata.

9. Post-stratification Example: Sex  Age Weight of a young female:
0.209 / = 1.393
Population
Sample
Male
Female
Total
Young
226
209
435 23 15 38
Middle
152
144
296 16 17 33
Elderly
133
136
269 13 29
511
480
1000 52 48
100
Weights
0.983
1.393
0.950
0.847
1.023
0.850

10. Linear weighting Who not post-stratification?
Many auxiliary variables: too few (or no) observations per strata.
Lack of sufficient population information.
Solution
Linear or multiplicative weighting
Population
Sample
Male
Female
Total
Young ?
435 23 15 38
Middle
296 16 17 33
Elderly
269 13 29
511
480
1000 52 48
100
Weights

11. Linear weighting Generalised regression estimator – full response
with
vector of population means of auxiliary variables
vector of sample means of auxiliary variables
b vector of regression coefficients:
The estimator is asymptotically design unbiased (ADU)
Variance

12. Linear weighting Generalised regression estimator - nonresponse
Bias:
with
The bias is small if
Residuals are small, i.e. regression model fits well.
There is little correlation between residuals and response behaviour (MAR).

13. Linear weighting Generalised regression estimator = Weighting
Estimator can be re-written as:
(under general conditions)
Consequently
with
and v a vector of weight coefficients:

14. Linear weighting Post-stratification
Replace each qualitative auxiliary variable by a set of dummy variables.
Use these dummy variables in regression model
Example: one qualitative variable with L categories (strata)
Introduce L dummy variables X1, X2, …, XL.
Xh = 1 for an observation in stratum h, and 0 otherwise
Vector of population means
Vector of weight coefficients

15. Linear weighting Post-stratification - example
Two auxiliary variables: Sex  AgeClass
Weight for young female = 1.393
Sex
AgeClass X1 X2 X3 X4 X5 X6
Male
Young 1
Middle
Elderly
Female
Population means
0.226
0.152
0.133
0.209
0.144
0.136
Weight coefficients
0.983
0.950
1.023
1.393
0.847
0.850

16. Linear weighting Use only marginal distributions
Two auxiliary variables: Sex + AgeClass
Weight for young female = = 1.185
Sex
AgeClass X1 X2 X3 X4 X5 X6
Male
Young 1
Middle
Elderly
Female
Population means
1.000
0.511
0.489
0.435
0.296
0.269
Weight coefficients
0.991
-0.033
0.033
0.161
-0.095
-0.066

17. Linear weighting
Many possible weighting models with more than two variables
For example: three variables Sex, AgeClass, MarStat
Models:
Sex  AgeClass  MarStat
(Sex  AgeClass) + (AgeClass  MarStat) + (Sex  MarStat)
Sex + AgeClass + MarStat
And many more …
(Sex  MarStat) + AgeClass …

18. Linear weighting Qualitative and quantitative auxiliary variables
Examples: Age, Age + Sex, Age  Sex
Age
Sex X1 X2 X3 X4 X5 X6 65
Male 1 36 73
Female 6 33 82 2 32 66
Population means
1.000
34.369
0.511
0.489
33.509
35.268
Weight coefficients 1
1.101
-0.003
Weight coefficients 2
-0.032
0.032
Weight coefficients 3
1.087
-0.001
-0.004

19. Multiplicative weighting
Alternative for linear weighting.
Qualitative auxiliary variables only.
Difference: weight is not a linear combination of weight coefficients but a product of weight factors.
Iterative process:
Weight model: (A  B  C) + (D  E) + (F  G  H)
Step 1: Introduce weight factors for each stratum in each cross-classification term. Set factors to 1.
Step 2: Adjust factors for term 1, so that weighted sample distribution is equal to population distribution for variables involved.
Step 3: Adjust factors for next term. This may disturb previous factors.
Step 4: To this for all terms in the model.
Step 5: Repeat steps 2-4 until factors do not change any more.

20. Multiplicative weighting
Example: Sex + AgeClass
Start situation
Weight for young female = 1.000 = 1.000
Starting situation
Male
Female
Weight factor
Weighted sum
Population distribution
Young
0.230
0.150
1.000
0.380
0.435
Middle
0.160
0.170
0.330
0.296
Elderly
0.130
0.290
0.269
0.520
0.480
Popul. distr.
0.511
0.489

21. Multiplicative weighting
Example: Sex + AgeClass
Adjustment for Age
Weight for young female = 1.145 = 1.145
Step 1
Male
Female
Weight factor
Weighted sum
Population distribution
Young
0.230
0.150
1.145
0.435
Middle
0.160
0.170
0.897
0.296
Elderly
0.130
0.928
0.269
1.000
0.527
0.473
Popul. distr.
0.511
0.489

22. Multiplicative weighting
Example: Sex + AgeClass
Adjustment for Sex
Weight for young male = 1.145 = 1.185
Step 2
Male
Female
Weight factor
Weighted sum
Population distribution
Young
0.230
0.150
1.145
0.433
0.435
Middle
0.160
0.170
0.897
0.297
0.296
Elderly
0.130
0.928
0.270
0.269
0.969
1.035
0.511
0.489
1.000
Popul. distr.

23. Multiplicative weighting
Example: Sex + AgeClass
Final situation after convergence
Weight for young female = 1.035 = 1.191
Step …
Male
Female
Weight factor
Weighted sum
Population distribution
Young
0.230
0.150
1.151
0.435
Middle
0.160
0.170
0.895
0.296
Elderly
0.130
0.923
0.269
0.968
1.035
0.511
0.489
1.000
Popul. distr.

24. Linear or multiplicative weighting?
Advantages of linear weighting:
Linear weighting based on regression model.
Analytic formula for variances of estimators.
Both qualitative and quantitative auxiliary variables.
Disadvantage of linear weighting:
Resulting weights may be negative.
However:
Both weighting methods produce estimates that are asymptotically equal.

25. Calibration estimation
How to compare different weighting methods?
Calibration estimation offers general framework for weighting methods
General formulas for properties of methods like asymptotic distributions.
Linear and multiplicative weighting are special cases
Idea:
Calibrate known auxiliary characteristics while affecting response as little as necessary
Ingredients
A distance measure D.
Calibration of auxiliary variables X.

26. Calibration estimation
Strategy
Minimize
under the constraint
Examples
Linear weighting:
Multiplicative weighting:
Without nonresponse both methods have the same asymptotic properties.
With nonresponse the effectiveness depends on the validity of the underlying model .

27. Other weighting issues
Consistency between person and household weights
A survey may be used to make statistics about persons as well as about households.
The person weights need not sum up to the household weights.
Two sets of weights are impractical.
Unsatisfactory solutions
The household weight equals the weight of a reference person or randomly selected person in the household.
Use the average person weight for the household.
Generalised regression estimation

28. Other weighting issues
Generalised regression estimation
Auxiliary information at person level, X
Household membership, H
Auxiliary information at household level, Z = H’X
Person
0-20
20-60
> 60
Male
Female 1 2 3 4
Person H1 H2 H3
. . . 1 2 3 4
Household
0-20
20-60
> 60
Male
Female 1 2

29. A practical example Selection of a weighting model
Identification of auxiliary variables
Collection of population totals
Selection of weighting variables
Weighting
Example - Step 1
Available auxiliary variables: sex, age, marital status, province of residence and degree of urbanisation

30. A practical example Collection of population totals
Ideal: complete crossing of all auxiliary variables.
In practice totals of complete crossing often not available.
Empty cells.
Example - Step 2
Available: Age  Sex  Marital status, Age  Province and Age  Degree of urbanisation
Age
Male
Female
Unmarried
Married
Widowed
Divorced
12-19
752.4
0.4
0.0
716.5
3.5
20-29
981.5
185.7
0.2
10.2
785.0
330.6
0.7
22.7
30-39
445.4
795.1
1.9
72.1
283.5
879.3
5.7
93.8
40-49
164.7
899.0
6.9
113.9
103.1
882.9
21.5
138.4
50-59
67.3
732.9
15.8
86.3
44.4
675.9
56.1
98.8
60-69
42.0
519.2
31.7
42.6
41.4
458.9
140.0
51.5
70-79
21.4
308.4
52.5
16.6
43.0
239.9
254.3
27.9
80+
8.0
84.0
50.4
4.0
35.0
49.6
243.9
12.4

31. A practical example How to select weighting variables?
Relation to nonresponse
Relation to key survey topics
Variables that are used as marginal variables in statistical tables
Compute contingency tables (and possibly test for independence)
Build model for nonresponse using auxiliary variables
Select survey variables that cover range of topics in survey
Compute contingency tables
Build models for survey variables using auxiliary variables
Take union of sets of auxiliary variables in models

32. A practical example Example - Step 3 Age Resp Pop Diff Province 12-19
12.8
11.1
1.7
Groningen
2.7
3.5
-0.8
20-29
15.9
17.5
-1.6
Friesland
4.3
3.9
0.4
39-39
20.5
19.4
1.1
Drenthe
2.3
3.0
-0.7
40-49
17.9
17.6
0.3
Overijssel
6.8
6.7
0.1
50-59
14.0
13.4
0.6
Flevoland
1.8
60-69
10.0
0.0
Gelderland
15.4
12.1
3.3
70-79
6.5
7.3
Utrecht
5.4
6.9
-1.5
80+
2.5
3.7
-1.2
N-Holland
16.1
-2.1
Z-Holland
18.0
21.5
-3.5
Zeeland
2.4
N-Brabant
14.8
2.8
Limburg
9.1
7.4

33. A practical example Example - Step 3 Mar. status Resp Pop Diff
Urbanisation
Unmarried
32.7
34.2
-1.5
Very strong
11.8
18.0
-6.2
Married
57.2
53.2
4.0
Strong
24.0
23.8
0.2
Widowed
5.2
6.0
-0.8
Moderate
23.2
20.5
2.7
Divorced
4.9
6.7
-2.8
Little
23.3
21.1
2.2
Non
17.7
16.5
1.2
Sex
Male
48.6
49.1
-0.5
Female
51.4
50.9
0.5

34. A practical example Computation of weights
Construct weighting models with candidate auxiliary variables as ingredients for a number of key survey topics
Control variance of estimates
Example - Step 4
Weighting model
Parameters
Estimate
Standard error 1
No weighting
43.4
1.2 2
Sex 3
Prov 12
43.3 4
MarStat
42.9 5
Urban
1.0 6
Age8 8
42.8 7
Age5  Province 60
(Sex  Age8) + (Sex  MarStat) 22
42.3
1.1 9
Age5  Urban 25
42.5 10
Sex + Age8 + MarStat + Urban + Province 23
42.1 11
(Sex  Age8) + (Sex  MarStat) (Age5  Urban) + Province 53
42.0
0.9