1. The European Statistical Training Programme (ESTP)

2. Chapter 3: Response models
Handbook: chapter 2
Introduction
Fixed Response Model
Random Response Model
Effects on confidence intervals

3. Response Models Fixed Response Model
The population consists of only two types of elements.
Respondents are in the response stratum. They always respond.
Nonrespondents are in the nonresponse stratum. They never respond.
Random Response Model
Each element has a certain probability to respond.
The response probability is different for each element.
The response probabilities are unknown.

4. Response Models Fixed Response Model
Population divided into two strata
Response stratum: Persons in this stratum always respond.
Nonresponse stratum: Persons in this stratum never respond.
Response indicators R1, R2, …, RN, with
Rk = 1 if element k is in Response stratum
Rk = 0 if element k is in Nonresponse stratum
Response stratum
Nonresponse stratum

5. Response Models Fixed Response Model
Simple random sample of size n from population.
Sample indicators a1, a2, …, aN, with
ak = 1 if element k is in the sample
ak = 0 if element k is not in the sample
Only nR elements in the response stratum are observed.
Question: Is the response mean a good estimator for the mean of the complete target population?
Response stratum
Nonresponse stratum

6. Fixed Response Model Estimator Expected value Bias
The bias of the estimator is determined by
Differences (on average) between respondents and nonrespondents.
Relative size of nonresponse stratum, i.e. the expected nonresponse rate.

7. Fixed Response Model Example Dutch Housing Demand Survey 1981
Target variable: Intention to move within two years
Information about nonresponse from follow-up survey
Estimate based on response: 29.7%
Estimate based on nonresponse: 12.8%
Contrast K = 16.9%
Relative size of nonresponse stratum Q = 0.288
Bias of estimator = 16.9  = 4.9%
Move within 2 years
Response
Nonresponse
Total
Yes
17,515
3,056
20,571 No
41,457
20,821
62,278
58,972
23,877
82,849

8. Response Models Random Response Model
Each element k has an unknown response probability ρk
Response indicators R1, R2, …, RN, with
Rk = 1 if element k is in responds
Rk = 0 if element k does not respond
P(Rk = 1) = ρk, P(Rk = 0) = 1 – ρk
Sampling
Simple random sample of size n from population.
Sample indicators a1, a2, …, aN, with
ak = 1 if element k is in the sample
ak = 0 if element k is in in the sample
Number of respondents

9. Random Response Model Estimator Expected value Bias
The bias of the estimator is determined by
Correlation RρY between response behaviour and target variable
Variation of response probabilities
Mean of response probabilities (expected response rate)

10. Random Response Model Example
Estimating the mean income of working population of Samplonia.
Simulation 1: 1000 samples, size n = 40, no nonresponse
Simulation 2: 1000 samples, size n = 40, nonresponse increases with income
Simulation 3: 1000 samples, size n = 80, nonresponse increases with income

11. Effect of nonresponse on confidence interval
Full response
95% confidence interval:
Confidence level
Nonresponse
Confidence interval
Confidence level not equal to 0.95.

12. Effect of nonresponse on confidence interval
Confidence level as a function of the relative bias
0.0
0.95
0.2
0.4
0.93
0.6
0.91
0.8
0.87
1.0
0.83
1.2
0.78
1.4
0.71
1.6
0.64
1.8
0.56
2.0
0.48

13. Effect of nonresponse on confidence interval
Example
Confidence intervals for the mean income of working population of Samplonia.
Simulation 1: 30 samples, size n = 40, no nonresponse Confidence level = 97%
Simulation 2: 30 samples, size n = 40, nonresponse increases with income Confidence level = 60%

14. Response models Some conclusions
There is only a bias if the target variable of the survey is related to the response behaviour.
The magnitude of the bias of estimators is only partly determined by the response rate.
The magnitude of the bias of estimators is also determined by the variation in response probabilities.
In case of nonresponse, the confidence interval cannot be used any more as a measure of accuracy.
The nonresponse bias is not reduced by increasing the sample size.