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

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.

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

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

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.

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  0.288 = 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

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

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)

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

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

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

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%

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.