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Post-stratification and Response Bias

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IntroductionSources of Selection Bias in Sample Survey DataNonresponse BiasNumerical IllustrationsPost-stratification to Increase PrecisionPost-stratification and WeightingZhang's General FrameworkAn Example: The Youth Circle SurveyExample: Post-stratification and Binary Regression Analysis in Youth Vote's National Youth SurveyDiscussionPost-stratification and Response Bias in Survey Datawith Applications in Political ScienceIffigenia Barboza and Rohan WilliamsDepartment of Statistics and Political Science, Michigan State University∗April 8, 2005AbstractPost-stratification is a common technique implemented to obtain more precise estimates ofsample statistics in survey data. If used correctly, this technique increases the representativenessof the sample so we have greater confidence in the validity of our inferences about populationparameters of interest. Methodologically speaking, however, using p os t-stratification to correctfor the effects of differential non-response in the poststrata for the purpose of increasing the non-reflectiveness of the population is problematic. Moreover, when proportional allocation is used todesign the sample, it is not necessarily the case that the number of units in the simple randomsample for stratum h is proportional to the total number of sampling units in each stratum, ornh∝ Nh. In this paper, we illustrate the pros and cons of using post-stratification as a methodto increase representation due to non-response by simulation, using race as our post-stratifyingvariable. The population information is drawn from the Census’ Current Population Survey March2004 Supplement. Our intent is to design criteria that promote careful use of post-stratificationwhen the survey instrument may not be reflective of certain segments of the population.keywords: Post-stratification, differential nonresponse, nonignorable nonre-sponse1 IntroductionStratifying sample data on known variables is often fruitless for several reasons, the most commonreason being the difficulty involved in compiling a sufficient sampling frame. Post-stratification issometimes used when a good stratification criterion is known but it is not feasible to sample fromthe strata or it is simply too costly to do so. Moreover, when the sample is not representative of thepopulation, post-stratification is employed to mimic the stratification process. The main difference isthat post-stratification methods rely on a process of stratification that occurs only after the sample istaken. T his statistical methodology must be carefully implemented to avoid the dangers and pitfallsthat occur with incorrect use.In order to use post-stratification effectively the ratio of the candidate stratum proportions, i.e.,NhN, must be known. In many social and political surveys it may be difficult to stratify such factorsas income and age by race. The reason is that the stratum proportions are not known beforehand.Methodologically however we can get post-stratification proportions for example by turning to Censusdata via the following process: take a simple random sample of size n, split the data into the Hstrata and proceed as if we had originally stratified the random sample. Note, however, that underthe stratification procedure, the nhare fixed whereas when we post-stratify the nhare random. Thismay cause difficulty for estimating population parameters.A primary use of post-stratification is to reduce nonresponse bias in surveys. (Bethlehem 1988;Smith 1991; Zhang 1999). With respect to nonresponse, post-stratification estimation is most usefulwhen the respondents are not reflective of the population. If the nonreflectiveness of the respondents∗Acknowledgements: The authors are g rateful to Drs. Jim Stapleton (“Lao-Shi”) and Connie Page for theirsupport and guidance.12 Post-stratification and Nonresponse Bias in Political Surveysdepends on the variable of interest, the sample mean estimator will be biased. If this is true, the post-stratification estimator, which incorporates known information about the size of components of thepopulation, gives “better” results. In fact, post-stratification w ill completely remove nonresponse biasif the nonresponse is conditionally independent of the variable of interest within each post-stratum.(Zhang 1999). For example, it is common in political surveys for response rates to be lower amongvarious ethnic subpopulations. If we are asking a simple random sample of individuals about politicalactivities that are more common among whites and we use the basic sample mean estimator then ahigher response rate for whites will lead to a biased over-estimate of the political behaviors of thepopulation.A major drawback of post-stratification, however, is that the nonresponse is rarely conditionallyindependent of the variable of interest within post-strata, and this is especially true for s urvey data inpolitical science. This happens if the nonresponse is more severe among one subsample than the otherin which case the response is not strictly independent of the post-strata conditional on the variableof interest. (Zhang 1999). To illustrate, consider estimating the proportion of registered voters whoparticipated in the last primary election. Assuming that the nonresponse is conditionally independentof the variable of interest within each post-stratum would require assuming that the nonresponse doesnot depend on whether or not the individual voted given the individual’s race. A more reasonableassumption would be that, for example, certain minority groups are less likely to respond if they didnot vote in the last election, in which case the nonresponse is more severe among one subsample. Inthis paper, we first discuss the relative effect of post-stratification when the variable of interest dependson the response variable. Next, we show the relative reduction in bias and efficiency given that thedependency between the response and the variable of interest is moderate. Through simulation, we areable to evaluate two primary assumptions made by Zhang (1999), who showed that the bias-reductiondue to post-stratification can be estimated from the respondents alone, with respect to variables ofinterest in political science. In this way we hope not only to extend Zhang’s results, but also to makethis method available to political scientists.1.1 Sources of Selection Bias in Sample Survey DataSelection bias o c curs when the sampled population is not representative of the target


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