Statistics 522: Sampling and Survey Techniques. Topic 12. See Example 11.2 on page 350 and discussion on page 353.

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1 Topic Overview Statistics 522: Sampling and Survey Techniques This topic will cover Regression with complex surveys Topic 12 Simple linear regression for an SRS Model with parameters Estimates of the parameters least squares or ordinary least squares (OLS) normal equations Plot the data Examine the residuals Regression in Complex Surveys Can we ignore the design when analyzing the data? If the probability of selection π i is related to the response y i then we can have bias in our regression coefficient estimates. Design effect can be important. Example Bias See Example 11.2 on page 350 and discussion on page 353. The file anthrop.dat has data on 3000 criminals. Variables of interest are height (y) and length of left middle finger (x). Regression for an SRS is performed on page 350 Suppose we select a sample with probabilities related to height (probabilities are higher for shorter men; see page 353 for details). The estimated regression line will be biased; the slope is smaller. 1

2 Point Estimation Consider the regression equation for the entire population y = B 0 + B 1 x. These parameters are functions of the population totals t x,y, t x, t x,x,andt y. We use methods from previous chapters to estimate these totals (weights). And substitute into the formulas for B 0 and B 1. Summary of Example Results SRS Unequal probability, ignore design Unequal probability using weights Population Comment Ŷ = x Ŷ = x Ŷ = x Ŷ = x We had bias because the probabilities of selection were related to the response variable y. If, for example, they were related to the explanatory variable x, and not to y, then we would not have bias. Note connection with design of experiments Software Software that allows weight options will give the correct estimates of the regression parameters. Note that the standard errors, confidence intervals and results of significance tests will not be correct. 2

3 Plot Always plot the data. Separate plots or symbols for strata can be useful. Cluster information can be indicated by color. Plotting symbol size can be equal to the number of observations at that point. Standard errors The regression coefficient estimates are functions of the estimate totals. These functions are nonlinear. We can use the methods in Chapter 9 to obtain approximations for the standard errors of the coefficient estimates. Linearization Variance formula for the slope is given as (11.8) on page 357. Note that this does not reduce to the usual formula when we have an SRS. Explanatory variables can be fixed or random. Linearization estimate is more robust with respect to homogeneity of variance. Confidence Intervals Use MOE =1.96SE The CI is the estimate plus or minus the MOE. The value 1.96 (normal approximation) can be replaced by a critical value from a t distribution (see text page 356). Significance Tests Use the Wald method. Divide the estimate by the standard error. Compare with a standard normal. 3

4 Multiple Regression Model: Y = XB + E E is assumed by to be N(0,σ 2 I) Normal equations: X XB = X Y. Estimates: ˆB =(X X) 1 X Y Modification for Weights W is an (n n) matrix with the weights on the diagonal. Normal equations: X WXB = X WY. Estimates: ˆB =(X WX) 1 X WY. These can be obtained from software that allows weights. Standard errors will be wrong. Linearization A formula for the estimated covariance matrix of the regression coefficient estimates is given as (11.11) on page 360. Regression using weights versus Weighted least squares We have described regression using weights. When there is heterogeneity of variance and or covariance among the errors, we use weighted least squares. Estimator is the same but framework is different. To weight or not to weight (Section 11.3) Uses of regression in traditional settings To describe the relationship between two (or more) variables. To predict y for a future observation. To control y by changing x. 4

5 Data Observational vs experimental Longitudinal vs cross-sectional Association vs causation Design based vs model-based inference Design vs model Design-based estimate the population relationship don t know model if there is one Model-based random mechanism generated the population This mechanism is what we want to study. Models vs Data summaries Models are an abstraction. They allow us to make statements about things that we have not observed. Regression vs anova Recommendation Perform analysis with and without weights. Compare the results. If they differ substantially, then explore alternative models; this suggests that there is a portion of the population for which the model does not hold. Clustering If we have a design with a large amount of clustering, the design characteristics should not be ignored. The issue here is appropriate estimation of standard errors. 5

6 Mixed Models Egypt project example Measure weight and height for 100 toddlers from age 1.5 to 2.5 years every month. Data problems Fit regression line for each child. Analyze the slopes. Comments In the previous example, the child is a cluster in the survey sampling terminology. See Example 11.7 on page 368. Models are called mixed linear models, random-coefficient regression models, andmultilevel or hierarchical models. SAS proc mixed can be used. Logistic Regression A binary (or categorical) response variable with one or more continuous or categorical explanatory variables. When the explanatory variables are continuous, alternative approach is discriminant analysis. Probit analysis (Finney) is another alternative. Model We assume that the response variable is Bernoulli (binomial with n =1),sayYesor No (0 or 1). The probability of a Yes depends on one or more explanatory variables. The relationship between this probability and the explanatory variables is specified by a link function. Logistic details For logistic regression, we assume that the log odds is a linear function of the explanatory variables (we can use indicator variables for categorical variables). Old examples use grouped values for the explanatory variable. 6

7 Estimation Use maximum likelihood for estimates. Iterative methods fare needed for solution of the MLE equations. Use linearization (delta method) for standard errors. Use Wald statistics for tests for individual regression coefficients. Modifications for complex designs Bias Use weights to obtain estimates. Use methods in Chapter 9 to obtain standard errors. In contrast to linear regression, the bias introduced by ignoring the design affects only the intercept in logistic regression. This has important consequences for epidemiology; for case-control studies we stratify based on y. Generalized regression estimation for population totals In Chapter 3, we studied ratio estimators and regression estimators in the setting of an SRS. We can combine weights based on the design with weights based on unequal variances. See Section 11.6, page for details. Software SUDAAN PC-CARP SAS proc surveyreg Beware of missing values (casewise deletion). Design effects can be used to adjust standard errors. 7