14. A public health survey of elderly Americans includes many questions, including “How many hours per week did you exercise in your most active years as a young adult?” and also several questions about current mobility and health status. Response rates are high for the questions about recent activities and status, but there is a lot of nonresponse for the question on past activity. You are considering imputing the missing values on the question, “How many hours per week did you exercise in your most active years as a young adult?” Which of the following statements are basically correct? (Indicate all that apply.)

(a) If done reasonably well, imputation is preferred to available-case and complete-case analysis.

(b) If you do impute, you should also present the available-case and complete-case analysis and analyze how the imputed estimates differ.

(c) It is OK to include current health status variables as predictors in a model imputing past activities: anything that adds information is good when imputing.

(d) It is probably not a good idea to include current health status variables as predictors in a model imputing past activities: current health is possibly influenced by past activities, and including a casual outcome can bias estimates of a treatment variable.

(e) If you fit a regression model and impute your best prediction for each person (rather than imputing random draws from the predictive distribution), you can have problems because you will be more likely to impute extreme values.

(f) It is a good idea to fit a logistic regression predicting response/nonresponse to the question of interest as a way to look for systematic differences between respondents and nonrespondents on this question.

Solution to question 13

From yesterday:

13. A survey of American adults is conducted that includes too many women and not enough men in the sample. In the resulting weighting, each female respondent is given a weight of 1 and each male respondent is given a weight of 1.5. The sample includes 600 women and 380 men, of whom 400 women and 100 men respond Yes to a particular question of interest. Give an estimate and standard error for the proportion of American adults who would answer Yes to this question if asked.

Solution: Define W1 = 600*1/(600*1+380*1.5) = 0.51, W2 = 380*1.5/(600*1+380*1.5) = 0.49, p1.hat = 400/600 = 0.67, p2.hat = 100/380 = 0.26. The desired estimate is then W1*p1.hat + W2*p2.hat = 0.47 and the standard error is sqrt(W1^2*p1.hat*(1-p1.hat)/600 + W2^2*p2.hat*(1-p2.hat)/380)=0.015.