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# Researchers are interested in identifying if completion of a summer individualized remedial program for 160 eighth graders, which is the outcome, is related to several predictor variables.

INSTRUCTIONS TO CANDIDATES
###### ANSWER ALL QUESTIONS

LOGISTIC REGRESSION PROBLEM USING SAS

1. Researchers are interested in identifying if completion of a summer individualized remedial program for 160 eighth graders (coded 1 for completion, 0 if not), which is the outcome, is related to several predictor variables. The predictor variables include student aptitude, an award for good behavior given by teachers during the school year (coded 1 if received, 0 if not), and age. Use these results to address the questions that appear at the end of the output.

For the model with the Intercept only: −2LL = 219.300

For the model with predictors: −2LL = 160.278

 Logistic Regression Estimates Odds ratio Variable (coefficient) β(SE) Wald chi-square test p value Estimate 95% CI Aptitude (β1) .138(.028) 23.376 .000 1.148 [1.085, 1.213] Award (β2) 3.062(.573) 28.583 .000 21.364 [6.954, 65.639] Age (β3) 1.307(.793) 2.717 .099 3.694 [.781, 17.471] Constant -22.457(8.931) 6.323 .012 .000

 Cases Having Standardized Residuals > |2| Case Observed Outcome Predicted Probability Residual Pearson 22 0 .951 -.951 -4.386 33 1 .873 -.873 -2.623 90 1 .128 .872 2.605 105 0 .966 -.966 -5.306

 Classification Results (With Cut Value of .05) Predicted Observed Dropped out Completed Total Percent correct Dropped out 50 20 70 71.4 Completed 11 79 90 87.8 Total 80.6

Complete the following:

1. Report and interpret the test result for the overall null hypothesis.
2. Compute and interpret the odds ratio for a 10-point increase in aptitude.
3. Interpret the odds ratio for the award variable.
4. Determine the number of outliers that appear to be present.
5. Describe how you would implement the Box–Tidwell procedure with these data.
6. Assuming that classification is a study goal, list the percent of cases correctly classified by the model, compute and interpret the proportional reduction in classification errors due to the model, and compute the binomial d test to determine if a reduction in classification errors is present in the population.
7. What statistical assumptions must be met to use logistic regression?

Principal Component and Factor Analysis Problem Using SAS

1. Consider the following principal components solution with five variables using no rotation and then a varimax rotation. Only the first two components are given, because the eigenvalues corresponding to the remaining components were very small (< .3).
 Unrotated Solution Varimax Solution Variables Comp 1 Comp 2 Comp 1 Comp 2 1 .581 .806 .016 .994 2 .767 -.545 .941 -.009 3 .672 .726 .137 .980 4 .932 -.104 .825 .447 5 .791 -.558 .968 -.006
1. Find the amount and percent of variance accounted for by each unrotated component.
2. Find the amount and percent of variance accounted for by each varimax rotated component.
3. Compare the variance accounted for by each unrotated component with the variance accounted for by each corresponding rotated component.
4. Compare (to 2 decimal places) the total amount and percent of variance accounted for by the two unrotated components with the total amount and percent of variance accounted for by the two rotated components. Does rotation change the variance accounted for by the two components?
5. Compute the communality (to two decimal places) for the first observed variable using the loadings from the (i) unrotated loadings and (ii) loadings following rotation. Do communalities change with rotation?
6. Run an exploratory factor analysis using principal axis extraction using the correlations shown below using the first nine items (exclude the bodily symptom items). Run a two- and three-factor solution for the remaining nine items.
1. Which solution(s) have empirical support?
2. Which solution seems more conceptually meaningful?
 Items Correlations for the Reactions-to-Tests Scales 1 2 3 4 5 6 7 8 9 10 11 12 Ten1 1.000 Ten2 .657 1.000 Ten3 .652 .660 1.000 Wor1 .279 .338 .300 1.000 Wor2 .290 .330 .350 .644 1.000 Wor3 .358 .462 .440 .659 .566 1.000 Tirt1 .076 .093 .120 .317 .313 .367 1.000 Tirt2 .003 .035 .097 .308 .305 .329 .612 1.000 Tirt3 .026 .100 .097 .305 .339 .313 .674 .695 1.000 Body1 .287 .312 .459 .271 .307 .351 .122 .137 .185 1.000 Body2 .355 .377 .489 .261 .277 .369 .196 .191 .197 .367 1.000 Body3 .441 .414 .522 .320 .275 .383 .170 .156 .101 .460 .476 1.000

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