1. Stat 512 – Lecture 14 Analysis of Variance (Ch. 12)

2. Recap Comparing means  Male/female body temperatures  Average improvement with/without enough sleep Comparing proportions  Newspaper believability in 1998/2002  Survival with and without letrozole Comparing more than two proportions  Probability of selecting a woman juror among 7 judges

3. Recap – Chi-Square Procedures Why use Chi-Square procedures?  Each test (at the 5% level of significance) has a 5% chance of making a Type I Error  With 21 tests, have a 66% chance of at least one false rejection of the null hypothesis  With the overall chi-square procedure, have a 5% chance of making a type I error Downside?  Only get to say “at least one judge differs”

4. Recap – Chi-Square Procedures How carry out a chi-square test?  Minitab  Enter two-way table of observed counts Not row and column totals  Is test statistic too large?  Interpret p-value as usual if all expected counts are at least 5 and have randomness in study design  Can do some follow-up analysis on chi-square contributions

5. Recap – Chi-Square Procedures When use chi-square procedures?  Answer 1: Whenever have two qualitative variables on each observational unit

6. Answer 2 Chi-square tests arise in several situations 1. Comparing 2 or more population proportions H 0 :       H a : at least one  i differs 2. Comparing 2 or more population distributions on categorical response variable H 0 : the population distributions are the same H a : the population distributions are not all the same

7. Answer 2 (cont.) 3. Association between 2 categorical variables H o : no association between var 1 and var 2 (independent) H a : is an association between the variables Technical conditions: Random Case 1 and 2: Independent random samples from each population or randomized experiment Case 3: Random sample from population of interest Large sample(s) All expected cell counts >5

8. PP 11 – Problem 1 (a) Technical conditions “valid,” “appropriate” vs. “can we” Population size? Populations normal? Both sample sizes? Both samples random? (b)-(c) Inference procedures Include all the steps! Population vs. sample values The confidence interval is from 14.94 to 65.05 (d) Would you pay money to improve your scores this much?

9. PP 11 – Problem 2 Let  w represent the probability of a winner living this long, with  N for the nominees and  C ­ for the controls. H 0 :  W =  N =  C (the long-term survival rate is the same for the 3 processes) H a : at least one  differs We are considering these three groups to be independent random samples from the award winning process. The expected counts (smallest = 124.98) are large enough for the chi-square approximation to be accurate.

10. PP 11 – Problem 2 The chi-square value (13.229) is large and the p-value (.001) small so we reject the null hypothesis. We have very strong evidence that the three population probabilities are not the same. The largest contributions to the chi-square sum arise from the deaths among controls, where we observed more than we would have expected had the three probabilities all been equal. died alive

11. HW Questions? Note HW 7 is posted online

12. Example 1: Handicap Discrimination In 1984, handicapped individuals in the labor force had an unemployment rate of 7% compared to 4.5% in the non-impaired labor force. Cesare, S.J., Tannenbaum, R.J., and Dalessio, A. (1990), “Interviewers’ Decisions Related to Applicant Handicap Type and Rater Empathy,” Human Performance 3(3): 157-71.

13. Example 1: Handicap Discrimination Observational units?  Undergraduate students Explanatory variable  Which type of handicap in video (qualitative) Response variable  Qualification rating (quantitative) Type of study  Experiment since randomly assigned them to different videos

14. Example 1: Handicap Discrimination Is it possible that there is no treatment effect but the observed treatment group means are 4.429, 5.921, 4.050, 4.900, 5.343? How decide? Let  i = underlying true mean treatment response H 0 :  none =  leg amp =  crutches =  hearing =  wheel H a : at least one  differs

16. Example 1: Handicap Discrimination

17. Inference Procedure Want one procedure for comparing the 5 treatment means simultaneously  Take into account the distances between the sample means relative to the variability in the data  Comparisons of “between group” variability to “within group” variability (“by chance”)

18. “Analysis of Variance” (ANOVA) F statistic = discrepancy in group means variability in data If F statistic is large, have evidence against the null hypothesis p-value = probability of observing an F statistic at least this large when H 0 is true

19. Minitab Stat > ANOVA > One-Way

20. Notes The test statistic takes the sample sizes into account, giving more weight to the group with larger sizes. Producing a pooled estimate of the overall variability in the data requires us to assume that each population/treatment group has the same variability  2.

21. Checking the Technical Conditions Normal populations Equal variances  Ratio of largest SD/smallest SD < 2 Independence  Random samples/randomization

22. Example 1: Handicap Discrimination The samples look reasonably symmetric with similar standard deviations, so it is appropriate to apply the Analysis of Variance procedure. There is moderate evidence that the mean qualification ratings differ depending on the type of handicap (p- value =.030). Descriptively, the candidates with crutches appear to have higher ratings and the candidates with hearing impairments slightly lower ratings (other procedures could be used to follow-up to test the significance of these individual differences). This was a randomized experiment so we can attribute these differences to the handicap status but we must be cautious in thinking the students in this study are representative of a larger population, particularly, a population of employers who make hiring decisions.

23. Example 2: Restaurant Spending Hypotheses Technical conditions? How do different factors affect the size of the p- value?  when the population means are further apart, the p-value is usually smaller (more evidence they aren’t equal)  when the within group variability is larger, the p-value is larger (less evidence didn’t happen by chance)  when the sample sizes are larger, and there is a true difference between the population means, then the p-value is smaller

24. Example 3: Follow-up Analysis Multiple comparison procedures control overall Type I Error rate  Are several different such procedures Bonferroni, Tukey, Scheffe’… Let Minitab do all the work

25. Example 4: Lifetimes of Notables Which professions appear to differ?

26. For Thursday Submit PP 12 in Blackboard Continue reading Ch. 12 Preview Example 5, complete (a) and (b)