Anova and contingency tables Week 12 Anova and contingency tables

Two categorical variables Joint probabilities px,y = P(X=x and Y=y) proportion of popn with values (x, y) School performance and wt of children

Conditional probabilities Proportion within row

Conditional probabilities School performance and wt of children Weight & performance are independent

Independence from sample? 214 child skiers classified by skiing ability and whether they got injured Are ability and injury independent in underlying population?

Independence from sample? Conditional sample proportions Is there independence in underlying population?

Testing for independence Can a relationship observed in the sample data be inferred to hold in the population represented by the data? Could observed sample relationship have occurred by chance?

Expected counts — independence 31 out of 214 injured overall Expect 31/214 of the 80 beginners to be injured i.e. expect injured beginners

Expected counts — independence General formula Injured Uninjured Beginner 80 Intermediate 93 Advanced 41 31 183 214

Observed and estimated counts Injured Uninjured Beginner 20 (11.59) 60 (68.41) 80 Intermediate 9 (13.47) 84 (79.53) 93 Advanced 2 (5.94) 39 (35.06) 41 31 183 214 Are the differences more than would be expected by chance?

Chi-squared test of independence H0: independence of injury & experience HA: association between injury & experience or equivalently H0: P(injury|beginner) = P(injury|intermediate) = ... HA: P(injury | experience) depends on experience Test statistic:

Chi-squared test of independence Small values consistent with independence Big values arise when observed are very different from what would be expected under independence. p-value = Prob(2 as big as obtained) if indep Tail area of chi-squared distribution d.f. of chi-squared = (rows–1)(cols–1)

Chi-squared distributions Skewed to the right distributions. Minimum value is 0. Indexed by the degrees of freedom.

Skiing injury and experience Chi-Square Test: Injured, Uninjured Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts Injured Uninjured Total 1 20 60 80 11.59 68.41 6.105 1.034 2 9 84 93 13.47 79.53 1.484 0.251 3 2 39 41 5.94 35.06 2.613 0.443 Total 31 183 214 Chi-Sq = 11.930, DF = 2, P-Value = 0.003 p-value = 0.003 Strong evidence that the chance of injury is related to experience.

Ear Infections and Xylitol Experiment: n = 533 children randomized to 3 groups Group 1: Placebo Gum; Group 2: Xylitol Gum; Group 3: Xylitol Lozenge Response = Did child have an ear infection?

Ear Infections and Xylitol Moderately strong evidence of differences between probs of infection

Making friends With whom do you find it easiest to make friend — opposite sex, same sex or no difference?

Making friends H0: No difference in distribution of responses of men and women (no relationship between gender & response) HA: Difference in distribution of responses of men and women (association between gender & response) Chi-Square Test: Opposite sex, Same sex, No difference Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts Opposite sex Same sex No difference Total 1 58 16 63 137 48.79 19.38 68.83 1.740 0.590 0.494 2 15 13 40 68 24.21 9.62 34.17 3.507 1.188 0.996 Total 73 29 103 205 Chi-Sq = 8.515, DF = 2, P-Value = 0.014 Fairly strong evidence of difference between Females(1) & Males(2) Females more likely to choose opposite sex

Comparing means of 3+ groups Do best students sit in the front of a classroom? Seat location and GPA for n = 384 students Students sitting in the front generally have slightly higher GPAs than others. Chance?

Seat location and GPA H0: m1 = m2 = m3 HA: The means are not all equal. p-value = 0.0001. Such big differences between sample means unlikely if popn means were same Extremely strong evidence that means are not all same.

Seat location and GPA 95% CIs for separate means: Main difference seems to be between front and others

Assumptions for F-test Independent random samples. Normal distribution within each population. Perhaps different population means. Same standard deviation,  in each group. Can still proceed if n is big or assumptions approx hold

F ratio More evidence of a real difference when: How do you measure: Group means are far apart Variability within groups is small How do you measure: Variation between means? Variation within groups?

Variation between means Between-groups sum of squares Mean sum of squares for groups (k groups):

Variation within groups Within-groups sum of squares Residual sum of squares Also called residual sum of squares Mean residual sum of squares: Best estimate of error st devn, :

Total variation Total sum of squares = SSTotal SSTotal = SSGroups + SSError

Analysis of variance table Anova table F test is based on F ratio p-value = Prob of such a high F ratio if all means same (p-value found from an ‘F distribution’)

Seat location and GPA (again) H0: m1 = m2 = m3 HA: The means are not all equal. p-value = P(F ≥ 6.69) under H0 = 0.0001. Such a big F ratio unlikely if popn means were same Extremely strong evidence that means are not all same.