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Review. I volunteer in my son’s first grade class on library day. Each kid gets to check out one book. Here are the types of books they picked this week: Astronauts Ninjas Ponies Birds Total Boys 4 3 1 1 9 Girls 2 1 6 3 12 Total 6 4 7 4 21
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Review I volunteer in my son’s first grade class on library day. Each kid gets to check out one book. Here are the types of books they picked this week: Astronauts Ninjas Ponies Birds Total Boys 4 3 1 1 9 Girls 2 1 6 3 12 Total 6 4 7 4 21 Suppose we want to know whether sex and book type are independent. Which of the following is NOT a correct statement of the null hypothesis? • The distribution of book preferences is the same for boys and girls. • Boys like all book types equally, and so do girls. • Knowing whether a kid is male or female gives no information about his or her likely book preference. • Knowing a kid’s book preference gives no information about the kid’s sex.
Review I volunteer in my son’s first grade class on library day. Each kid gets to check out one book. Here are the types of books they picked this week: AstronautsNinjasPoniesBirds Total Boys43119 Girls216312 Total 647421 If book type is independent of sex, how many boys would be expected to pick pony books? • 2.25 • 2.63 • 3.00 • 3.50
Review Here are the expected frequencies, in red: AstronautsNinjasPoniesBirds Total Boys4 2.63 1.71 3.01 1.79 Girls2 3.41 2.36 4.03 2.312 Total 647421 Calculate the c2 statistic for testing independence. • 2.32 • 5.89 • 9.04 • 9.69
Non-Parametric Tests 12/5
Parametric vs. Non-parametric Statistics • Parametric statistics • Most common type of inferential statistics • r, t, F • Make strong assumptions about the population • Mathematically fully described, except for a few unknown parameters • Powerful, but limited to situations consistent with assumptions • When parametric statistics fail • Assumptions not met • Ordinal data: Assumptions not meaningful • Non-parametric statistics • Alternatives to parametric statistics • "Naive" approach: Far fewer assumptions about data • Work in wider variety of situations • Not as powerful as parametric statistics (when applicable)
Assumption Violations • Parametric statistics only work if data obey certain properties • Normality • Shape of population distribution • Determines shape of sampling distributions • Tells how likely extreme results should be; critical for correct p-values • More important with small sample sizes (Central Limit Theorem) • Homogeneity of variance • Variance of groups is equal (t-test or ANOVA) • Variance from regression line does not depend on values of predictors • Linear relationships • Pearson correlation cannot recognize nonlinear relationships
Assumption Violations • Parametric statistics only work if data obey certain properties • If these assumptions are true: • Population is almost fully described in advance • Goal is simply to estimate a few unknown parameters • If assumptions violated: • Parametric statistics will not give correct answer • Need more conservative and flexible approach Normal(m1, s2) Normal(m2, s2)
Ordinal Data • Some variables have ordered values but are not as well-defined as interval/ratio variables • Preferences • Rankings • Nonlinear measures, e.g. money as indicator of value • Can't do statistics based on differences of scores • Mean, variance, r, t, F • More structure than nominal data • Scores are ordered • Chi-square goodness of fit ignores this structure • Want to answer same types of questions as with interval data, but without parametric statistics • Are variables correlated? • Do central tendencies differ?
Non-parametric Tests • Can use without parametric assumptions and with ordinal data • Basic idea • Convert raw scores to ranks • Do statistics on the ranks • Answer similar questions as parametric tests • Your job: Understand what each is used for and in what situations
Spearman Correlation • Alternative to Pearson correlation • Produces correlation between -1 and 1 • Convert data on each variable to ranks • For each subject, find rank on X and rank on Y within sample • Compute Pearson correlation from ranks • Works for • Ordinal data • Monotonic nonlinear relationships (consistently increasing or decreasing) Y RY X RX
Mann-Whitney Test • Alternative to independent-samples t-test • Do two groups differ? • Combine groups and rank-order all scores • If groups differ, high ranks should be mostly in one group and low ranks in the other • Test statistic (U) measures how well the groups' ranks are separated • Compare U to its sampling distribution • Is it smaller than expected by chance? • Compute p-value in usual way • Works for • Ordinal data • Non-normal populations and small sample sizes 0 Perfectseparation U Noseparation
Wilcoxon Test • Alternative to single- or paired-samples t-test • Does median differ from m0? • Does median difference score differ from 0? • Subtract m0 from all scores • Can skip this step for paired samples or if m0 = 0 • Rank-order the absolute values • Sum the ranks separately for positive and negative difference scores • If m > m0, positive scores should be larger • If m < m0, negative scores should be larger • If sums of ranks are more different than likely by chance, reject H0 • Works for • Ordinal data • Non-normal populations and small sample sizes
Kruskal-Wallis Test • Alternative to simple ANOVA • Do groups differ? • Extends Mann-Whitney Test • Combine groups and rank-order all scores • Sum ranks in each group • If groups differ, then their sums of ranks should differ • Test statistic (H) essentially measures variance of sums of ranks • If H is larger than likely by chance, reject null hypothesisthat populations are equal • Works for • Ordinal data • Non-normal populations and small sample sizes
Friedman Test • Alternative to repeated-measures ANOVA • Do measurements differ? • Look at each subject separately and rank-order his/her scores • Best to worst for that subject, or favorite to least favorite • For each measurement, sum ranks from all subjects • If measurements differ, then their sums of ranks should differ • Test statistic (c2r) essentially measures variance of sums of ranks • If larger than likely by chance, reject H0 • Works for • Ordinal data • Non-normal populations and small sample sizes
Review Comparing average score between two groups, on an ordinal-scale variable. What test should you use? • Friedman • Mann-Whitney • Spearman • Kruskal-Wallis
Review Five subjects each measured in three conditions, on an interval-scale variable with a non-normal distribution. What test should you use? • Repeated-measures ANOVA • Kruskal-Wallis • Friedman • Spearman
Review Comparing average scores among five groups, on an interval-scale variable with a non-normal distribution. 100 subjects per group. What test should you use? • Kruskal-Wallis • Friedman • One-way ANOVA • Mann-Whitney