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STATISTICS FOR LAWYERS

STATISTICS FOR LAWYERS. “Nonparametric” Statistics “Distribution-Free” Statistics. Distributional Violations. Treat data as nominal Chi Square Tests Binomial Sign test (for matched samples) Use specialized tests that do not make assumptions about the underlying distribution of the data

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STATISTICS FOR LAWYERS

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  1. STATISTICS FOR LAWYERS “Nonparametric” Statistics “Distribution-Free” Statistics

  2. Distributional Violations • Treat data as nominal • Chi Square Tests • Binomial • Sign test (for matched samples) • Use specialized tests that do not make assumptions about the underlying distribution of the data • If sample size is sufficient, replace original values with ranks, and run traditional tests • Variance is known for uniform distribution

  3. Specialized Tests • Large samples • statistics themselves will conform to central limit theorem, and have a known standard deviation • Can use normal theory to test hypotheses • Small samples • Exact tables available in specialized text books • Spiegel, Nonparametric Statistics • Some tables may be available online

  4. “Binomial” Testcoin flip analogy • Dichotomize data • Treat as coin flip • p=? • Split at median is a common approach • Example • Students who took the stats workshop • in number in top half (top quarter?) of their class

  5. Runs Test Large Sample: R = 17 critical value from table is 9 (or less) (16 males and 12 Females) E{R} = 15 σ{R} = sqrt(27)/2 = 2.6 Z = (12-15)/2.6) = -1.15

  6. Tests Based On Ranks Uniform Distribution Central Limit Theorem applies and variance is known

  7. Normality Violation Number of Hours Devoted to Civil Cases by Hourly Fee Lawyers All Cases Drop cases requiring more than 500 hours Drop cases requiring more than 100 hours

  8. Lawyer Effort by Type of Courtt-test State Federal

  9. Lawyer Effort by Type of Courtt-test on ranks • Assign ranks • Run t-test using ranks as the variable Compare Lawyer Effort for Federal and State Cases State Federal

  10. Wilcoxin Rank Sum TestMann-Whitney U Test Alternative to the two sample t-test • Identify which group is smaller, and rank from low to high or high to low so that group has the smaller ranks • Compute W by summing the ranks of smaller group • If n is large enough (i.e., both samples are 10 or more, W will have a normal distribution • If n is small, exact tables are available.

  11. Lawyer Effort by Type of CourtWilcoxin Rank-Sum Test State Federal

  12. Transformation to Cure Distributional Violation Taking the logarithm will sometimes cure nonnomality issues If there are values of 0, need to add 1 to do log

  13. Lawyer Effort by Type of Courtt-test on log(hours) State Federal t-test on original data produced a t of -5.590 Rank tests would not change using log transform because the log transformed data are monotonically identical to the original data.

  14. Lawyer Effort by Type of CourtMedian test State Federal

  15. Outlier Issues • Tests that rely on means can be substantially influenced by a small number of extreme values • The log transform is one way to reduce the influence of outliers • A second approach is to use the ranks rather than the original values

  16. Example of Grouped OrdinalLawyers’ Assessment of Case Complexity

  17. Wilcoxin & t-testComplexity State Federal State Federal

  18. Grouped OrdinalSimple Chi Square Test State Court Federal Court

  19. Wilcoxin Matched-Paired Signed Ranks Test Alternative to the matched pair t-test before after differ rank 76 65 -11 (-)6 32 24 -8 (-)5 65 70 +5 (+)4 87 85 -2 (-)1 22 25 +3 (+)2.5 9 12 +3 (+)2.5 37 25 -12 (-)7 T = 2.5 + 2.5 + 4 = 9 critical value at .05 level is 2

  20. Kruskal-Wallis multi-group comparison Alternative to one analysis of variance Ni = number of observations in ith group M = number of sets of ties Tj = tj3 - tj tj = number of observations tied for the jth set of ties

  21. Lawyer Hours by ComplexityANOVA

  22. Lawyer Hours by ComplexityKruskal-Wallis Test

  23. Lawyer Hours by ComplexityANOVA on Ranks

  24. Lawyer Hours by ComplexityANOVA on Log(hours)

  25. Lawyer Hours by ComplexityMedian Test

  26. Spearman Rank Order Correlation (rho) Replace original values of each variable with ranks, and then compute Pearson’s product moment correlation using the ranks as the data.

  27. Hours by sqrt(Stakes)Pearson & Spearman’s Correlations

  28. Condordant & Discordant Pairs • Rank observations separately on each variable • Look at each pair of observations • call one observation a and the other b • if observation a has a lower rank than observation b for both variables, pair is concordant • if observation a has a higher rank than observation b for both variables, pair is concordant • Otherwise pair is discordant

  29. Kendall’s Taurank order correlation C is number of concordant pairs; D is number of discordant pairs Specialized version to use with contingency table formed from two ordinal variables

  30. Hours by StakesSpearman’s and Kendall’s Tau Correlations

  31. Other Rank Order Correlations gamma Somer’s assymetric D

  32. Tau for Grouped OrdinalComplexity by “Difficulty”

  33. Ordinal Variables in Regression • “Grouped Ordinal” vs. ranks • True ranks get used as if they were interval subject of distributional assumptions of maximum likelihood • Grouped ordinal (i.e., with a small number of values, e.g., 1,2,3,4,5) can be dealt with differently • Ordinal predictors • Ordinal dependent variables

  34. Ordinal Predictors • Test for “linearity” • fit as an interval variable (e.g., values 1 to 5) • fit as a set of dummy variables • actually, add k-2 (not k-1) dummies to the model with the interval version • test whether dummies significantly improve fit • compare fit (i.e., does the set of dummies fit significantly better) • Choose form based on test of linearity

  35. Predicting Lawyer Hours IComplexity as Quantitative

  36. Predicting Lawyer Hours IIComplexity as Dummies (ordinal)

  37. Predicting Lawyer Hours IIIAre Dummies (Ordinal) Better?

  38. Ordinal Dependent VariableOrdinal Probit and Ordinal Logit • Assume that there is an interval scale Y* underlying an observed grouped ordinal variable Y • e.g., “complexity” measured on a 5 point scale • Estimate regression model on the underlying scale along with the cut points (τ’s) that define the grouping - inf + inf τ1 τ2 τ3 τ4 Y* Y=1 Y=2 Y=3 Y=4 Y=5

  39. Ordinal Probit • Assume that Y* has a standard normal distribution • β’s can be interpreted as change in standard deviations in Y* • Estimation is done by finding the β’s and τ’s that maximize the probability of observing the sample • Constraint: - inf + inf τ1 τ2 τ3 τ4 Y* Y=1 Y=2 Y=3 Y=4 Y=5

  40. Explaining Case ComplexityOrdered Probit & Ordinary Regression

  41. Explaining Case ComplexityOrdered Probit & Ordered Logit

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