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Nonparametric Statistics Timothy C. Bates

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Parametric Statistics 1 Assume data are drawn from samples with a certain distribution (usually normal) Compute the likelihood that groups are related/unrelated or same/different given that underlying model t-test, Pearson’s correlation, ANOVA…

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Parametric Statistics 2 Assumptions of Parametric statistics 1. Observations are independent 2. Your data are normally distributed 3. Variances are equal across groups Can be modified to cope with unequal ∂ 2

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Non-parametric Statistics? Non-parametric statistics do not assume any underlying distribution They estimate the distribution AND compute the probability that your groups are the related/the same or unrelated/different

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Nonparametric ≠ No parameters Model structure is not specified a priori but is instead determined from data. The data are parameterised by the analysis AKA: “distribution free”

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Non-parametric Statistics Assumptions of non-parametric statistics 1. Observations are independent

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Non-parametric Statistics? Non-parametric statistics do not assume any underlying distribution Estimating or modeling this distribution reduces their power to detect effects… So never use them unless you have to

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Why use a Non-parametric Statistic? Very small samples (<20 replicates) High probability of violating the assumption of normality Leads to spurious Type-1 (false alarm) errors

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Why use a Non-parametric Statistic? Outliers more often lead to spurious Type- 1 (false alarm) errors in parametric statistics. Nonparametric statistics reduce data to an ordinal rank, which reduces the impact or leverage of outliers.

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Error Type-I error: False Alarm for a bogus effect reject the null hypothesis when it is really true Type-II error: Miss a real effect fail to reject our null hypothesis when it is really false Type-III error: :-) lazy, incompetent, or willful ignorance of the truth

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Power 1-alpha

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Non-parametric Choices Data type? χ2χ2 discrete Question? continuous Number of groups? Spearman’s Rank associationDifferent central value Mann-Whitney U Wilcoxon’s Rank Sums Kruskal-Wallis test two-groupsmore than 2 Brown- Forsythe Difference in ∂ 2

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Non-parametric Choices Data type? χ2χ2 discrete Question? continuous Number of groups? Spearman’s Rank Like a Pearson’s R Mann-Whitney U Wilcoxon’s Rank Sums Kruskal-Wallis test two-groupsmore than 2 Like ANOVA Like Student’s t No alternative Different central value Brown- Forsythe Difference in ∂ 2 Like F-test association

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Chi-Squared (Χ 2 ) χ2 tests the null hypothesis that observed events occur with an expected frequency in large samples frequencies are distributed as Χ 2 e.g. Ho: “This six-sided dice is fair ” Expect all 6 outcomes to occur equally often Assumptions Observations are independent Outcomes mutually exclusive Sample is not small Small samples require exact test:, i.e., binomial test

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Chi-Squared Χ 2 formula Χ 2 = the sum of each squared difference between the observed and expected frequencies divided its expected frequency

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Χ 2 and contingency tables Χ 2 essentially tests if each cell in a contingency table has its expected value In a 2-way table, this expectation will be the value of an adjacent cell

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Example: coin toss Random sample of 100 coin tosses, of a coin believed to be fair We observed number of 45 heads, and and 55 tails Is the coin fair?

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Coin toss If h o is true, our test statistic is drawn from a Χ 2 distribution with df = 1 (45-50) 2 + (55-50) 2 = = Χ 2 (1) = 1, p > 0.3

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Coin toss Χ 2 in R chisq.test(c(45,55), p=c(.5,.5)) Chi-squared test for given probabilities Χ 2 = 1, df = 1, p =

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Spearman Rank test (ρ (rho)) Named after Charles Spearman, Non-parametric measure of correlation Assesses how well an arbitrary monotonic function describes the relationship between two variables, Does not require the relationship be linear Does not require interval measurement

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Spearman Rank test (ρ (rho)) Mathematically, it is simply a Pearson’s r computed on ranked data d = difference in rank of a given pair n = number of pairs Alternative test = Kendall's Tau (Kendall's τ)

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Mann-Whitney U AKA: “Wilcoxon rank-sum test Mann & Whitney, 1947; Wilcoxon, 1945 Non-parametric test for difference in the medians of two independent samples Assumptions: Samples are independent Observations can be ranked (ordinal or better)

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Mann-Whitney U U tests the difference in the medians of two independent samples n 1 = number of obs in sample 1 n 2 = number of obs in sample 2 R = sum of ranks of the lower-ranked sample

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Mann-Whitney U or t-test? Should you use it over the t-test? Yes if you have a very small sample (<20) (central limit assumptions not met) Possibly if your data are inherently ordinal Otherwise, probably not. It is less prone to type-I error (spurious significance) due to outliers. But does not in fact handle comparisons of samples whose variances differ very well (Use unequal variance t-test with rank data)

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Aesop: Mann-Whitney U Example Suppose that Aesop is dissatisfied with his classic experiment in which one tortoise was found to beat one hare in a race. He decides to carry out a significance test to discover whether the results could be extended to tortoises and hares in general…

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Aesop 2: Mann-Whitney U He collects a sample of 6 tortoises and 6 hares, and makes them all run his race. The order in which they reach the finishing post (their rank order) is as follows: tort = c(1, 7, 8, 9, 10,11) hare = c(2, 3, 4, 5, 6, 12) Original tortoise still goes at warp speed, original hare is still lazy, but the others run truer to stereotype.

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Aesop 3: Mann-Whitney U wilcox.test(tort, hare) Wilcoxon = W = 25, p-value = 0.31 Tortoises are not faster (but neither are hares) tort = c(1, 7, 8, 9, 10,11) (n 2 = 6) hare = c(2, 3, 4, 5, 6, 12) (n 1 = 6, R 1 =32)

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Aesop 4: Mann-Whitney U Wilcoxon = W = 25, p-value = 0.31 Tortoises are not faster (but neither are hares). Welch Two Sample t-test t = , df = 10, p-value = 0.28 Alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: ~ 6.91 sample estimates: mean of x = 7.6 mean of y = 5.3

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Power comparison with continuous normal data tort = hare = Wilcoxon W = 25, p = 0.31 t.test t.test(tort, hare, var.equal = TRUE) t(10) = 1.5, p = 0.16

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Wilcoxon signed-rank test (related samples) Same idea as MW U, generalized to matched samples Equivalent to non-independent sample t- test

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Kruskall-Wallis Non-parametric one-way analysis of variance by ranks (named after William Kruskal and W. Allen Wallis) tests equality of medians across groups. It is an extension of the Mann-Whitney U test to 3 or more groups. Does not assume a normal population, Assumes population variances among groups are equal.

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