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Nonparametric tests I Back to basics

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Lecture Outline What is a nonparametric test? Rank tests, distribution free tests and nonparametric tests Which type of test to use

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MTB > dotplot 'Male' 'Female'; SUBC> same.. : :: :..:::.. :..:: :....:..... : MALE..:. : : :..: ::::::.::.:. ::.: :. : FEMALE

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MTB > dotplot 'Male' 'Female'; SUBC> same.. : :: :..:::.. :..:: :....:..... : MALE..:. : : :..: ::::::.::.:. ::.: :. : FEMALE MTB > desc 'Male' 'Female’ Variable N Mean Median TrMean StDev SEMean MALE FEMALE Variable Min Max Q1 Q3 MALE FEMALE

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Lecture Outline What is a nonparametric test? –What is a parameter? –What are examples of non-parametric tests? Rank tests, distribution free tests and nonparametric tests Which type of test to use

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Parameters are central to inference in GLM and ANOVA and represent assumptions about the underlying processes

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LET K1=4.7 # Group 1 mean minus grand mean LET K2=-2.5 # Group 2 mean minus grand mean LET K3=10.4 # The grand mean LET K4=1.9 # Standard deviation of the error RANDOM 30 'Error' LET 'Y'=K3+K1*'DUM1'+K2*'DUM2'+K4*'Error'

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LET K1=4.7 # Group 1 mean minus grand mean LET K2=-2.5 # Group 2 mean minus grand mean LET K3=10.4 # The grand mean LET K4=1.9 # Standard deviation of the error RANDOM 30 'Error' LET 'Y'=K3+K1*'DUM1'+K2*'DUM2'+K4*'Error' Fitted value = + Group 1 1 2 2 3- 1 - 2 Error has Normal Distribution with zero mean and standard deviation

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LET K1=4.7 # Group 1 mean minus grand mean LET K2=-2.5 # Group 2 mean minus grand mean LET K3=10.4 # The grand mean LET K4=1.9 # Standard deviation of the error RANDOM 30 'Error' LET 'Y'=K3+K1*'DUM1'+K2*'DUM2'+K4*'Error' Fitted value = + Group 1 1 2 2 3- 1 - 2 Error has Normal Distribution with zero mean and standard deviation

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Parameters are central to inference in GLM and ANOVA but represent assumptions about the underlying processes

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Parameters are central to inference in GLM and ANOVA but represent assumptions about the underlying processes can be done without in some simple situations

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Parameters are central to inference in GLM and ANOVA but represent assumptions about the underlying processes can be done without in some simple situations – BUT HOW?

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RnkWtSex

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RnkWtSex Remember ties

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Mean Rank

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The ‘Male’ mean rank = The ‘Female’ mean rank = Mean Rank

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MTB > mann-whitney male female

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Mann-Whitney Test and CI: MALE, FEMALE

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median =

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200)

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200) W =

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200) W = Sum of ranks of 2763 corresponds to a mean rank of 2763/50 = 55.26

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The ‘Male’ mean rank = The ‘Female’ mean rank = Mean Rank

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The ‘Male’ mean rank = The ‘Female’ mean rank = Mean Rank

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200) W = Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200) W = Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at The test is significant at (adjusted for ties)

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200) W = Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at The test is significant at (adjusted for ties) Cannot reject at alpha = 0.05

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200) W = Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at The test is significant at (adjusted for ties) Cannot reject at alpha = 0.05

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MTB > mann-whitney male female Mann-Whitney Test and CI: MALE, FEMALE MALE N = 50 Median = FEMALE N = 50 Median = Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.1200) W = Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at The test is significant at (adjusted for ties) Cannot reject at alpha = 0.05 The null hypothesis is better expressed as “the distributions of male and female weights are the same”.

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Parameters are central to inference in GLM and ANOVA but represent assumptions about the underlying processes can be done without in some simple situations

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Nonparametric vs Parametric

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Sign TestOne-sample t-test

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Nonparametric vs Parametric Sign Test Mann-Whitney Test One-sample t-test Two-sample t-test

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Nonparametric vs Parametric Sign Test Mann-Whitney Test Spearman Rank Test One-sample t-test Two-sample t-test Correlation/Regression

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Nonparametric vs Parametric Sign Test Mann-Whitney Test Spearman Rank Test Kruskal-Wallis Test One-sample t-test Two-sample t-test Correlation/Regression One-way ANOVA

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Nonparametric vs Parametric Sign Test Mann-Whitney Test Spearman Rank Test Kruskal-Wallis Test Friedman Test One-sample t-test Two-sample t-test Correlation/Regression One-way ANOVA One-way blocked ANOVA

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Lecture Outline What is a nonparametric test? Rank tests, distribution free tests and nonparametric tests Which type of test to use

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A rose by any other name.. Non-parametric tests lack parameters Rank tests start by ranking the data Distribution-free tests don’t assume a Normal distribution (or any other) These are mainly but not completely overlapping sets of tests (and some are scale-invariant too).

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Lecture Outline What is a nonparametric test? Rank tests, distribution free tests and nonparametric tests Which type of test to use

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Fewer assumptions but... still some assumptions (including independence) limited range of situations –no more than 2 x-variables –can’t mix continuous and categorical x-variables provide p-values but estimation is dodgy loss of efficiency if parametric assumptions are upheld there is a grand scheme for parametric statistics (GLM) but a lot of separate strange names for nonparametrics

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When is there a choice? when there is a non-parametric test –fewer than two or three variables altogether and prediction is not required

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How to choose: If the assumptions of parametric test are upheld, use it – on grounds of efficiency If not upheld, consider fixing the assumptions (e.g. by transforming the data, as in the practical) If assumptions not fixable, use nonparametric test

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MTB > dotplot 'LogM' 'LogF'; SUBC> same ::: :... :::.. :..::.:....: : :. : LogM.:. :... : ::.:: : :. ::.::. ::.:. :. : LogF

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MTB > dotplot 'LogM' 'LogF'; SUBC> same ::: :... :::.. :..::.:....: : :. : LogM.:. :... : ::.:: : :. ::.::. ::.:. :. : LogF MTB > desc 'LogM' 'LogF' Variable N Mean Median TrMean StDev SEMean LogM LogF Variable Min Max Q1 Q3 LogM LogF

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Lecture Outline What is a nonparametric test? Rank tests, distribution free tests and nonparametric tests Which type of test to use

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Last remarks Nonparametric tests are an opportunity to revise the basic ideas of statistical inference They are sometimes useful in biology They are often used in biology NEXT WEEK: more nonparametrics, including confidence intervals and randomisation tests. READ the handout

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