Parametric vs. Parametric vs. Nonparametric Nonparametric

Statistical InferenceStatistical Inference

BIOL457BIOL45727 January 201627 January 2016

Normal vs. Non-normalNormal vs. Non-normalFig. 4-6, p. 89Fig. 4-6, p. 89

More Non-normal DistributionsMore Non-normal DistributionsFig. 2-7, p. 26Fig. 2-7, p. 26

Converting Data to the Converting Data to the Standard Normal (“Bell”) CurveStandard Normal (“Bell”) Curve

Curve with mean of 0 and SD of 1Curve with mean of 0 and SD of 1 Convert sample mean to 0 via subtraction of Convert sample mean to 0 via subtraction of

mean from each data pointmean from each data point Divide transformed data points by SDDivide transformed data points by SD

Fig. 2-9, p. 30Fig. 2-9, p. 30

Fig. 4-3, p. 83Fig. 4-3, p. 83

Characteristics of the Normal DistributionCharacteristics of the Normal DistributionFig. 4-4, p. 85Fig. 4-4, p. 85

SymmetricSymmetric 68.26% of data within 1 68.26% of data within 1

SD of meanSD of mean Curve inflection points Curve inflection points

are at are at ± 1 SD± 1 SD 95.44% within 2 SDs95.44% within 2 SDs 99.74% within 3 SDs99.74% within 3 SDs

Parametric or Non-parametric Test?Parametric or Non-parametric Test?

Normality is often assumed without examination of Normality is often assumed without examination of data distributiondata distribution

Under some circumstances, non-normality can be Under some circumstances, non-normality can be assumedassumed ExEx:: Texas map turtle Texas map turtle

habitat data (depth, habitat data (depth, distance from shore)distance from shore)

for 60 turtles but just for 60 turtles but just 18 trapping locations18 trapping locations

Distribution DiagnosticsDistribution Diagnostics Why be concerned?Why be concerned? Many statistical tests assume data come from a Many statistical tests assume data come from a

parametric distributionparametric distribution Many parametric tests have nonparametric Many parametric tests have nonparametric

equivalents (but with less statistical power)equivalents (but with less statistical power) ExsExs: : Mann-Whitney Mann-Whitney UU rather than rather than tt

Kruskal-Wallis rather than ANOVAKruskal-Wallis rather than ANOVASpearman’s rho rather than Pearson’s Spearman’s rho rather than Pearson’s

rr

Distribution DiagnosticsDistribution Diagnostics 1) Kurtosis1) Kurtosis

Leptokurtic—tails smaller than those of a normal Leptokurtic—tails smaller than those of a normal distributiondistribution

Mesokurtic—normal bell curveMesokurtic—normal bell curve Platykurtic—tails larger than those of a normal Platykurtic—tails larger than those of a normal

distributiondistribution

Leptokurtic, Mesokurtic, Leptokurtic, Mesokurtic, and Platykurtic Distributionsand Platykurtic Distributions

Distribution DiagnosticsDistribution Diagnostics 2) Skew2) Skew

Positive skew—modal hump toward left (lower Positive skew—modal hump toward left (lower values)values)

Symmetric—modal hump in middle (normal)Symmetric—modal hump in middle (normal) Negative skew—modal hump toward right (higher Negative skew—modal hump toward right (higher

values)values)

Positive and Negative SkewPositive and Negative Skew

The Kolmogorov-Smirnov TestThe Kolmogorov-Smirnov Test Compares sample distribution to any of a Compares sample distribution to any of a

variety of theoretical distributionsvariety of theoretical distributions Most commonly, the normal distributionMost commonly, the normal distribution

pp-value evaluates the probability data are -value evaluates the probability data are drawn from the specified distributiondrawn from the specified distribution

Transforming Non-Normal Data Transforming Non-Normal Data to Achieve Normalityto Achieve Normality

Allows use of more powerful parametric tests Allows use of more powerful parametric tests rather than nonparametric testsrather than nonparametric tests

Each data point (Each data point (xx) transformed, as…) transformed, as… ……ln(ln(xx)) ……xx22

…√…√xx ……log(log(xx+1)+1) ……eexx

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Transformation to Achieve NormalityTransformation to Achieve NormalityFig. 6-1, p. 140Fig. 6-1, p. 140

Transformation to Achieve NormalityTransformation to Achieve NormalityFig. 6-2, p. 142Fig. 6-2, p. 142