Statistics One

Lecture 22 Non-parametric statistics

Two segments

•  Parametric vs. non-parametric statistics •  Examples

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Lecture 22 ~ Segment 1

Parametric vs. non-parametric statistics

Parametric vs. non-parametric

•  Parametric statistics are used to make inferences about population parameters

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Parametric vs. non-parametric

•  Examples covered in this course •  Regression coefficient •  Regression model R-squared •  Group mean •  Difference between group means •  Proportion of cases distributed across

categories

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Parametric vs. non-parametric

•  In all these cases inferences were based on the sample statistics as well as NHST and/or confidence intervals

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Parametric vs. non-parametric

•  As well, in all these cases we assumed a particular sampling distribution •  t •  F •  Chi-square

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Parametric vs. non-parametric

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Parametric vs. non-parametric

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Parametric vs. non-parametric

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Parametric vs. non-parametric

•  Inferences about population parameters are not valid if all assumptions are not met

•  When assumptions are violated sometimes a “quick fix” is possible and the parametric approach is still considered to be valid

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Parametric vs. non-parametric

•  However, sometimes it is best to abandon the parametric approach and use a non-parametric procedure

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Parametric vs. non-parametric

•  Non-parametric statistics do not assume that the data or population have any characteristic structure

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Parametric vs. non-parametric

•  Correlation •  Regression •  t-tests •  ANOVA •  Chi-square

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Parametric vs. non-parametric

•  Correlation •  Spearman’s rank correlation coefficient •  Kendall’s tau

•  Regression •  Non-parametric regression

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Parametric vs. non-parametric

•  t-tests •  Independent: Mann-Whitney U test •  Dependent: Wilcoxan signed-rank test

•  ANOVA •  Kruskal-Wallis one-way •  Friedman two-way

•  Chi-square •  McNemar’s test •  Fisher’s exact test

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Segment summary

•  Parametric statistics are used to make inferences about population parameters

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Segment summary

•  Inferences about population parameters are not valid if all assumptions are not met

•  When assumptions are violated sometimes a “quick fix” is possible and the parametric approach is still considered to be valid

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Segment summary

•  However, sometimes it is best to abandon the parametric approach and use a non-parametric procedure

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Segment summary

•  Non-parametric statistics do not assume that the data or population have any characteristic structure •  Many procedures available

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END SEGMENT

Lecture 22 ~ Segment 2

Non-parametric statistics Examples

Non-parametric examples

•  Two simple and popular tests covered in this course •  Paired samples t-test •  Independent t-test

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Non-parametric examples

•  Here we will illustrate the non-parametric equivalent of these tests •  Wilcoxan’s ranking method •  Mann-Whitney U test

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Wine tasting!

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Non-parametric examples

•  Wilcoxan ranking method (Wilcoxan, 1945) – Null hypothesis •  Median difference between pairs = 0

– Alternative hypothesis •  Median difference between pairs != 0

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Non-parametric examples

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Wilcoxan: Null hypothesis true Red   White   Sign(S)   ABS   Rank(R)   S*R  

1   65   60   -­‐1   5   1   -­‐1  

2   60   70   +1   10   2   2  

3   65   80   +1   15   3   3  

4   85   65   -­‐1   20   4   -­‐4  

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Wilcoxan: Null hypothesis false Red   White   Sign(S)   ABS   Rank(R)   S*R  

1   65   70   +1   5   1   1  

2   60   70   +1   10   2   2  

3   65   80   +1   15   3   3  

4   65   85   +1   20   4   4  

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Wilcoxan: R output

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Non-parametric examples

•  Mann Whitney U (Mann & Whitney, 1947) – Adapted Wilcoxan’s ranking method to compare

independent groups – Arrange all observations, regardless of group,

into a single ranked series

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Non-parametric examples

•  Mann Whitney U (Mann & Whitney, 1947) – The sum of all ranks, Sum = N(N+1) / 2 •  For example, Sum = 4(5) / 2 = 10

– R = Sum of Ranks from one group – U = Sum - R

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Mann Whitney: Null hypothesis true

Ra3ng   Group   Rank(R)  

1   60   A   1  

2   70   B   2  

3   80   B   3  

4   90   A   4  

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Mann Whitney: Null hypothesis false

Ra3ng   Group   Rank(R)  

1   60   A   1  

2   70   A   2  

3   80   B   3  

4   90   B   4  

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Non-parametric examples

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Non-parametric examples

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Segment summary

•  Two simple and popular tests covered in this course •  Paired samples t-test •  Independent t-test

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Segment summary

•  Here we illustrated the non-parametric equivalent of these tests •  Wilcoxan’s ranking method •  Mann-Whitney U test

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END SEGMENT

END LECTURE 22