lecture slides stats1.13.l21.air

Statistics One

Lecture 21 Assumptions Revisited

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Two segments

•  Assumptions •  Transformations

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Lecture 21 ~ Segment 1

Assumptions

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Review of main assumptions

•  Correlation •  Normal distribution in X and Y •  Linear relationship between X and Y •  Homoscedasticity

•  Plot histograms and scatterplots •  Print summary statistics

Review of main assumptions

•  Regression •  Normal distribution in Y •  Linear relationship between X and Y •  Homoscedasticity •  Correlations among predictor variables all < .80 •  No multicolinearity

•  Plot histograms and scatterplots and residuals •  Print summary statistics

Review of main assumptions

•  t-tests •  Normal distribution in Y (DV) •  Homogeneity of variance •  Equivalent sample size

•  Levene’s test

Review of main assumptions

•  Between groups and Factorial ANOVA •  Normal distribution in Y (DV) •  Homogeneity of variance •  Equivalent sample size

•  Levene’s test

Review of main assumptions

•  Repeated measures ANOVA •  Normal distribution in Y (DV) •  Sphericity assumption •  Homogeneity of variance •  Homogeneity of covariance

•  Equivalent sample size

•  Levene’s test •  Mauchly’s test

Review of main assumptions

•  Chi-square •  Independence •  Adequate expected cell counts

Review of main assumptions

•  Two primary constraints of the assumptions •  Normal distribution in Y (DV) •  Linear relationship between predictor variables

and outcome variable (see Lecture 23)

Review of main assumptions

•  Normal distribution in Y (DV) •  How to test •  Histograms and summary statistics •  Q-Q plots •  Empirical tests, such as D’Agostino’s K2 test

Review of main assumptions

•  Normal distribution in Y (DV) •  How to test •  Histograms and summary statistics

•  Look for extreme skew and/or kurtosis and/or outliers (for example, cases +/- 3 SDs from the mean)

•  Rule of thumb is (skew > 3) and/or (kurtosis > 10) indicates a non-normal distribution

Positive skew with outliers

Positive skew

Review of main assumptions

•  Normal distribution in Y (DV) •  How to test •  Q-Q plot

•  A plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution.

•  If the distribution is normal then the plotted points should approximately lie on a straight line.

Normal distribution

Positive skew

Segment summary

•  The inferential statistics covered in this course involve several assumptions

•  Two primary assumptions – Normal distribution in the outcome Y (DV) – Linear relationship between predictors and Y

Segment summary

•  Normal distribution in Y (DV) •  How to test •  Histograms and summary statistics •  Q-Q plots •  Empirical tests, such as D’Agostino’s K2 test

END SEGMENT

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Lecture 21 ~ Segment 2

Transformations

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Transformations •  If a distribution is not normal then it is sometimes possible

to transform the data in an attempt to make the distribution normal

•  A transformation is a single function applied to all data points in the distribution (for example, square root)

•  The rank order of cases should remain the same but the distance between cases may change

Transformations

•  Most common transformations for positive skew •  Square root •  Logarithm •  Inverse

Transformations

Transformations

•  Most common transformations for negative skew •  Reflect and Square root •  Reflect and Logarithm •  Reflect and Inverse

Transformations

Positive skew

Skew = 3.05 Kurtosis = 9.26

Square root transform

Skew = 2.86 Kurtosis = 7.91

Log transform

Skew = 2.69 Kurtosis = 6.75

Negative skew

Skew = -3.36 Kurtosis = 11.73

Reflect and square root

Skew = 2.58 Kurtosis = 6.62

Reflect and log

Skew = 1.59 Kurtosis = 2.92

Segment summary

•  If a distribution is not normal then it is sometimes possible to transform the data in an attempt to make the distribution normal

Segment summary

•  Most common transformations for positive skew •  Square root •  Logarithm •  Inverse

Segment summary

•  Most common transformations for negative skew •  Reflect and Square root •  Reflect and Logarithm •  Reflect and Inverse

END SEGMENT

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END LECTURE 21

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