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Assumptions

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“Essentially, all models are wrong, but some are useful” George E.P. Box Your model has to be wrong… … but that’s o.k. if it’s illuminating!

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Linear Model Assumptions Absence of Collinearity Normality of Errors Homoskedasticity of Errors No influential data points Independence

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Linear Model Assumptions Absence of Collinearity Normality of Errors Homoskedasticity of Errors No influential data points Independence

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Absence of Collinearity Baayen (2008: 182)

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Absence of Collinearity Baayen (2008: 182)

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Where does collinearity come from? …most often, correlated predictor variables Demo

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What to do?

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Linear Model Assumptions Absence of Collinearity Normality of Errors Homoskedasticity of Errors No influential data points Independence

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Baayen (2008: 189-190)

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DFbeta (…and much more) Leave-one-out Influence Diagnostics

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Winter & Matlock (2013)

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Linear Model Assumptions Absence of Collinearity Normality of Errors Homoskedasticity of Errors No influential data points Independence

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Normality of Error The error (not the data!) is assumed to be normally distributed So, the residuals should be normally distributed

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xmdl = lm(y ~ x) hist(residuals(xmdl)) ✔

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qqnorm(residuals(xmdl)) qqline(residuals(xmdl)) ✔

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qqnorm(residuals(xmdl)) qqline(residuals(xmdl)) ✗

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Linear Model Assumptions Absence of Collinearity Normality of Errors Homoskedasticity of Errors No influential data points Independence

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Homoskedasticity of Error The error (not the data!) is assumed to have equal variance across the predicted values So, the residuals should have equal variance across the predicted values

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✔

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✗

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✗

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WHAT TO IF NORMALITY/HOMOSKEDASTI CITY IS VIOLATED? Either: nothing + report the violation Or: report the violation + transformations

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Two types of transformations Linear Transformations Nonlinear Transformations Leave shape of the distribution intact (centering, scaling) Do change the shape of the distribution

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Before transformation

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After transformation Still bad…. …. but better!! Still bad…. …. but better!!

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Assumptions Absence of Collinearity Normality of Errors Homoskedasticity of Errors No influential data points Independence

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Normality of Errors Homoskedasticity of Errors (Histogram of Residuals) Q-Q plot of Residuals Residual Plot Assumptions

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Absence of Collinearity No influential data points Independence Normality of Errors Homoskedasticity of Errors Assumptions

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Absence of Collinearity Normality of Errors Homoskedasticity of Errors No influential data points Independence Assumptions

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What is independence?

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Rep 1 Rep 2 Rep 3 Item #1 Subject Common experimental data Item...

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Rep 1 Rep 2 Rep 3 Item #1 Subject Common experimental data Pseudoreplication = Disregarding Dependencies Pseudoreplication = Disregarding Dependencies Item...

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Subject1Item1 Subject1Item2 Subject1Item3… Subject2Item1 Subject2Item2 Subject3Item3 ….… Machlis et al. (1985) “ pooling fallacy ” Hurlbert (1984) “pseudoreplication”

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Hierarchical data is everywhere Typological data (e.g., Bell 1978, Dryer 1989, Perkins 1989; Jaeger et al., 2011) Organizational data Classroom data

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Germa n French English Spanish Italian Swedish Norwegian Finnish Hungarian Turkish Romanian

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Germa n French English Spanish Italian Swedish Norwegian Finnish Hungarian Turkish Romanian

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Class 1Class 2 Hierarchical data is everywhere

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Class 1Class 2 Hierarchical data is everywhere

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Class 1Class 2 Hierarchical data is everywhere

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Intraclass Correlation (ICC) Hierarchical data is everywhere

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Simulation for 16 subjects pseudoreplication items analysis Type I error rate

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Interpretational Problem: What’s the population for inference? Interpretational Problem: What’s the population for inference?

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Violating the independence assumption makes the p-value… …meaningless

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S1 S2

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S1 S2

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That’s it (for now)

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. More About Regression Chapter 14.

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. More About Regression Chapter 14.

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