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Assumptions. “Essentially , all models are wrong, but some are useful”. Your model has to be wrong… … but that’s o.k. if it’s illuminating!. George E.P. Box. Linear Model Assumptions. Absence of Collinearity. No influential data points. Normality of Errors. Homoskedasticity of Errors.
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“Essentially, all models are wrong, but some are useful” Your model has to bewrong… … but that’s o.k.if it’s illuminating! George E.P. Box
Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence
Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence
Absence of Collinearity Baayen(2008: 182)
Absence of Collinearity Baayen(2008: 182)
Demo Where does collinearitycome from? …most often, correlated predictor variables
Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence
Leverage Baayen(2008: 189-190)
Leave-one-outInfluence Diagnostics DFbeta (…and much more)
Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence
Normality of Error The error (not the data!) is assumed to be normally distributed So, the residuals should be normally distributed
xmdl = lm(y ~ x) hist(residuals(xmdl)) ✔
qqnorm(residuals(xmdl)) qqline(residuals(xmdl)) ✔
qqnorm(residuals(xmdl)) qqline(residuals(xmdl)) ✗
Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence
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
WHAT TO IF NORMALITY/HOMOSKEDASTICITY IS VIOLATED? Either: nothing + report the violation Or: report the violation + transformations
Two types of transformations LinearTransformations NonlinearTransformations Leave shape of the distributionintact (centering, scaling) Do change the shape of the distribution
After transformation Still bad…. …. but better!!
Assumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence
Assumptions Normality of Errors Homoskedasticity of Errors (Histogram of Residuals) Q-Q plot of Residuals Residual Plot
Assumptions Normality of Errors Homoskedasticity of Errors Absence ofCollinearity No influentialdata points Independence
Assumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence
Common experimental data Subject Item... Item #1 Item... Rep 1 Rep 3 Rep 2
Common experimental data Pseudoreplication = Disregarding Dependencies Subject Item... Item #1 Item... Rep 1 Rep 3 Rep 2
Subject1 Item1 Subject1 Item2 Subject1 Item3 … … Subject2 Item1 Subject2 Item2 Subject3 Item3 …. … “pooling fallacy” Machlis et al. (1985) “pseudoreplication” Hurlbert (1984)
Hierarchicaldataiseverywhere • Typological data(e.g., Bell 1978, Dryer 1989, Perkins 1989; Jaeger et al., 2011) • Organizational data • Classroom data
Finnish Norwegian Swedish English German Hungarian French Romanian Italian Spanish Turkish
Finnish Norwegian Swedish English German Hungarian French Romanian Italian Spanish Turkish
Hierarchicaldataiseverywhere Class 1 Class 2
Hierarchicaldataiseverywhere Class 1 Class 2
Hierarchicaldataiseverywhere Class 1 Class 2
Hierarchicaldataiseverywhere IntraclassCorrelation (ICC)
