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Assumptions

Assumptions

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Assumptions

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  1. Assumptions

  2. “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

  3. Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence

  4. Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence

  5. Absence of Collinearity Baayen(2008: 182)

  6. Absence of Collinearity Baayen(2008: 182)

  7. Demo Where does collinearitycome from? …most often, correlated predictor variables

  8. What to do?

  9. Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence

  10. Leverage Baayen(2008: 189-190)

  11. Leave-one-outInfluence Diagnostics DFbeta (…and much more)

  12. Winter & Matlock (2013)

  13. Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence

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

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

  16. qqnorm(residuals(xmdl)) qqline(residuals(xmdl)) ✔

  17. qqnorm(residuals(xmdl)) qqline(residuals(xmdl)) ✗

  18. Linear ModelAssumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence

  19. 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

  20. WHAT TO IF NORMALITY/HOMOSKEDASTICITY IS VIOLATED? Either: nothing + report the violation Or: report the violation + transformations

  21. Two types of transformations LinearTransformations NonlinearTransformations Leave shape of the distributionintact (centering, scaling) Do change the shape of the distribution

  22. Before transformation

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

  24. Assumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence

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

  26. Assumptions Normality of Errors Homoskedasticity of Errors Absence ofCollinearity No influentialdata points Independence

  27. Assumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence

  28. What isindependence?

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

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

  31. Subject1 Item1 Subject1 Item2 Subject1 Item3 … … Subject2 Item1 Subject2 Item2 Subject3 Item3 …. … “pooling fallacy” Machlis et al. (1985) “pseudoreplication” Hurlbert (1984)

  32. Hierarchicaldataiseverywhere • Typological data(e.g., Bell 1978, Dryer 1989, Perkins 1989; Jaeger et al., 2011) • Organizational data • Classroom data

  33. Finnish Norwegian Swedish English German Hungarian French Romanian Italian Spanish Turkish

  34. Finnish Norwegian Swedish English German Hungarian French Romanian Italian Spanish Turkish

  35. Hierarchicaldataiseverywhere Class 1 Class 2

  36. Hierarchicaldataiseverywhere Class 1 Class 2

  37. Hierarchicaldataiseverywhere Class 1 Class 2

  38. Hierarchicaldataiseverywhere

  39. Hierarchicaldataiseverywhere IntraclassCorrelation (ICC)