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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. - PowerPoint PPT Presentation
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Assumptions
“Essentially, all models are wrong, but some are useful”
George E.P. Box
Your model has to bewrong…… but that’s o.k.if it’s illuminating!
Linear ModelAssumptions
Absence ofCollinearity
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence
Linear ModelAssumptions
Absence ofCollinearity
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence
Absence of Collinearity
Baayen(2008: 182)
Absence of Collinearity
Baayen(2008: 182)
Where does collinearitycome from?
…most often, correlated predictor variables
Demo
What to do?
Linear ModelAssumptions
Absence ofCollinearity
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence
Baayen(2008: 189-
190)
Leverage
DFbeta(…and much
more)
Leave-one-outInfluence Diagnostics
Winter & Matlock (2013)
Linear ModelAssumptions
Absence ofCollinearity
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence
Normality of ErrorThe 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
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence
Homoskedasticity of ErrorThe 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/HOMOSKEDAS
TICITY IS VIOLATED? Either: nothing + report the
violation Or: report the violation
+ transformations
Two types of transformations
LinearTransformation
s
NonlinearTransformation
s
Leave shape of the distribution
intact (centering, scaling)
Do change the shape of the distribution
Before transformation
After transformation
Still bad….…. but better!!
Assumptions
Absence ofCollinearity
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence
Normality of Errors
Homoskedasticity of Errors
(Histogram of Residuals)
Q-Q plot of Residuals
Residual Plot
Assumptions
Absence ofCollinearity
No influentialdata points
Independence
Normality of Errors
Homoskedasticity of Errors
Assumptions
Absence ofCollinearity
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence
Assumptions
What isindependence?
Rep 1
Rep 2
Rep 3
Item #1
Subject
Common experimental data
Item...
Item...
Rep 1
Rep 2
Rep 3
Item #1
Subject
Common experimental data
Pseudoreplication= DisregardingDependenciesItem
...
Item...
Subject1 Item1Subject1 Item2Subject1 Item3… …
Subject2 Item1Subject2 Item2Subject3 Item3…. …
Machlis et al. (1985)“pooling fallacy”
Hurlbert (1984)“pseudoreplication”
Hierarchical data is everywhere• Typological data
(e.g., Bell 1978, Dryer 1989, Perkins 1989; Jaeger et al., 2011)
• Organizational data
• Classroom data
German
French
English
Spanish Italian
Swedish
NorwegianFinnish
Hungarian
Turkish
Romanian
German
French
English
Spanish Italian
Swedish
NorwegianFinnish
Hungarian
Turkish
Romanian
Class 1 Class 2
Hierarchical data is everywhere
Class 1 Class 2
Hierarchical data is everywhere
Class 1 Class 2
Hierarchical data is everywhere
Hierarchical data is everywhere
IntraclassCorrelation (ICC)
Hierarchical data is everywhere
Simulation for 16 subjects
pseudoreplication
items analysis
Type Ierrorrate
Interpretational Problem:What’s the population
for inference?
Violating the independence assumption makesthe p-value…
…meaningless
S1
S2
S1
S2
That’s it(for now)
