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Conditions of applications. Key concepts. Testing conditions of applications in complex study design Residuals Tests of normality Residuals plots Residuals vs. fitted QQ plots Cook’s distance. Conditions of applications.

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Key concepts
Key concepts

  • Testing conditions of applications in complex study design

  • Residuals

  • Tests of normality

  • Residuals plots

    • Residuals vs. fitted

    • QQ plots

    • Cook’s distance


Conditions of applications1
Conditions of applications

  • RM ANOVA and multilevel modeling have 2 conditions of application in common:

    • Normality of the DV by cell of the IV

      • Few outliers

    • Homoscedasticity (equality of variance)

    • (Linearity: trivial in ANOVA since we only estimate mean differences)


Problems with checking normality by cell
Problems with checking normality by cell

  • Number of cells grow with number of IV

  • What about continuous IV

  • How to deal with number of tests


Problems with checking homoscedasticity by pair of cells
Problems with checking homoscedasticity by pair of cells

  • Number of cells grow with number of IV

  • What about continuous IV

  • How to deal with number of tests


Residuals definition
Residuals: definition

  • Yi = b0 + b1X + e

  • Thus,

  • Where e are the residuals, and correspond to the distance between the observed value and the best predicted value


Residuals what to look for
Residuals: what to look for

  • Residuals should have a normal distribution across (or irrespective of) groups since differences in IV have been subtracted.

  • Residuals should have equal variances, similarly to observed DV by cell

  • There should be no remaining structure in the residuals (allow to check for linearity


Normality tests
Normality tests

  • Many normality tests exist. By order of type I and type II error:

    • Shapiro-Wilk:

    • Where a depends on the parameters of a normal distribution and xi are the value of x from the smallest to the largest

    • Anderson-Darling: same idea of ordering data

    • Kolmogorov-smirnov


Conditions of applications
But…

  • All of these tests are known to be incorrect.

    • When data are in fact from a normal distribution, they reject the null too often or too rarely

    • When data are in fact not from a normal distribution, they do not reject the null often enough (low power)


Residual plots residuals vs fitted or vs each iv
Residual plots: residuals vs. fitted or vs. each IV

  • Scatterplot of the predicted values (Yi hat) against the residuals or against each IV.

  • There are different versions of this type of plot (e.g., residuals can be divided by their estimated standard deviation or not)

  • They allow to examine

    • homoscedasticity,

    • Linearity of relationship between IV and DV,

    • Normality of residuals (should have ellipsoid shape),

    • outliers


Residual plot quantile quantile plot
Residual plot: Quantile-Quantileplot

  • Graphical method for comparing two probability distributions

  • Compare the quantiles of the normal distribution with mean 0 and variance s2 to the values (ordered) of the residuals

  • All the points should align on the diagonal from bottom left to top right


Outliers
Outliers

  • Outliers are extreme values either on the IV or on the DV or both.

  • Leverage observations are extreme on the X-axis (IV). But may not influence too much the estimation of the parameters.

  • Influential observations are extreme on the X and Y axes, and influence greatly the estimation of the parameters


Cook s distances
Cook’s distances

Where Yj are the predicted values of Y, and Yj(i) are the predicted values of Y if observation i was removed and the model was estimated again.

p is the number of parameters of the model and MSE is the mean square error.

Cutoff: 1 or 4/n or Fp,n-p


An example of a residual analysis
An example of a residual analysis

  • Back to autism data again.

    • Step 1: obtain the residualsuse the option save in the mixed linear model

    • Step 2: check normality (analysisexplore)

    • Step 3: look at residuals plot

      • Residuals vs fitted

      • Residuals vs time

      • (Standardized residuals vs fitted)

      • QQ plots

      • (Cook’s distance)