Model Fit Assessment
Residual Fit Spread Plot
Plot to detect non-linearity, influential observations and outliers. Consists of side-by-side quantile plots of the centered fit and the residuals. It shows how much variation in the data is explained by the fit and how much remains in the residuals. For inappropriate models, the spread of the residuals in such a plot is often greater than the spread of the centered fit.
Part & Partial Correlations
Correlations
Relative importance of independent variables in determining Y. How much each variable uniquely contributes to \(R^{2}\) over and above that which can be accounted for by the other predictors.
Zero Order
Pearson correlation coefficient between the dependent variable and the independent variables.
Part
Unique contribution of independent variables. How much \(R^{2}\) will decrease if that variable is removed from the model?
Partial
How much of the variance in Y, which is not estimated by the other independent variables in the model, is estimated by the specific variable?
## Correlations
## -------------------------------------------
## Variable Zero Order Partial Part
## -------------------------------------------
## disp -0.848 0.048 0.019
## hp -0.776 -0.224 -0.093
## wt -0.868 -0.574 -0.285
## qsec 0.419 0.219 0.091
## -------------------------------------------Observed vs Predicted Plot
Plot of observed vs fitted values to assess the fit of the model. Ideally, all your points should be close to a regressed diagonal line. Draw such a diagonal line within your graph and check out where the points lie. If your model had a high R Square, all the points would be close to this diagonal line. The lower the R Square, the weaker the Goodness of fit of your model, the more foggy or dispersed your points are from this diagonal line.
Lack of Fit F Test
Assess how much of the error in prediction is due to lack of model fit. The residual sum of squares resulting from a regression can be decomposed into 2 components:
- Due to lack of fit
- Due to random variation
If most of the error is due to lack of fit and not just random error, the model should be discarded and a new model must be built. The lack of fit F test works only with simple linear regression. Moreover, it is important that the data contains repeat observations i.e. replicates for at least one of the values of the predictor x. This test generally only applies to datasets with plenty of replicates.
## Lack of Fit F Test
## -----------------
## Response : mpg
## Predictor: disp
##
## Analysis of Variance Table
## ----------------------------------------------------------------------
## DF Sum Sq Mean Sq F Value Pr(>F)
## ----------------------------------------------------------------------
## disp 1 808.8885 808.8885 314.0095 1.934413e-17
## Residual 30 317.1587 10.57196
## Lack of fit 25 304.2787 12.17115 4.724824 0.04563623
## Pure Error 5 12.88 2.576
## ----------------------------------------------------------------------
