MANE 6313
Week 12, Module E
Student Learning Outcome
- Select an appropriate experimental design with one or more factors,
- Select an appropriate model with one or more factors,
- Evaluate statistical analyses of experimental designs,
- Assess the model adequacy of any experimental design, and
- Interpret model results.
Module Learning Outcome
Assessing linear regression model diagnostics.
Resources for the Week 12, Module E micro-lecture are:
Assessing Linear Regression Model Diagnostics
provide table of contents
\(R^2\)
- The coefficient of multiple determination is defined to be
\[
R^2=\frac{SS_R}{SS_T}=1-\frac{SS_E}{SS_T}
\]
- \(R^2\) is the reduction in variability (in the data) due to using the regressor variables \(x_1,x_2,\ldots,x_k\) (the model).
-
A larger \(R^2\) value indicates more of the total variability is explained by the model; however it does not imply that the model is a GOOD model
-
\(R^2\) always increases as the number of terms in the model is increased
-
Adjusted \(R^2\) is
\[
R^2_{\mbox{adj}}=1-\frac{SS_E/(n-p)}{SS_T/(n-1)}=1-\left(\frac{n-1}{n-p}\right)\left(1-R^2\right)
\]
Model Assumptions and Residuals
- Least squares estimation requires that \(E(\mathbf{\varepsilon})=0\) and \(V(\mathbf{\varepsilon})=\sigma^2\) and the \(\left\{\mathbf{\varepsilon}_i\right\}\) are uncorrelated
- To perform statistical hypothesis tests, we further assume that \(\mathbf{\varepsilon}\sim \mbox{NID}(0,\sigma^2)\)
- These assumptions are validated by examining the residuals
- Perform same analyses previously used for the fixed effects model