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MANE 6313

Week 12, Module B

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 assumptions.

Fitting Linear Regression Models

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

Test for Significance of Regression

  • Test for significance of regression is a test to determine if there is a linear relationship between \(y\) and a subset of the regressors
\[ \begin{aligned} H_0:\beta_1&=\beta_2=\cdots=\beta_k=0\\ H_a:\beta_j&\neq 0 \mbox{ for at least one }j\end{aligned} \]
  • The test statistic is
\[ F_0=\frac{SS_R/k}{SS_E/(n-k-1)}=\frac{MS_R}{MS_E} \]
  • Reject \(H_0\) if \(F_0>F_{\alpha,k,n-k-1}\)

Example 12.8

Ex 12.8 lm() Output

Residuals - Normality Assumption

Ex 12.8 Normality Assumption

Residuals vs.Fitted Values

Ex. 12.8 Residuals vs. Fitted Values

Residuals vs. Powder Temperature

Residuals vs. Powder Temperature

Residuals vs. Extrusion

Residuals vs. Extrusion

Residuals vs. Die Temperature

Residuals vs. Die Temperature