MANE 6313
Week 7, Module G
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
Analyze a Single Replicate of a \(2^k\) with Center Points using R, part 2
Lack of Fit Test
- Addition of center points also enables a test for lack of fit
- Null Hypothesis is \(H_0:\mbox{There is no lack of fit in model}\)
- Alternative Hypothesis is \(H_1:\mbox{ There is lack of fit in model}\)
RSM ANOVA
Test for Pure Quadratic Curvature
- A second-order response model is defined to be
\[
y=\beta_0+\sum_{j=1}^k\beta_jx_j+\sum_{i<j}\sum\beta_{ij}x_ix_j+\sum_{j=1}^k\beta_{jj}x_j^2+\varepsilon
\]
-
Notice the model contains interaction terms and pure quadratic terms
-
Tests the hypothesis that \(H_0:\sum_{j=1}^k\beta_{jj}=0\) vs. \(H_1:\sum_{j=1}^k\beta_{jj}\neq 0\)
-
Calculate one degree of freedom sum of squares for pure quadratic curvature
\[
SS_{\mbox{pure quadratic}}=\frac{n_cn_f(\bar{y}_f-\bar{y}_c)^2}{n_f+n_c}
\]
Test for Pure Quadratic Curvature, continued
- Perform \(F\) test
\[
F_0 =\frac{SS_{\mbox{pure quadratic}}}{MS_{\mbox{residuals}}}
\]
- Rejection Region
Reject \(H_0\) if \(F_0>F_{\alpha,1,df_{\mbox{residuals}}}\)
Test for Pure Quadratic Curvature
Residuals - Normality
Residuals versus Time
Residuals versus Fitted Values
Residuals versus Factor A
Residuals versus Factor B
Residuals versus Factor C
Residuals versus Factor D
