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MANE 6313

Week 12, Module D

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

Employing test of hypothesis test on groups of variables.

Applying Tests of Hypothesis for Groups of Variables

  • There is a concern regarding the overall error rate for multiple decisions
  • Very similar to family-wise error rate in multiple comparisons
  • In Ex 12.8, when examining the slope terms, three decisions are made at some value of alpha (usually 0.05). The overall error rate will be larger
\[ \alpha_{overall}=1-\left(1-\alpha_{individual}\right)^n \]
  • For Ex 12.8 using \(\alpha_{individual}=0.05\),
\[ \alpha_{overall}=1-\left(1-.05\right)^3=0.1426 \]

Tests for groups of regression coefficients

  • We can examine the contribution of the regression sum of squares for a particular variable, say \(x_j\), given that other variables \(x_i(i\neq j)\) are included in the model

  • We can also determine if the subset of regressor variables \(x_1,x_2,\ldots,x_r (r<k)\) contribute to the model

  • This test is necessary to control the overall error rate

  • The full model is

\[ \mathbf{y=X\beta+\varepsilon} \]
  • We partition the regressors in two groups
\[ \mathbf{\beta=\left[\begin{array}{c}\beta_1\\ \beta_2\end{array}\right]} \]

where \(\mathbf{\beta_1}\) is \((r\times 1)\) and \(\mathbf{\beta_2}\) is \([(p-r)+1]\)

  • We can rewrite the full model as
\[ \mathbf{y=X_1\beta_1+X_2\beta_2+\varepsilon} \]

  • The regression sum of squares for the full model is denoted \(SS_R(\mathbf{\beta})\) with \(p\) degrees of freedom

  • We define the reduced model to be

\[ \mathbf{y=X_2\beta_2\varepsilon} \]

  • The regression sum of squares for the reduced model is \(SS_r(\mathbf{\beta_2})\) with \(p-r\) degrees of freedom.

  • The regression sum of squares due to \(\mathbf{\beta_1}\) given that \(\mathbf{\beta}_2\) is already in the model is

\[ SS_R(\mathbf{\beta_1|\beta_2})=SS_R(\mathbf{\beta})-SS_R(\mathbf{\beta}_2) \]
  • The null hypothesis \(\mathbf{\beta_1=0}\) can be tested with the statistic
\[ F_0=\frac{SS_R(\mathbf{\beta_1|\beta_2})/r}{MS_E} \]

  • Reject \(H_0\) if \(F_0>F_{\alpha,r,n-p}\)

  • This is sometimes called the partial F test or extra sum of squares method

Step 1 - Fit Full Model

Full Model

Step 2 - Fit Reduced Model

Reducedl Model

Step 3 - Analysis

Analysis

Partial F-test in R

Source: https://www.statology.org/partial-f-test/#:~:text=A%20partial%20F-test%20is%20used%20to%20determine%20whether,the%20predictor%20variables%20in%20the%20overall%20regression%20model.