Question

For the given model \(x_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + e_{ijk}; i=1, \dots,p; j=1, \dots,q; k=1, \dots,m\), under the assumption of the normality of the parent population, the p.d.f. of \(y = \frac{S_{AB}^2}{\sigma_e^2}\), is

• A. Gamma with parameters \((q/2, 2)\)
• B. Gamma with parameters \((pq/2, 2)\)
• C. Gamma with parameters \(((p-1)(q-1)/2, 2)\)
• D. Gamma with parameters \(((p-1)/2, 2)\)

Hint:

Remember the fundamental theorem of ANOVA: For a normal linear model, any Sum of Squares (SS) divided by the error variance \(\sigma^2\) follows a Chi-squared distribution with degrees of freedom equal to the degrees of freedom of that SS. Also, remember the relationship: \(\chi^2_v \equiv \text{Gamma}(v/2, 2)\).

Answer 1

The Correct Option is C

Step 1: Concept Overview:
We are given a two-way ANOVA model with interaction effects (\(\gamma_{ij}\)). \(S_{AB}^2\) represents the interaction sum of squares between factors A (\(\alpha_i\)) and B (\(\beta_j\)). The goal is to determine the distribution of \(S_{AB}^2\) when scaled by the error variance \(\sigma_e^2\).

Step 2: Core Principles:
Under standard ANOVA assumptions (normality, independence, and equal variances), the ratio of a sum of squares (SS) to the error variance, \( \frac{\text{SS}}{\sigma_e^2} \), follows a Chi-squared distribution with \(v\) degrees of freedom (\( \chi^2_v \)), where \(v\) is the degrees of freedom for that SS.A Chi-squared distribution is a special case of the Gamma distribution: \( \chi^2_v \equiv \text{Gamma}(\text{shape}=v/2, \text{scale}=2) \). The rate parameter is the inverse of the scale, so rate = 1/2.We need to determine the degrees of freedom associated with the interaction sum of squares, \(S_{AB}^2\).

Step 3: Detailed Explanation:
The ANOVA model is defined as \(x_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + e_{ijk}\), where:- Factor A has \(p\) levels.- Factor B has \(q\) levels.- There are \(m\) replicates per cell.The degrees of freedom are calculated as follows:- df(A) = \(p-1\)- df(B) = \(q-1\)- df(Interaction AB) = \((p-1)(q-1)\)- df(Error) = \(pq(m-1)\)- df(Total) = \(pqm-1\)The interaction sum of squares, \(S_{AB}^2\), has \(v = (p-1)(q-1)\) degrees of freedom. Therefore, the variable \( y = \frac{S_{AB}^2}{\sigma_e^2} \) follows a Chi-squared distribution with \((p-1)(q-1)\) degrees of freedom:\[ y \sim \chi^2_{(p-1)(q-1)} \]We can express this Chi-squared distribution as a Gamma distribution. The relationship between Chi-squared and Gamma is \( \chi^2_v \equiv \text{Gamma}(\text{shape}=k, \text{scale}=\theta) \) where \(k = v/2\) and \(\theta=2\).Thus, the shape parameter is \( k = \frac{(p-1)(q-1)}{2} \), and the scale parameter is \( \theta = 2 \).The probability density function (PDF) presented in the options is a form of the Gamma distribution.The PDF for \( Z \sim \text{Gamma}(k, \theta) \) is \( f(z) = \frac{1}{\Gamma(k)\theta^k} z^{k-1} e^{-z/\theta} \).The PDF in the options, \( f(y) = \frac{e^{-y/2} y^{k-1}}{2^k \Gamma(k)} \), matches a Gamma distribution with shape \(k\) and scale \(\theta=2\).For our variable \(y\), the shape parameter is \( k = \frac{(p-1)(q-1)}{2} \).Therefore, the distribution is a Gamma distribution with parameters \( (\frac{(p-1)(q-1)}{2}, 2) \), corresponding to option (C).
Step 4: Conclusion:
The probability density function (p.d.f.) of \(y\) is that of a Gamma distribution with shape = \(\frac{(p-1)(q-1)}{2}\) and scale = 2.