Two-Way ANOVA Model and Estimability

The problem describes a two-way ANOVA model with fixed effects and specific constraints on the parameters. The linear model is given by:

\(y_{ij} = \alpha + \beta_i + \gamma_j + \epsilon_{ij}\)

where \(1 \le i \le I\), \(1 \le j \le J\), \(I > 5\), \(J > 5\).

The error terms \(\epsilon_{ij}\) are assumed to have \(E(\epsilon_{ij}) = 0\) and \(Var(\epsilon_{ij}) = \sigma^2\).

The model includes the following constraints:

• \(\sum_{i=1}^I \beta_i = 0\)
• \(\sum_{j=1}^J \gamma_j = 0\)

We are asked to determine which of the given linear combinations of parameters is estimable in this setup.

Estimability in Linear Models

A linear combination of parameters, say \(c^T \theta\), is estimable if there exists a linear combination of the observations, say \(a^T y\), such that the expected value of \(a^T y\) is equal to \(c^T \theta\) for all possible values of \(\theta\). In simpler terms, an estimable function is one that can be linearly estimated by the observations. In ANOVA models with standard constraints like the sum-to-zero constraints given here, certain functions of parameters are estimable while others are not.

The expected value of an observation \(y_{ij}\) is:

\(E(y_{ij}) = E(\alpha + \beta_i + \gamma_j + \epsilon_{ij}) = \alpha + \beta_i + \gamma_j + E(\epsilon_{ij}) = \alpha + \beta_i + \gamma_j\)

Any linear combination of these expected values is an estimable function of the parameters.

Analyzing the Options for Estimability

Let's examine each option:

Option 1: \(\beta_1\) is estimable

Consider the expectation \(E(y_{1j}) = \alpha + \beta_1 + \gamma_j\). If we average over \(j\), we get \(\frac{1}{J} \sum_{j=1}^J E(y_{1j}) = \frac{1}{J} \sum_{j=1}^J (\alpha + \beta_1 + \gamma_j) = \alpha + \beta_1 + \frac{1}{J} \sum_{j=1}^J \gamma_j\). Due to the constraint \(\sum_{j=1}^J \gamma_j = 0\), this simplifies to \(\alpha + \beta_1\). We cannot isolate \(\beta_1\) itself because \(\alpha\) is also in this expression. The constraint \(\sum \beta_i = 0\) implies that the parameters \(\beta_i\) are deviations from the overall mean effect of factor A, not the effects themselves relative to zero. Therefore, individual \(\beta_i\) values are not estimable under these constraints.

Option 2: \(\gamma_1\) is estimable

Similar to \(\beta_1\), consider \(E(y_{i1}) = \alpha + \beta_i + \gamma_1\). Averaging over \(i\) gives \(\frac{1}{I} \sum_{i=1}^I E(y_{i1}) = \frac{1}{I} \sum_{i=1}^I (\alpha + \beta_i + \gamma_1) = \alpha + \frac{1}{I} \sum_{i=1}^I \beta_i + \gamma_1\). Due to the constraint \(\sum_{i=1}^I \beta_i = 0\), this simplifies to \(\alpha + \gamma_1\). We cannot isolate \(\gamma_1\). Individual \(\gamma_j\) values are not estimable under these constraints.

Option 3: \(\beta_1 - \beta_2\) is estimable

Consider the difference between the expected values of observations for levels 1 and 2 of the first factor, holding the second factor constant (any \(j\)).

\(E(y_{1j}) - E(y_{2j}) = (\alpha + \beta_1 + \gamma_j) - (\alpha + \beta_2 + \gamma_j) = \beta_1 - \beta_2\)

This difference \(\beta_1 - \beta_2\) is a linear combination of expected values and thus is estimable. A linear combination of observations like \(y_{1j} - y_{2j}\) has the expected value \(\beta_1 - \beta_2\). Averaging over \(j\), \(\frac{1}{J}\sum_{j=1}^J (y_{1j} - y_{2j})\) is an unbiased estimator for \(\beta_1 - \beta_2\). This is a contrast of the \(\beta_i\) parameters. Contrasts like \(\beta_i - \beta_{i'}\) are generally estimable in fixed-effects ANOVA models with sum-to-zero constraints.

Option 4: \(\beta_1 + \gamma_2\) is estimable

Consider a linear combination of expected values that might yield \(\beta_1 + \gamma_2\). For example, \(E(y_{12}) = \alpha + \beta_1 + \gamma_2\). This expectation includes \(\alpha\). It is not possible to form a linear combination of expected values \(E(y_{ij})\) that isolates the sum \(\beta_1 + \gamma_2\) alone without involving \(\alpha\). Functions that combine parameters from different effects in this way are generally not estimable under these constraints unless they form part of the overall mean for a cell \(E(y_{ij})\).

Conclusion

Based on the analysis, contrasts of parameters within the same effect (like \(\beta_1 - \beta_2\)) are estimable in this two-way ANOVA model with sum-to-zero constraints. Individual parameters (\(\beta_i\), \(\gamma_j\)) or sums of parameters from different effects (\(\beta_i + \gamma_j\)) are typically not estimable.

Therefore, \(\beta_1 - \beta_2\) is estimable.