Math Problems Vault

Problem 8c.1

(a)(b)

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(a)

Prove that

\sum _{i = 1}^{I} \sum _{j = 1}^{J} \sum _{k = 1}^{K}\left(\bar{Y}_{i . .}-\bar{Y}_{\ldots }\right)^{2} = \sum _{i = 1}^{I} \frac{Y_{i .}^{2}}{J K}-\frac{Y_{. .}^{2}}{I J K}

Obtain a similar expression for

\sum _{i = 1}^{I} \sum _{j = 1}^{J} \sum _{k = 1}^{K}\left(\bar{Y}_{i j .}-\bar{Y}_{i . .}-\bar{Y}_{. j .}+\bar{Y}_{\ldots }\right)^{2}

(b)

Using Table 8.2, prove the following:

\begin{aligned} E\left[(I-1) S_{A}^{2}\right] & = \sigma ^{2}(I-1)+J K \sum _{i} \alpha _{i}^{* 2} \\ E\left[(J-1) S_{B}^{2}\right] & = \sigma ^{2}(J-1)+I K \sum _{j} \beta _{i}^{* 2} \\ E\left[(I-1) S_{A B}^{2}\right] & = \sigma ^{2}(I-1)(J-1)+K \sum _{i} \sum _{j}(\alpha \beta )_{i j}^{* 2} \end{aligned}

Problem 8c.2

(a)(b)(c)

Given the population means $\mu_{i j}\left(i = 1,2, \ldots , I ; j = 1,2, \ldots , J\right)$ , let

\begin{aligned} A_{i} & = \sum _{j} v_{j} \mu _{i j} \quad \left(\sum _{j} v_{j} = 1\right), \\ B_{i} & = \sum _{i} u_{i} \mu _{i j} \quad \left(\sum _{j} u_{j} = 1\right), \end{aligned}

and

\mu = \sum _{i} u_{i} A_{i} = \sum _{j} v_{j} B_{j} = \sum _{i} \sum _{j} u_{i} v_{j} \mu _{i j}

Define $\alpha_{i} = A_{i}-\mu , \beta_{j} = B_{j}-\mu$ , and

(\alpha \beta )_{i j} = \mu _{i j}-A_{i}-B_{j}+\mu

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(a)

Show that $\sum_{i} u_{i} \alpha_{i} = \sum_{j} v_{j} \beta_{j} = 0, \sum_{i} u_{i}(\alpha \beta )_{i j} = 0$ (all $j$ ), and $\sum_{j} v_{j}(\alpha \beta )_{i j} = 0$ (all $i$ ).

(b)

Conversely, given

\mu _{i j} = \mu +\alpha _{i}+\beta _{j}+(\alpha \beta )_{i j}

show that the parameters in the equation above are uniquely determined by the constraints in (a).

(c)

Prove that if the interactions $\left\{ (\alpha \beta )_{i j}\right\}$ are all zero for some system of weights $\left\{ u_{i}\right\}$ and $\left\{ v_{i}\right\}$ , then they are zero for every system of weights. In that case show that every contrast in the $\left\{ \alpha_{i}\right\}$ , or $\left\{ \beta_{i}\right\}$ , has a value that does not depend on the system of weights.

Problem 8c.3

Let $Y_{i j k} = \mu +\alpha_{i}+\beta_{j}+(\alpha \beta )_{i j}+\varepsilon_{i j k}$ , where $i = 1,2, \ldots , I ; j = 1,2, \ldots , J ; k = 1,2, \ldots , K_{i j}$ ; and the $\varepsilon_{i j k}$ are independently distributed as $N\left(0, \sigma^{2}\right)$ . Given $K_{i j} = K_{i .} K_{. j} / K_{. .}$ for all $i, j$ , find a test statistic for testing the hypothesis $H:(\alpha \beta )_{i j} = 0$ (all $i, j$ ). Hint: By Exercise 2, the validity of $H$ does not depend on the weights used in the identifiability constraints $\sum_{i} u_{i} \alpha_{i} = \sum_{j} v_{j} \beta_{j} = 0$ . We can therefore use $u_{i} = K_{i .} / K_{.}$ and $v_{j} = K_{. j} / K_{. .}$ and find the least squares estimates of $\alpha_{i}$ and $\beta_{j}$ when $H$ is true.

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