STATS-Consider the two-way random effects ANOVA mo

Consider the two-way random effects ANOVA model

\begin{array}{c} y_{i j k}=\mu_{j k}+\epsilon_{i j k}, \text { with } \\ \mu_{j k}=\mu+a_{j k}, \end{array}

for

k=1, \cdots, K

schools,

j=1, \cdots, g_{k}

classes, and

i=1, \cdots, n_{j k}

students, where for each given

k,\left(a_{1 k}, \cdots, a_{g_{k} k}\right)^{\top}

is multivariate normal

N\left(\mathbf{0}_{g_{k}}, \boldsymbol{\Sigma}_{k}\right)

with

\mathbf{0}_{g_{k}}

being a

g_{k}

-dimensional vector of zeros,

\Sigma_{k}

being a

g_{k} \times g_{k}

covariance matrix and the

\left(j_{1}, j_{2}\right)

-th element of

\boldsymbol{\Sigma}_{k}

is

\sigma_{j_{1} j_{2}, k},\left\{a_{11}, \cdots, a_{g_{1} 1}\right\}, \cdots,\left\{a_{1 K}, \cdots, a_{g_{K} K}\right\}

are independent,

\left\{a_{j k}\right\}

are independent of

\left\{\epsilon_{i j k}\right\}, \epsilon_{i j k}

are IID

N\left(0, \sigma^{2}\right), \mu

is a constant, and

n=\sum_{k=1}^{K} \sum_{j=1}^{g_{k}} n_{j k}

. Note: In this course, we have introduced the reference parameterisation of the two-way fixed effects ANOVA model. The above can be treated as the centred parametrisation but under the two-way random effects ANOVA model. Based on the above model, please answer the following questions. (b)

[1

mark

]

Please derive

\operatorname{Corr}\left[y_{i_{1} j k}, y_{i_{2} j k}\right]

for any

i_{1} \neq i_{2}

. What is the interpretation of your derivation outcome based on the background given in this question?

Solution by Steps

step 1

To derive the correlation between

y_{i_{1} j k}

and

y_{i_{2} j k}

, we first express these variables using the given model:

y_{i_{1} j k} = \mu + a_{j k} + \epsilon_{i_{1} j k}

y_{i_{2} j k} = \mu + a_{j k} + \epsilon_{i_{2} j k}

step 2

The correlation between

y_{i_{1} j k}

and

y_{i_{2} j k}

is given by:

\operatorname{Corr}[y_{i_{1} j k}, y_{i_{2} j k}] = \frac{\operatorname{Cov}[y_{i_{1} j k}, y_{i_{2} j k}]}{\sqrt{\operatorname{Var}[y_{i_{1} j k}] \cdot \operatorname{Var}[y_{i_{2} j k}]}}

step 3

Since

\epsilon_{i_{1} j k}

and

\epsilon_{i_{2} j k}

are independent and identically distributed, their covariance is zero. The covariance between

y_{i_{1} j k}

and

y_{i_{2} j k}

is therefore:

\operatorname{Cov}[y_{i_{1} j k}, y_{i_{2} j k}] = \operatorname{Cov}[a_{j k}, a_{j k}] = \operatorname{Var}[a_{j k}]

step 4

The variance of

y_{i_{1} j k}

(and similarly

y_{i_{2} j k}

) is the sum of the variances of its components, since

a_{j k}

and

\epsilon_{i_{1} j k}

are independent:

\operatorname{Var}[y_{i_{1} j k}] = \operatorname{Var}[a_{j k}] + \operatorname{Var}[\epsilon_{i_{1} j k}] = \operatorname{Var}[a_{j k}] + \sigma^2

step 5

Substituting the results from steps 3 and 4 into the formula from step 2, we get:

\operatorname{Corr}[y_{i_{1} j k}, y_{i_{2} j k}] = \frac{\operatorname{Var}[a_{j k}]}{\operatorname{Var}[a_{j k}] + \sigma^2}

Answer

The correlation between

y_{i_{1} j k}

and

y_{i_{2} j k}

is

\frac{\operatorname{Var}[a_{j k}]}{\operatorname{Var}[a_{j k}] + \sigma^2}

.

Key Concept

The correlation between two observations within the same class and school in a two-way random effects ANOVA model is determined by the ratio of the class variance to the total variance.

Explanation

The derived correlation reflects the proportion of variance explained by the class effect relative to the total variance, which includes both the class effect and the individual student's variance within the class.

What is the interpretation of vour derivation outcome based on the background given in this question?

Solution by Steps

step 1

To interpret the correlation between

y_{i_{1} j k}

and

y_{i_{2} j k}

, we must understand the components of the two-way random effects ANOVA model

step 2

The correlation formula given is

\frac{\text{Var}[a_{j k}]}{\text{Var}[a_{j k}] + \sigma^2}

. This represents the proportion of variance explained by the random effect

a_{j k}

relative to the total variance

step 3

The numerator

\text{Var}[a_{j k}]

is the variance due to the random effect for the

j

th level of one factor and the

k

th level of another factor

step 4

The denominator

\text{Var}[a_{j k}] + \sigma^2

is the sum of the variance due to the random effect and the variance of the error term

\sigma^2

step 5

The correlation indicates the strength of the relationship between two observations sharing the same levels of the random effects. A higher correlation suggests a stronger relationship due to the random effects

Answer

The correlation of

\frac{\text{Var}[a_{j k}]}{\text{Var}[a_{j k}] + \sigma^2}

indicates the proportion of the total variance that is attributable to the random effect shared by

y_{i_{1} j k}

and

y_{i_{2} j k}

. A higher value suggests that the random effect has a stronger influence on the observed values.

Key Concept

Correlation in two-way random effects ANOVA

Explanation

The derived correlation measures the extent to which the random effect contributes to the total variance observed in two measurements at the same levels of the factors in the model.

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