Refer to the matched-pairs experiment of Section 12.3 and...

Refer to the matched-pairs experiment of Section 12.3 and assume that the measurement receiving treatment $i,$ where $i=1,2,$ in the $j$ th pair, where $j=1,2, \ldots, n,$ is $$Y_{i j}=\mu_{i}+P_{j}+\varepsilon_{i j}$$, where $\mu_{i}=$ expected response for treatmenti, for $i=1,2$, $P_{j}=$ additive random effect (positive or negative) contribution by thejth pair of experimental units, for $j=1,2, \ldots, n$, $\varepsilon_{i j}=$ random error associated with the experimental unit in thejth pair that receives treatmenti. Assume that the $\varepsilon_{i j}$ 's are independent normal random variables with $E\left(\varepsilon_{i j}\right)=0, V\left(\varepsilon_{i j}\right)=\sigma^{2} ;$ and assume that the $P_{j}$ 's are independent normal random variables with $E\left(P_{j}\right)=0, V\left(P_{j}\right)=\sigma_{p}^{2} .$ Also, assume that the $P_{j}$ 's and $\varepsilon_{i j}$ 's are independent. a. Find $E\left(Y_{i j}\right)$. b. Find $E\left(\bar{Y}_{i}\right), V\left(\bar{Y}_{i}\right),$ where $\bar{Y}_{i}$ is the mean of the $n$ observations receiving treatment $i,$ where $i=1,2$. c. Let $\bar{D}=\bar{Y}_{1}-\bar{Y}_{2} .$ Find $E(\bar{D}), V(\bar{D}),$ and the probability distribution for $\bar{D}$.

Refer to the matched-pairs experiment of Section 12.3 and assume that the measurement receiving treatment $i,$ where $i=1,2,$ in the $j$ th pair, where $j=1,2, \ldots, n,$ is $$Y_{i j}=\mu_{i}+P_{j}+\varepsilon_{i j}$$, where $\mu_{i}=$ expected response for treatmenti, for $i=1,2$, $P_{j}=$ additive random effect (positive or negative) contribution by thejth pair of experimental units, for $j=1,2, \ldots, n$, $\varepsilon_{i j}=$ random error associated with the experimental unit in thejth pair that receives treatmenti.
Assume that the $\varepsilon_{i j}$ 's are independent normal random variables with $E\left(\varepsilon_{i j}\right)=0, V\left(\varepsilon_{i j}\right)=\sigma^{2} ;$ and assume that the $P_{j}$ 's are independent normal random variables with $E\left(P_{j}\right)=0, V\left(P_{j}\right)=\sigma_{p}^{2} .$ Also, assume that the $P_{j}$ 's and $\varepsilon_{i j}$ 's are independent.
a. Find $E\left(Y_{i j}\right)$.
b. Find $E\left(\bar{Y}_{i}\right), V\left(\bar{Y}_{i}\right),$ where $\bar{Y}_{i}$ is the mean of the $n$ observations receiving treatment $i,$ where $i=1,2$.
c. Let $\bar{D}=\bar{Y}_{1}-\bar{Y}_{2} .$ Find $E(\bar{D}), V(\bar{D}),$ and the probability distribution for $\bar{D}$.

Key Concepts

Matched Pairs Experimental Design

A matched pairs design is an experimental framework in which subjects are paired or grouped based on certain similarities, and each subject or member of the pair receives a different treatment. The design helps control for variability between subjects by comparing treatment effects within each pair instead of across unrelated subjects, making it easier to detect differences attributable to the treatments.

Linear Mixed Models

Linear mixed models incorporate both fixed effects, which are consistent and repeatable influences such as treatment effects, and random effects, which represent random variability from sources like subjects or experimental units. In this context, treatment means are considered fixed, while pair effects are modeled as random, allowing the analysis to account for correlation within pairs.

Expectation and Variance Calculation in Linear Models

Calculating the expectation and variance of observations in a linear model involves linearity properties of expectation and the rules for variance of independent and correlated components. This includes summing the means for fixed effects and accounting for additional variability from random effects. These computations are essential for understanding how the data's overall variability is partitioned between treatment effects and random noise.

Properties of the Normal Distribution

The normal distribution is crucial in this context because the model assumes that both the random pair effects and the experimental errors are normally distributed. The normality assumption facilitates statistical inference including hypothesis testing and construction of confidence intervals, as the sum and difference of normally distributed variables is also normally distributed.

Difference of Treatment Means

Analyzing the difference of treatment means involves comparing the average response under two treatments. This difference is a key statistic in determining treatment effect, and its expectation and variance are derived by using linear combinations of the individual treatment means. Under the normality assumption, the difference also follows a normal distribution, which simplifies the construction of tests and confidence intervals for treatment comparisons.

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