How should these two models be interpreted differently? Specifically, what is the circumstance where you would run one over the other?
aov(Temperature~Day+Error(Subject)) aov(Temperature~Day+Error(Subject/Day)) We'll use an example where I measured the temperature of 10 people once every day for a week. My main interest is to see if the temperature measurements change significantly day-to-day, and I am not interested in the longitudinal trend from Monday to Sunday.
Best Answer
Depending on the contrasts you are using, the R command
aov(Temperature~Day+Error(Subject)) fits a model like
$$y_{ij} = mu + beta_j + b_i + epsilon_{ij},$$
where $y_{ij}$ is the response value for the $i$th individual at the $j$th period (day), $mu$ is global mean, $beta_j$ is the effect of $j$th day, $b_isim N(0,sigma_b^2)$ is the Gaussian random effect or random intercept for the $i$th individual and $epsilon_{ij}sim N(0,sigma^2)$ is the Gaussian residual term. The unknown parameters are $(mu, beta_j,sigma_b^2, sigma^2)$.
On the other hand, the command
aov(Temperature~Day+Error(Subject/Day)) fits the model
$$y_{ijk} = mu + beta_j + b_i + b_{ij} + epsilon_{ijk},$$
where $b_{ij}sim N(0, sigma_1^2)$ is a Gaussian random individual-period interaction term. As you can see from the expression, to estimate also $sigma_1^2$ you need to have replications for each $i$ and $j$, that's the reason for the third index $k$.
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