What is the meaning of EMS and BMS in an ANOVA?

Question

For an ANOVA, what is the meaning of the Error Mean Squares (EMS, or similarly TMS) and also the Between Mean Squares (BMS)? I know how they are calculated, but what meaning do they actually hold?

Answer 1

I assume you are working with a balanced one-way ANOVA design, with model $$ Y_{ij} = \mu + \alpha_i + e_{ij},$$ where $\sum \alpha_i = 0$ and $e_{ij} \stackrel{iid}{\sim} N(0, \sigma^2),$ and $i = 1, \dots, g$ groups and $j = 1, \dots, n$ replications per group.

Then MSE is an unbiased estimate of $\sigma^2$ regardless whether the null hypothesis $H_0: \alpha_1 = \cdots \alpha_g$ is true. It is a generalization of the 'pooled' estimate of $\sigma^2$ in a pooled 2-sample t test.

By contrast, BSE is an estimate of $\sigma^2$ that is unbiased if $H_0$ is true, and has positive bias if $H_0$ is false. (The bias is a funcation of $\sum \alpha_i^2.$)

Thus the variance ratio statistic $F = BMS/EMS$ tends to be 'near' 1 when $H_0$ is true and larger than 1 when $H_0$ is false.

The 'sufficient statistics' for such an ANOVA (in addition to the known sample sizes), are the $g$ sample means $\bar Y_{i \cdot}$ and the $g$ sample variances $S_i^2.$ BMS depends on the data only via these means, and EMS depends on the data only through the variances. So BMS and EMS are independent estimates of $\sigma^2$ as required by the derivation of Snedecor's F-distribution.

The DF's in an ANOVA table can be interpreted as dimensions of sub-spaces in the $gn$-dimensional space of the data $Y_{ij}.$ One dimension is 'used' to estimate $\mu$ by $\bar Y_{\cdot \cdot} = \frac{1}{gn} \sum_i \sum_j Y_{ij}.$ The $g-1$-dimensional sub-space of BMS and the $g(n-1)$-dimensional sub-space of EMS are orthogonal to each other and to the space of the grand mean $\bar Y_{\cdot \cdot}$.