ANOVA with one observation per treatment (optional). If a two-way layout has just one observation for each treatment, the mean square for error (MSE) is 0 because there is no variation within each treatment group. The usual two-way ANOVA F tests can't be done, because they have MSE in their denominators. But if we are willing to assume that there is no interaction between the two factors, the interaction mean square can be used as the denominator of F statistics for testing the two main effects. Software usually allows you to choose "no interaction." Here is an example. Does the addition of sodium chloride change brick quality? One measure of quality is "resistance anisotropy," which combines the results of ultrasound and mechanical tests. Researchers tested two types of bricks, one without sodium chloride (MPW) and one with sodium chloride (MPS), at three firing temperatures (850, 1000, and 1100 degrees Celsius). Here are the data for one brick per treatment:^{18}

(a) Make a plot of the group means (which are the same as the individual observations). Is the interaction between type of brick and firing temperature small? If so, it may be reasonable to assume that in the population there is no interaction.

(b) The researchers did fit a two-way ANOVA model without interaction. Use statistical software to obtain this two-way ANOVA table. What do you conclude about the main effects?

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