Could chance cause the differences between factor levels?

The least squares estimates of the model parameters describe differences between the factor levels. However the point estimates do not give any indication of whether the apparent factor effects could be simply caused by the random (unexplained) variability in the data.

A confidence intervals for each parameters help us to assess the parameter estimates, but with several parameters, it can be difficult to come to an overall conclusion about the significance of the two factors.

This section develops hypothesis tests for whether the two factors affect the response.

The tests are based on an analysis of variance table that is closely related to the analysis of variance table for a single factor.

Warping of copper plates

An experiment was conducted to investigate warping of copper plates. The two factors studied in the experiment were the temperature and the copper content of the plates. The response variable measured the amount of warping.

The diagram above shows the least squares estimates for a model with main effects for temperature and copper content. The '±' values give 95% confidence intervals for the estimates. From them, we estimate that:

From the first of these CIs, we can be fairly sure that the difference is not zero, and that the percentage copper does affect warping. However there is more doubt about the effect of temperature since the 95% CI for 100°C does include zero.

A single test for each factor would give a clearer message than several confidence intervals.