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.
Soybean yield and trace elements
An experiment was conducted to assess how different applications of manganese (Mn) and copper (Cu) affect the yield of soybeans. A large field was subdivided into 32 plots and two were randomly allocated to each combination of Mn rate and Cu rate — i.e. there were 2 replicates for each of the 16 treatments. Soybeans were planted in rows 1 metre apart and the yield of soybeans (in tonnes per hectare) was recorded from each plot.
The diagram above shows the least squares estimates for a model with main effects for Mn and Cu. 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 Mn does affect yield. However there is more doubt about the effect of Cu since all 95% CIs for the differences between the Cu rates and rate 1 include zero.
A single test for each factor would give a clearer message than several confidence intervals.