Splitting a factor sum of squares

As in other types of experiment, it is often insufficient to simply conclude that the levels of a factor are different. If there is structure to the levels, it is possible to test further hypotheses about the levels using a sequence of simpler models. This is illustrated in the example below.

Fertiliser and wheat yield

An experiment was conducted to assess the effect of different fertilisers on the yield of wheat. Treatment O is a control level in which no fertiliser is applied. In treatment S, a single dressing of sulphate of ammonia was applied in March, whereas treatment M applied the same fertiliser in regular monthly dressings over a six-month period. In treatment C, an equivalent amount of nitrogen was applied as cyanamide in a single dressing in October, whereas in D, half was applied in October as cyanamide and half as dicyanadiomide. The table below shows the layout of the plots (each of area 0.025 acres) in the field, the treatments applied to each plot and the resulting yields in pounds of wheat.

  Column
Row 1 2 3 4 5
1  (D) 72.2   (M) 55.4   (O) 36.6   (C) 67.9   (S) 73.0 
2 (O) 36.4 (C) 46.9 (M) 46.8 (S) 54.9 (D) 68.5
3 (M) 71.5 (S) 55.6 (D) 71.6 (O) 67.5 (C) 78.4
4 (S) 68.9 (O) 53.2 (C) 69.8 (D) 79.6 (M) 77.2
5 (C) 82.0 (D) 81.0 (S) 76.0 (M) 87.9 (O) 70.9

This is a Latin square design since each letter appears once in every row and column so Rows, Columns and Treatments are pairwise orthogonal. The analysis of variance table for the data is shown below.

From the p-value associated with the treatments, it is clear that they do affect wheat yield. Click Split treatments to consider a sequence of simpler models:

From the p-values associated with these, there are clearly differences between Control, Ammonia and Cyanamide, but no evidence that the application time of Ammonia affects yield and only slight evidence of a difference between the two type of Cyanamide application.