Question 9.11: Three fertilizers are studied for their effect on yield in a...
Three fertilizers are studied for their effect on yield in an orange grove. Nine plots of land are used, divided into blocks of three plots each. A randomized complete block design is used, with each fertilizer applied once in each block. The results, in pounds of harvested fruit, are presented in the following table, followed by MINITAB output for the two-way ANOVA. Can we conclude that the mean yields differ among fer1ilizers? What assumption is made about interactions between fertilizer and plot? How is the sum of squares for error computed?
\begin{array}{cccc}\hline \text{Fertilizer}& \text{Plot 1}& \text{Plot 2}& \text{Plot 3}\\\hline \text{A}& 430 & 542 & 287 \\\text{B}& 367 & 463 & 253 \\\text{C}& 320 & 421 & 207 \\\hline\end{array}
Two -way ANOVA: Yield versus Block. Fertilizer
\begin{array}{lrrrrr}\text{Source}& \text{DF}& \text{SS}& \text{MS}& F & P \\\text{Fertilizer}& 2 & 16213.6 & 8106.778 & 49.75 & 0.001 \\\text{Block}& 2 & 77046.9 & 38523.44 & 236.4 & 0.000 \\\text{Error}& 4 & 651.778 & 162.9444 & & \\\text{Total}& 8 & 93912.2 & & &\end{array}The P-value for the fertilizer factor is 0.001, so we conclude that fertilizer does have an effect on yield. The assumption is made that there is no interaction between the fertilizer and the blocking factor (plot), so that the main effects of fertilizer can be interpreted. Since there is only one observation for each treatment-block combination (i.e., K = 1), the sum of squares for error (SSE) reported in the output is really SSAB, the sum of squares for interaction, and the error mean square (MSE) is actually MSAB. (See the discussion on page 435.)