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6166 UNIT 4 Section 2 Answers
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Two Factor Factorial in a CRD
You are interested in the effect on biomass reductions in lettuce shoots after exposure to a pesticide. You further suspect that the temperature at time of application may also have an impact.
Starting with 42 lettuce plants you randomly assign them to treatments constructed as the combination of Pesticide concentration (%) at 7 levels and Temperature applied at three levels, with two replicates for each treatment combination. Biomass is reported as a natural log of biomass to account for suspected heterogeneity of variances. Using the data below, determine whether there are main effects due to pesticide concentration or temperature and whether there is an interaction between the two. [This is an example of stacked data ready for running in most statistics packages that will perform a two way analysis of variance.]
Conc Temp Rep Ln_Biomass 0 10 1 0.343 0 10 2 1.511 0 15 1 0.140 0 15 2 1.456 0 20 1 0.530 0 20 2 1.099 0.33 10 1 -0.246 0.33 10 2 1.049 0.33 15 1 1.140 0.33 15 2 0.617 0.33 20 1 0.436 0.33 20 2 0.462 0.5 10 1 -0.333 0.5 10 2 0.356 0.5 15 1 0.294 0.5 15 2 0.008 0.5 20 1 0.518 0.5 20 2 0.628 1 10 1 -0.843 1 10 2 -1.680 1 15 1 0.352 1 15 2 0.043 1 20 1 -0.296 1 20 2 -0.206 2 10 1 -0.561 2 10 2 -0.629 2 15 1 0.285 2 15 2 -0.284 2 20 1 -0.102 2 20 2 -0.421 5 10 1 -1.348 5 10 2 -0.232 5 15 1 -0.550 5 15 2 -1.057 5 20 1 -0.079 5 20 2 0.234 10 10 1 -1.809 10 10 2 -1.617 10 15 1 -1.266 10 15 2 -0.066 10 20 1 -0.378 10 20 2 -0.618
The analysis will be preformed in SAS. The program to perform this analysis is given here (SAS Program). The important output from the SAS program is below. SAS automatically creates the appropriate linear contrasts to test the main effects for both factors as well as the interaction effect. The p-values for the associated F-statistics for the main and interaction hypotheses suggest that there are strong pesticide concentration effects, strong temperature differences and no interaction between the two. We could follow up this analysis with an appropriate multiple comparison procedure to determine which effect means are different from which other effect means.
Source DF Squares Mean Square F Value Pr > F
Model 20 21.09385957 1.05469298 4.30 0.0008
Error 21 5.15446100 0.24545052
Corrected Total 41 26.24832057
R-Square Coeff Var Root MSE Ln_Biomass Mean
0.803627 -666.9245 0.495430 -0.074286
Source DF Type I SS Mean Square F Value Pr > F
Conc 6 15.11276690 2.51879448 10.26 <.0001
Temp 2 2.69475100 1.34737550 5.49 0.0121
Conc*Temp 12 3.28634167 0.27386181 1.12 0.3980
Source DF Type III SS Mean Square F Value Pr > F
Conc 6 15.11276690 2.51879448 10.26 <.0001
Temp 2 2.69475100 1.34737550 5.49 0.0121
Conc*Temp 12 3.28634167 0.27386181 1.12 0.3980
SNK Grouping Mean N Conc
A 0.8465 6 0
A 0.5763 6 0.33
B A 0.2452 6 0.5
B C -0.2853 6 2
B C -0.4383 6 1
B C -0.5053 6 5
C -0.9590 6 10
SNK Grouping Mean N Temp
A 0.1291 14 20
A 0.0794 14 15
B -0.4314 14 10
Note that both factors concentration and temperature are quantitative factors. Hence we could examine hypotheses regrading the form of the trends (response surface) relating to changes in MPG as a regression on concentration and temperature.