STA 6166 UNIT 4 Section 2 Answers
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# Toxicology and Health

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.