PLS205
Winter 2015
Homework Topic 9
Due TUESDAY, February 24, at the beginning of lecture. Answer all parts of the questions
completely, and clearly document the procedures used in each exercise. To ensure maximum points for
yourself, invest some time in presenting your answers in a concise, organized, and clear manner.
Question 1
[50 points]
An investigator would like to test if different sources of protein have different effects on weight gain
when consumed at different protein levels. The investigator prepared 60 rat cages with 15 rats in each
cage. Ten cages were randomly assingned to each of the 6 treatments. Treatments included all possible
combinations of two factors: protein level (high and low) and protein source (beef, cereal, pork). After a
month of providing each cage with the respective feed type the investigator weighed the 15 rats in each
cage and calculated the average weight gain. The data is reported in the table below:
:
Data hw9_1;
Do rep = 1 to 10;
Do Level = 'High', 'Low';
Do Source = 'Beef', 'Cereal', 'Pork';
Input weightgain @@;
Output;
End;
End;
End;
Cards;
60
84
80
76
90
35
88
60
65
62
81
68
104
42
81
76
83
59
90
97
84
50
66
72
67
81
88
72
84
67
93
74
88
37
60
83
86
68
94
58
60
92
73
63
77
76
53
56
103
72
106
81
75
51
97
78
91
64
44
68
;
*Proc Print data = hw9_1;
Proc GLM data = hw9_1;
title 'ANOVA';
Class Level Source;
Model weightgain = Level Source Level*Source;
Output Out = hw9_1PR r = res p = pred;
Means Level*Source;
run;
quit;
[A POST-MIDTERM GIFT: DO NOT WORRY ABOUT TESTING ASSUMPTIONS]
1.1
Describe in detail the design of this experiment [see appendix at the end of this problem set].
Design:
HW Topic 9
CRD with a 2x3 Factorial
1
Response Variable:
Experimental Unit:
Class
Variable
1
2
Weight gain of rats after a month
Rat cage
Block or
Treatment
Treatment
Treatment
No. of
Levels
2
3
Subsamples?
YES
Description
Level of protein
Protein Source
15 rats were measured in each cage but the means are reported.
1.2
Perform the appropriate ANOVA and report the results below. Briefly discuss what effects are
significant.
ANOVA results:
Source
DF Type III SS Mean Square F Value Pr > F
Level
1 3153.750000
3153.750000
15.09 0.0003
Source
2
296.133333
148.066667
0.71 0.4968
Level*Source
2 1117.200000
558.600000
2.67 0.0781
The ANOVA results suggest that the level of protein content had significant effects on weight gain
(P = 0.0003). The source of the protein had no significant effect on weight gain (P = 0.4968), and
there were no significant interactions between the level of protein and the source of the protein (P =
0.0781)
HW Topic 9
2
1.3
Use SAS, Excel, or R to create a plot of the interactions between level of protein and source and
interpret the results of the plot.
Weight Gain of Rats with Different Combinations of Protein
Level and Protein Source
90
85
Weight gain (g)
80
75
Beef
70
Cereal
65
Pork
60
55
50
Low
High
Protein Level
The plot demonstrates that when the rats were fed beef or pork at a high level of protein the weight
gain was larger than when using cereal as a protein source. The weight gain when fed a high level of
protein from a cereal source was not significant.
1.4 Perform the appropriate contrast to answer the following questions:
To make it easier to answer the questions below it is possible to assing a treatment ID to each level
by source combination as shown below.
Data hw9_12;
Do rep = 1 to 10;
Do trtmt = 1 to 6;
Input weightgain2 @@;
Output;
End;
End;
Cards;
60
84
80
76
90
35
88
60
65
62
81
68
104
42
81
76
83
59
90
97
84
50
66
72
67
81
88
72
84
67
93
74
88
37
60
83
86
68
94
58
60
92
73
63
77
76
53
56
103
72
106
81
75
51
97
78
91
64
44
68
;
Proc Print data = hw9_12;
Proc GLM data = hw9_12;
title 'nested';
HW Topic 9
3
Class Trtmt;
Model weightgain2 = Trtmt;
Output Out = hw9_1PR2 r = res2 p = pred2;
Means Trtmt;
contrast 'animal vs veg' Trtmt
-1
2
contrast 'beef v pork'
Trtmt
1
0
contrast 'high v low'
Trtmt
1
1
contrast 'an v veg * level' Trtmt
-1
2
contrast 'b v p * level'
Trtmt
1
0
run; quit;
-1
-1
1
-1
-1
-1
1
-1
1
-1
2
0
-1
-2
0
-1;
-1;
-1;
1;
1;
1.4.1 Is there a difference in weight gain when the mice were fed a high level of protein compared
to a low level of protein?
Contrast
DF Contrast SS Mean Square F Value Pr > F
1 3153.750000
High v Low
3153.750000
15.09 0.0003
The contrast suggest that there are significant differences between the feed with high protein
content and the feeed with low protein content (P = 0.0003).
1.4.2 Is there a difference in weight gain when mice were fed a vegetable source of protein (cereal)
compared to an animal source of protein (beef and pork)?
Contrast DF Contrast SS Mean Square F Value Pr > F
1
a v veg
294.533333
294.533333
1.41 0.2403
The contrast suggest that there are not significant differences between the feed with an animal
protein source and a feed with vegetable protein source (P = 0.2403).
1.4.3 Is there a difference in weight gain when mice were fed a protein source made of beef
compared to a protein source made of pork?
Contrast
DF Contrast SS Mean Square F Value Pr > F
1
beef v pork
1.600000
1.600000
0.01 0.9306
The contrast suggest that there are not significant differences between the feed with a beef protein
source and a feed with a pork protein source (P = 0.9306).
1.4.4 Is the difference between animal and vegetable sources of protein different at high levels of
protein compared to low levels of protein?
Contrast
DF Contrast SS Mean Square F Value Pr > F
1 1116.300000
an v veg*level
1116.300000
5.34 0.0247
The contrast suggest that the difference between animal and vegetable sources of protein are
different between the low protein food and the high protein food (P = 0.0247).
1.4.5 Is the difference between beef and pork sources of protein different at high levels of protein
compared to low levels of protein?
Contrast
b v p*level
DF Contrast SS Mean Square F Value Pr > F
1
0.900000
0.900000
0.00 0.9479
The contrast suggest that the difference between the beef and pork sources of protein are the same
between the low protein food and the high protein food (P = 0.9479).
HW Topic 9
4
1.5
Take the sum of the SS of contrast 1.4.2 and 1.4.3 and compare it to the SS in the ANOVA in 1.2 to
test for differences between the sources of protein.
Contrast
DF Contrast Mean
F
Pr > F
SS
Square Value
a v veg
beef v pork
1
1
294.533 294.533
1.6
1.6
1.41
0.01
0.2403
0.9306
The sum of the SS of contrast 1.3.2 and 1.3.3 is 296.13 the SS of source in the ANOVA model is also
296.13. The two contrasts have partitioned the SS treatment into two non-overlapping
(independent) questions
1.6
Take the sum of the SS of contrast 1.3.4 and 1.3.5 and compare it to the SS in the ANOVA in 1.2
to test for interactions between the source of protein and the level of protein.
Contrast
DF Contrast SS Mean Square F Value Pr > F
an v veg*level
1 1116.300000
b v p*level
1
0.900000
1116.300000
5.34 0.0247
0.900000
0.00 0.9479
The sum of the SS of contrast 1.3.4 and 1.3.5 is 1117.2 the SS of the source by level interaction in
the ANOVA model is also 1117.2. The two interaction contrasts have partitioned the SS interaction
into two non-overlapping (independent) questions
1.7
Discuss why the significance of the the interaction in the overall ANOVA is not the same as in the
interaction contrasts.
In the ANOVA, the sums of squares for the interaction are divided evenly into the degress of
freedom, which in this case is two. In the contrast the sums of squares are not partitioned equally.
The interaction that is the source of more variance is given a larger portion of the sums of squares,
and that portion is significant.
Question 2
[50 points]
An investigator in a seed company would like to test how seed germination in the field of 4 new varieties
of carrot is affected by different formulations of pelleting (FYI: some seed is coated with a combination
of pesticide, nutrients, and inert ingredients to improve germination and make it easier to plant,
particularly very small seed). The 12 different seed treatments include 4 different pellet formulations in
combination with 3 different pesticide formulations. To conduct the experiment the investigator prepared
two fields in two different farms. The research has no particular interest in these two farms that can be
considered as blocks. In each field the investigator prepared 48 plots and randomly assigned one of the
48 combinations of variety, pellet formulation, and pesticide combinations to each plot. Each plot
received the same number of coated seed. After three weeks the investigator visited the field and scoreed
stand establishment using a scale that varied from 0 to 100, where 0 is no stand establishment and 100 is
maximum stand establishment. The data are summarized below:
Data hw9_2;
Do Farm = 1 to 2;
Do Variety = 1 to 4;
Do Pellet = 1 to 4;
Do Pesticide = 1 to 3;
Input vigor @@;
HW Topic 9
5
Output;
End;
End;
End;
End;
Cards;
72
76
76
63
67
87
64
67
77
87
78
68
67
52
62
80
79
89
64
65
64
67
70
60
57
66
69
72
72
73
63
66
67
56
75
87
57
56
78
60
67
68
61
79
68
73
86
72
48
76
65
62
70
69
73
66
78
55
67
73
44
58
54
77
79
81
62
61
79
60
78
82
53
50
56
81
86
86
55
56
66
56
58
64
46
55
64
56
59
67
64
66
62
59
58
86
;
*Proc Print data = hw9_2;
Proc GLM data = hw9_2;
Class Farm Variety Pellet Pesticide;
Model vigor = Farm
Variety
Pellet
Pesticide
Variety*Pellet Variety*Pesticide Pellet*Pesticide
Variety*Pellet*Pesticide;
Output Out = hw9_2PR r = res p = pred;
Means Pesticide / REGWQ;
Means Variety*Pellet;
Means Variety*Pesticide;
Means Pellet*Pesticide;
Proc Gplot data = hw9_2;
Title "Pesticide Main Effects";
symbol1 i=std1mtj v=none color=BLUE;
Plot vigor*Pesticide /
Description = "Pesticide Main Effects";
Proc Gplot data = hw9_2;
title "Variety by Pesticide Formulation Interactions";
symbol1 i=std1mtj v=none color=BLUE;
symbol2 i=std1mtj v=none color=BLACK;
symbol3 i=std1mtj v=none color=GREEN;
symbol4 i=std1mtj v=none color=BROWN;
Plot vigor*Variety = Pellet /
Description = "Variety by Pesticide Formulation Interactions";
run;
Proc Sort Data = hw9_2;
by Pellet;
Proc GLM Data = hw9_2;
Class Farm Variety Pesticide;
Model vigor = Farm Variety Pesticide Variety*Pesticide;
Means Variety / REGWQ;
by Pellet;
Proc Sort Data = hw9_2;
by Variety;
HW Topic 9
6
Proc GLM Data = hw9_2;
Class Farm Pellet Pesticide;
Model vigor = Farm Pellet Pesticide Pellet*Pesticide;
Means Pellet / REGWQ;
by Variety;
run; quit;
(THE GIFT CONTINUES: DO NOT WORRY ABOUT TESTING ASSUMPTIONS)
2.1
Describe in detail the design of this experiment [see appendix at the end of this problem set].
Design:
Response Variable:
Experimental Unit:
Class
Variable
1
2
3
4
2.2
RCBD with a 4x4x3 Factorial
Score for seedling vigor
Plot
Block or
Treatment
Block
Treatment
Treatment
Treatment
No. of
Levels
2
4
4
3
Subsamples?
NO
Description
The experiment was done in two different farms
Four different varieties of carrot
Four different pellet formulations
Three different formulations of pesticide
Conduct the correct ANOVA analysis of the above data and report the results below. Briefly
mention what effects are significant.
Source
DF Type III SS Mean Square F Value Pr > F
Farm
1
518.010417
518.010417
7.55 0.0085
Variety
3
333.114583
111.038194
1.62 0.1978
Pellet
3 1967.364583
655.788194
9.56 <.0001
Pesticide
2 1252.895833
626.447917
9.13 0.0004
Variety*Pellet
9 1820.427083
202.269676
2.95 0.0074
Variety*Pesticide
6
57.604167
9.600694
0.14 0.9902
Pellet*Pesticide
6
59.604167
9.934028
0.14 0.9892
Variet*Pellet*Pestic 18
857.229167
47.623843
0.69 0.7994
The ANOVA results suggest that there are no significant differences between varieties (P = 0.1978).
However there are significant variety by pellet interactions (P = 0.0074), which suggest that
different the response of seedling vigor was not the same for all varieties under the different pellet
formulations. We need to analyse the simple effects to correctly represent the effect of variety and
pellet formulation. There were significant differences between pellet formulations (P < 0.0001).
There was significant differences between the different pesticide formulations (P = 0.0004). There
was no significant interactions between variety and pesticide formulations and pellet by pesticide
HW Topic 9
7
formulations (P =0.9902 and P = 0.9892, respectively). The three-way interaction was also not
significant (P = 0.7994).
2.3
Present and analyze a plot of the main effects for pesticide. Explain why is the analysis of the main
effects of pesticide justified. Conduct a REGWQ multiple comparison test on the main effects of
pesticide.
The analysis of the main effect of pesticide is justified because the two interactions involving
pesticide are not significant
To plot the main effects of pesticide the mean seedling vigor for pesticides 1, 2, and 3 can be
calculated in SAS by using the following code:
Proc Gplot data = hw9_2;
Title "Pesticide Main Effects";
symbol1 i=std1mtj v=none color=BLUE;
Plot vigor*Pesticide /
Description = "Pesticide Main Effects";
run;
Or
The GLM Procedure
HW Topic 9
8
The plot demonstrates that the pesticide formulations resulted in different seedling vigor scores.
The multiple comparison test for pesticide can also be done in SAS by simply specifying to do a
REGWQ comparison under the ANOVA model in question 2.2.
Means with the same letter
are not significantly different.
REGWQ Grouping Mean N Pesticide
A
71.781 32 3
A
67.625 32 2
B
62.938 32 1
The multiple comparison test suggest that the pesticide formulation that resulted in the highest seed
vigor was pesticide 3 and is significantly different from pesticide 1, which is the pesticide
formulation with the lowest seedling vigor. Pesticide formulation 2 was significantly different from
pesticide 1 but not from pesticide 2. Since there is no interaction with variety or pellet formulation
it is safe to recommend the use of pesticide 3 in all combinations, although 2 can be recommended
too since the differences with 3 are not significant (which does not necessarily mean they are not
real! But another study with more power will be required to test that).
2.4
Present and analyze a plot of the interaction between the varieties and the pellet formulation. Based
on the plot and on the ANOVA results, explain why is the analysis of the simple effects of variety
and pellet formulation justified.
To plot the interactions between varieties and pellet formulation we can use the following SAS
code:
Proc Gplot data = hw9_2;
title "Variety by Pesticide Formulation Interactions";
symbol1 i=std1mtj v=none color=BLUE;
symbol2 i=std1mtj v=none color=BLACK;
symbol3 i=std1mtj v=none color=GREEN;
symbol4 i=std1mtj v=none color=BROWN;
Plot vigor*Variety = Pellet /
Description = "Variety by Pesticide Formulation Interactions";
HW Topic 9
9
The graph showns no parallel lines supporting the result of the ANOVA of a significant interaction
between these two factors (P = 0.0074);. Since there is a significant interaction between pellet and
variety, it is necessary to describe the effect of pellet by variety and the effect of variety by pellet
because the responses are different.
2.5
Based on your response to question 2.4 conduct all 8 simple effect ANOVAs, and for the
significant ones perform an REGWQ analysis. For each test describe which combination of variety/
pellet formulation would you recommend if high vigor is desired? For each recommendation
indicate which other combinations are not significantly different from the recommended ones.
To compare varieties at the different pellet combinationsthe data must be sorted by pellet.
Proc Sort Data = hw9_2;
by Pellet;
Proc GLM Data = hw9_2;
Class Farm Variety Pesticide;
Model vigor = Farm Variety Pesticide Variety*Pesticide;
Means Variety / REGWQ; by Pellet;
The GLM Procedure
HW Topic 9
10
Level of Level of N
Variety Pellet
HW Topic 9
vigor
Mean
Std Dev
1
1
6 68.8333333 11.0709831
1
2
6 69.6666667
9.0700974
1
3
6 70.8333333
5.9805239
1
4
6 71.3333333 10.8566416
2
1
6 56.1666667
8.0601902
2
2
6 80.8333333
4.2150524
2
3
6 65.8333333
6.6156380
2
4
6 69.5000000
9.1159201
3
1
6 58.5000000
7.4498322
3
2
6 78.3333333
6.8313005
11
Level of Level of N
Variety Pellet
vigor
Mean
Std Dev
5.3447794
3
3
6 62.1666667
3
4
6 66.0000000 12.5698051
4
1
6 59.3333333 10.8012345
4
2
6 62.8333333
5.1153364
4
3
6 66.6666667
6.5625198
4
4
6 72.3333333 12.3071795
Pellet=1
Source
DF Type III SS Mean Square F Value Pr > F
Farm
1 590.0416667
590.0416667
14.34 0.0030
Variety
3 560.4583333
186.8194444
4.54 0.0264
Pesticide
2 402.0833333
201.0416667
4.89 0.0303
Variety*Pesticide
6 353.9166667
58.9861111
1.43 0.2857
The ANOVA results suggest that the varieties are significantly different when using pellet
formulation 1 (P = 0.0264).
Means with the same letter
are not significantly different.
REGWQ Grouping
Mean N Variety
A
68.833
6 1
A
59.333
6 4
B
58.500
6 3
B
56.167
6 2
B
The multiple comparison test suggest that the variety 1 was significantly differen from varieties 3
and 2 but not from variety 4. Varieties 4, 3, and 2 were not significantly different.
Pellet=2
Source
DF Type III SS Mean Square F Value Pr > F
Farm
1
0.666667
0.666667
0.02 0.9025
Variety
3 1225.500000
408.500000
9.64 0.0021
Pesticide
2
301.083333
150.541667
3.55 0.0646
Variety*Pesticide
6
96.250000
16.041667
0.38 0.8780
The ANOVA results suggest that there are significant differences in seedling vigor between the
different varieties (P = 0.0021) when using pellet formulation 2.
HW Topic 9
12
Means with the same letter
are not significantly different.
REGWQ Grouping
Mean N Variety
A
80.833
6 2
B
A
78.333
6 3
B
C
69.667
6 1
C
62.833
6 4
The multiple comparison test suggest that variety 2 has the most seed vigor when pellet 2 is used.
Variety 3 was not significantly different from variety 2.
Pellet=3
Source
DF Type III SS Mean Square F Value Pr > F
Farm
1
12.0416667
12.0416667
0.39 0.5461
Variety
3 227.7916667
75.9305556
2.45 0.1188
Pesticide
2 193.7500000
96.8750000
3.12 0.0844
Variety*Pesticide
6 208.5833333
34.7638889
1.12 0.4109
The ANOVA results suggest that there are no significant differences between variety when using
pellet 3 formulation.
Pellet=4
Source
DF Type III SS Mean Square F Value Pr > F
Farm
1 287.0416667
287.0416667
1.98 0.1869
Variety
3 139.7916667
46.5972222
0.32 0.8097
Pesticide
2 415.5833333
207.7916667
1.43 0.2795
Variety*Pesticide
6 256.0833333
42.6805556
0.29 0.9271
There are no significant differences between variety when using the pellet 4 formulations.
The multiple comparison test does not report significant differences.
ANALYSIS BY VARIETY
Proc Sort Data = hw9_2;
by Variety;
Proc GLM Data = hw9_2;
Class Farm Pellet Pesticide;
Model vigor = Farm Pellet Pesticide Pellet*Pesticide;
Means Pellet / REGWQ; by Variety;
Variety=1
HW Topic 9
Source
DF Sum of Squares Mean Square F Value Pr > F
Model
12
940.000000
78.333333
0.98 0.5139
13
Source
DF Sum of Squares Mean Square F Value Pr > F
Error
11
875.333333
Corrected Total 23
1815.333333
79.575758
R-Square Coeff Var Root MSE vigor Mean
0.517811
Source
12.71334 8.920525
70.16667
DF Type III SS Mean Square F Value Pr > F
Farm
1 266.6666667 266.6666667
3.35 0.0944
Pellet
3
7.6666667
0.10 0.9604
Pesticide
2 303.5833333 151.7916667
1.91 0.1944
Pellet*Pesticide
6 346.7500000
0.73 0.6381
23.0000000
57.7916667
WHEN USING VARIETY 1 THERE IS NO EFFECT OF PELLET OR PESTICIDE
COMBINATIONS
Dependent Variable: vigor
Variety=2
Source
DF Sum of Squares Mean Square F Value Pr > F
Model
12
2146.500000
178.875000
Error
11
771.333333
70.121212
Corrected Total 23
2917.833333
2.55 0.0658
R-Square Coeff Var Root MSE vigor Mean
0.735649
Source
12.29940 8.373841
68.08333
DF Type III SS Mean Square F Value Pr > F
Farm
1
0.666667
0.666667
0.01 0.9241
Pellet
3 1869.833333
623.277778
8.89 0.0028
Pesticide
2
157.583333
78.791667
1.12 0.3597
Pellet*Pesticide
6
118.416667
19.736111
0.28 0.9339
Ryan-Einot-Gabriel-Welsch Multiple Range Test for vigor
Variety=2
Means with the same letter
are not significantly different.
REGWQ Grouping
Mean
N Pellet
A
80.833
6 2
B
A
69.500
6 4
B
C
65.833
6 3
C
56.167
6 1
WHEN USING VARIETY 2 it is better to use pellet 2 (4 is not significantly different but is a far
second)
HW Topic 9
14
Variety=3
Source
DF Sum of Squares Mean Square F Value Pr > F
Model
12
1976.166667
164.680556
Error
11
804.333333
73.121212
Corrected Total 23
2780.500000
2.25 0.0947
R-Square Coeff Var Root MSE vigor Mean
0.710723
Source
12.90731 8.551094
66.25000
DF Type III SS Mean Square F Value Pr > F
Farm
1
130.666667
130.666667
1.79 0.2083
Pellet
3 1336.833333
445.611111
6.09 0.0107
Pesticide
2
351.750000
175.875000
2.41 0.1360
Pellet*Pesticide
6
156.916667
26.152778
0.36 0.8909
Ryan-Einot-Gabriel-Welsch Multiple Range Test for vigor
Variety=3
Means with the same letter
are not significantly different.
REGWQ Grouping
Mean
N Pellet
A
78.333
6 2
A
66.000
6 4
B
62.167
6 3
B
58.500
6 1
B
WHEN USING VARIETY 3 it is better to use pellet 2 (4 is not significantly different but is a far
second)
Dependent Variable: vigor
Variety=4
Source
DF Sum of Squares Mean Square F Value Pr > F
Model
12
1637.500000
136.458333
Error
11
607.458333
55.223485
Corrected Total 23
2244.958333
2.47 0.0724
R-Square Coeff Var Root MSE vigor Mean
0.729412
Source
HW Topic 9
11.38162 7.431251
65.29167
DF Type III SS Mean Square F Value Pr > F
Farm
1 287.0416667 287.0416667
5.20 0.0435
Pellet
3 558.1250000 186.0416667
3.37 0.0585
15
Source
DF Type III SS Mean Square F Value Pr > F
Pesticide
2 497.5833333 248.7916667
4.51 0.0372
Pellet*Pesticide
6 294.7500000
0.89 0.5342
49.1250000
Ryan-Einot-Gabriel-Welsch Multiple Range Test for vigor
Variety=4
Means with the same letter
are not significantly different.
REGWQ Grouping
Mean
N Pellet
A
72.333
6 4
B
A
66.667
6 3
B
A
62.833
6 2
59.333
6 1
B
WHEN USING VARIETY 4 it is better to use pellet 4 (1 and 3 are not significantly different but 1
is significantly lower)
2.7
How would the ANOVA model differ if the experiment was done in only one farm (i.e. you only
have one replication of all the treatment combinations)? Perform the ANOVA by only using the
data from farm 2 and report the results below. Provide a brief discussion of the overall ANOVA
results (you are not asked to do mean comparisons here).
If we lose one of the farms we lose the second replication and the model will no longer have enough
degrees of freedom to test for the three-way interactions. To conduct the analysis that the three-way
interactions are not significant and excluded from the model to use as the MSE.
Data hw9_22;
Do Variety = 1 to 4;
Do Pellet = 1 to 4;
Do Pesticide = 1 to 3;
Input vigor @@;
Output;
End;
End;
End;
Cards;
48
76
65
62
70
69
73
66
78
55
67
73
44
58
54
77
79
81
62
61
79
60
78
82
53
50
56
81
86
86
55
56
66
56
58
64
46
55
64
56
59
67
64
66
62
59
58
86
;
Proc Print data = hw9_22;
Proc GLM data = hw9_22;
Class Variety Pellet Pesticide;
Model vigor = Variety
Pellet
Pesticide
Variety*Pellet Variety*Pesticide Pellet*Pesticide;
run; quit;
Source
Variety
HW Topic 9
DF Type III SS Mean Square F Value Pr > F
3
276.083333
92.027778
2.72 0.0753
16
Source
DF Type III SS Mean Square F Value Pr > F
Pellet
3 1773.416667
591.138889
17.44 <.0001
Pesticide
2 1023.875000
511.937500
15.11 0.0001
Variety*Pellet
9 1593.083333
177.009259
5.22 0.0014
Variety*Pesticide
6
161.791667
26.965278
0.80 0.5855
Pellet*Pesticide
6
354.958333
59.159722
1.75 0.1677
The ANOVA results suggest that there are no significant differences between varieties (P = 0.0753).
But the values are close to significant, and since the Variety x Pellet is highly significant, it would be
worth to analyze the simple effect of varieties by Pellet as exemplified above (not requested in this
question).
There are significant differences between pellet and pesticide formulations (P < 0.0001 and P =
0.0014, respectively). There are significant interactions between variety and pellet formulations (P =
0.0014) but no significant interactions between variety and pesticide (P = 0.5855) or pellet and
pesticide (P = 0.1677), so the main effect of pesticide can be analyzed separately.
Appendix
When you are asked to "describe in detail the design of this experiment," please do so by
completing the following template:
Design:
Response Variable:
Experimental Unit:
Class
Variable
1
2
↓
n
Block or
Treatment
No. of
Levels
Subsamples?
YES / NO
Description
NOTICE: There is a new column in the above table ("Block or Treatment"). Now, for each class
variable in your model, you need to indicate if it is a "Block" variable or a "Treatment" variable.
HW Topic 9
17