PLS205 2014 6.1 Lab 6 (Topic 9) PLS205 Lab 6 February 13, 2014 Laboratory Topic 9 ∙ A word about factorials ∙ Specifying interactions among factorial effects in SAS ∙ The relationship between factors and treatment ∙ Interpreting results of an experiment with a factorial treatment structure ∙ Visualizing simple and main effects ∙ Visualizing three-way interactions ∙ APPENDIX: The Almost Practically Complete Analysis of Example 6.1 ∙ APPENDIX 2: Graphing in Excel A word about factorials A factorial is not an experimental design. Why? Because the term "factorial" merely describes the structure of the treatment effects (i.e. the factors), not how they are randomized. Specifically, a factorial treatment structure is one in which all levels of every factor are present in all possible combinations with all the levels of every other factor in the experiment (i.e. the crossing of factors is complete and orthogonal). It is this complete, orthogonal structure that allows an experimenter to gain insight into interactions among factors. Seen in this way, it becomes clear that any of the true experimental designs (i.e. randomization strategies) we have discussed so far (CRD, RCBD, Latin Square) can be factorials, provided the treatments are structured correctly. A factorial is a complete, orthogonal structure of treatment effects intended to provide insight into their interactions. Specifying Interactions Among Factorial Effects in SAS Specifications about designs with factorial treatment structures are entered through the Model statement of the Proc GLM, and this syntax can assume one of two forms: Stars (*) or bars (|). Stars are used to partition out specific interactions from the Treatment SS and are useful when certain interactions must be used as error terms in custom F tests. Examples: Model Resp = a b a*b specifies partitioning of SST into main effect A, me B and interaction AxB Model Resp = a a*b a*b*c specifies me A and interactions AxB and AxBxC Bars are used as a nice shortcut to partition the Treatment SS into all possible combinations of the included factors. On a standard PC keyboard, the bar symbol (|) is typed as Shift-\. Examples: Model Resp = a|b is equivalent to Model Resp = a b a*b Model Resp = a|b|c is equivalent to Model Resp = a b c a*b a*c b*c a*b*c An additional nice trick to know is the use of "@" in factorial model statements. The "@" symbol in conjunction with bars (|) allows you to specify all possible combinations of model factors up to a certain level (e.g. two-way effects), saving you lots of typing. An example: Model Resp = Block a|b|c@2 is equivalent to Model Resp = Block a b c a*b a*c b*c notice this excludes the three-way effect a*b*c
Transcript
PLS205 2014 6.1 Lab 6 (Topic 9)
PLS205 Lab 6 February 13, 2014
Laboratory Topic 9
∙ A word about factorials
∙ Specifying interactions among factorial effects in SAS
∙ The relationship between factors and treatment
∙ Interpreting results of an experiment with a factorial treatment structure
∙ Visualizing simple and main effects
∙ Visualizing three-way interactions
∙ APPENDIX: The Almost Practically Complete Analysis of Example 6.1
∙ APPENDIX 2: Graphing in Excel
A word about factorials
A factorial is not an experimental design. Why? Because the term "factorial" merely describes the
structure of the treatment effects (i.e. the factors), not how they are randomized. Specifically, a factorial
treatment structure is one in which all levels of every factor are present in all possible combinations with
all the levels of every other factor in the experiment (i.e. the crossing of factors is complete and
orthogonal). It is this complete, orthogonal structure that allows an experimenter to gain insight into
interactions among factors. Seen in this way, it becomes clear that any of the true experimental designs
(i.e. randomization strategies) we have discussed so far (CRD, RCBD, Latin Square) can be factorials,
provided the treatments are structured correctly.
A factorial is a complete, orthogonal structure of treatment effects
intended to provide insight into their interactions.
Specifying Interactions Among Factorial Effects in SAS
Specifications about designs with factorial treatment structures are entered through the Model statement
of the Proc GLM, and this syntax can assume one of two forms: Stars (*) or bars (|).
Stars are used to partition out specific interactions from the Treatment SS and are useful when certain
interactions must be used as error terms in custom F tests. Examples:
Model Resp = a b a*b specifies partitioning of SST into main effect A, me B and interaction AxB
Model Resp = a a*b a*b*c specifies me A and interactions AxB and AxBxC
Bars are used as a nice shortcut to partition the Treatment SS into all possible combinations of the
included factors. On a standard PC keyboard, the bar symbol (|) is typed as Shift-\. Examples:
Model Resp = a|b is equivalent to Model Resp = a b a*b
Model Resp = a|b|c is equivalent to Model Resp = a b c a*b a*c b*c a*b*c
An additional nice trick to know is the use of "@" in factorial model statements. The "@" symbol in
conjunction with bars (|) allows you to specify all possible combinations of model factors up to a certain
level (e.g. two-way effects), saving you lots of typing. An example:
Model Resp = Block a|b|c@2 is equivalent to Model Resp = Block a b c a*b a*c b*c
notice this excludes the three-way effect a*b*c
PLS205 2014 6.2 Lab 6 (Topic 9)
The Relationship Between Factors and Treatment
Until now, we have had only a single 'treatment' (the effect of which we are trying to understand) with
zero (CRD), one (RCBD), or two (LS) blocking variables (the effects of which we are trying to account
for but not really investigate). With factorials, we now have two or more 'factors' that are experimentally
equivalent to the single 'treatment' variable from the first half of the course. To illustrate this equivalence,
reconsider Example 1 from Lab 3 (Topics 4-5): An experiment with 6 treatments (L08, L12, L16, H08,
H12, H16), where L/H refers to Low/High temperatures and 8/12/16 refers to hours of light. This is
exactly equivalent to having temperature as one factor and light as another, organized as a factorial:
Model Growth = Treatment; df Treatment = 5
Model Growth = Temp Light Temp*Light; df Temp = 1
df Light = 2
df Temp*Light = 2
Sum = 5
What this is meant to show is that the old classification variable ‘Treatment’ is simply a combination of
two factors (light and temperature). Rewriting the model in terms of the factors does not affect the Model
df at all; it simply expands the class variable ‘Treatment’ into ‘Temp Light Temp*Light’. Before, we
accomplished this ‘opening up’ of the treatment through orthogonal contrasts. The insights gained
through each approach are equivalent.
Example 6.1 Two-Way ANOVA with interactions [Lab6ex1.sas]
In a study comparing the relative growth of five varieties of turfgrass (VARIETY) in three experimental
soil mixtures (SOIL), six pots were prepared with each VARIETY-SOIL combination. The 90 pots were
randomly allocated to six growth chambers (BLOCKS) and the dry matter yields were measured by
clipping the plants at the end of four weeks. In this experiment, the researchers are interested only in
these five varieties and three soil mixtures; so VARIETY and SOIL can be regarded as fixed factors.
Data RCBDFactorial;
Do Soil = 1 to 3;
Do Variety = 1 to 5;
Do Block = 1 to 6;
Input Yield @@;
Output;
End;
End;
End;
Cards;
22.1 24.1 19.1 22.1 25.1 18.1
27.1 15.1 20.6 28.6 15.1 24.6
22.3 25.8 22.8 28.3 21.3 18.3
19.8 28.3 26.8 27.3 26.8 26.8
20.0 17.0 24.0 22.5 28.0 22.5
13.5 14.5 11.5 6.0 27.0 18.0
16.9 17.4 10.4 19.4 11.9 15.4
15.7 10.2 16.7 19.7 18.2 12.2
PLS205 2014 6.3 Lab 6 (Topic 9)
15.1 6.5 17.1 7.6 13.6 21.1
21.8 22.8 18.8 21.3 16.3 14.3
19.0 22.0 20.0 14.5 19.0 16.0
20.0 22.0 25.5 16.5 18.0 17.5
16.4 14.4 21.4 19.9 10.4 21.4
24.5 16.0 11.0 7.5 14.5 15.5
11.8 14.3 21.3 6.3 7.8 13.8
;
Proc GLM Data = RCBDFactorial;
Class Soil Variety Block;
Model Yield = Soil|Variety Block; * This Model includes all main effects
as well as the Method*Variety interaction;
Proc GLM Data = RCBDFactorial;
Class Soil Variety Block;
Model Yield = Soil|Variety|Block@2; * Exploratory model to examine the
one-way block interactions (see discussion below);
Run;
Quit;
NOTE: This initial analysis enables us to see if the interaction is significant and decide:
Main or simple effects?
Take a look at the resultant ANOVA table:
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 19 1361.153778 71.639673 3.45 <.0001
Error 70 1451.637778 20.737683
Corrected Total 89 2812.791556
R-Square Coeff Var Root MSE Yield Mean
0.483916 24.69855 4.553865 18.43778
Source DF Type III SS Mean Square F Value Pr > F
Soil 2 953.1562222 476.5781111 22.98 <.0001
Variety 4 11.3804444 2.8451111 0.14 0.9680
Soil*Variety 8 374.4882222 46.8110278 2.26 0.0330
Block 5 22.1288889 4.4257778 0.21 0.9557
This is an RCBD with 6 blocks. Even though there are six replications per Method-Variety combination
(which allows us to include their interaction in the model), there is only one replication per Method-
Variety-Block combination. The upshot of this is that the Block*Factor interactions are inside the
experimental error for this ANOVA. In other words, if the model statement had been:
Model Yield = Soil|Variety|Block;
there would have been no variation left to estimate the error (dfe = 0), because:
Block * Soil = 10 df
Block * Variety = 20 df
Block * Soil * Variety = 40 df
70 df = dfe
PLS205 2014 6.4 Lab 6 (Topic 9)
We exclude the one-way Block interactions from the model because, in general, we don't care about them
(remember, we block to reduce the error term, not to gain understanding of the effect of blocking). In
other words, this is a choice we make. Excluding the two-way Block interaction is not a choice, however;
it cannot be a part of the model because it is the only term we have for our error. Of course, we still want
to check these Block*Treatment interactions to see if they are significant. It they are not significant it is
justifiable to relegate these interactions to the error term. If they are significant, you can attempt a
transformation or be aware that they will contribute to a larger MSE when taken out of the model.. To
test the Block*Treatment interactions they can simply be placed into an exploratory model (the second
Neither 2-way block interaction is significant, so we're justified in merging them into the error (and
gaining 30 df by doing so).
Step 2: Analyze the simple effect of Soil (i.e. for each Variety separately) Data RCBDFactorial;
Do Soil = 1 to 3;
Do Variety = 1 to 5;
Do Block = 1 to 6;
Input Yield @@;
Output;
End;
End;
End;
Cards;
PLS205 2014 6.12 Lab 6 (Topic 9)
22.1 24.1 19.1 22.1 25.1 18.1
27.1 15.1 20.6 28.6 15.1 24.6
22.3 25.8 22.8 28.3 21.3 18.3
19.8 28.3 26.8 27.3 26.8 26.8
20.0 17.0 24.0 22.5 28.0 22.5
13.5 14.5 11.5 6.0 27.0 18.0
16.9 17.4 10.4 19.4 11.9 15.4
15.7 10.2 16.7 19.7 18.2 12.2
15.1 6.5 17.1 7.6 13.6 21.1
21.8 22.8 18.8 21.3 16.3 14.3
19.0 22.0 20.0 14.5 19.0 16.0
20.0 22.0 25.5 16.5 18.0 17.5
16.4 14.4 21.4 19.9 10.4 21.4
24.5 16.0 11.0 7.5 14.5 15.5
11.8 14.3 21.3 6.3 7.8 13.8
;
Proc Sort Data = RCBDFactorial;
By Variety;
Proc GLM Data = RCBDFactorial;
Class Soil Block;
Model Yield = Soil Block;
Means Soil / Tukey;
By Variety;
Output Out = PR r = res p = pred;
Proc Print Data = PR;
Proc Univariate normal data = PR;
Var res;
By Variety;
Proc GLM data = RCBDFactorial;
Class Soil;
Model Yield = Soil;
Means Soil / hovtest = Levene;
By Variety;
Proc GLM Data = PR;
Class Soil Block;
Model Yield = Soil Block pred*pred;
By Variety;
Proc Plot Data = PR;
Plot res*pred;
By Variety;
Run;
Quit;
The first Proc GLM carries out five separate ANOVA's, one for each Variety; it also generates predicted
and residual values. The Proc Univariate tests for normality of residuals within each ANOVA. The
second Proc GLM conducts Levene's Tests for Soil within each level of variety. And the last Proc GLM
tests for nonadditivity within each of the five models. The output is extensive but can be organized as
shown on the next page:
Normality of residuals (Variety 1 – Variety 5)
Test --Statistic--- -----p Value------
Shapiro-Wilk W 0.962536 Pr < W 0.6512
Shapiro-Wilk W 0.954449 Pr < W 0.4990 Shapiro-Wilk W 0.954754 Pr < W 0.5043 Shapiro-Wilk W 0.991391 Pr < W 0.9996 Shapiro-Wilk W 0.945644 Pr < W 0.3608
