Laboratory for Topics 4 & 5 - UC Davis Plants
PLS205 Lab 3 January 23, 2014
Laboratory Topics 4 & 5
∙ Orthogonal contrasts
Class comparisons in SAS
Trend analysis in SAS
∙ Multiple mean comparisons
Orthogonal contrasts
Planned, single degree-of-freedom orthogonal contrasts are powerful means of perfectly partitioning the ANOVA model sum of squares to gain greater insight into your data; and this method of analysis is available in SAS via Proc GLM.
Whenever you program contrasts, be sure to use the "Order = Data" option in Proc GLM so that the coefficients featured in the subsequent Contrast statements will correspond accurately to the levels of the indicated classification variable. For example:
Proc GLM Order = Data;
The Contrast statements can come anywhere after the Model statement in Proc GLM. These statements specify the independent F-test to be conducted. Its syntax:
Contrast 'ID' ClassVariable Coefficients;
where ID, enclosed in single quotes, is the label you assign to the contrast (just a title, it can be anything); ClassVariable is the classification variable whose means are being compared; and Coefficients is the set of orthogonal coefficient values, separated by spaces or tabs. Note that in a nested design, it is imperative that the coefficients be followed by a declaration of the appropriate error term:
Contrast 'A vs. B' Trtmt 1 1 1 -1 -1 -1 / e = Pot(Trtmt);
Before looking at our first example, remember that Proc GLM uses several different methods for determining SS, the details of which will be covered later in the course [For more details, refer to Topic 11.4 in your class notes]. For now, let's reiterate the following rule of thumb:
Use the Type I SS (sum of squares) for regressions
The Type I SS measures incremental sums of squares for the model as each variable is added.
Use the Type III SS for F-tests
Type III is the sum of squares for each effect adjusted for every other effect and is used for both balanced and unbalanced designs.
Class comparisons using contrasts
Example 4.1 ST&D pg. 159 [Lab3ex1.sas]
This is a CRD in which 18 mint plants were randomly assigned to 6 different treatments (i.e. all combinations of two temperature [High and Low] and three light [8, 12, and 16 hour days] conditions) and their growth measured.
Data MintMean;
Input Trtmt $ Growth @@;
Cards;
L08 15.0
L08 17.5
L08 11.5
L12 18.0
L12 14.0
L12 17.5
L16 19.0
L16 21.5
L16 22.0
H08 32.0
H08 28.0
H08 28.0
H12 22.0
H12 26.5
H12 29.0
H16 33.0
H16 27.0
H16 35.0
;
* L08 means Low Temp and 8 hours of light, H12 means High Temp and 12 hours of light, etc.;
Proc GLM Order = Data; * To maintain the order in which we entered data;
Class Trtmt;
Model Growth = Trtmt; * L08 L12 L16 H08 H12 H16;
Contrast 'Temp' Trtmt 1 1 1 -1 -1 -1;
Contrast 'Light linear' Trtmt 1 0 -1 1 0 -1;
Contrast 'Light quadratic' Trtmt 1 -2 1 1 -2 1;
Contrast 'Temp * Light linear' Trtmt 1 0 -1 -1 0 1;
Contrast 'Temp * Light quadratic' Trtmt 1 -2 1 -1 2 -1;
Run;
Quit;
What questions we are asking here exactly? To answer this, it is helpful to articulate the null hypothesis for each contrast:
Contrast ‘Temp’ H0: Mean plant growth under low temperature conditions is the same as under high temperature conditions.
Contrast ‘Light Linear’ H0: Mean plant growth under 8 hour days is the same as under 16 hour days (OR: The response of growth to light has no linear component).
Contrast ‘Light Quadratic’ H0: Mean plant growth under 12 hour days is the same as the average mean growth under 8 and 16 hour days combined (OR: The growth response to light is perfectly linear; OR: The response of growth to light has no quadratic component).
Contrast ‘Temp * Light Linear’ H0: The linear component of the response of growth to light is the same at both temperatures.
Contrast ‘Temp * Light Quadratic’ H0: The quadratic component of the response of growth to light is the same at both temperatures.
So what would it mean to find significant results and to reject each of these null hypotheses?
Reject contrast ‘Temp’ H0 = There is a significant response of growth to temperature.
Reject contrast ‘Light linear’ H0 = The response of growth to light has a significant linear component.
Reject contrast ‘Light quadratic’ H0 = The response of growth to light has a significant quadratic component.
Reject contrast ‘Temp * Light Linear’ H0 = The linear component of the response of growth to light depends on temperature.
Reject contrast ‘Temp * Light Quadratic’ H0 = The quadratic component of the response of growth to light depends on temperature.
Results of the GLM procedure
Dependent Variable: Growth
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 5 718.5694444 143.7138889 16.69 F
Trtmt 5 718.5694444 143.7138889 16.69 F
Temp 1 606.6805556 606.6805556 70.45 F
Model 4 125.6613333 31.4153333 9.90 F
Sp 4 125.6613333 31.4153333 9.90 F
Linear 1 91.26666667 91.26666667 28.76 F
Model 4 125.6613333 31.4153333 9.90 F
Sp 1 91.26666667 91.26666667 28.76 F
Sp 1 0.41016441 0.41016441 0.13 0.7222
Sp*Sp 1 0.27910540 0.27910540 0.09 0.7692
Sp*Sp*Sp 1 0.22140395 0.22140395 0.07 0.7938
Sp*Sp*Sp*Sp 1 0.19716667 0.19716667 0.06 0.8052
Again, since this is a regression analysis, use the Type I SS, not the Type III SS. Notice in this case that the Type I SS results match perfectly those from our earlier analysis by contrasts.
For the interested:
When you carry out a trend analysis using a regression approach, SAS also provides estimates of the parameters for your model:
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept 92.91666667 132.6083560 0.70 0.4900
Sp -6.97245370 19.3932598 -0.36 0.7222
Sp*Sp 0.30495756 1.0282517 0.30 0.7692
Sp*Sp*Sp -0.00620499 0.0234905 -0.26 0.7938
Sp*Sp*Sp*Sp 0.00004876 0.0001956 0.25 0.8052
In this case, the equation of the trend line that best fits out data would be:
Yield = 0.30 * Sp2 – 6.97 * Sp + 92.92
Multiple Mean Comparisons
Orthogonal contrasts are planned, a priori tests that partition the experimental variance cleanly. They are a powerful tool for analyzing data, but they are not appropriate for all experiments. Less restrictive comparisons among treatment means can be performed using Proc GLM by way of the Means statement. Any number of Means statements may be used within a given Proc GLM, provided they appear after the Model statement. The syntax:
Means Class-Variables / Options;
This statement tells SAS to:
1. Compute the means of the response variable for each level of the specified classification variable(s), all of which were featured in the original Model statement; then
2. Perform multiple comparisons among these means using the stated Options.
Some of the available Options are listed below:
Fixed Range Tests
∙ DUNNETT ('control') Dunnett's test [NOTE: If no control is specified, the first treatment is used.]
∙ T or LSD Fisher's least significant difference test
∙ TUKEY Tukey's studentized range test (HSD: Honestly significant difference)
∙ SCHEFFE Scheffé’s test
Multiple Range Tests
∙ DUNCAN Duncan's test
∙ SNK Student-Newman-Keuls test
∙ REGWQ Ryan-Einot-Gabriel-Welsch test
The default significance level for comparisons among means is α = 0.05, but this can be changed easily using the option Alpha = α, where α is the desired significance level. The important thing to keep in mind is the EER (experimentwise error rate); we want to keep it controlled while keeping the test as sensitive as possible, so our choice of test should reflect that.
Example 4.4 (One-Way Multiple Comparison) [Lab3ex4.sas]
Here’s the clover experiment again, a CRD in which 30 different clover plants were randomly inoculated with six different strains of rhizobium are the resulting level of nitrogen fixation measured.
Data Clover;
Input Culture $ Nlevel;
Cards;
3DOk1 24.1
3DOk1 32.6
3DOk1 27
3DOk1 28.9
3DOk1 31.4
3DOk5 19.1
3DOk5 24.8
3DOk5 26.3
3DOk5 25.2
3DOk5 24.3
3DOk4 17.9
3DOk4 16.5
3DOk4 10.9
3DOk4 11.9
3DOk4 15.8
3DOk7 20.7
3DOk7 23.4
3DOk7 20.5
3DOk7 18.1
3DOk7 16.7
3DOk13 14.3
3DOk13 14.4
3DOk13 11.8
3DOk13 11.6
3DOk13 14.2
Comp 17.3
Comp 19.4
Comp 19.1
Comp 16.9
Comp 20.8
;
Proc GLM;
Class Culture;
Model Nlevel = Culture;
Means Culture / LSD;
Means Culture / Dunnett ('Comp'); * The control treatment is 'Comp';
Means Culture / Tukey;
Means Culture / Scheffe;
Means Culture / Duncan;
Means Culture / SNK;
Means Culture / REGWQ;
Proc Boxplot;
Title 'Boxplot Comparing Treatment Means';
Plot NLevel*Culture / cboxes = black;
Run;
Quit;
In this experiment, there is no obvious structure to the treatment levels and therefore no way to anticipate the relevant questions to ask. We want to know how the different rhizobial strains performed; and to do this, we must systematically make all pair-wise comparisons among them.
In the output on the following pages, keep an eye on the
least (or minimum) significant difference(s) used for each test.
What is indicated by changes in these values from test to test?
Also notice how the comparisons change significance with the different tests.
t Tests (LSD) for Nlevel
This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 24
Error Mean Square 6.668833
Critical Value of t 2.06390
Least Significant Difference 3.3709
Means with the same letter are not significantly different.
t Grouping Mean N Culture
A 28.800 5 3DOk1
B 23.940 5 3DOk5
C 19.880 5 3DOk7
C 18.700 5 Comp
D 14.600 5 3DOk4
D 13.260 5 3DOk13
Dunnett's t Tests for Nlevel
This test controls the Type I experimentwise error for comparisons of all treatments against a control.
Alpha 0.05
Error Degrees of Freedom 24
Error Mean Square 6.668833
Critical Value of Dunnett's t 2.69540
Minimum Significant Difference 4.4023
Comparisons significant at the 0.05 level are indicated by ***.
Difference
Culture Between Simultaneous 95%
Comparison Means Confidence Limits
3DOk1 - Comp 10.100 5.698 14.502 ***
3DOk5 - Comp 5.240 0.838 9.642 ***
3DOk7 - Comp 1.180 -3.222 5.582
3DOk4 - Comp -4.100 -8.502 0.302
3DOk13 - Comp -5.440 -9.842 -1.038 ***
Tukey's Studentized Range (HSD) Test for Nlevel
This test controls the Type I experimentwise error rate (MEER), but it generally has a higher Type II error rate than REGWQ.
Alpha 0.05
Error Degrees of Freedom 24
Error Mean Square 6.668833
Critical Value of Studentized Range 4.37265
Minimum Significant Difference 5.0499
Means with the same letter are not significantly different.
Tukey Grouping Mean N Culture
A 28.800 5 3DOk1
B A 23.940 5 3DOk5
B C 19.880 5 3DOk7
D C 18.700 5 Comp
D E 14.600 5 3DOk4
E 13.260 5 3DOk13
Scheffe's Test for Nlevel [For group comparisons with Scheffe, see Section 5.3.1.4]
This test controls the Type I MEER.
Alpha 0.05
Error Degrees of Freedom 24
Error Mean Square 6.668833
Critical Value of F 2.62065
Minimum Significant Difference 5.9121
Means with the same letter are not significantly different.
Scheffe Grouping Mean N Culture
A 28.800 5 3DOk1
B A 23.940 5 3DOk5
B C 19.880 5 3DOk7
B C D 18.700 5 Comp
C D 14.600 5 3DOk4
D 13.260 5 3DOk13
Duncan's Multiple Range Test for Nlevel
This test controls the Type I comparisonwise error rate, not the MEER.
Alpha 0.05
Error Degrees of Freedom 24
Error Mean Square 6.668833
Number of Means 2 3 4 5 6
Critical Range 3.371 3.540 3.649 3.726 3.784
Means with the same letter are not significantly different.
Duncan Grouping Mean N Culture
A 28.800 5 3DOk1
B 23.940 5 3DOk5
C 19.880 5 3DOk7
C 18.700 5 Comp
D 14.600 5 3DOk4
D 13.260 5 3DOk13
Student-Newman-Keuls (SNK) Test for Nlevel
This test controls the Type I experimentwise error rate under the complete null hypothesis (EERC) but not under partial null hypotheses (EERP).
Alpha 0.05
Error Degrees of Freedom 24
Error Mean Square 6.668833
Number of Means 2 3 4 5 6
Critical Range 3.3708858 4.0787156 4.5055234 4.8116298 5.0499266
Means with the same letter are not significantly different.
SNK Grouping Mean N Culture
A 28.800 5 3DOk1
B 23.940 5 3DOk5
C 19.880 5 3DOk7
C 18.700 5 Comp
D 14.600 5 3DOk4
D 13.260 5 3DOk13
Ryan-Einot-Gabriel-Welsch (REGWQ) Multiple Range Test for Nlevel
This test controls the Type I MEER.
Alpha 0.05
Error Degrees of Freedom 24
Error Mean Square 6.668833
Number of Means 2 3 4 5 6
Critical Range 4.1910831 4.5900067 4.8041049 4.8116298 5.0499266
Means with the same letter are not significantly different.
REGWQ Grouping Mean N Culture
A 28.800 5 3DOk1
B 23.940 5 3DOk5
C B 19.880 5 3DOk7
C D 18.700 5 Comp
E D 14.600 5 3DOk4
E 13.260 5 3DOk13
And to make the relationships among the tests easier to see (i.e. to make sure the dead horse is thoroughly beaten), here is a nice little summary table of all the above results:
| |Significance Groupings |
|Culture |LSD |Dunnett |Tukey |Scheffe |Duncan |SNK |REGWQ |
Least Sig't |3.371 |4.402 |5.05 |5.912 |3.371 |3.371 |4.191 | |Difference |fixed |fixed |fixed |fixed |3.784 |5.05 |5.05 | |EER Control |no |yes |yes |yes |no |EERC
only |yes | |
Notice where the non-EER-controlling tests get you into potential Type I trouble, namely by their readiness to declare significant differences between 3DOk5 and 3DOk7 and between Comp and 3DOk4.
On the other hand, regarding potential Type II trouble, notice where the relatively insensitive Scheffe's test (insensitive due to its ability to make unlimited pair-wise and group comparisons) failed to pick up a difference detected by other EER-controlling tests (e.g. between 3DOk7 and 3DOk4). Notice, too, how the multiple-range REGWQ was able to detect the difference between 3DOk1 and 3DOk5 when the fixed-range Tukey test was not (both control for EER).
Remember, while you should steer clear of tests that do not control for EER, there's no "right" test or "wrong" test. There's only knowing the characteristics of each and choosing the most appropriate one for your experiment (and the culture of your discipline).
It is instructive to consider the above table of comparisons with the boxplot below in hand:
[pic]
Something to think about:
Does the boxplot above raise any red flags for you about your data?
How would go about investigating such concerns?
-----------------------
'Linear' H0: There is no linear component
'Quadratic' H0: There is no quadratic component
'Cubic' H0: There is no cubic component
'Quartic' H0: There is no quartic component
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