Examples
Example 17.1. Randomized Complete Block With Factorial Treatment Structure
This example uses statements for the analysis of a randomized block with two treatment factors occuring in a factorial structure. The data, from Neter, Wasserman, and Kutner (1990, p. 941), are from an experiment examining the effects of codeine and acupuncture on post- operative dental pain in male subjects. Both treatment factors have two levels. The codeine levels are a codeine capsule or a sugar capsule . The acupuncture levels are two inactive acupuncture points or two active acupuncture points. There are four distinct treatment combinations due to the factorial treatment structure. The 32 subjects are assigned to eight blocks of four subjects each based on an assessment of pain tolerance.
The data for the analysis are balanced, so PROC ANOVA is used. The data are as follows :
title1 'Randomized Complete Block With Two Factors'; data PainRelief; input PainLevel Codeine Acupuncture Relief @@; datalines; 1 1 1 0.0 1 2 1 0.5 1 1 2 0.6 1 2 2 1.2 2 1 1 0.3 2 2 1 0.6 2 1 2 0.7 2 2 2 1.3 3 1 1 0.4 3 2 1 0.8 3 1 2 0.8 3 2 2 1.6 4 1 1 0.4 4 2 1 0.7 4 1 2 0.9 4 2 2 1.5 5 1 1 0.6 5 2 1 1.0 5 1 2 1.5 5 2 2 1.9 6 1 1 0.9 6 2 1 1.4 6 1 2 1.6 6 2 2 2.3 7 1 1 1.0 7 2 1 1.8 7 1 2 1.7 7 2 2 2.1 8 1 1 1.2 8 2 1 1.7 8 1 2 1.6 8 2 2 2.4 ;
The variable PainLevel is the blocking variable, and Codeine and Acupuncture represent the levels of the two treatment factors. The variable Relief is the pain relief score (the higher the score, the more relief the patient has).
The following code invokes PROC ANOVA. The blocking variable and treatment factors appear in the CLASS statement. The bar between the treatment factors Codeine and Acupuncture adds their main effects as well as their interaction Codeine * Acupuncture to the model.
proc anova data=PainRelief; class PainLevel Codeine Acupuncture; model Relief = PainLevel CodeineAcupuncture;
The results from the analysis are shown in Output 17.1.1, Output 17.1.2, and Output 17.1.3.
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Randomized Complete Block With Two Factors The ANOVA Procedure Class Level Information Class Levels Values PainLevel 8 1 2 3 4 5 6 7 8 Codeine 2 1 2 Acupuncture 2 1 2 Number of Observations Read 32 Number of Observations Used 32
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Dependent Variable: Relief Sum of Source DF Squares Mean Square F Value Pr > F Model 10 11.33500000 1.13350000 78.37 <.0001 Error 21 0.30375000 0.01446429 Corrected Total 31 11.63875000 R-Square Coeff Var Root MSE Relief Mean 0.973902 10.40152 0.120268 1.156250
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The Class Level Information and ANOVA table are shown in Output 17.1.1 and Output 17.1.2. The class level information summarizes the structure of the design. It is good to check these consistently in search of errors in the data step. The overall F test is significant, indicating that the model accounts for a significant amount of variation in the dependent variable.
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Dependent Variable: Relief Source DF Anova SS Mean Square F Value Pr > F PainLevel 7 5.59875000 0.79982143 55.30 <.0001 Codeine 1 2.31125000 2.31125000 159.79 <.0001 Acupuncture 1 3.38000000 3.38000000 233.68 <.0001 Codeine*Acupuncture 1 0.04500000 0.04500000 3.11 0.0923
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Output 17.1.3 shows tests of the effects. The blocking effect is significant; hence, it is useful. The interaction between codeine and acupuncture is significant at the 90% level but not at the 95% level. The significance level of this test should be determined before the analysis. The main effects of both treatment factors are highly significant.
Example 17.2. Alternative Multiple Comparison Procedures
The following is a continuation of the first example in the the 'One-Way Layout with Means Comparisons' section on page 424. You are studying the effect of bacteria on the nitrogen content of red clover plants, and the analysis of variance shows a highly significant effect. The following statements create the data set and compute the analysis of variance as well as Tukey's multiple comparisons test for pairwise differences between bacteria strains; the results are shown in Figure 17.1, Figure 17.2, and Figure 17.3
title1 'Nitrogen Content of Red Clover Plants'; data Clover; input Strain $ Nitrogen @@; datalines; 3DOK1 19.4 3DOK1 32.6 3DOK1 27.0 3DOK1 32.1 3DOK1 33.0 3DOK5 17.7 3DOK5 24.8 3DOK5 27.9 3DOK5 25.2 3DOK5 24.3 3DOK4 17.0 3DOK4 19.4 3DOK4 9.1 3DOK4 11.9 3DOK4 15.8 3DOK7 20.7 3DOK7 21.0 3DOK7 20.5 3DOK7 18.8 3DOK7 18.6 3DOK13 14.3 3DOK13 14.4 3DOK13 11.8 3DOK13 11.6 3DOK13 14.2 COMPOS 17.3 COMPOS 19.4 COMPOS 19.1 COMPOS 16.9 COMPOS 20.8 ; proc anova data=Clover; class Strain; model Nitrogen = Strain; means Strain / tukey; run;
The interactivity of PROC ANOVA enables you to submit further MEANS statements without re-running the entire analysis. For example, the following command requests means of the Strain levels with Duncan's multiple range test and the Waller- Duncan k -ratio t test.
means Strain / duncan waller; run;
Results of the Waller-Duncan k -ratio t test are shown in Output 17.2.1.
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Nitrogen Content of Red Clover Plants The ANOVA Procedure Waller-Duncan K-ratio t Test for Nitrogen NOTE: This test minimizes the Bayes risk under additive loss and certain other assumptions. Kratio 100 Error Degrees of Freedom 24 Error Mean Square 11.78867 F Value 14.37 Critical Value of t 1.91873 Minimum Significant Difference 4.1665 Means with the same letter are not significantly different. Waller Grouping Mean N Strain A 28.820 5 3DOK1 B 23.980 5 3DOK5 B C B 19.920 5 3DOK7 C C D 18.700 5 COMPOS D E D 14.640 5 3DOK4 E E 13.260 5 3DOK13 Nitrogen Content of Red Clover Plants The ANOVA Procedure Duncan's Multiple Range Test for Nitrogen NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
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The Waller-Duncan k -ratio t test is a multiple range test. Unlike Tukey's test, this test does not operate on the principle of controlling Type I error. Instead, it compares the Type I and Type II error rates based on Bayesian principles (Steel and Torrie 1980).
The Waller Grouping column in Output 17.2.1 shows which means are significantly different. From this test, you can conclude the following:
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The mean nitrogen content for strain 3DOK1 is higher than the means for all other strains.
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The mean nitrogen content for strain 3DOK5 is higher than the means for COMPOS, 3DOK4, and 3DOK13.
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The mean nitrogen content for strain 3DOK7 is higher than the means for 3DOK4 and 3DOK13.
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The mean nitrogen content for strain COMPOS is higher than the mean for 3DOK13.
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Differences between all other means are not significant based on this sample size .
Output 17.2.2 shows the results of Duncan's multiple range test. Duncan's test is a result-guided test that compares the treatment means while controlling the comparison-wise error rate. You should use this test for planned comparisons only (Steel and Torrie 1980). The results and conclusions for this example are the same as for the Waller-Duncan k -ratio t test. This is not always the case.
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Waller-Duncan K-ratio t Test for Nitrogen NOTE: This test minimizes the Bayes risk under additive loss and certain other assumptions. Duncan's Multiple Range Test for Nitro gen NOTE: 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 11.78867 Number of Means 2 3 4 5 6 Critical Range 4.482 4.707 4.852 4.954 5.031 Means with the same letter are not significantly different. Duncan Grouping Mean N Strain A 28.820 5 3DOK1 B 23.980 5 3DOK5 B C B 19.920 5 3DOK7 C C D 18.700 5 COMPOS D E D 14.640 5 3DOK4 E E 13.260 5 3DOK13
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Tukey and Least Significant Difference (LSD) tests are requested with the following MEANS statement. The CLDIFF option requests confidence intervals for both tests.
means strain/ lsd tukey cldiff ; run;
The LSD tests for this example are shown in Output 17.2.3, and they give the same results as the previous two multiple comparison tests. Again, this is not always the case.
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The ANOVA Procedure t Tests (LSD) for Nitrogen NOTE: 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 11.78867 Critical Value of t 2.06390 Least Significant Difference 4.4818 Comparisons significant at the 0.05 level are indicated by ***. Difference Strain Between 95% Confidence Comparison Means Limits 3DOK1 - 3DOK5 4.840 0.358 9.322 *** 3DOK1 - 3DOK7 8.900 4.418 13.382 *** 3DOK1 - COMPOS 10.120 5.638 14.602 *** 3DOK1 - 3DOK4 14.180 9.698 18.662 *** 3DOK1 - 3DOK13 15.560 11.078 20.042 *** 3DOK5 - 3DOK1 -4.840 -9.322 -0.358 *** 3DOK5 - 3DOK7 4.060 -0.422 8.542 3DOK5 - COMPOS 5.280 0.798 9.762 *** 3DOK5 - 3DOK4 9.340 4.858 13.822 *** 3DOK5 - 3DOK13 10.720 6.238 15.202 *** 3DOK7 - 3DOK1 -8.900 -13.382 -4.418 *** 3DOK7 - 3DOK5 -4.060 -8.542 0.422 3DOK7 - COMPOS 1.220 -3.262 5.702 3DOK7 - 3DOK4 5.280 0.798 9.762 *** 3DOK7 - 3DOK13 6.660 2.178 11.142 *** COMPOS - 3DOK1 -10.120 -14.602 -5.638 *** COMPOS - 3DOK5 -5.280 -9.762 -0.798 *** COMPOS - 3DOK7 -1.220 -5.702 3.262 vCOMPOS - 3DOK4 4.060 -0.422 8.542 COMPOS - 3DOK13 5.440 0.958 9.922 *** 3DOK4 - 3DOK1 -14.180 -18.662 -9.698 *** 3DOK4 - 3DOK5 -9.340 -13.822 -4.858 *** 3DOK4 - 3DOK7 -5.280 -9.762 -0.798 *** 3DOK4 - COMPOS -4.060 -8.542 0.422 3DOK4 - 3DOK13 1.380 -3.102 5.862 3DOK13 - 3DOK1 -15.560 -20.042 -11.078 *** 3DOK13 - 3DOK5 -10.720 -15.202 -6.238 *** 3DOK13 - 3DOK7 -6.660 -11.142 -2.178 *** 3DOK13 - COMPOS -5.440 -9.922 -0.958 *** 3DOK13 - 3DOK4 -1.380 -5.862 3.102 The ANOVA Procedure Tukey's Studentized Range (HSD) Test for Nitrogen NOTE: This test controls the Type I experimentwise error rate.
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If you only perform the LSD tests when the overall model F -test is significant, then this is called Fisher's protected LSD test. Note that the LSD tests should be used for planned comparisons.
The TUKEY tests shown in Output 17.2.4 find fewer significant differences than the other three tests. This is not unexpected, as the TUKEY test controls the Type I experimentwise error rate. For a complete discussion of multiple comparison methods , see the 'Multiple Comparisons' section on page 1806 in Chapter 32, 'The GLM Procedure.'
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t Tests (LSD) for Nitrogen NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Tukey's Studentized Range (HSD) Test for Nitrogen NOTE: This test controls the Type I experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 24 Error Mean Square 11.78867 Critical Value of Studentized Range 4.37265 Minimum Significant Difference 6.7142 Comparisons significant at the 0.05 level are indicated by ***. Difference Strain Between Simultaneous 95% Comparison Means Confidence Limits 3DOK1 - 3DOK5 4.840 -1.874 11.554 3DOK1 - 3DOK7 8.900 2.186 15.614 *** 3DOK1 - COMPOS 10.120 3.406 16.834 *** 3DOK1 - 3DOK4 14.180 7.466 20.894 *** 3DOK1 - 3DOK13 15.560 8.846 22.274 *** 3DOK5 - 3DOK1 -4.840 -11.554 1.874 3DOK5 - 3DOK7 4.060 -2.654 10.774 3DOK5 - COMPOS 5.280 -1.434 11.994 3DOK5 - 3DOK4 9.340 2.626 16.054 *** 3DOK5 - 3DOK13 10.720 4.006 17.434 *** 3DOK7 - 3DOK1 -8.900 -15.614 -2.186 *** 3DOK7 - 3DOK5 -4.060 -10.774 2.654 3DOK7 - COMPOS 1.220 -5.494 7.934 3DOK7 - 3DOK4 5.280 -1.434 11.994 3DOK7 - 3DOK13 6.660 -0.054 13.374 COMPOS - 3DOK1 -10.120 -16.834 -3.406 *** COMPOS - 3DOK5 -5.280 -11.994 1.434 COMPOS - 3DOK7 -1.220 -7.934 5.494 COMPOS - 3DOK4 4.060 -2.654 10.774 COMPOS - 3DOK13 5.440 -1.274 12.154 3DOK4 - 3DOK1 -14.180 -20.894 -7.466 *** 3DOK4 - 3DOK5 -9.340 -16.054 -2.626 *** 3DOK4 - 3DOK7 -5.280 -11.994 1.434 3DOK4 - COMPOS -4.060 -10.774 2.654 3DOK4 - 3DOK13 1.380 -5.334 8.094 3DOK13 - 3DOK1 -15.560 -22.274 -8.846 *** 3DOK13 - 3DOK5 -10.720 -17.434 -4.006 *** 3DOK13 - 3DOK7 -6.660 -13.374 0.054 3DOK13 - COMPOS -5.440 -12.154 1.274 3DOK13 - 3DOK4 -1.380 -8.094 5.334
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Example 17.3. Split Plot
In some experiments, treatments can be applied only to groups of experimental observations rather than separately to each observation. When there are two nested groupings of the observations on the basis of treatment application, this is known as a split plot design . For example, in integrated circuit fabrication it is of interest to see how different manufacturing methods affect the characteristics of individual chips. However, much of the manufacturing process is applied to a relatively large wafer of material, from which many chips are made. Additionally, a chip's position within a wafer may also affect chip performance. These two groupings of chips-by wafer and by position-within-wafer-might form the whole plots and the subplots , respectively, of a split plot design for integrated circuits.
The following statements produce an analysis for a split-plot design. The CLASS statement includes the variables Block , A , and B , where B defines subplots within BLOCK * A whole plots. The MODEL statement includes the independent effects Block , A , Block * A , B , and A * B . The TEST statement asks for an F test of the A effect, using the Block * A effect as the error term . The following statements produce Output 17.3.1 and Output 17.3.2:
title1 'Split Plot Design'; data Split; input Block1A2B3Response; datalines; 142 40.0 141 39.5 112 37.9 111 35.4 121 36.7 122 38.2 132 36.4 131 34.8 221 42.7 222 41.6 212 40.3 211 41.6 241 44.5 242 47.6 231 43.6 232 42.8 ; proc anova data=Split; class Block A B; model Response = Block A Block*A B A*B; test h=A e=Block*A; run;
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Split Plot Design The ANOVA Procedure Class Level Information Class Levels Values Block 2 1 2 A 4 1 2 3 4 B 2 1 2 Number of Observations Read 16 Number of Observations Used 16 Split Plot Design The ANOVA Procedure Dependent Variable: Response Sum of Source DF Squares Mean Square F Value Pr > F Model 11 182.0200000 16.5472727 7.85 0.0306 Error 4 8.4300000 2.1075000 Corrected Total 15 190.4500000 R-Square Coeff Var Root MSE Response Mean 0.955736 3.609007 1.451723 40.22500
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First, notice that the overall F test for the model is significant.
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Dependent Variable: Response Source DF Anova SS Mean Square F Value Pr > F Block 1 131.1025000 131.1025000 62.21 0.0014 A 3 40.1900000 13.3966667 6.36 0.0530 Block*A 3 6.9275000 2.3091667 1.10 0.4476 B 1 2.2500000 2.2500000 1.07 0.3599 A*B 3 1.5500000 0.5166667 0.25 0.8612 Tests of Hypotheses Using the Anova MS for Block*A as an Error Term Source DF Anova SS Mean Square F Value Pr > F A 3 40.19000000 13.39666667 5.80 0.0914
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The effect of Block is significant. The effect of A is not significant: look at the F test produced by the TEST statement, not at the F test produced by default. Neither the B nor A * B effects are significant. The test for Block * A is irrelevant, as this is simply the main-plot error.
Example 17.4. Latin Square Split Plot
The data for this example is taken from Smith (1951). A Latin square design is used to evaluate six different sugar beet varieties arranged in a six-row ( Rep ) by six-column ( Column ) square. The data are collected over two harvests. The variable Harvest then becomes a split plot on the original Latin square design for whole plots. The following statements produce Output 17.4.1, Output 17.4.2,andOutput 17.4.3:
title1 'Sugar Beet Varieties'; title3 'Latin Square Split-Plot Design'; data Beets; do Harvest=1 to 2; do Rep=1 to 6; do Column=1 to 6; input Variety Y @; output; end; end; end; datalines; 3 19.1 6 18.3 5 19.6 1 18.6 2 18.2 4 18.5 6 18.1 2 19.5 4 17.6 3 18.7 1 18.7 5 19.9 1 18.1 5 20.2 6 18.5 4 20.1 3 18.6 2 19.2 2 19.1 3 18.8 1 18.7 5 20.2 4 18.6 6 18.5 4 17.5 1 18.1 2 18.7 6 18.2 5 20.4 3 18.5 5 17.7 4 17.8 3 17.4 2 17.0 6 17.6 1 17.6 3 16.2 6 17.0 5 18.1 1 16.6 2 17.7 4 16.3 6 16.0 2 15.3 4 16.0 3 17.1 1 16.5 5 17.6 1 16.5 5 18.1 6 16.7 4 16.2 3 16.7 2 17.3 6 16.0 2 15.3 4 16.0 3 17.1 1 16.5 5 17.6 1 16.5 5 18.1 6 16.7 4 16.2 3 16.7 2 17.3 2 17.5 3 16.0 1 16.4 5 18.0 4 16.6 6 16.1 4 15.7 1 16.1 2 16.7 6 16.3 5 17.8 3 16.2 5 18.3 4 16.6 3 16.4 2 17.6 6 17.1 1 16.5 ; proc anova data=Beets; class Column Rep Variety Harvest; model Y=Rep Column Variety Rep*Column*Variety Harvest Harvest*Rep Harvest*Variety; test h=Rep Column Variety e=Rep*Column*Variety; test h=Harvest e=Harvest*Rep; run;
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Sugar Beet Varieties Latin Square Split-Plot Design The ANOVA Procedure Class Level Information Class Levels Values Column 6 1 2 3 4 5 6 Rep 6 1 2 3 4 5 6 Variety 6 1 2 3 4 5 6 Harvest 2 1 2 Number of Observations Read 72 Number of Observations Used 72
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Dependent Variable: Y Sum of Source DF Squares Mean Square F Value Pr > F Model 46 98.9147222 2.1503200 7.22 <.0001 Error 25 7.4484722 0.2979389 Corrected Total 71 106.3631944 R-Square Coeff Var Root MSE Y Mean 0.929971 3.085524 0.545838 17.69028 Source DF Anova SS Mean Square F Value Pr > F Rep 5 4.32069444 0.86413889 2.90 0.0337 Column 5 1.57402778 0.31480556 1.06 0.4075 Variety 5 20.61902778 4.12380556 13.84 <.0001 Column*Rep*Variety 20 3.25444444 0.16272222 0.55 0.9144 Harvest 1 60.68347222 60.68347222 203.68 <.0001 Rep*Harvest 5 7.71736111 1.54347222 5.18 0.0021 Variety*Harvest 5 0.74569444 0.14913889 0.50 0.7729
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First, note from Output 17.4.2 that the overall model is significant.
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Dependent Variable: Y Tests of Hypotheses Using the Anova MS for Column*Rep*Variety as an Error Term Source DF Anova SS Mean Square F Value Pr > F Rep 5 4.32069444 0.86413889 5.31 0.0029 Column 5 1.57402778 0.31480556 1.93 0.1333 Variety 5 20.61902778 4.12380556 25.34 <.0001 Tests of Hypotheses Using the Anova MS for Rep*Harvest as an Error Term Source DF Anova SS Mean Square F Value Pr > F Harvest 1 60.68347222 60.68347222 39.32 0.0015
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Output 17.4.3 shows that the effects for Rep and Harvest are significant, while the Column effect is not. The average Y s for the six different Variety s are significantly different. For these four tests, look at the output produced by the two TEST statements, not at the usual ANOVA procedure output. The Variety * Harvest interaction is not significant. All other effects in the default output should either be tested using the results from the TEST statements or are irrelevant as they are only error terms for portions of the model.
Example 17.5. Strip-Split Plot
In this example, four different fertilizer treatments are laid out in vertical strips , which are then split into subplots with different levels of calcium. Soil type is stripped across the split-plot experiment, and the entire experiment is then replicated three times. The dependent variable is the yield of winter barley. The data come from the notes of G. Cox and A. Rotti.
The input data are the 96 values of Y , arranged so that the calcium value ( Calcium ) changes most rapidly , then the fertilizer value ( Fertilizer ), then the Soil value, and, finally, the Rep value. Values are shown for Calcium (0 and 1); Fertilizer (0, 1, 2, 3); Soil (1, 2, 3); and Rep (1, 2, 3, 4). The following example produces Output 17.5.1, Output 17.5.2, Output 17.5.3,andOutput 17.5.4.
title1 'Strip-split Plot'; data Barley; do Rep=1 to 4; do Soil=1 to 3; /* 1=d 2=h 3=p */ do Fertilizer=0 to 3; do Calcium=0,1; input Yield @; output; end; end; end; end; datalines; 4.91 4.63 4.76 5.04 5.38 6.21 5.60 5.08 4.94 3.98 4.64 5.26 5.28 5.01 5.45 5.62 5.20 4.45 5.05 5.03 5.01 4.63 5.80 5.90 6.00 5.39 4.95 5.39 6.18 5.94 6.58 6.25 5.86 5.41 5.54 5.41 5.28 6.67 6.65 5.94 5.45 5.12 4.73 4.62 5.06 5.75 6.39 5.62 4.96 5.63 5.47 5.31 6.18 6.31 5.95 6.14 5.71 5.37 6.21 5.83 6.28 6.55 6.39 5.57 4.60 4.90 4.88 4.73 5.89 6.20 5.68 5.72 5.79 5.33 5.13 5.18 5.86 5.98 5.55 4.32 5.61 5.15 4.82 5.06 5.67 5.54 5.19 4.46 5.13 4.90 4.88 5.18 5.45 5.80 5.12 4.42 ; proc anova data=Barley; class Rep Soil Calcium Fertilizer; model Yield = Rep Fertilizer Fertilizer*Rep Calcium Calcium*Fertilizer Calcium*Rep(Fertilizer) Soil Soil*Rep Soil*Fertilizer Soil*Rep*Fertilizer Soil*Calcium Soil*Fertilizer*Calcium Soil*Calcium*Rep(Fertilizer); test h=Fertilizer e=Fertilizer*Rep; test h=Calcium calcium*fertilizer e=Calcium*Rep(Fertilizer); test h=Soil e=Soil*Rep; test h=Soil*Fertilizer e=Soil*Rep*Fertilizer; test h=Soil*Calcium Soil*Fertilizer*Calcium e=Soil*Calcium*Rep(Fertilizer); means Fertilizer Calcium Soil Calcium*Fertilizer; run;
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Strip-split Plot The ANOVA Procedure Class Level Information Class Levels Values Rep 4 1 2 3 4 Soil 3 1 2 3 Calcium 2 0 1 Fertilizer 4 0 1 2 3 Number of Observations Read 96 Number of Observations Used 96
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Dependent Variable: Yield Sum of Source DF Squares Mean Square F Value Pr > F Model 95 31.89149583 0.33569996 . . Error 0 0.00000000 . Corrected Total 95 31.89149583 R-Square Coeff Var Root MSE Yield Mean 1.000000 . . 5.427292 Source DF Anova SS Mean Square F Value Pr > F Rep 3 6.27974583 2.09324861 . . Fertilizer 3 7.22127083 2.40709028 . . Rep*Fertilizer 9 6.08211250 0.67579028 . . Calcium 1 0.27735000 0.27735000 . . Calcium*Fertilizer 3 1.96395833 0.65465278 . . Rep*Calcium(Fertili) 12 1.76705833 0.14725486 . . Soil 2 1.92658958 0.96329479 . . Rep*Soil 6 1.66761042 0.27793507 . . Soil*Fertilizer 6 0.68828542 0.11471424 . . Rep*Soil*Fertilizer 18 1.58698125 0.08816563 . . Soil*Calcium 2 0.04493125 0.02246562 . . Soil*Calcium*Fertili 6 0.18936042 0.03156007 . . Rep*Soil*Calc(Ferti) 24 2.19624167 0.09151007 . .
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Dependent Variable: Yield Tests of Hypotheses Using the Anova MS for Rep*Fertilizer as an Error Term Source DF Anova SS Mean Square F Value Pr > F Fertilizer 3 7.22127083 2.40709028 3.56 0.0604 Tests of Hypotheses Using the Anova MS for Rep*Calcium(Fertili) as an Error Term Source DF Anova SS Mean Square F Value Pr > F Calcium 1 0.27735000 0.27735000 1.88 0.1950 Calcium*Fertilizer 3 1.96395833 0.65465278 4.45 0.0255 Tests of Hypotheses Using the Anova MS for Rep*Soil as an Error Term Source DF Anova SS Mean Square F Value Pr > F Soil 2 1.92658958 0.96329479 3.47 0.0999 Tests of Hypotheses Using the Anova MS for Rep*Soil*Fertilizer as an Error Term Source DF Anova SS Mean Square F Value Pr > F Soil*Fertilizer 6 0.68828542 0.11471424 1.30 0.3063 Tests of Hypotheses Using the Anova MS for Rep*Soil*Calc(Ferti) as an Error Term Source DF Anova SS Mean Square F Value Pr > F Soil*Calcium 2 0.04493125 0.02246562 0.25 0.7843 Soil*Calcium*Fertili 6 0.18936042 0.03156007 0.34 0.9059
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Level of ------------Yield------------ Fertilizer N Mean Std Dev 0 24 5.18416667 0.48266395 1 24 5.12916667 0.38337082 2 24 5.75458333 0.53293265 3 24 5.64125000 0.63926801 Level of ------------Yield------------ Calcium N Mean Std Dev 0 48 5.48104167 0.54186141 1 48 5.37354167 0.61565219 Level of ------------Yield------------ Soil N Mean Std Dev 1 32 5.54312500 0.55806369 2 32 5.51093750 0.62176315 3 32 5.22781250 0.51825224 Level of Level of ------------Yield------------ Calcium Fertilizer N Mean Std Dev 0 0 12 5.34666667 0.45029956 0 1 12 5.08833333 0.44986530 0 2 12 5.62666667 0.44707806 0 3 12 5.86250000 0.52886027 1 0 12 5.02166667 0.47615569 1 1 12 5.17000000 0.31826233 1 2 12 5.88250000 0.59856077 1 3 12 5.42000000 0.68409197
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As the model is completely specified by the MODEL statement, the entire top portion of output (Output 17.5.2) should be ignored. Look at the following output produced by the various TEST statements.
The only significant effect is the Calcium * Fertilizer interaction.
The final portion of output shows the results of the MEANS statement. This portion shows means for various effects and combinations of effects, as requested. Because no multiple comparison procedures are requested, none are performed. You can examine the Calcium * Fertilizer means to understand the interaction better.
In this example, you could reduce memory requirements by omitting the Soil * Calcium * Rep ( Fertilizer ) effect from the model in the MODEL statement. This effect then becomes the ERROR effect, and you can omit the last TEST statement (in the code shown earlier). The test for the Soil * Calcium effect is then given in the Analysis of Variance table in the top portion of output. However, for all other tests, you should look at the results from the TEST statement. In large models, this method may lead to significant reductions in memory requirements.
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