proc print data=raw; /* print the data */
run;
Tensile Strength Experiment from Table 4-5 pg. 145 1
OBS STRENGTH WGHT_PCT ORDER
1 7 15 15
2 7 15 19
3 15 15 25
4 11 15 12
5 9 15 6
6 12 20 8
7 17 20 14
8 12 20 1
9 18 20 11
10 18 20 3
11 14 25 18
12 18 25 13
13 18 25 20
14 19 25 7
15 19 25 9
16 19 30 22
17 25 30 5
18 22 30 2
19 19 30 24
20 23 30 10
21 7 35 17
22 10 35 21
23 11 35 4
24 15 35 16
25 11 35 23
proc npar1way data=raw anova wilcoxon; /* Kruskal-Wallis Test for ANOVA */
class wght_pct;
var strength;
run;
Tensile Strength Experiment from Table 4-5 pg. 145 2
N P A R 1 W A Y P R O C E D U R E
Analysis of Variance for Variable STRENGTH
Classified by Variable WGHT_PCT
WGHT_PCT N Mean Among MS Within MS
118.940000 8.06000000
15 5 9.8000000
20 5 15.4000000 F Value Prob > F
25 5 17.6000000 14.757 0.0001
30 5 21.6000000
35 5 10.8000000
note: The above output pertains to a classical ANOVA assuming normality, equal
variances, etc. It is requested by suppling the "anova" option in the proc
npar1way procedure.
Tensile Strength Experiment from Table 4-5 pg. 145 3
N P A R 1 W A Y P R O C E D U R E
Wilcoxon Scores (Rank Sums) for Variable STRENGTH
Classified by Variable WGHT_PCT
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
15 5 27.500000 65.0 14.6344343 5.5000000
20 5 66.000000 65.0 14.6344343 13.2000000
25 5 85.000000 65.0 14.6344343 17.0000000
30 5 113.000000 65.0 14.6344343 22.6000000
35 5 33.500000 65.0 14.6344343 6.7000000
Average Scores Were Used for Ties
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 19.064 DF = 4 Prob > CHISQ = 0.0008
note: The above output pertains to the Kruskal-Wallis ANOVA. In this output,
CHISQ = H (pg. 144) except for some roundoff error and "Prob > CHISQ"
represents the corresponding p-value for the test. Since p-value=0.0008
we reject H0: and conclude that the treatments are different.
note: To determine which treatments are different we can perform the
Kruskal-Wallis Test on each of the 10 pairwise comparisons. This is analagous
to using the Protected LSD method.
* Pairwise comparisons using the Kruskal-Wallis Test;
proc npar1way data=raw wilcoxon;
class wght_pct;
var strength;
where wght_pct=15 or wght_pct=20;
run;
proc npar1way data=raw wilcoxon;
class wght_pct;
var strength;
where wght_pct=15 or wght_pct=25;
run;
.
.
.
proc npar1way data=raw wilcoxon;
class wght_pct;
var strength;
where wght_pct=30 or wght_pct=35;
run;
Tensile Strength Experiment from Table 4-5 pg. 145 4
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
15 5 17.0 27.5000000 4.74341649 3.40000000
20 5 38.0 27.5000000 4.74341649 7.60000000
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 4.9000 DF = 1 Prob > CHISQ = 0.0269
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
15 5 16.0 27.5000000 4.74341649 3.20000000
25 5 39.0 27.5000000 4.74341649 7.80000000
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 5.8778 DF = 1 Prob > CHISQ = 0.0153
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
15 5 15.0 27.5000000 4.75803414 3.0
30 5 40.0 27.5000000 4.75803414 8.0
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 6.9018 DF = 1 Prob > CHISQ = 0.0086
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
15 5 24.5000000 27.5000000 4.65474668 4.90000000
35 5 30.5000000 27.5000000 4.65474668 6.10000000
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 0.41538 DF = 1 Prob > CHISQ = 0.5192
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
20 5 20.0 27.5000000 4.60977223 4.0
25 5 35.0 27.5000000 4.60977223 7.0
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 2.6471 DF = 1 Prob > CHISQ = 0.1037
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
20 5 15.0 27.5000000 4.74341649 3.0
30 5 40.0 27.5000000 4.74341649 8.0
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 6.9444 DF = 1 Prob > CHISQ = 0.0084
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
20 5 38.0 27.5000000 4.74341649 7.60000000
35 5 17.0 27.5000000 4.74341649 3.40000000
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 4.9000 DF = 1 Prob > CHISQ = 0.0269
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
25 5 17.0 27.5000000 4.62481231 3.40000000
30 5 38.0 27.5000000 4.62481231 7.60000000
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 5.1545 DF = 1 Prob > CHISQ = 0.0232
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
25 5 39.0 27.5000000 4.74341649 7.80000000
35 5 16.0 27.5000000 4.74341649 3.20000000
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 5.8778 DF = 1 Prob > CHISQ = 0.0153
Sum of Expected Std Dev Mean
WGHT_PCT N Scores Under H0 Under H0 Score
30 5 40.0 27.5000000 4.75803414 8.0
35 5 15.0 27.5000000 4.75803414 3.0
Kruskal-Wallis Test (Chi-Square Approximation)
CHISQ = 6.9018 DF = 1 Prob > CHISQ = 0.0086
note: After looking at all 10 p-values we can summarize to findings with the
following diagram.
15 35 20 25 30
--------- ---------
note: This is the same conclusion that was reached using the traditional
analysis. Since the classical analysis and the Kruskal-Wallis analysis
produced the same results it is customary to report only the traditional
analysis.