STAT 330 Lecture 24
Reading for Today's Lecture: 10.2, 10.3.
Goals of Today's Lecture:
Today's notes
Summary of Confidence Intervals
One at a time confidence intervals for 
:
where 
, 
is sample size i and 
.
Tukey's Intervals
FACT: If
then
which is
Now we look at the SAS analysis of the co-agulation data. I want you to see:
which is
which becomes
or just
which matches the SAS output.
SAS example: ANOVA for a one way layout
The data consist of blood coagulation times for 24 animals fed one of 4 different diets. I use proc anova to test the hypothesis of no differences between mean coagulation times and to get simulataneous confidence intervals for all possible differences between mean times.
The data
A 62 A 60 A 63 A 59 B 63 B 67 B 71 B 64 B 65 B 66 C 68 C 66 C 71 C 67 C 68 C 68 D 56 D 62 D 60 D 61 D 63 D 64 D 63 D 59
I ran the following SAS code:
options pagesize=60 linesize=70; data coagul; infile 'coagul.dat'; input diet $ time ; label time='Coagulation Time'; proc sort data=coagul; by diet; proc anova data=coagul; class diet; model time = diet; means diet / bon regwq tukey cldiff; means diet / bon regwq tukey ; run;
The two lines labelled means request confidence intervals for differences between the 4 population means. The line class diet is required saying that the variable class defines categories for the analysis of variance. The model line declares that I am interested only in the influence of diet on time; this line is also required.
The output from proc anova is
The SAS System 8
15:40 Thursday, October 26, 1995
Analysis of Variance Procedure
Class Level Information
Class Levels Values
DIET 4 A B C D
Number of observations in data set = 24
The SAS System 9
15:40 Thursday, October 26, 1995
Analysis of Variance Procedure
Dependent Variable: TIME Coagulation Time
Source DF Sum of Squares F Value Pr > F
Model 3 228.00000000 13.57 0.0001
Error 20 112.00000000
Corrected Total 23 340.00000000
R-Square C.V. TIME Mean
0.670588 3.697550 64.0000000
Source DF Anova SS F Value Pr > F
DIET 3 228.00000000 13.57 0.0001
The SAS System 10
15:40 Thursday, October 26, 1995
Analysis of Variance Procedure
Tukey's Studentized Range (HSD) Test for variable: TIME
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 20 MSE= 5.6
Critical Value of Studentized Range= 3.958
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
DIET Confidence Between Confidence
Comparison Limit Means Limit
C - B -1.824 2.000 5.824
C - A 2.725 7.000 11.275 ***
C - D 3.423 7.000 10.577 ***
B - C -5.824 -2.000 1.824
B - A 0.725 5.000 9.275 ***
B - D 1.423 5.000 8.577 ***
A - C -11.275 -7.000 -2.725 ***
A - B -9.275 -5.000 -0.725 ***
A - D -4.056 0.000 4.056
D - C -10.577 -7.000 -3.423 ***
D - B -8.577 -5.000 -1.423 ***
D - A -4.056 0.000 4.056
The SAS System 11
15:40 Thursday, October 26, 1995
Analysis of Variance Procedure
Bonferroni (Dunn) T tests for variable: TIME
NOTE: This test controls the type I experimentwise error rate
but generally has a higher type II error rate than Tukey's
for all pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 20 MSE= 5.6
Critical Value of T= 2.92712
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
DIET Confidence Between Confidence
Comparison Limit Means Limit
C - B -1.999 2.000 5.999
C - A 2.529 7.000 11.471 ***
C - D 3.259 7.000 10.741 ***
B - C -5.999 -2.000 1.999
B - A 0.529 5.000 9.471 ***
B - D 1.259 5.000 8.741 ***
A - C -11.471 -7.000 -2.529 ***
A - B -9.471 -5.000 -0.529 ***
A - D -4.242 0.000 4.242
D - C -10.741 -7.000 -3.259 ***
D - B -8.741 -5.000 -1.259 ***
D - A -4.242 0.000 4.242
The SAS System 12
15:40 Thursday, October 26, 1995
Analysis of Variance Procedure
Ryan-Einot-Gabriel-Welsch Multiple Range Test for variable: TIME
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 df= 20 MSE= 5.6
WARNING: Cell sizes are not equal.
Harmonic Mean of cell sizes= 5.647059
Number of Means 2 3 4
Critical Range 3.4039958 3.5629957 3.9417679
Means with the same letter are not significantly different.
REGWQ Grouping Mean N DIET
A 68.000 6 C
A
A 66.000 6 B
B 61.000 4 A
B
B 61.000 8 D
The SAS System 13
15:40 Thursday, October 26, 1995
Analysis of Variance Procedure
Tukey's Studentized Range (HSD) Test for variable: TIME
NOTE: This test controls the type I experimentwise error rate,
but generally has a higher type II error rate than REGWQ.
Alpha= 0.05 df= 20 MSE= 5.6
Critical Value of Studentized Range= 3.958
Minimum Significant Difference= 3.9418
WARNING: Cell sizes are not equal.
Harmonic Mean of cell sizes= 5.647059
Means with the same letter are not significantly different.
Tukey Grouping Mean N DIET
A 68.000 6 C
A
A 66.000 6 B
B 61.000 4 A
B
B 61.000 8 D
The SAS System 14
15:40 Thursday, October 26, 1995
Analysis of Variance Procedure
Bonferroni (Dunn) T tests for variable: TIME
NOTE: This test controls the type I experimentwise error rate,
but generally has a higher type II error rate than REGWQ.
Alpha= 0.05 df= 20 MSE= 5.6
Critical Value of T= 2.93
Minimum Significant Difference= 4.1223
WARNING: Cell sizes are not equal.
Harmonic Mean of cell sizes= 5.647059
Means with the same letter are not significantly different.
Bon Grouping Mean N DIET
A 68.000 6 C
A
A 66.000 6 B
B 61.000 4 A
B
B 61.000 8 D
The ANOVA table shows overwhelmingly significant evidence against the hypothesis of equal means. The multiple comparison procedures say that diets B and C are indistinguishable, as are A and D but that these two groups definitely differ from each other. The procedure called TUKEY is the only one under consideration in this course. The keyword CLDIFF on the first means line asked for actual confidence intervals while the second line produced the grouping just discussed.