252mvarex 4/06/05 Minitab Examples of Tests for Equality of Variances
————— 3/31/2004 8:27:58 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > Retrieve "C:\Documents and Settings\RBOVE.WCUPANET\My Documents\Drive
D\MINITAB\bartlett1.MTW".
Retrieving worksheet from file: C:\Documents and Settings\RBOVE.WCUPANET\My
Documents\Drive D\MINITAB\bartlett1.MTW
# Worksheet was saved on Wed Mar 31 2004
Results for: bartlett1.MTW
MTB > describe c1-c4
Descriptive Statistics: s1, s2, s3, s4
Variable N Mean Median TrMean StDev SE Mean s1 31 -0.962 -1.145 -1.044 2.339 0.420 s2 15 0.052 -0.088 -0.051 2.154 0.556 s3 20 1.998 1.984 2.131 3.445 0.770 s4 42 0.066 0.008 -0.041 3.360 0.518
Variable Minimum Maximum Q1 Q3 s1 -5.408 5.268 -2.102 0.316 s2 -2.468 3.920 -2.093 1.394 s3 -5.201 6.797 -1.007 5.165 s4 -6.330 8.186 -2.129 1.587
1

252mvarex1 3/31/04
MTB > print c1-c4
Data Display This is the reconstructed data for
Row s1 s2 s3 s4
1 -1.14481 -2.35661 -2.37257 -0.68164
2 -3.75468 1.28112 2.48574 -4.47822
3 0.31609 -2.38417 4.34545 0.12907
4 -3.32796 3.28997 6.79694 1.08077
5 -0.90303 -2.09316 4.56030 2.23035
6 -2.31875 0.56144 5.49388 1.14518
7 3.48176 -1.93507 6.06126 4.69290
8 -0.41916 0.45391 1.42648 1.71574
9 -4.98559 -2.46837 -2.03075 -1.58772
10 -1.59144 -0.08803 1.70938 -3.74374
11 -4.98725 -0.20482 5.23415 -4.56758
12 -1.40986 1.39435 -2.55932 -3.93202
13 0.14245 2.73224 2.25960 1.27105
14 0.52388 -1.31843 -0.51042 -0.66111
15 -1.39077 3.91964 1.15361 -4.50816
16 -1.54722 1.29051 2.49753
17 5.26812 6.02585 -1.97453
18 1.73404 -5.20091 -0.71953
19 -5.40784 4.95880 -4.27275
20 1.14133 -1.17221 2.83155
21 -1.17195 8.18588
22 -3.26684 1.07365
23 -2.01815 0.27719
24 -0.54529 -0.52695
25 -2.10166 -0.40461
26 0.19862 -6.32974
27 -0.51006 7.88240
28 -1.96248 -0.11337
29 0.55073 -5.41302
30 1.48555 -0.74814
31 0.10133 4.94961
32 -3.48670
33 0.38791
34 -0.32043
35 -2.59297
36 -0.31446
37 1.28942
38 6.13556
39 1.54460
40 0.33019
41 1.38072
42 3.10783
MTB > Stack c1-c4 c5; Data is put into a single column in c5 with labels in c6.
SUBC> UseNames.
SUBC> Subscripts c6;
2

252mvarex1 3/31/04
Note: The new version of Minitab has a change here. The subcommand is not needed unless you change the confidence level from 95% and the instruction below would simply read
Vartest c5 c6
But it really is unnecessary to stack the data and you could use vartest c2-c5; unstacked.
MTB > %Vartest c5 c6;
SUBC> Confidence 95.0.
Executing from file: W:\wminitab13\MACROS\Vartest.MAC
Macro is running ... please wait
Test for Equal Variances
Response stack
Factors labels
ConfLvl 95.0000
Bonferroni confidence intervals for standard deviations
Lower Sigma Upper N Factor Levels
1.76293 2.33860 3.40578 31 s1
1.45640 2.15411 3.90877 15 s2
2.44170 3.44504 5.64186 20 s3
2.62678 3.35969 4.59923 42 s4
Bartlett's Test (normal distribution)
Test Statistic: 7.438
P-Value : 0.059 Since the p-value is over 5%, do not reject the null hypothesis of equal variances at the
5% level.
Levene's Test (any continuous distribution)
Test Statistic: 2.083
P-Value : 0.107 Same Conclusion
Test for Equal Variances: stack vs labels
MTB > Retrieve "C:\Documents and Settings\RBOVE.WCUPANET\My Documents\Drive
D\MINITAB\bartlett2.MTW".
Retrieving worksheet from file: C:\Documents and Settings\RBOVE.WCUPANET\My
Documents\Drive D\MINITAB\bartlett2.MTW
# Worksheet was saved on Wed Mar 31 2004
Results for: bartlett2.MTW
MTB > describe c1-c4
Descriptive Statistics: s1, s2, s3, s4
Variable N Mean Median TrMean StDev SE Mean s1 3 -0.24 0.26 -0.24 2.52 1.45 s2 3 0.730 1.349 0.730 1.153 0.666 s3 3 -0.57 0.32 -0.57 2.08 1.20 s4 4 1.93 1.67 1.93 2.08 1.04
3

Variable Minimum Maximum Q1 Q3 s1 -2.97 1.99 -2.97 1.99 s2 -0.600 1.442 -0.600 1.442 s3 -2.95 0.91 -2.95 0.91 s4 -0.13 4.50 0.07 4.04
252mvarex1 3/31/04
MTB > print c1-c4
Data Display This is the reconstructed data from another of Kanji’s examples.
Row s1 s2 s3 s4
1 1.98909 1.44173 0.32108 -0.13422
2 0.25843 1.34921 0.90819 2.67526
3 -2.96760 -0.59979 -2.95139 0.66330
4 4.49583
MTB > %Vartest c5 c6;
SUBC> Confidence 95.0.
Executing from file: W:\wminitab13\MACROS\Vartest.MAC
Macro is running ... please wait
Test for Equal Variances
Response stack
Factors labels
ConfLvl 95.0000
Bonferroni confidence intervals for standard deviations
Lower Sigma Upper N Factor Levels
1.11667 2.51566 31.7710 3 s1
0.51176 1.15289 14.5602 3 s2
0.92314 2.07967 26.2648 3 s3
1.02579 2.08202 12.4856 4 s4
Bartlett's Test (normal distribution)
Test Statistic: 0.947
P-Value : 0.814 Since the p-value is over 5%, do not reject the null hypothesis of equal variances at the
5% level.
Levene's Test (any continuous distribution)
Test Statistic: 0.379 See next example for computation of the test statistic. Note that conclusion is the same as for Bartlett’s test.
P-Value : 0.771
4

252mvarex1 3/31/04
Test for Equal Variances: stack vs labels
Graph for example 1
Test for Equal Variances for stack
95% Confidence Intervals for Sigmas Factor Levels s1
Bartlett's Test
Test Statistic: 7.438
P-Value : 0.059
s2 s3
Levene's Test
Test Statistic: 2.083
P-Value : 0.107
s4
1 2 3 4 5 6
Graph for example 2.
Test for Equal Variances for stack
95% Confidence Intervals for Sigmas Factor Levels s1
Bartlett's Test
Test Statistic: 0.947
P-Value : 0.814
s2 s3
Levene's Test
Test Statistic: 0.379
P-Value : 0.771
s4
0 10 20 30
5

252mvarex1 3/31/04
————— 3/31/2004 8:44:40 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > Retrieve "C:\Documents and Settings\RBOVE.WCUPANET\My Documents\Drive
D\MINITAB\bartlett2.MTW".
Retrieving worksheet from file: C:\Documents and Settings\RBOVE.WCUPANET\My
Documents\Drive D\MINITAB\bartlett2.MTW
# Worksheet was saved on Wed Mar 31 2004
Results for: bartlett3.MTW
MTB > Save "C:\Documents and Settings\RBOVE.WCUPANET\My Documents\Drive
D\MINITAB\bartlett3.MTW";
SUBC> Replace.
Saving file as: C:\Documents and Settings\RBOVE.WCUPANET\My Documents\Drive
D\MINITAB\bartlett3.MTW
* NOTE * Existing file replaced.
MTB > describe c1-c4
Descriptive Statistics: s1, s2, s3, s4 This is original Bartlett 2 statistics
Variable N Mean Median TrMean StDev SE Mean s1 3 -0.24 0.26 -0.24 2.52 1.45 s2 3 0.730 1.349 0.730 1.153 0.666 s3 3 -0.57 0.32 -0.57 2.08 1.20 s4 4 1.93 1.67 1.93 2.08 1.04
Variable Minimum Maximum Q1 Q3 s1 -2.97 1.99 -2.97 1.99 s2 -0.600 1.442 -0.600 1.442 s3 -2.95 0.91 -2.95 0.91 s4 -0.13 4.50 0.07 4.04
MTB > let c1=c1-0.26
MTB > let c2=c2-1.349
MTB > let c3=c3-0.32
MTB > let c4=c4-1.67
MTB > describe c1-c4
Subtract medians from original data.
Descriptive Statistics: s1, s2, s3, s4
Variable N Mean Median TrMean StDev SE Mean s1 3 -0.50 -0.00 -0.50 2.52 1.45 s2 3 -0.619 0.000 -0.619 1.153 0.666 s3 3 -0.89 0.00 -0.89 2.08 1.20 s4 4 0.26 -0.00 0.26 2.08 1.04
Variable Minimum Maximum Q1 Q3 s1 -3.23 1.73 -3.23 1.73 s2 -1.949 0.093 -1.949 0.093 s3 -3.27 0.59 -3.27 0.59 s4 -1.80 2.83 -1.60 2.37
6

252mvarex1 3/31/04
MTB > let c1=abso(c1)
MTB > let c2=abso(c2)
MTB > let c3=abso(c3)
MTB > let c4=abso(c4)
MTB > print c1-c4
Data Display
Take absolute value of numbers after the median was subtracted.
This is what the Example 2 data looks like after medians are subtracted and absolute values taken.
Row s1 s2 s3 s4
1 1.72909 0.09273 0.00108 1.80422
2 0.00157 0.00021 0.58819 1.00526
3 3.22760 1.94879 3.27139 1.00670
4 2.82583
MTB > AOVOneway c1 c2 c3 c4. Do a one-way ANOVA on the revised data.
One-way ANOVA: s1, s2, s3, s4
Analysis of Variance
Source DF SS MS F P
Factor 3 2.02 0.67 0.38 0.770 Note that except for
Error 9 15.94 1.77
Total 12 17.96 rounding error this F and p are the same as in the
Previous example.
Individual 95% CIs For Mean
Based on Pooled StDev
Level N Mean StDev --------+---------+---------+-------- s1 3 1.653 1.614 (-----------*-----------) s2 3 0.681 1.099 (-----------*----------) s3 3 1.287 1.744 (-----------*----------) s4 4 1.661 0.863 (---------*---------)
--------+---------+---------+--------
Pooled StDev = 1.331 0.0 1.5 3.0
MTB >
7