Copyright (c) Bani K. mallick1 STAT 651 Lecture #14.

Copyright (c) Bani K. mallick 1

STAT 651

Lecture #14


Topics in Lecture #14 The Kruskal Wallis Test

Review of ANOVA

Theory for the ANOVA Table


Book Sections Covered in Lecture #14

Chapter 8.6


Relevant SPSS Tutorials Kruskal-Wallis


What you need for the ANOVA Table

Error sum of squares (SSE)

Corrected Total sum of squares (CTSS)

Between sum of squares: SSB = CTSS – SSE

Number of populations t: df1 = t – 1

Total number of observations nT: df2 = nT –t

Critical value from Table 8: F for and df1

and df2


The ANOVA Table: General Linear Model

Sum of Squares

Degrees of freedom

Mean Square

F for equal means

Variable Name

SSB df1 = t – 1 SSB/(t-1) SSB/(t-1)

------------

SSE/(nT –t)

Error SSE df2 = nT –t

SSE/(nT –t)

Corrected Total

CTSS nT – 1


What you need for the ANOVA Table

df1 = t – 1

df2 = nT –t

Critical value from Table 8: F for and df1 and df2

You compute the F statistic, and reject the hypothesis that the population means are equal at Type I error probability if the F statistic exceeds this tabulated value


Illustration: What you need for the ANOVA Table

df1 = t – 1 = 4

df2 = nT –t = 25

= 0.05

Critical value from Table 8: F for and df1 and df2 = 2.76.

You reject the hypothesis that the population means are equal at Type I error probability if the F statistic exceeds 2.76


Nonparametric Methods

As we found for 1-sample comparisons and 2-sample comparisons, there are also nonparametric methods for ANOVA

These are often called Kruskal-Wallis methods

The idea is the same as in the 2-sample problem

Remember it?



Replace each observation by its rank in the pooled data

Do the usual ANOVA F-test


Female Concho Water Snakes, Ages 2-4, Tail Length

Normal Q-Q Plot of Residual for TAILL

Observed Value

3020100-10-20-30-40

Exp

ect

ed

No

rma

l Va

lue

30

20

10

0

-10

-20

-30

We need a method that allows for non-normal data!


Concho Water Snake Data for Females

Ranks

11 8.82

17 19.29

9 30.89

37

Age2.00

3.00

4.00

Total

Tail LengthN Mean Rank

Kruskal-Wallis Test


Concho Water Snake Data for Females

Test Statisticsa,b

20.637

2

.000

Chi-Square

df

Asymp. Sig.

Tail Length

Kruskal Wallis Testa.

Grouping Variable: Ageb.



Once you have decided that the populations are different in their means, there is no version of a LSD

You simply have to do each comparison in turn

This is a bit of a pain in SPSS, because you physically must do each 2-population comparison, defining the groups as you go



Illustrate Kruskal Wallis in SPSS and then remind ourselves about how to do the 2-population comparisons


Lipids Research Study

Four populations:

Healthy, non-smokers

Healthy, smokers

CHD, non-smokers

CHD, Smokers

Compared on basis of cholesterol levels


Lipid Research Study

Between-Subjects Factors

Healthy,NonSmoker

143

Healthy,Smoker

44

HeartDisease,NonSmoker

113

HeartDisease,Smoker

37

.00

1.00

2.00

3.00

CHD/SmokingCategory

Value Label N



3711344143N =


CHD/Smoking Category

CHD, Smoke

CHD, NonSmoker

Healthy, Smoker

Healthy, NonSmoker

Ch

ole

ste

rol

400

300

200

100

0

255305

85

173


Lecture 14 Review: Residuals

Testing for Normality in ANOVA

I use the General Linear Model to define these residuals

Form the residuals, which are simply the differences of the data with their group mean

Then do a q-q plot

Useful if you have many groups with a small number of observations per group


Lipid Research StudyNormal Q-Q Plot of Cholesterol Residuals

Observed Value

400300200100

Exp

ect

ed

No

rma

l Va

lue

400

300

200

100


Lipid Research Study: Note Table Entries

Tests of Between-Subjects Effects

Dependent Variable: Cholesterol

13450.180a 3 4483.393 2.738 .043

12637872.4 1 12637872.42 7716.573 .000

13450.180 3 4483.393 2.738 .043

545373.126 333 1637.757

17709567.0 337

558823.306 336

SourceCorrected Model

Intercept

NCHDSMOK

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .024 (Adjusted R Squared = .015)a.



Kruskal-Wallis Test

Test Statisticsa,b

7.360

3

.061

Chi-Square

df

Asymp. Sig.

Cholesterol


Grouping Variable: CHD/Smoking Categoryb.



The p-values are 0.043 for ANOVA, 0.061 for Kruskal Wallis.

Weakish evidence of population differences

The Q-Q plot was pretty normal, so I would probably go with the smaller p-value and publish, but with some warnings.

BTW, what hypothesis were we testing??



The p-values are 0.043 for ANOVA, 0.061 for Kruskal Wallis.

Weakish evidence of population differences

BTW, what hypothesis were we testing??

That the population means for the 4 populations were all simultaneously equal.


Lipids Research Study: Fisher LSD

Multiple Comparisons


LSD

-.9983 6.9767 .886 -14.7222 12.7257

-9.0113 5.0937 .078 -19.0313 1.0087

-19.1168* 7.4644 .011 -33.8000 -4.4336

.9983 6.9767 .886 -12.7257 14.7222

-8.0131 7.1913 .266 -22.1592 6.1331

-18.1186* 9.0269 .046 -35.8755 -.3616

9.0113 5.0937 .078 -1.0087 19.0313

8.0131 7.1913 .266 -6.1331 22.1592

-10.1055 7.6653 .188 -25.1840 4.9731

19.1168* 7.4644 .011 4.4336 33.8000

18.1186* 9.0269 .046 .3616 35.8755

10.1055 7.6653 .188 -4.9731 25.1840

(J) CHD/SmokingCategory

Healthy, Smoker

Heart Disease,NonSmoker

Heart Disease, Smoker

Healthy, NonSmoker



Healthy, NonSmoker

Healthy, Smoker


Healthy, NonSmoker

Healthy, Smoker


(I) CHD/SmokingCategoryHealthy, NonSmoker

Healthy, Smoker



MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

Based on observed means.

The mean difference is significant at the .05 level.*.

Healthy Nonsmokers differ from Heart Disease Smokers

Healthy and CHD patients differ among smokers



Fisher’s LSD suggested that Healthy, Non-smokers has significantly less Healthy, Smokers (p = 0.011)

P-value for Wilcoxon rank sum test is 0.024

Ranks

143 86.04 12304.00

37 107.73 3986.00

180

CHD/Smoking CategoryHealthy, NonSmoker


Total

CholesterolN Mean Rank Sum of Ranks

Test Statisticsa

2008.000

12304.000

-2.257

.024

Mann-Whitney U

Wilcoxon W

Z

Asymp. Sig. (2-tailed)

Cholesterol

Grouping Variable: CHD/Smoking Categorya.


Variances

The ANOVA method is remarkably robust

Although it assumes that the populations have equal population variances, as long as the sample sizes are reasonably close, it is not much affected by unequal variances

Of course, the sample variances will be different: why?


Variances

It is still possible to compare variances

Realistically, if you are intrinsically interested in whether populations have the same variances or not, you should consult a statistician

However, there is a version of the Levene test that can be computed from SPSS. It uses the same algorithm as in the 2-population case


Lipid Research Study: Variances?

3711344143N =


CHD/Smoking Category

CHD, Smoke

CHD, NonSmoker

Healthy, Smoker

Healthy, NonSmoker

Ch

ole

ste

rol

400

300

200

100

0

255305

85

173


Variances

The IQR are 57, 44, 56, 74

I don’t see a massive inequality of variability

The medians are 220, 225, 225, 232


Lipid Research Study: Variances?

Levene's Test of Equality of Error Variancesa


1.657 3 333 .176F df1 df2 Sig.

Tests the null hypothesis that the error variance ofthe dependent variable is equal across groups.

Design: Intercept+NCHDSMOKa.

You get this under “Options” in the ANOVA Run


Concho Water Snakes

91711N =

Female Concho Water Snakes

Age

4-year olds3-year olds2-year olds

Ta

il L

en

gth

220

200

180

160

140

120

35

27


Concho Water Snakes

Test Statisticsa,b

20.637

2

.000

Chi-Square

df

Asymp. Sig.

Tail Length


Grouping Variable: Ageb.


Concho Water Snakes

Levene's Test of Equality of Error Variancesa

Dependent Variable: Tail Length

2.903 2 34 .069F df1 df2 Sig.

Tests the null hypothesis that the error variance ofthe dependent variable is equal across groups.

Design: Intercept+AGEa.

Copyright (c) Bani K. mallick1 STAT 651 Lecture #14.

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