1 hypothesis testing: about more than two (k) related populations
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HYPOTHESIS TESTING:HYPOTHESIS TESTING:ABOUT ABOUT MORE THAN MORE THAN TWOTWO (K) (K)
RELATEDRELATED POPULATIONS POPULATIONS
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Often, one wants to administer the same test to the same
subjects repeatedly over a period of time or under different
circumstances.
In essence, one is interested in examining differences within
each subject, for example, subjects' improvement over time.
Such designs are referred to as within-subjects designs or
repeated measures designs.
Repeated Measures ANOVA
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For example, imagine that one wants to monitor the
improvement of students' algebra skills over three months
of instruction. A standardized algebra test is administered
after one month (level 1 of the repeated measures factor),
and comparable tests are administered after two months
(level 2 of the repeated measures factor) and after three
months (level 3 of the repeated measures factor). Thus,
the repeated measures factor (Time) has three levels.
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Mean
n
xitxijxi1i
x1tx1jx111
tj1 MeanTime
Subject
1.x jx. tx. ..x
1nx
njx ntx
.1x
.ix
.nx
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Repeated measures can occure in different ways:
• Repeated measures can be taken at different time points in a single group.
•Repeated measures can be taken at different time points in several groups.
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subject time1 time2 time3 time41 30,9 30,7 30,9 30,92 31,9 31,6 31,6 31,73 31,3 31,1 31,0 31,34 32,1 31,0 31,7 31,35 30,9 31,2 30,5 30,86 31,3 31,7 31,4 31,27 31,3 31,8 31,8 31,78 32,1 33,0 31,7 31,59 30,3 30,9 30,8 30,610 32,2 32,1 32,2 32,4
Example Temperatures of the forehead (in degrees Celsius) measured at 30-minute intervals in a single group of subjects are given in the table.
H0:There is no difference between time periods
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Sources of variation SS df MS F Sig.Times 0,178 3 0,059 0,57 0,64Subjects 10,086 9 1,121Error 2,812 27 0,104
There is no difference between time periods.
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Example Temperatures of the forehead (in degrees Celsius) measured at 30-minute intervals in two groups of subjects are given in the table.
Subject Group time1 time2 time3 time41 1 30,90 30,70 30,90 30,902 1 31,90 31,60 31,60 31,703 1 31,30 31,10 31,00 31,304 1 32,10 31,00 31,70 31,305 1 30,90 31,20 30,50 30,806 1 31,30 31,70 31,40 31,207 1 31,30 31,80 31,80 31,708 1 32,10 33,00 31,70 31,509 1 30,30 30,90 30,80 30,60
10 1 32,20 32,10 32,20 32,4011 2 31,50 30,60 30,80 31,0012 2 31,20 31,20 31,10 31,3013 2 31,30 31,30 31,50 31,4014 2 30,40 30,80 30,40 30,2015 2 30,70 30,90 30,90 30,9016 2 29,80 30,80 30,90 30,8017 2 31,40 32,00 31,70 31,6018 2 30,90 32,40 31,80 31,9019 2 31,10 31,30 31,20 31,2020 2 31,30 31,50 31,60 31,70
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Source of variation df SS MS F Sig.Groups 1 1,275 1,275 1,336 0,263Times 3 0,41 0,137 1,412 0,249Interaction 3 0,336 0,112 1,158 0,334Subjects 18 17,176 0,954Residual 54 5,231 0,097
Three pairs of hypotheses can be tested:
1. Hypothesis on groups
2. Hypothesis on time points
3. Hypothesis on interaction
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Time
4321
Est
ima
ted
Ma
rgin
al M
ea
ns
31,6
31,5
31,4
31,3
31,2
31,1
31,0
30,9
Group
1,00
2,00
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Cochran’s Q Test
Cochran’s Q Test extends McNemar Test to examine
change in a dichotomous variable (0-1) across more
than two observations. It is a particularly appropriate test
when subjects are used as their own controls and
dichotomous outcome variable is measured across
multiple time periods or under several types of
conditions.
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N
ii
N
ii
k
jj
k
jj
LLk
GGkk
Q
1
2
1
2
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2)1(
where
k: # of conditions or time periods
N: # of subjects
Gj: The total # of 1s in the jth column
Li: The total # of 1s in the ith row
Post hoc tests are necessary to determine where the differences lie.
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ExampleThe children in the anxiety reduction intervention groups were evaluated for the presence of certain symptom before and after the intervention.
Presence of symptom
Subject Preint. Postint.
1 1 0
2 1 0
3 1 1
4 0 0
5 1 0
6 1 0
7 1 1
8 1 0
9 1 1
10 1 0
Is the clinical intervention effective in reducing the symptom?
Since the pretest-posttest measure is dichotomous and the data are paired, the use of McNemar test is appropriate to answer this question.
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Suppose one month after the end of the intervention program a
third measurement is taken from all children. Is the proportion of
children who presents the symptom (yes,no) same across all three
data collection periods?
Since there are now three points of data collection, the McNemar
test can no longer be used. Cochran’s Q test, however, woud be
appropriate.
H0: the proportion of “yes” responses with regard to the presence
of the symptom same across all three time periods for those
children.
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Presence of symptom
Subj Preint. Postint One month later.
Li
1 1 0 1 2
2 1 0 0 1
3 1 1 1 3
4 0 0 0 0
5 1 0 1 2
6 1 0 1 2
7 1 1 1 3
8 1 0 0 1
9 1 1 1 3
10 1 0 0 1
Gj 9 3 6 18
912
108
)112()112(3
18639(3)13(
)1(
222
2222
1
2
1
2
11
2
N
ii
N
ii
k
jj
k
jj
LLk
GGkk
Q
2(2,0.05)=5.99 <Qcal, p<0.05
Reject H0.
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Friedman TestThe Friedman Test extends the Wilcoxon Signed Ranks Test
to include more than two time periods of data collection or
conditions
)1(3)1(
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1
2
kNRkNk
Fk
jjr
Where
Rj: The sum of the ranks for column j
N: The # of subjects
k: # of time periods or conditions
This statistic is distributed as a chi-square with df=k-1.
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Multiple Comparisons TestOnce a determination has been made that the overall Friedman test is significant, post hoc tests can be undertaken that compare the differences in average ranks for all possible pairs to determine where the differences lie.
NkkzRR kkji 6/)1()1(/
The null hypothesis of no differences in mean ranks of the pairs being examined will be rejected if absolute value of these differences is greater than a specified critical value. That is, we would reject the null hypothesis if the following condition holds true:
Ri: the mean rank for time or condition i
k: the number of time periods or conditions
N: the number of cass
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Example Suppose we had collected information concerning the 10 children’s anxiety levels not only at pretest and immediately following the anxiety reduction intervention but also just prior to the administration of the preoperative medication. What are the differences in the anxiety levels of the 10 children who took part in the intervention across the three time periods?
H0: there will be no differences among the median anxiety scores at preintervention, at postintervention, and at preoperative medication for the 10 children who took part in the intervention.
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Children’s anxiety levels
Subj Preint Postint Preop
1 7 5 6
2 4 4 4
3 3 5 5
4 3 6 4
5 6 3 4
6 7 3 6
7 6 5 5
8 7 5 6
9 5 4 4
10 7 5 6
Rank Rank Rank
3 1 2
2 2 2
3 1.5 1.5
1 3 2
3 1 2
3 1 2
3 1.5 1.5
3 1 2
3 1.5 1.5
3 1 2
Sum Rj 27 14.5 18.5
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15.8
)13)(10(35.185.1427)13)(3(10
12
)1(3)1(
12
222
1
2
kNRkNk
Fk
jjr
2(2,0.05)=5.99 <Fr, p<0.05
Reject H0.
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08.1
)10(6/)4(339.2
6/)1()1(/
Nkkz kk
Groups Critical value Decision
1-2 1.25 1.08 Reject H0
1-3 0.85 <1.08 Accept H0
2-3 0.40 <1.08 Accept H0
ji RR