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Epidemiology 9509 oneway (continued)
Epidemiology 9509Principles of Biostatistics
Chapter 15- One way analysis of variance (continued)
John Koval
Department of Epidemiology and BiostatisticsUniversity of Western Ontario
1
Epidemiology 9509 oneway (continued)
What is being covered
1. multiple comparisons (posthoc tests)
1.1 p-value1.2 confidence intervals for differences
2. using SAS for oneway ANOVA
2.1 SAS Proc ANOVA2.2 options for unequal group size
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Epidemiology 9509 oneway (continued)
p-value method
test statistic
q =|y1 − y2|
√
s2p2
(
1n1
+ 1n2
)
when sample size equal
q =|y1 − y2|√
s2pn
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Epidemiology 9509 oneway (continued)
example
we calculate, from the data,q12 = q23 =
0.5√
(0.183)5
= 2.613
Similarly, q13 = 5.227
by table A.5 on page A.37, p-value for q12 and q23= Pr(Q3,12 > 2.613) > 0.10
Similarly, for q13p-value = Pr(Q3,12 > 5.227) < 0.01
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Epidemiology 9509 oneway (continued)
Conclusion
Hence, at α = 0.05
1. the effect of Placebo is different from that of ASA
2. but the effect of AC is not different from Placebo or ASA
sometimes described pictoriallyPlacebo AC ASA
SAS uses this in its LINES option
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Epidemiology 9509 oneway (continued)
multiple confidence intervals
using Tukey methodHSD (Honestly Significance Difference)
for equal group size
|yi − yj | ±Qk,N−k,0.05
√
s2p
n
for unequal group size
|yi − yj | ± Qk,N−k,0.05
√
s2p
2
(
1
ni+
1
nj
)
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Epidemiology 9509 oneway (continued)
multiple confidence intervals - example
√
(
s2pn
)
=√
0.1835 = 0.1915
From the tables (page A.31)Q3,12,0.05 = 3.783
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Epidemiology 9509 oneway (continued)
multiple confidence intervals - example (continued)
six pairs of differencescompute the confidence intervals for only three
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Epidemiology 9509 oneway (continued)
multiple confidence intervals - example (continued)
six pairs of differencescompute the confidence intervals for only three
1. δ21 = µ2 − µ1 = −δ12(positive difference)
2. δ32 = µ3 − µ2 = −δ23
3. δ31 = µ3 − µ1 = −δ13
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Epidemiology 9509 oneway (continued)
multiple confidence intervals - example (again)
1. 0.5± 3.783(0.1915) = 0.5 +−0.724= (−0.224, 1.224)for the first two
2. 1.0± 3.783(0.1915) = 1.0 +−0.724= (0.276, 1.724)for the last
using the confidence intervals for testinggives same conclusion as critical value and p-value
10
Epidemiology 9509 oneway (continued)
using SAS for ANOVA
balanced data onlyequal group size
1. ask for multiple comparisons
MEANS groupvar/TUKEY;
2. critical value is defaultask for confidence intervals for differences
MEANS groupvar/CLDIFF TUKEY;
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Epidemiology 9509 oneway (continued)
data
1 0.6
1 0.4
1 0.0
1 -0.4
1 -0.6
2 1.0
2 0.5
2 0.0
2 0.3
2 0.7
3 1.0
3 1.5
3 0.5
3 1.2
3 0.8
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Epidemiology 9509 oneway (continued)
program
filename one ’onew.dat’;
data marj;
infile one;
input group difft ;
label group = ’treatment group’ ;
label difft = ’change in body temperature’;
proc anova ;
class group;
model difft= group;
means group/tukey;
means group/cldiff tukey;
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Epidemiology 9509 oneway (continued)
SAS output
oneway analysis of variance
The ANOVA Procedure
Class Level Information
Class Levels Values
group 3 1 2 3
Number of Observations Read 15
Number of Observations Used 15
Dependent Variable: difft change in body temperature
Sum of
Source DF Squares Mean Square FValue Pr > F
Model 2 2.50000000 1.25000000 6.82 0.0105
Error 12 2.20000000 0.18333333
Corrected Total 14 4.70000000
14
Epidemiology 9509 oneway (continued)
output II
Tukey’s Studentized Range (HSD) Test for difft
NOTE: This test controls the Type I experimentwise
error rate, but it generally has a higher Type II
error rate than REGWQ.
15
Epidemiology 9509 oneway (continued)
output III
Alpha 0.05
Error Degrees of Freedom 12
Error Mean Square 0.183333
Critical Value of Studentized Range 3.77278
Minimum Significant Difference 0.7224
Means with the same letter
are not significantly different.
Tukey Grouping Mean N group
A 1.0000 5 3
A
B A 0.5000 5 2
B
B 0.0000 5 1
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Epidemiology 9509 oneway (continued)
output IV
Comparisons significant at the 0.05 level
are indicated by ***.
Difference
group Between Simultaneous 95%
Comparison Means Confidence Limits
3 - 2 0.5000 -0.2224 1.2224
3 - 1 1.0000 0.2776 1.7224 ***
2 - 3 -0.5000 -1.2224 0.2224
2 - 1 0.5000 -0.2224 1.2224
1 - 3 -1.0000 -1.7224 -0.2776 ***
1 - 2 -0.5000 -1.2224 0.2224
17
Epidemiology 9509 oneway (continued)
Comment
For equal sized groups (balance)the critical value methodand use of multiple confidence interval
GIVE THE SAME INFERENTIAL RESULTS
18
Epidemiology 9509 oneway (continued)
unbalanced data
remove one ASA score from previous data
1 0.6
1 0.4
1 0.0
1 -0.4
1 -0.6
2 1.0
2 0.5
2 0.0
2 0.3
2 0.7
3 1.5
3 0.5
3 1.2
3 0.8
19
Epidemiology 9509 oneway (continued)
Using SAS
1. The SAS manual says to use Proc GLMinstead of Proc ANOVAbut, for our example,Procs give identical output
2. with unequal group sizefor multiple comparisons
2.1 the default is OPTION CLDIFFuses the actual group sizes
2.2 OPTION LINESuses the harmonic mean of the groups sizes
these two approachesCAN lead to different inferencesI recommend OPTION CLDIFFand the actual group sizes
20
Epidemiology 9509 oneway (continued)
SAS program for unbalanced data
title1 ’oneway analysis of variance’;
title2 ’unbalanced data’;
filename one ’onew2.dat’;
data marj;
infile one;
input group difft ;
label group = ’treatment group’ ;
label difft = ’change in body temperature’;
proc anova;
class group;
model difft= group;
means group/lines tukey;
means group/cldiff tukey;
21
Epidemiology 9509 oneway (continued)
SAS output for unbalanced data
oneway analysis of variance
unequal sample size
proc anova
The ANOVA Procedure
Class Level Information
Class Levels Values
group 3 1 2 3
Number of Observations Read 14
Number of Observations Used 14
Dependent Variable: difft change in body temperature
Sum of
Source DF Squares Mean Square FValue Pr > F
Model 2 2.23214286 1.11607143 5.58 0.0212
Error 11 2.20000000 0.20000000
Corrected Total 13 4.43214286
22
Epidemiology 9509 oneway (continued)
SAS output for unbalanced data II
Tukey’s Studentized Range (HSD) Test for difft
NOTE: This test controls the Type I experimentwise
error rate, but it generally has a higher Type II
error rate than REGWQ.
Alpha 0.05
Error Degrees of Freedom 11
Error Mean Square 0.2
Critical Value of Studentized Range 3.81952
Minimum Significant Difference 0.7951
Harmonic Mean of Cell Sizes 4.615385
NOTE: Cell sizes are not equal.
23
Epidemiology 9509 oneway (continued)
SAS output for unbalanced data III
Tukey’s Studentized Range (HSD) Test for difft
Means with the same letter
are not significantly different.
Tukey Grouping Mean N group
A 1.0000 4 3
A
B A 0.5000 5 2
B
B 0.0000 5 1
24
Epidemiology 9509 oneway (continued)
SAS output for unbalanced data
Tukey’s Studentized Range (HSD) Test for difft
NOTE: This test controls the Type I experimentwise
error rate.
Alpha 0.05
Error Degrees of Freedom 11
Error Mean Square 0.2
Critical Value of Studentized Range 3.81952
25
Epidemiology 9509 oneway (continued)
SAS output for unbalanced data IV
Comparisons significant at the 0.05 level
are indicated by ***.
Difference
group Between Simultaneous 95%
Comparison Means Confidence Limits
3 - 2 0.5000 -0.3102 1.3102
3 - 1 1.0000 0.1898 1.8102 ***
2 - 3 -0.5000 -1.3102 0.3102
2 - 1 0.5000 -0.2639 1.2639
1 - 3 -1.0000 -1.8102 -0.1898 ***
1 - 2 -0.5000 -1.2639 0.2639
26
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