Download - Comparing k > 2 Groups - Numeric Responses
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Comparing k > 2 Groups - Numeric Responses
• Extension of Methods used to Compare 2 Groups• Parallel Groups and Crossover Designs• Normal and non-normal data structures
DataDesign
Normal Non-normal
ParallelGroups(CRD)
F-Test1-WayANOVA
Kruskal-Wallis Test
Crossover(RBD)
F-Test2-WayANOVA
Friedman’sTest
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Parallel Groups - Completely Randomized Design (CRD)
• Controlled Experiments - Subjects assigned at random to one of the k treatments to be compared
• Observational Studies - Subjects are sampled from k existing groups
• Statistical model Yij is a subject from group i:
ijiijiijY
where is the overall mean, i is the effect of treatment i , ij is a random error, and i is the population mean for group i
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1-Way ANOVA for Normal Data (CRD)
• For each group obtain the mean, standard deviation, and sample size:
1
)( 2
i
jiij
ii
jij
i n
yys
n
yy
• Obtain the overall mean and sample size
n
y
nynyn
ynnn i j ijkkk
11
1
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Analysis of Variance - Sums of Squares
• Total Variation
1)(1 1
2 ndfyyTotalSS Total
k
i
n
j iji
• Between Group Variation
k
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i Tiiii kdfyynyySST
1 1 122 1)()(
• Within Group Variation
ETTotal
Ek
i iik
i
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j iij
dfdfdfSSESSTTotalSS
kndfsnyySSE i
12
1 12 )1()(
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Analysis of Variance Table and F-TestSource ofVariation Sum of Squares
Degrres ofFreedom Mean Square F
Treatments SST k-1 MST=SST/(k-1) F=MST/MSEError SSE n-k MSE=SSE/(n-k)Total Total SS n-1
• H0: No differences among Group Means (k=0)
• HA: Group means are not all equal (Not all i are 0)
)(:
)4.(:..
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,1,
obs
knkobs
obs
FFPvalP
ATableFFRRMSEMSTFST
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Example - Relaxation Music in Patient-Controlled Sedation in Colonoscopy
• Three Conditions (Treatments): – Music and Self-sedation (i = 1)– Self-Sedation Only (i = 2)– Music alone (i = 3)
• Outcomes– Patient satisfaction score (all 3 conditions)– Amount of self-controlled dose (conditions 1 and 2)
Source: Lee, et al (2002)
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Example - Relaxation Music in Patient-Controlled Sedation in Colonoscopy
• Summary Statistics and Sums of Squares Calculations:
Trt (i) ni Mean Std. Dev.1 55 7.8 2.12 55 6.8 2.33 55 7.4 2.3
Total 165 overall mean=7.33 ---
164162275.84046.80929.31162316546.809)3.2)(155()3.2)(155()1.2)(155(
21329.31)33.74.7(55)33.78.6(55)33.78.7(55222
222
Total
E
T
dfTotalSSdfSSE
dfSST
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Example - Relaxation Music in Patient-Controlled Sedation in Colonoscopy
• Analysis of Variance and F-Test for Treatment effects
Source ofVariation Sum of Squares
Degrres ofFreedom Mean Square F
Treatments 31.29 2 15.65 3.13Error 809.46 162 5.00Total 840.75 164
•H0: No differences among Group Means (3=0)
• HA: Group means are not all equal (Not all i are 0)
05.0)13.3(:
)4.(055.3:..
13.300.565.15:..
162,2,05.
FPvalP
ATableFFRR
FST
kobs
obs
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Post-hoc Comparisons of Treatments
• If differences in group means are determined from the F-test, researchers want to compare pairs of groups. Three popular methods include:– Dunnett’s Method - Compare active treatments with a
control group. Consists of k-1 comparisons, and utilizes a special table.
– Bonferroni’s Method - Adjusts individual comparison error rates so that all conclusions will be correct at desired confidence/significance level. Any number of comparisons can be made.
– Tukey’s Method - Specifically compares all k(k-1)/2 pairs of groups. Utilizes a special table.
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Bonferroni’s Method (Most General)
• Wish to make C comparisons of pairs of groups with simultaneous confidence intervals or 2-sided tests
• Want the overall confidence level for all intervals to be “correct” to be 95% or the overall type I error rate for all tests to be 0.05
• For confidence intervals, construct (1-(0.05/C))100% CIs for the difference in each pair of group means (wider than 95% CIs)
• Conduct each test at =0.05/C significance level (rejection region cut-offs more extreme than when =0.05)
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Bonferroni’s Method (Most General)
• Simultaneous CI’s for pairs of group means:
jikncji nn
MSEtyy 11,2/
• If entire interval is positive, conclude i > j
• If entire interval is negative, conclude i < j
• If interval contains 0, cannot conclude i j
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Example - Relaxation Music in Patient-Controlled Sedation in Colonoscopy
• C=3 comparisons: 1 vs 2, 1 vs 3, 2 vs 3. Want all intervals to contain true difference with 95% confidence
• Will construct (1-(0.05/3))100% = 98.33% CIs for differences among pairs of group means
)42.0,62.1(02.1)4.78.6(:32)42.1,62.0(02.1)4.78.7(:31)02.2,02.0(02.1)8.68.7(:21
02.1551
55100.540.211
5500.540.2
162),3(2/05.
3210083.162),3(2/05.
vsvsvs
nnMSEt
nnnMSEzt
ji
Note all intervals contain 0, but first is very close to 0 at lower end
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CRD with Non-Normal Data Kruskal-Wallis Test
• Extension of Wilcoxon Rank-Sum Test to k>2 Groups
• Procedure:– Rank the observations across groups from smallest (1)
to largest (n = n1+...+nk), adjusting for ties
– Compute the rank sums for each group: T1,...,Tk . Note that T1+...+Tk = n(n+1)/2
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Kruskal-Wallis Test• H0: The k population distributions are identical (1=...=k)
• HA: Not all k distributions are identical (Not all i are equal)
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Post-hoc comparisons of pairs of groups can be made by pairwise application of rank-sum test with Bonferroni adjustment
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Example - Thalidomide for Weight Gain in HIV-1+ Patients with and without TB
• k=4 Groups, n1=n2=n3=n4=8 patients per group (n=32)
• Group 1: TB+ patients assigned Thalidomide
• Group 2: TB- patients assigned Thalidomide
• Group 3: TB+ patients assigned Placebo
• Group 4: TB- patients assigned Placebo
• Response - 21 day weight gains (kg) -- Negative values are weight losses
Source: Klausner, et al (1996)
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Example - Thalidomide for Weight Gain in HIV-1+ Patients with and without TB
TB+/Thal TB-/Thal TB+/Plac TB-/Plac9.0 (32) 2.5 (23) 0.0 (9) -0.5 (7)6.0 (31) 3.5 (26.5) 1.0 (15.5) 0.0 (9)4.5 (30) 4.0 (28.5) -1.0 (6) 2.5 (23)
2.0 (20.5) 1.0 (15.5) -2.0 (4) 0.5 (12)2.5 (23) 0.5 (12) -3.0 (1.5) -1.5 (5)3.0 (25) 4.0 (28.5) -3.0 (1.5) 0.0 (9)
1.0 (15.5) 1.5 (18.5) 0.5 (12) 1.0 (15.5)1.5 (18.5) 2.0 (20.5) -2.5 (3) 3.5 (26.5)T1=195.5 T2=173.0 T3=52.5 T4=107.0
815.7:..
98.17)33(38
)0.107(8
)5.52(8
)0.173(8
)5.195()33(32
12:..
214,05.
2222
HRR
HST
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Weight Gain Example - SPSS OutputF-Test and Post-Hoc Comparisons
ANOVA
WTGAIN
109.688 3 36.563 10.206 .000100.313 28 3.583210.000 31
Between GroupsWithin GroupsTotal
Sum ofSquares df Mean Square F Sig.
GROUP
TB-/PlaceboTB+/PlaceboTB-/ThalidomideTB+/Thalidomide
Mea
n of
WTG
AIN
4
3
2
1
0
-1
-2
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Weight Gain Example - SPSS OutputF-Test and Post-Hoc Comparisons
Multiple Comparisons
Dependent Variable: WTGAIN
1.313 .9464 .518 -1.271 3.8964.938* .9464 .000 2.354 7.5213.000* .9464 .018 .416 5.584
-1.313 .9464 .518 -3.896 1.2713.625* .9464 .003 1.041 6.2091.688 .9464 .302 -.896 4.271
-4.938* .9464 .000 -7.521 -2.354-3.625* .9464 .003 -6.209 -1.041-1.938 .9464 .195 -4.521 .646-3.000* .9464 .018 -5.584 -.416-1.688 .9464 .302 -4.271 .8961.938 .9464 .195 -.646 4.5211.313 .9464 1.000 -1.374 3.9994.938* .9464 .000 2.251 7.6243.000* .9464 .022 .313 5.687
-1.313 .9464 1.000 -3.999 1.3743.625* .9464 .004 .938 6.3121.688 .9464 .512 -.999 4.374
-4.938* .9464 .000 -7.624 -2.251-3.625* .9464 .004 -6.312 -.938-1.938 .9464 .301 -4.624 .749-3.000* .9464 .022 -5.687 -.313-1.688 .9464 .512 -4.374 .9991.938 .9464 .301 -.749 4.624
(J) GROUPTB-/ThalidomideTB+/PlaceboTB-/PlaceboTB+/ThalidomideTB+/PlaceboTB-/PlaceboTB+/ThalidomideTB-/ThalidomideTB-/PlaceboTB+/ThalidomideTB-/ThalidomideTB+/PlaceboTB-/ThalidomideTB+/PlaceboTB-/PlaceboTB+/ThalidomideTB+/PlaceboTB-/PlaceboTB+/ThalidomideTB-/ThalidomideTB-/PlaceboTB+/ThalidomideTB-/ThalidomideTB+/Placebo
(I) GROUPTB+/Thalidomide
TB-/Thalidomide
TB+/Placebo
TB-/Placebo
TB+/Thalidomide
TB-/Thalidomide
TB+/Placebo
TB-/Placebo
Tukey HSD
Bonferroni
MeanDifference
(I-J) Std. Error Sig. Lower Bound Upper Bound95% Confidence Interval
The mean difference is significant at the .05 level.*.
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Weight Gain Example - SPSS OutputKruskal-Wallis H-Test
Ranks
8 24.448 21.638 6.568 13.38
32
GROUPTB+/ThalidomideTB-/ThalidomideTB+/PlaceboTB-/PlaceboTotal
WTGAINN Mean Rank
Test Statisticsa,b
18.0703
.000
Chi-SquaredfAsymp. Sig.
WTGAIN
Kruskal Wallis Testa.
Grouping Variable: GROUPb.
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Crossover Designs: Randomized Block Design (RBD)
• k > 2 Treatments (groups) to be compared• b individuals receive each treatment (preferably in
random order). Subjects are called Blocks.• Outcome when Treatment i is assigned to Subject j
is labeled Yij
• Effect of Trt i is labeled i
• Effect of Subject j is labeled j
• Random error term is labeled ij
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Crossover Designs - RBD
• Model:
ijjiijjiijY
• Test for differences among treatment effects:
• H0: 1k 0 (1k )
• HA: Not all i = 0 (Not all i are equal)
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RBD - ANOVA F-Test (Normal Data)• Data Structure: (k Treatments, b Subjects)
• Mean for Treatment i:
• Mean for Subject (Block) j:
• Overall Mean:
• Overall sample size: n = bk
• ANOVA:Treatment, Block, and Error Sums of Squares
.iyjy .
y
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1
1
1
1
2
.
1
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.
1 1
2
kbdfSSBSSTTotalSSSSE
bdfyykSSB
kdfyybSST
bkdfyyTotalSS
E
Bb
j j
Tk
i i
k
i
b
j Totalij
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RBD - ANOVA F-Test (Normal Data)• ANOVA Table:Source SS df MS F
Treatments SST k-1 MST = SST/(k-1) F = MST/MSEBlocks SSB b-1 MSB = SSB/(b-1)Error SSE (b-1)(k-1) MSE = SSE/[(b-1)(k-1)]Total TotalSS bk-1
•H0: 1k 0 (1k )
• HA: Not all i = 0 (Not all i are equal)
)(:
:..
:..
)1)(1(,1,
obs
kbkobs
obs
FFPvalP
FFRRMSEMSTFST
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Example - Theophylline Interaction• Goal: Determine whether Cimetidine or Famotidine interact with Theophylline
• 3 Treatments: Theo/Cim, Theo/Fam, Theo/Placebo
• 14 Blocks: Each subject received each treatment
• Response: Theophylline clearance (liters/hour) Subject T/C T/F T/P BLK Mean
1 3.69 5.13 5.88 4.902 3.61 7.04 5.89 5.513 1.15 1.46 1.46 1.364 4.02 4.44 4.05 4.175 1.00 1.15 1.09 1.086 1.75 2.11 2.59 2.157 1.45 2.12 1.69 1.758 2.59 3.25 3.16 3.009 1.57 2.11 2.06 1.91
10 2.34 5.20 4.59 4.0411 1.31 1.98 2.08 1.7912 2.43 2.38 2.61 2.4713 2.33 3.53 3.42 3.0914 2.34 2.33 2.54 2.40
TRT Mean 2.26 3.16 3.08 2.83Source: Bachmann, et al (1995)
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Example - Theophylline InteractionTests of Between-Subjects Effects
Dependent Variable: CLRNCE
78.817a 15 5.254 15.888 .000336.713 1 336.713 1018.119 .000
7.005 2 3.503 10.591 .00071.811 13 5.524 16.703 .0008.599 26 .331
424.129 4287.415 41
SourceCorrected ModelInterceptTRTSUBJECTErrorTotalCorrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .902 (Adjusted R Squared = .845)a.
• The test for differences in mean theophylline clearance is given in the third line of the table
•T.S.: Fobs=10.59
• R.R.: Fobs F.05,2,26 = 3.37 (From F-table)
• P-value: .000 (Sig. Level)
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Example - Theophylline InteractionPost-hoc Comparisons
Multiple Comparisons
Dependent Variable: CLRNCE
-.9036* .21736 .001 -1.4437 -.3635-.8236* .21736 .002 -1.3637 -.2835.9036* .21736 .001 .3635 1.4437.0800 .21736 .928 -.4601 .6201.8236* .21736 .002 .2835 1.3637
-.0800 .21736 .928 -.6201 .4601-.9036* .21736 .001 -1.4598 -.3474-.8236* .21736 .002 -1.3798 -.2674.9036* .21736 .001 .3474 1.4598.0800 .21736 1.000 -.4762 .6362.8236* .21736 .002 .2674 1.3798
-.0800 .21736 1.000 -.6362 .4762
(J) TRTTheophylline/FamotidineTheophylline/PlaceboTheophylline/CimetidineTheophylline/PlaceboTheophylline/CimetidineTheophylline/FamotidineTheophylline/FamotidineTheophylline/PlaceboTheophylline/CimetidineTheophylline/PlaceboTheophylline/CimetidineTheophylline/Famotidine
(I) TRTTheophylline/Cimetidine
Theophylline/Famotidine
Theophylline/Placebo
Theophylline/Cimetidine
Theophylline/Famotidine
Theophylline/Placebo
Tukey HSD
Bonferroni
MeanDifference
(I-J) Std. Error Sig. Lower Bound Upper Bound95% Confidence Interval
Based on observed means.The mean difference is significant at the .05 level.*.
514.31:
57.22:
)1)(1(,,05.
)1)(1(,2/
qb
MSEqyyTukey
tb
MSEtyyBonferroni
kbkji
kbcji
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Example - Theophylline InteractionPlot of Data (Marginal means are raw data)
Estimated Marginal Means of CLRNCE
SUBJECT
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Est
imat
ed M
argi
nal M
eans
7
6
5
4
3
2
1
0
TRT
Theophylline/Cimetid
ine
Theophylline/Famotid
ine
Theophylline/Placebo
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RBD -- Non-Normal DataFriedman’s Test
• When data are non-normal, test is based on ranks• Procedure to obtain test statistic:
– Rank the k treatments within each block (1=smallest, k=largest) adjusting for ties
– Compute rank sums for treatments (Ti) across blocks
– H0: The k populations are identical (1=...=k)
– HA: Differences exist among the k group means
)(:
:..
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12:..
2
21,
12
r
kr
k
i ir
FPvalP
FRR
kbTkbk
FST
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Example - tmax for 3 formulation/fasting states
• k=3 Treatments of Valproate: Capsule/Fasting (i=1), Capsule/nonfasting (i=2), Enteric-Coated/fasting (i=3)
• b=11 subjects
• Response - Time to maximum concentration (tmax)Subject C/F C/NF EC/F
1 3.5 (2) 4.5 (3) 2.5 (1)2 4.0 (2) 4.5 (3) 3.0 (1)3 3.5 (2) 4.5 (3) 3.0 (1)4 3.0 (1.5) 4.5 (3) 3.0 (1.5)5 3.5 (1.5) 5.0 (3) 3.5 (1.5)6 3.0 (1) 5.5 (3) 3.5 (2)7 4.0 (2.5) 4.0 (2.5) 2.5 (1)8 3.5 (2) 4.5 (3) 3.0 (1)9 3.5 (1.5) 5.0 (3) 3.5 (1.5)
10 3.0 (1) 4.5 (3) 3.5 (2)11 4.5 (2) 6.0 (3) 3.0 (1)
Rank sum T1=19.0 T2=32.5 T3=14.5Source: Carrigan, et al (1990)
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Example - tmax for 3 formulation/fasting states
99.5:..
95.15)13)(11(35.145.320.19)13)(3(11
12:..
213,05.
222
r
r
FRR
FST
H0: The k populations are identical (1=...=k)HA: Differences exist among the k group means