analysis of variance: case study bios 662mhudgens/bios/662/2008fall/anova_case.pdfanalysis plan •...
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![Page 1: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/1.jpg)
Analysis of Variance: Case Study
Bios 662
Michael G. Hudgens, Ph.D.
http://www.bios.unc.edu/∼mhudgens
2008-10-28 15:15
BIOS 662 1 ANOVA Case Study
![Page 2: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/2.jpg)
Strategy for Analysis
1. Consider data generation mechanism
2. Analysis plan: specify model/assumptions; hyps to be
tested; diagnostics to be performed
3. Summary statistics, tables, figures
4. Model fitting and diagnostics
5. Inference
6. Sensitivity analysis
7. Conclusions; summary; limitations (eg lack of power)
BIOS 662 2 ANOVA Case Study
![Page 3: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/3.jpg)
Hypothetical Example
• Survival times of patients following heart transplant surgery
based on degree of mismatch (low, med, high) of the tis-
sue type between donor and recipient
BIOS 662 3 ANOVA Case Study
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Analysis Plan
• ANOVA
Yij = µi + εij
i = 1, 2, 3 denoting low, med, high groups
j = 1, 2, . . . , ni denoting jth patient in ith group
n1 = 14, n2 = 13, n3 = 12
Yij survival time in days
µi mean survival time in patients with type i mismatch
BIOS 662 4 ANOVA Case Study
![Page 5: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/5.jpg)
Analysis Plan
• Primary hypotheses of interest: all pairwise compar-
isons
H0 : µ1 = µ2, H0 : µ1 = µ3, H0 : µ2 = µ3
• Based on power considerations, decide α = .1
• All pairwise comparisons using Tukey
• Further contrasts, group level means, etc will be consid-
ered hyp generating/exploratory; thus no multiplicity
adjustment
• Diagnostics, remedial measures, sensitivity analyses
BIOS 662 5 ANOVA Case Study
![Page 6: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/6.jpg)
Data
j Low Medium High
1 44 15 3.0
2 551 280 136.0
3 127 1024 65.0
4 1400 836 400.0
5 1000 51 10.4
6 700 600 39.4
7 550 250 33.4
8 300 200 48.4
9 47 22 13.5
10 26 71 34.5
11 50 62 35.5
12 10 69 45.5
13 70 13
14 20
BIOS 662 6 ANOVA Case Study
![Page 7: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/7.jpg)
Figure 1: Box plot with raw data
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l m h
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Degree of mismatch
Sur
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BIOS 662 7 ANOVA Case Study
![Page 8: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/8.jpg)
Table 1: Summary Statistics
Low Medium High
n 14 13 12
Mean 349.6 268.7 72.1
Median 98.5 71.0 37.5
SD 435.31 337.51 108.82
(Min, Max) (10,1400) (13, 1042) (3,400)
BIOS 662 8 ANOVA Case Study
![Page 9: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/9.jpg)
Homogeneity of Variance Plot
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100 150 200 250 300 350
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150
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400
Group sample means
With
in g
roup
sam
ple
SD
s
BIOS 662 9 ANOVA Case Study
![Page 10: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/10.jpg)
Residual Plot
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100 150 200 250 300 350
−20
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400
600
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1000
fit$fitted.values
fit$r
esid
uals
BIOS 662 10 ANOVA Case Study
![Page 11: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/11.jpg)
Modified Levene Test
Brown and Forsythe’s Test for Homogeneity of surv Variance
ANOVA of Absolute Deviations from Group Medians
Sum of Mean
Source DF Squares Square F Value Pr > F
degree 2 452324 226162 2.46 0.0993
Error 36 3304203 91783.4
BIOS 662 11 ANOVA Case Study
![Page 12: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/12.jpg)
QQ Plot
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−2 −1 0 1 2
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Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
BIOS 662 12 ANOVA Case Study
![Page 13: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/13.jpg)
Box-Cox Transformations
Yλ =
{k1(Y
λ − 1) for λ 6= 0
k2 log(Y ) for λ = 0
where
k2 =
∏i,j
Yij
1/N
and k1 =1
λkλ−12
BIOS 662 13 ANOVA Case Study
![Page 14: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/14.jpg)
Box-Cox Transformation
Box-Cox Root mean .95
Power Log squared Confidence
(lambda) Likelihood error Interval
-1.0 -232.441 387.62
-0.8 -216.424 257.07
-0.6 -203.272 183.48
-0.4 -193.809 143.95
-0.2 -188.568 125.85 *
0.0 -187.530 122.54 <+
0.2 -190.213 131.27
0.4 -195.970 152.15
0.6 -204.212 187.96
0.8 -214.467 244.48
1.0 -226.364 331.69
BIOS 662 14 ANOVA Case Study
![Page 15: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/15.jpg)
Box-Cox Transformation
• Choose log transformation
• Run diagnostics again
BIOS 662 15 ANOVA Case Study
![Page 16: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/16.jpg)
Residual Plot
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3.6 3.8 4.0 4.2 4.4 4.6 4.8
−2
−1
01
2
fit$fitted.values
fit$r
esid
uals
BIOS 662 16 ANOVA Case Study
![Page 17: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/17.jpg)
Modified Levene Test
Brown and Forsythe’s Test for Homogeneity of lsurv Variance
ANOVA of Absolute Deviations from Group Medians
Sum of Mean
Source DF Squares Square F Value Pr > F
degree 2 2.2384 1.1192 1.52 0.2324
Error 36 26.5038 0.7362
BIOS 662 17 ANOVA Case Study
![Page 18: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/18.jpg)
QQ Plot
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−2 −1 0 1 2
−2
−1
01
2
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
BIOS 662 18 ANOVA Case Study
![Page 19: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/19.jpg)
Tests for Normality
Pearson’s product-moment correlation
data: qq$x and qq$y
sample estimates:
cor
0.9882992
Lilliefors (Kolmogorov-Smirnov) normality test
data: fit$residuals
D = 0.0799, p-value = 0.766
Shapiro-Wilk normality test
data: fit$residuals
W = 0.9682, p-value = 0.3294
BIOS 662 19 ANOVA Case Study
![Page 20: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/20.jpg)
Box-Cox Transformation
• Diagnostics suggest adequate fit after log transformation
• Employ Tukey for all (3) pairwise comparisons of factor
level means
BIOS 662 20 ANOVA Case Study
![Page 21: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/21.jpg)
Tukey Simultaneous 90% CIs
Tukey’s Studentized Range (HSD) Test for lsurv
NOTE: This test controls the Type I experimentwise error rate.
Comparisons significant at the 0.1 level are indicated by ***.
Difference
degree Between Simultaneous 90%
Comparison Means Confidence Limits
l - m 0.1476 -1.0444 1.3397
l - h 1.2805 0.0630 2.4980 ***
m - h 1.1329 -0.1061 2.3718
BIOS 662 21 ANOVA Case Study
![Page 22: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/22.jpg)
Tukey Simultaneous 90% CIs
−2 −1 0 1 2 3
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●l−m
l−h
m−h
BIOS 662 22 ANOVA Case Study
![Page 23: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/23.jpg)
Sensitivity Analyses: Bonferroni
Bonferroni (Dunn) t Tests for lsurv
NOTE: This test controls the Type I experimentwise error rate, but it generally
has a higher Type II error rate than Tukey’s for all pairwise comparisons.
Comparisons significant at the 0.1 level are indicated by ***.
Difference
degree Between Simultaneous 90%
Comparison Means Confidence Limits
l - m 0.1476 -1.0969 1.3922
l - h 1.2805 0.0094 2.5516 ***
m - h 1.1329 -0.1606 2.4263
BIOS 662 23 ANOVA Case Study
![Page 24: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/24.jpg)
Sensitivity Analyses: Wilcoxon Ranksum
Wilcoxon rank sum test
data: low.surv and med.surv
W = 93, p-value = 0.943
alternative hypothesis: true mu is not equal to 0
Wilcoxon rank sum test
data: low.surv and high.surv
W = 122, p-value = 0.05264
alternative hypothesis: true mu is not equal to 0
Wilcoxon rank sum test
data: med.surv and high.surv
W = 115, p-value = 0.04571
alternative hypothesis: true mu is not equal to 0
BIOS 662 24 ANOVA Case Study
![Page 25: Analysis of Variance: Case Study Bios 662mhudgens/bios/662/2008fall/anova_case.pdfAnalysis Plan • ANOVA Yij = µi + ij i = 1,2,3 denoting low, med, high groups j = 1,2,...,ni denoting](https://reader036.vdocuments.mx/reader036/viewer/2022070821/5f1f9fa06f75285c095cce4a/html5/thumbnails/25.jpg)
Conclusions
• Conclude there is evidence of a marginally sig difference
in mean log survival times between between low and
high (although not totally consistent across sensitivity
analyses)
• How to quantify/interpret on log scale?
• Power of study to detect what size effects?
Alternative analysis to test for trend?
BIOS 662 25 ANOVA Case Study