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TRANSCRIPT
Combining Confidence Distributions
Rare Event Meta-Analysis
Brinley Zabriskie and Chris Corcoran Utah State University, Logan, UT, USA
for
Pralay Senchaudhuri Cytel Software Corporation, Cambridge, MA, USA
What Is a Meta-Analysis?
What Is a Confidence
Distribution?
Confidence Distributions
in Meta-Analysis
Innovation
What Is a Meta-Analysis?
. . . Study 1 n=326�
𝑂𝑅 ↓1 =7.63�pval=0.004
Study 2 n=44�𝑂𝑅 ↓2 =2.43�pval=0.49
Study 3 n=570�𝑂𝑅 ↓3 =2.24�pval=0.14
Study 9 n=333�𝑂𝑅 ↓6 =4.74�pval=0.06
Combined Info
𝑂𝑅 ↓𝐶 =1.80�pval=0.009
Cerebral Microbleeds (CMBs) Example
Average Treatment Event Rate: 9.3% Average Control Event Rate: 3.6%
What Is a Meta-Analysis?
Random sample of 500 systematic reviews in 2009 demonstrated that over half included outcomes with event rates of < 5%
Cerebral Microbleeds (CMBs) Example
What Is a Meta-Analysis? Cerebral Microbleeds (CMBs) Example
θ 95% CI p-value
Peto Odds Ratio
1.83 (1.17, 2.87) 0.008 (significant)
Mantel-Haenszel Odds Ratio 1.71 (1.14, 2.57) 0.010
(significant)
Inverse Variance (0.5 continuity correction)
Odds Ratio 1.80 (1.16, 2.80) 0.009
(significant)
DerSimonian and Laird (0.5 continuity correction)
Odds Ratio 2.46 (1.22, 4.97) 0.012
(significant)
Conclude the presence of CMBs may increase the risk of sICH
Simulation with Rare Events (baseline probability: 0.7%)
Rates Differ Between TRT and CTRL
Can we lower the Type I Error rate by using methods better suited for
meta-analyses with rare events?
What Is a Meta-Analysis?
Point estimate
Interval estimate
Distribution estimate
𝝁 Population Parameter
What Is a Confidence Distribution?
. . . + + +
Singh et al. (2005), Tian et al. (2009), and Liu et. al (2014)
Confidence Distributions in Meta-Analysis Cerebral Microbleeds (CMBs) Example
p↓c (θ)=F(c)[w↓1 h(p↓1 (θ))+…+w↓6 h(p↓6 (θ))]
Combining Confidence Intervals
Tian et al. (2009)
Θ risk difference p↓i (⋅) an exact method h(⋅) logit w↓i based on sample size
of study
p↓c (θ)=F(c)[w↓1 h(p↓1 (θ))+…+w↓6 h(p↓6 (θ))]
Combining p-value Functions
Liu et al. (2014)
Θ log odds ratio p↓i (⋅) Fisher’s exact test
(mid-p) h(⋅) inverse Normal CDF w↓i based on event rates
Combining p-values Fisher (1932)
Θ - p↓i (⋅) Fisher’s exact test (mid-p) h(⋅) log w↓i none
Confidence Distributions in Meta-Analysis
Confidence Distributions in Meta-Analysis Cerebral Microbleeds (CMBs) Example
θ 95% CI p-value
Combining Confidence Intervals
Tian et al. (2009)
Risk Difference 0.03 (-0.0002, 0.058) 0.054
(non-significant)
Combining p-value Functions
Liu et al. (2014)
Odds Ratio 1.73 (1.12, 2.62) 0.014
(significant)
Combining p-values Fisher (1932)
- - 0.016 (significant)
Conflicting conclusions as to whether CMBs increase risk
Confidence Distributions in Meta-Analysis
Relatively Fast, ↑ Error Very Slow, ↓ Error Very Fast, ↑ Error
Rates Differ Between TRT and CTRL
Innovation
Modification 1 Liu et al. (2014)
with logit and weights
Θ log odds ratio p↓i (⋅) Fisher’s exact test (mid-p) h(⋅) logit w↓i based on event rates
p↓c (θ)=F(c)[w↓1 h(p↓1 (θ))+…+w↓6 h(p↓6 (θ))]
Modification 2 Liu et al. (2014)
with logit and no weights
Θ log odds ratio p↓i (⋅) Fisher’s exact test
(mid-p) h(⋅) logit w↓i none
Modification 3 Tian et al. (2009)
with Liu et al. (2014) weights
Θ risk difference p↓i (⋅) an exact method h(⋅) logit w↓i based on event rates
Innovation
θ 95% CI p-value
Modification 1 Liu et al. (2014)
with logit and weights
Odds Ratio 1.69 (1.16, 2.33) 0.010
(significant)
Modification 2 Liu et al. (2014)
with logit and no weights
Odds Ratio 1.22 (1.37, 3.07) 0.001
(significant)
Modification 3 Tian et al. (2009)
with Liu et al. (2014) weights
Risk Difference 0.01 (-0.008, 0.054) (non-significant)
Cerebral Microbleeds (CMBs) Example
Conclude CMBs may not increase the risk of sICH
Rates Differ Between TRT and CTRL
Can we lower the Type I Error rate by
using methods better suited for
meta-analyses with rare events?
YES!
Innovation
Fairly Fast,
↓ Error
Thank you!
K. Singh, M. Xie, and W. E. Strawderman. Combining Information from Independent Sources through Confidence Distributions. The Annals of Statistics, 33(1):159-183, 2005.
D. Liu, R. Y. Liu, and M. Xie. Exact Meta-Analysis Approach for Discrete Data and its Application to 2 x 2 Tables With Rare Events. Journal of the American Statistical Association, 109(508):1450-1465, 2014.
L. Tian, T. Cai, M. A. Pfeer, N. Piankov, P. Cremieux, and L. J. Wei. Exact and efficient inference procedure for meta-analysis and its application to the analysis of independent 2 x 2 tables with all available data but without artificial continuity correction. Biostatistics, 10(2):275-281, 2009.
R. A. Fisher. Statistical Methods for Research Workers. 4 edition, 1932.