Modern Methods of Clinical Investigation (1990) / Chapter Skim
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9. An Introduction to a Bayesian Method for Meta-Analysis: The Confidence Profile Method
9. An Introduction to a Bayesian Method for Meta-Analysis: The Confidence Profile Method
Pages 101-116
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From page 101... ...
EDDY, VIC HASSELBLAD, and ROSS SHACHTER The Confidence Profile Method is a form of meta-analysis. It is a Bayesian method for interpreting, adjusting, and combining evidence to estimate a probability distribution for a parameter.
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An important problem in the evaluation of evidence is the presence of biases. An important difference between the Confidence Profile Method and other meta-analysis techniques is the explicit modeling of biases and their incorporation in the distribution for the parameter of interest.
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From page 103... ...
The method requires prior distributions, likelihood functions, and functions that describe biases. It also requires functions that define the measures of effect (which will be introduced below)
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From page 104... ...
TABLE 9.1 Likelihood functions for various types of experimental designs, outcomes, and effect measures Outcomes Designs Dichotomous Categorical Count Continuous One-Arm Rate Rate Mean Count Mean Score Prospective Score Median Score l~vo-Arm Difference Difference Difference Difference Prospective Ratio Ratio Ratio Ratio Odds Ratio % Difference e-Arm Prospective Coefficients of Coefficients of Coefficients of Coefficients of Logistic Linear Linear Linear Regression Regression Regression Regression Equation, pi Equation, pi Equation, pi Equation, pi 2x2Case Odds Ratio NA NA NA Control 2 x n Case Coefficients of NA NA NA Control Logistic Regression Equation, pi Matched Case Odds Ratio NA NA NA Control Cross Coefficients of Coefficients of Coefficients of Coefficients of Sectional Logistic Linear Linear Linear Regression Regression Regression Regression Equation, pi Equation, pi Equation, pi Equation, pi NA, not applicable.
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From page 105... ...
The Confidence Profile Method includes likelihood functions for each type of outcome, experimental design, and effect measure (2~. ILLUSTRATION Imagine a randomized controlled trial with 100 patients in the control group and 104 patients in the group offered treatment (see Table 9.21.
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32 p1/2,l/2~ec' d`eCy (4) This likelihood function can be used in Bayes's formula to calculate a posterior distribution for £.
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If that is true, the likelihood function just derived (Equation 4) no longer estimates the parameter of interest, i.e., the effect of treatment in people who actually receive treatment.
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-0.5 0 0.5 FIGURE 9.2 Probability distribution B for an increase in five-year survival as a result of treatment. Based on a randomized controlled trial of 204 patients in which 20 percent of the patients offered treatment did not actually receive Reagent (dilution bias of 20 percent)
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From page 109... ...
The doped line represents the posterior distribution if the study is taken at face value; the dashed line takes into account a dilution factor of 0.2; the solid line incorporates uncertainty about the magnitude of that dilution. Additional biases and nested biases can be incorporated in the analysis.
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Based on a randomized controlled trial of 1,000 patients (solid line) , compared with a randomized controlled trial of 204 patients adjusted for dilution bias (dotted line)
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BASIC FORMULAS IN THE CONFIDENCE PROFILE METHOD The Confidence Profile Method contains likelihood functions for all the experimental designs, outcome measures, and effect measures shown in Table 9.1 (21. There is no requirement that all the studies to be combined have the same design.
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From page 112... ...
Diagram of indirect evidence: Technology ~ Intermediate Outcomes ~ Health Outcomes The Confidence Profile Method includes formulas for combining the two bodies of evidence, including the possibility that the intermediate outcome is not a perfect indicator of the health outcome (1~. For example, exercise might have an independent effect on the chance of a heart attack not mediated through a change in serum cholesterol.
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From page 113... ...
The distribution for the effect calculated by the Confidence Profile Method can be used in these calculations to obtain a power conditional on the existing evidence for the effect, rather than a hypothesized effect. Because the Confidence Profile Method delivers a distribution, it can also calculate the probability an experiment will yield results within a specified range (rather than simply a statistically significant result, as in a power calculation)
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From page 114... ...
RELATIONSHIP TO OTHER META-ANALYSIS TECHNIQUES The Confidence Profile Method differs from meta-analysis techniques based on classical statistics in several important ways. First, because it is based on Bayesian statistics, the Confidence Profile Method gives marginal probability distributions for the parameters of interest and, if the integrated approach is used, a joint probability distribution for all the parameters.
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From page 115... ...
For example, the production of probability distributions enables the Confidence Profile Method to analyze indirect evidence and technology families, neither of which can be analyzed by other meta-analysis techniques. A third distinguishing feature of the Confidence Profile Method, again enabled by the use of Bayesian statistics, is the explicit modeling of biases to internal and external validity.
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From page 116... ...
SUMMARY To summarize, the Confidence Profile Method can be used to assess technologies when the available evidence involves a variety of experimental designs, types of outcomes, and effect measures; a variety of biases; combinations of biases and nested biases; uncertainty about biases; an underlying variability in the parameter of interest; indirect evidence; and technology families. The result of an analysis with the Confidence Profile Method is a posterior distribution for the parameter of interest, posterior distributions for other parameters, and a covariance matrix for all the parameters in the model.
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