Extracting standard errors and effect estimates for meta-analysis: paging Rev Bayes

After a very long leave of absence I return to the issue of extracting the effect estimate (T) and standard error (se ) from reported and (rounded to a fixed number of decimal points) relative risk (t ), limits of 95% confidence intervals (t_L and t_U) and p-value (p_v) figures found in scientific publications. This solution is a Bayesian one, requiring nothing more than a straightforward application of the Bayes theorem for the posterior distribution of the A straightforward application of Bayes theorem for the quantities T, se given the t, t_L, t_U, p_v :

P(T,se|t, t_L, t_U, p_v) \propto P(t|T,se,t_L, t_U, p_v) \times P(T,se|t_L, t_U, p_v)

However P(t|T,se,t_L, t_U, p_v) = P(t|T) = \delta_{t,T_d} , because the rounding operation only depends on the value to be rounded (thus dropping conditioning on all other variables but T ). Since rounding to d digits is a deterministic operation, admitting only one value (T_d ) for a specific T , the conditional probability P(t|T) is essentially a point mass function at T_d given by the Kronecker \delta function.
When viewed as a parametric function of T, this conditional probability will be non-zero in the interval \left(-\frac{5}{10^{d+1}}+t\:,\frac{5}{10^{d+1}}+t\right) and zero otherwise.

Further application of the Bayes theorem yields:

P(T,se|t_L, t_U, p_v) \propto P(p_v|T,se,t_L, t_U) \times P(t,se|t_L,t_U)

The last factor in the right hand side of the formula is simply the (suitably transformed) joint probability function X,Y obtained from the pair H_L, H_U as discussed in the previous post . More specifically, it is the probability of T,se obtained by taking the logarithm of X and the ratio of \log(Y)/q_{0.975} conditioning on the pair H_L, H_U.

On the other hand, p-value calculations are based on the ratio z= T/se , allowing us to drop conditioning of P(p_v|T,se,t_L, t_U) on t_L,t_U. Applying the same line of reasoning as in the case of P(t|T) the conditional probability P(p_v|T,se) is mutatis mutandis equal to one when T/se is in the interval \left(-\frac{5}{10^{d+1}}+p_v\:,\frac{5}{10^{d+1}}+p_v\right) and zero otherwise.

To compute the Bayesian estimates of T,se we can  resort to a rejection type of algorithm:

  1. Simulate a pair of values from the joint density p(X,Y|H_L,H_U)
  2. Transform X,Y to T,se
  3. Reject T,se if T is not in the interval \left(-\frac{5}{10^{d+1}}+t\:,\frac{5}{10^{d+1}}+t\right)
  4. Reject T,se if the corresponding p-value 2 \times (1-CDF(|T/se|)) is not in the interval \left(-\frac{5}{10^{d+1}}+p_v\:,\frac{5}{10^{d+1}}+p_v\right)

Steps 1,2 ensure that the T,se are compatible with the limits of the confidence interval, while steps 3,4 guarantee compatibility with the reported relative risk and p-value respectively. The non-rejected Monte Carlo samples can be summarized by various statistics (e.g. the mean or the median) to yield point estimates for meta-analytic applications.

Example

  • True Values: T=0.403876 , se=0.11345
  • Rounding (to 2 figures):  t=1.50,t_L = 1.20,t_U=1.87 ,p_v=0.00
  • Result of steps 1,2 are shown in the figure below (the blue cross corresponds to the true values of T,se :Reject1st
  • Rejecting the pairs that are not compatible with the RR point estimate (step 3) yields the following two dimensional density:Reject1-3
  • Applying the criterion in step 4 does not reject further samples in this particular example so the corresponding image is virtually indistinguishable from the previous one:Reject1-4\

Point estimates for the two log-relative risk and the standard error are: 0.4043738 and  0.1131314 respectively. These are in close agreement the true values of 0.403876 and 0.11345 !

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One Response to “Extracting standard errors and effect estimates for meta-analysis: paging Rev Bayes”

  1. Extracting standard errors and treatment effects from medical journal tables (powered by R) | Statistical Reflections of a Medical Doctor Says:

    […] of interest (treatment effect, 95% confidence intervals and p-values). In a previous post I gave a Bayesian solution to this problem which obeys the precision of the treatment effect and confidence interval and the […]

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