Combining Information Across Diverse Sources: The II-CC-FF Paradigm

TL;DR

A general paradigm for combining information across diverse data sources is introduced, and the II‐CC‐FF strategy turns out to be competitive against state‐of‐the‐art methods.

Abstract

We introduce and develop a general paradigm for combining information across diverse data sources. In broad terms, suppose $φ$ is a parameter of interest, built up via components $ψ_1,\ldots,ψ_k$ from data sources $1,\ldots,k$. The proposed scheme has three steps. First, the Independent Inspection (II) step amounts to investigating each separate data source, translating statistical information to a confidence distribution $C_j(ψ_j)$ for the relevant focus parameter $ψ_j$ associated with data source $j$. Second, Confidence Conversion (CC) techniques are used to translate the confidence distributions to confidence log-likelihood functions, say $\ell_{{\rm con},j}(ψ_j)$. Finally, the Focused Fusion (FF) step uses relevant and context-driven techniques to construct a confidence distribution for the primary focus parameter $φ=φ(ψ_1,\ldots,ψ_k)$, acting on the combined confidence log-likelihood. In traditional setups, the II-CC-FF strategy amounts to versions of meta-analysis, and turns out to be competitive against state-of-the-art methods. Its potential lies in applications to harder problems, however. Illustrations are presented, related to actual applications.

Figures (14)

• Figure 1.1: Confidence curves for the treatment effect in the six continuous trials (dashed, black). In red, the confidence curve combining all the eleven studies. The horizontal red line marks the 95% confidence level. The median confidence estimate is $-83.7$ ml, with 95% interval $[-94.4, -73.1]$.
• Figure 8.1: Simulation results for the basic random effect model. The left plot gives the realised coverage rate of 95% confidence intervals, the right plot gives the median width of these intervals. The results in the top row are for a scenario with small between-study heterogeneity, while in the bottom row the between-study heterogeneity is large.
• Figure 8.2: Simulation results for fixed effect meta-analysis of $2 \times 2$ tables. The left column gives the realised coverage rate of 95% confidence intervals, the middle column gives the median width of these intervals and the right column gives the median bias of the point estimate coming from each of the methods. The top row gives the results for the (log) odds ratio, the middle row for the (log) risk ratio and the bottom row gives the results for the risk difference. When $k$ is small the confidence intervals can sometimes have infinite width (which explains the missing points for the log risk ratio).
• Figure 8.3: Simulation results for random effect meta-analysis of $2 \times 2$ tables. The left plot gives the realised coverage rate of 95% confidence intervals, the right plot gives the median width of these intervals.
• Figure 9.1: Left panel: Point estimates $\widehat{\psi}_j$ with 90% confidence intervals, for the skull stretch parameter $\psi$, across five time epochs (see Table \ref{['table:skulls']}). Right panel: Three confidence curves for the spread parameter $\tau$, with median confidence estimates 0.006 (the direct profile method), 0.272 (the corrected profile method), 0.390 (the $Q_k(\tau)$ method).