Combining diverse information sources with the II-CC-FF paradigm

Transcription

1 Combining diverse information sources with the II-CC-FF paradigm Céline Cunen (joint work with Nils Lid Hjort) University of Oslo 18/07/2017 1/25

2 The Problem - Combination of information We have independent data sources 1,..., k providing information about parameters ψ 1,..., ψ k. Our interest is in the overall focus parameter φ = φ(ψ 1,..., ψ k ). II-CC-FF: a general framework to provide inference for φ in cases like this. Similar to likelihood synthesis from Schweder and Hjort (1996) Beyond ordinary meta-analysis: not restricted to cases where the sources inform on the same parameter we can deal with complex functions of the parameters from each source: φ = φ(ψ 1,..., ψ k ) we can deal with cases where we only have summary statistics from some or all of the sources we can handle very diverse sources for example combining parametric and non-parametric analyses 2/25

3 Confidence distributions (CD) Confidence distribution Confidence density Confidence curve C(µ) c(µ) cc(µ) µ µ µ a posterior without having to specify a prior a sample-dependent distribution function on the parameter space can be used for inference (for example for constructing CIs of all levels) 3/25

4 Requirements for CDs Definition A function C(θ, Y ) is called a confidence distribution for a parameter θ if: C(θ, Y ) is a cumulative distribution function on the parameter space at the true parameter value θ = θ 0, C(θ 0, Y ) as a function of the random sample Y follows the uniform distribution U[0,1] The second requirement ensures that all confidence intervals have the correct coverage. More on CDs in Confidence, Likelihood, Probability. (Schweder and Hjort, 2016.) 4/25

5 Outline II-CC-FF - general procedure and some illustrations Examples illustrating what II-CC-FF can do Classic meta-analysis More complex meta-analysis: Blood loss Random effects: All Blacks Very diverse sources: First word confidence curves γ 5/25

6 II-CC-FF - overview Combining information, for inference about a focus parameter φ = φ(ψ 1,..., ψ k ): II: Independent Inspection: From data source y i to estimates and intervals, in the form of a confidence distribution/curve: y i = C i (ψ i ) CC: Confidence Conversion: From the confidence distribution to a confidence log-likelihood, C i (ψ i ) = l c,i (ψ i ) FF: Focused Fusion: Use the combined confidence log-likelihood l f = k i=1 l c,i(ψ i ) to construct a CD for the given focus φ = φ(ψ 1,..., ψ k ), often via profiling: l f (ψ 1,..., ψ k ) = C fusion (φ) 6/25

7 CC - Confidence Conversion The most difficult step? C i (ψ i ) = l c,i (ψ i ) In some cases we will already have a log-likelihood for ψ i from the II-step and then there are no problems. In other cases, the confidence curves from the II-step are not constructed via likelihoods. Then we need to do something else (and be more careful). A simple and general method - the normal conversion: l c (ψ) = 1 2 Γ 1 1 (cc(ψ, y)) = 1 2 {Φ 1 (C(ψ, y))} 2. [Note that the confidence log-likelihood is not equal to the log-confidence density (log C(ψ)/ ψ). Except under the normal model.] 7/25

8 CD theorem via Wilks approximation In many regular cases we have, at the true parameter value ψ 0 : 2{l n,prof ( ˆψ) l n,prof (ψ 0 )} d χ 2 1, as the sample size n increases. This gives us our favourite approximate confidence curve construction, cc(ψ) = Γ 1 (2{l n,prof ( ˆψ) l n,prof (ψ)}). 8/25

9 We can deal with 1: Classic meta-analysis Assume all sources inform on the exact same parameter ψ 1 = = ψ k = ψ, and that each source provide estimators ˆψ i that are normally distributed N(ψ, σ 2 j ) with known σ js. II: Data source y i leads to C i (ψ) = Φ((ψ ˆψ i )/σ i ). CC: From C i (ψ) to l c,i (ψ) = 1 2 (ψ ˆψ i ) 2 /σ 2 i. FF: Summing l f (ψ) = k i=1 l c,i(ψ) leads to the classic answer k i=1 ˆψ = ˆψ ( ) i /σi 2 k k N ψ, ( 1/σ 2 i=1 1/σ2 i ) 1. i i=1 9/25

10 We can deal with 2: more complicated meta-analysis We do not have access to the full dataset (only summaries), and the studies differ in their reported outcomes: some studies report continuous outcomes, others report counts of a binary outcome. Example from Whitehead et al. (1999): Blood loss during labor. Does treatment with oxytocic drugs help reduce blood loss? Total of 11 studies, 6 studies report summary statistics of the continuous outcome (the actual blood loss in ml): Treatment n Mean SD Study 1 Control Treatment Study studies report counts of a binary outcome (yes = blood loss greater than 500 ml): Treatment Yes No Total Study 7 Control Treatment Study /25

11 We can deal with 2: more complicated meta-analysis Model: y ij = α i + βz ij + ɛ ij ɛ ij N(0, σ 2 ) II and CC: If study i has a continuous outcome we have l c,i (α i, β, σ) = (n 1 + n 2 ) log(σ) {(n 1 1)s (n 2 1)s n 1 (ȳ 1 α i ) 2 + n 2 (ȳ 2 α i β) 2 }/(2σ 2 ). If study i has a binary outcome we have l b,i (α i, β, σ) = x 01 log{φ((500 α i )/σ)} + x 11 log{1 Φ((500 α i )/σ)} + x 02 log{φ((500 α i β)/σ)} + x 12 log{1 Φ((500 α i β)/σ)}. FF: Summing l f (α 1,..., α k, β, σ) = k c i=1 l c,i(α i, β, σ) + k b i=1 l b,i(α i, β, σ) and then profiling l f,prof (β) = l f (ˆα 1 (β),..., ˆα k (β), β, ˆσ(β)) and we get a combined confidence curve for β by using the Wilks approximation cc(β) = Γ 1 {2(l f,prof ( ˆβ) l f,prof (β))}. 11/25

12 We can deal with 2: more complicated meta-analysis confidence curve Oxytocic drugs is seen to decrease blood loss. β 12/25

13 We can deal with 3: Random effects! 13/25

14 We can deal with 3: Random effects We have measures of passage times in 10 Rugby games ( studies). There are 5 games before a certain change of rules and 5 after. Model: y ij Gamma(a i, b i ) Say we are interested in the standard deviation of passage times κ i = a i /b i. It is relatively straightforward to construct confidence curves for each κ i (by profiling and Wilks approximation). II: cc i (κ i ) = Γ 1 {2(l i,prof (ˆκ i ) l i,prof (κ i ))} 14/25

15 We can deal with 3: Random effects confidence curve Before After κ There seems to be smaller standard deviations in passage times (smaller κs) after the rule change. But there is substantial spread in the mean κs between different games - do we need random effects? 15/25

16 We can deal with 3: Random effects We assume : Before rule change κ 1,...κ 5 N(κ B, τ 2 B ) and After rule change κ 6,...κ 10 N(κ A, τ 2 A ). We are interested in making confidence curves for κ B, κ A, and the ratio between them δ = κ B /κ A. CC: We already have the l i,prof (κ i ) from the II-step, but now we need l i (κ B, τ B ) (and similarly for the parameters after the rule change): l i (κ B, τ B ) = log[ exp{l i,prof (κ i ) l i,prof (ˆκ i )} 1 τ B φ (We integrate numerically or use Laplace approximation) ( κi κ B τ B ) dκ i ] 16/25

17 We can deal with 3: Random effects FF: Summing l f (κ B, τ B ) = k B i=1 l i (κ B, τ B ), profile to get l f,prof (κ B ) and Wilks approximation to get cc(κ B ). Similarly for cc(κ A ). confidence curve Before After κ 17/25

18 We can deal with 3: Random effects The ratio δ = κ B /κ A. Sum, profile and Wilks: l ff (κ B, δ) = l f,prof,b (κ B ) + l f,prof,a (κ B /δ), then l ff (δ) = l ff (ˆκ B (δ), δ). confidence curve δ 18/25

19 We can deal with 4: Very diverse sources We have two sources: A large study: 1640 parents report the age (in months) at which their child said its first word. Ranges from 1 (!) to 25. A small study: 51 parents report the age (in months) at which their child said its first word. Here we have some covariate information: gender (of the child). Focus: When do girls start to speak? And when do boys start to speak? Model: proportional hazards model (no censoring here - but we could have dealt with that too!) Data from: Schneider, Yurovsky & Frank (2015). Large-scale investigations of variability in children s first words. In CogSci2015 Proceedings. 19/25

20 We can deal with 4: age at first word Focus: probability that a child with covariate information x 0 does not speak at the age of 12 months S(t 0 x0) = e H 0(t 0 )e xt 0 β = S 0 (t 0 ) ext 0 β = (1 F 0 (t 0 )) ext 0 β with t 0 = 12. Large study: will give information about F 0 at t 0 - = cc 1 (F 0 (t 0 )). Non-parametric! Small study: will give information about β = cc 2 (β). Cox model - Semi-parametric! with II-CC-FF we can combine these and obtain a cc for S(t 0 x0) with t 0 = /25

21 We can deal with 4: age at first word Obtaining a confidence curve for the baseline F 0 (t 0 ). An exact CD based on the binomial distribution. confidence curve F 0 (12) 21/25

22 We can deal with 4: age at first word Obtaining a confidence curve for the coefficient β (taking care to define gender as 1/-1, so that the value 0 corresponds to the overall mean) Approximate CD based on the normal distribution (here we only need the summary statistics: estimate and standard error). confidence curve /25

23 We can deal with 4: age at first word FF: summing and profiling l f,prof (S(t 0 x0)) = max{l f (F 0 (t 0 ), β) : (1 F 0 (t 0 )) ext 0 β = S(t 0 x 0 )} and then Wilks approximation. confidence curve Girl Boy S(12) 23/25

24 We can deal with 4: age at first word Comparing with results from small source only. confidence curve Girl Boy S(12) 24/25

25 Concluding remarks and references What can II-CC-FF do? Deal with summary statistics Deal with complex functions of the parameters from each source: φ = φ(ψ 1,..., ψ k ) Deal with very diverse sources (hard and soft data,...) References: Liu, Liu and Xie (2015): Multivariate meta-analysis of heterogeneous studies using only summary statistics: efficiency and robustness. JASA. Schneider & Frank (2015). Large-scale investigations of variability in children s first words. In CogSci2015 Proceedings. Schweder & Hjort (1996). Bayesian synthesis or likelihood synthesis what does Borel s paradox say? Reports of the International Whaling Commision, 46. Schweder & Hjort (2013). Integrating confidence intervals, likelihoods and confidence distributions. Proceedings 59th World Statistics Congress. Schweder & Hjort (2016). Confidence, Likelihood, Probability. Cambridge University Press. Whitehead, Bailey and Elbourne (1999). Combining summaries of binary outcomes with those of continuous outcomes in a meta-analysis. Journal of Biopharmaceutical Statistics. 25/25