Equivalence of random-effects and conditional likelihoods for matched case-control studies

Transcription

1 Equivalence of random-effects and conditional likelihoods for matched case-control studies Ken Rice MRC Biostatistics Unit, Cambridge, UK January 8 th 4

2 Motivation Study of genetic c-erbb- exposure and breast cancer (Rohan et al, JNCI, 998) Number of cases exposed Number of controls exposed Each exposure measured imperfectly; Rate of False Positive Exposure.49 Rate of False Negative Exposure. (External validation study, 87 subjects) Is any useful inference possible? /4 Random effects derivations of conditional likelihoods

3 Random effects derivations of conditional likelihoods Matched case-control studies W Some disease of interest, want to find if a binary exposure is associated W For each diseased case, find a control matched for other covariates; age, sex, etc W Then measure exposure of interest W The exposures are outcomes of interest, not the disease status - must build a model for Pr(exposure), not Pr(disease) W Common study design, efficient, simple, popular /4

4 Formal description Random effects derivations of conditional likelihoods W Control exposure ( or ) is Z k, Case exposure is Z k, for pair k W Z k ~Bern( p k ), Z k ~Bern( p k ) W Assume odds ratio identical in all pairs k; p ψ = p k i.e. logit(p k ) = log(ψ ) + logit(p k ) W Generates one nuisance parameter for each pair k p p k k 3/4

5 Problems! Random effects derivations of conditional likelihoods W Maximum likelihood estimates are badly inconsistent W Neyman-Scott problem number of nuisance parameters grows with size of dataset W Usual asymptotics not automatically valid W Sensible looking Bayes analysis can be even worse than MLEs! 4/4

6 Random effects derivations of conditional likelihoods Conditioning: a good solution W Assume T k = total number of exposures (,,) doesn t contain information about ψ W Condition on this (approx) ancillary statistic; conditional likelihood contributions are; Number of controls exposed Number of cases exposed +ψ ψ + ψ W Ratio of discordant pairs gives CMLE for ψ W Well behaved, standard likelihood asymptotics work, but very hard to generalize 5/4

7 A wish list Random effects derivations of conditional likelihoods W Analysis should reduce to conditional likelihood approach in standard situations W Flexible method, easy to accommodate data which is less than ideal W Allow use of prior information on ψ W Fully model based, for simple interpretation W Model criticism desirable, not currently well-supported 6/4

8 Random effects derivations of conditional likelihoods Random-effects: almost a dream solution W All nuisance parameters, e.g. p k, drawn independently from G Integrate likelihood w.r.t. p k, inference on ψ from marginal likelihood W Very similar to a fully Bayesian approach; mixing distribution G prior for p k marginal likelihood posterior for ψ (flat prior) W Flexible, priors on ψ allowed, model based, model criticism possible Just need to choose G but no default exists W To complete the wish list, we need G which equate marginal and conditional likelihoods, if possible 7/4

9 Random effects derivations of conditional likelihoods Random effects analysis W Suppose p k ~ G, the mixing distribution W Marginal likelihood contributions are; Number of cases exposed Number of (Pr( T = E G )) ψ + ψ E G (Pr( T = )) controls exposed + ψ E G (Pr( T = )) (Pr( T = E G )) W Define E G (Pr(T = t)) = m t ; the marginal probabilities 8/4

10 Equivalence Random effects derivations of conditional likelihoods Lemma: Conditional likelihood = marginal likelihood if and only if G makes all m t invariant with respect to ψ W G which satisfy this are called invariant mixing distributions Theorem: Invariant mixing distributions exist, for any matching ratio W Lindsay et al, JASA, 99, proved that for flexible G, CMLE and marginal MLE agree, but only for special datasets W Invariant G depend on ψ W Proofs follow by results on the Stieltjes Moment Problem 9/4

11 Random effects derivations of conditional likelihoods Invariant distributions: an example W An example, for : matched case-control; p k =/ with probability / p k =/ with probability / W Dependence on ψ is present but implicit W Nice coin-tossing interpretation W Get m={.5,.5,.5} other details about G don t affect analysis W Unchanged by relabelling case/controls, or exposure/non exposure; this property holds in some generality; is this non-informative? W This example is pretty but most aren t! Construction is essentially finding polynomial roots Most applications just require existence integrate over G to get m /4

12 Possible applications Random effects derivations of conditional likelihoods These results allow us to put together the conditional analysis with the (many) benefits of a full likelihood approach;. Using the conditional likelihood as a full likelihood; Combining conditional analyses with prior information (ISIS) No extra work. Fitting the conditional likelihood for ψ, and also fitting for m Goodness of fit measures for conditional likelihood analyses (follows) Allowing for misclassification in case-control studies (follows) Inference on complex function of parameters, e.g. ranks in Rasch models Involves complex likelihood function 3. MCMC algorithms for evaluating the conditional likelihood Specify invariant distribution explicitly (polynomial roots) /4

13 Random effects derivations of conditional likelihoods Priors and conditional likelihoods W Already used together, but necessary assumptions are now clear W ISIS case-control study of helicobacter infection and myocardial infarction gave a ballpark estimate incorporating prior beliefs - we can formalise this Odds ratio and 99% intervals /4

14 Random effects derivations of conditional likelihoods Goodness of fit () W A binomial mixture is a vector v with elements which can be written v r = E F n r n r ( θ ) θ ( θ ), r =,,. r W Previous {.5,.5,.5} corresponds to degenerate F; T =/ w.p. W All m which correspond to invariant mixing distributions are binomial mixtures W : correspondence holds in many (useful) special cases W Leads directly to a measure of fit for the conditional model do the observed marginal totals T k look like a binomial mixture?, n 3/4

15 Random effects derivations of conditional likelihoods Goodness of fit () What does this space look like? Sprott s example; conditional likelihood not appropriate Number of controls exposed Number of cases exposed 5 5 W Because m and ψ are orthogonal, some straightforward analyses aren t affected by this restriction 4/4

16 Misclassification Random effects derivations of conditional likelihoods W Define X as the multinomial representation of Z,Z W Usual measurement error model gives mixture for each data point; Pr( X = i) = * * Pr( X = j) Pr( X = i X = j j) W Assume true data X * from conditional model, in multinomial form W Observed data X follow a multinomial model, although complicated by error probabilities W Error probabilities can be known absolutely, or estimated W Derived from sensitivity and specificity of exposure measurement 5/4

17 Return to the motivating problem Study of genetic c-erbb- exposure and breast cancer (Rohan et al, JNCI, 998) Number of cases exposed Number of controls exposed Pr(Observed Exposed Unexposed).49 Pr(Observed Unexposed Exposed). (External validation study, 87 subjects) Assume: Common odds ratio ψ Invariant mixing distribution, with different vector m for :, :5 matching 6/4 Random effects derivations of conditional likelihoods

18 Random effects derivations of conditional likelihoods Extending the random effects model W Perfect data approach W Misclassified data W Actual calculation ψ G, m ψ G, m ψ error rates m p k, p k p k, p k X k X k X k * k= K k= K error rates X k m constrained to be a binomial mixture k= K 7/4

19 Application to breast cancer dataset Analysis allowing for errors; Random effects derivations of conditional likelihoods Odds ratio estimate False Positive Rate False Negative Rate Ignore errors in exposure.7 (.3,.69) NA NA Use plug-in error rates.66 (.3,.84).49. With uncertain error rates.6 (.7,.68).46 (.34,.59). (.,.4) W Odds ratio estimate decreases, interval widens on the log scale (attenuation towards the null) W Some inference is still possible, even with these error rates W Simulations show intervals have good coverage (approx 95%) W Estimates are slightly biased, on the log scale 8/4

20 Random effects derivations of conditional likelihoods Unusual estimate behavior () We do not impute the true data X * above our misclassified observations X, but the likely configurations characterise the analysis; Cell probabilities for true data Unobserved true data Number of cases exposed Number of controls exposed m m +ψ ψ m +ψ m W In this example, ratio of discordant pairs gives ψ > and everything is nice 9/4

21 Random effects derivations of conditional likelihoods Unusual estimate behavior () Even with sensible data and error rates, niceness is often absent; Unobserved true data W Most likely configuration is that all discordant pairs are same type W The maximum likelihood estimate of ψ is at infinity W Need confidence intervals which cope with extreme values of ψ W Most likely configuration is that we have no discordant pairs W The likelihood is maximized along a ridge; any value of ψ equally good W Need a mechanism for reporting no useful information /4

22 Random effects derivations of conditional likelihoods The (simplified) ecological problem W Assume a single x table, Exposed Not exposed Total Controls X n X n Cases X n X n Total X +X n +n X X n +n where only the marginal total T=X +X is observed W Using an invariant prior for the nuisance parameter, the marginal likelihood for ψ is L conditional ( ψ, X, X ) m T W We never learn about ψ - can also occur with standard priors, for special datasets Do we again need to report no useful information? Only if this model fits well? /4

23 Summary Random effects derivations of conditional likelihoods Conditional likelihood is a good approach for matched case-control studies An alternative derivation is available through random effects analysis The random effects derivation is easy to generalise and implement, allowing many new applications in matched case-control studies The random effects derivation uses the whole dataset, adding value to existing analysis at no cost of more data, and providing new inferences in situations beyond matched case-control studies /4

24 Other non-standard likelihoods Cox partial likelihood, already known to be approximately Bayesian; do the same ideas apply? Derivations of good priors our relabelling properties are not found in common non-informative priors; does this property guarantee sensible analysis? Other ideas Rasch models; grid of binary outcomes, Pr i, j ( success) αiβ j = + α β Want to estimate abilities α, condition out difficulties β Categorical exposures; Two nuisance parameters per pair Two odds ratio parameters of interest i j Random effects derivations of conditional likelihoods Student Student Case genotype : Q Control genotype : dd Q : dd Q3 : DD 3/4

25 References and acknowledgements Papers featuring work from this talk; W Rice, K, Equivalence between conditional and mixture approaches to the Rasch model and matched case-control studies, in press, JASA W Rice, K, Discussion of Wakefield, J, Ecological inference for x tables, in press, JRSS A W Rice, K and Holmans, P, Equivalence of conditional and marginal approaches to matched case control studies, with application to misclassification of a biallelic marker, in preparation W Rice, K, Full-likelihood approaches to misclassification of a binary exposure in matched case-control studies, Statistics in Medicine, 3; : W Duffy et al, Misclassification in a matched case-control study with variable matching ratio application to a study of c-erbb- overexpression and breast cancer, Statistics in Medicine, 3; : Thanks to; Random effects derivations of conditional likelihoods W Medical Research Council, ASA Epidemiology Section W David Spiegelhalter, Stephen Duffy, David Clayton, Vern Farewell, Jon Wakefield and several anonymous referees 4/4