Generalized t-statistic for two-group classification.

In the classic discriminant model of two multivariate normal distributions with equal variance matrices, the linear discriminant function is optimal both in terms of the log likelihood ratio and in terms of maximizing the standardized difference (the t-statistic) between the means of the two distributions. In a typical case-control study, normality may be sensible for the control sample but heterogeneity and uncertainty in diagnosis may suggest that a more flexible model is needed for the cases. We generalize the t-statistic approach by finding the linear function which maximizes a standardized difference but with data from one of the groups (the cases) filtered by a possibly nonlinear function U. We study conditions for consistency of the method and find the function U which is optimal in the sense of asymptotic efficiency. Optimality may also extend to other measures of discriminatory efficiency such as the area under the receiver operating characteristic curve. The optimal function U depends on a scalar probability density function which can be estimated non-parametrically using a standard numerical algorithm. A lasso-like version for variable selection is implemented by adding L1-regularization to the generalized t-statistic. Two microarray data sets in the study of asthma and various cancers are used as motivating examples.

Confidence intervals for random effects meta-analysis and robustness to publication bias.

The DerSimonian-Laird confidence interval for the average treatment effect in meta-analysis is widely used in practice when there is heterogeneity between studies. However, it is well known that its coverage probability (the probability that the interval actually includes the true value) can be substantially below the target level of 95 per cent. It can also be very sensitive to publication bias. In this paper, we propose a new confidence interval that has better coverage than the DerSimonian-Laird method, and that is less sensitive to publication bias. The key idea is to note that fixed effects estimates are less sensitive to such biases than random effects estimates, since they put relatively more weight on the larger studies and relatively less weight on the smaller studies. Whereas the DerSimonian-Laird interval is centred on a random effects estimate, we centre our confidence interval on a fixed effects estimate, but allow for heterogeneity by including an assessment of the extra uncertainty induced by the random effects setting. Properties of the resulting confidence interval are studied by simulation and compared with other random effects confidence intervals that have been proposed in the literature. An example is briefly discussed.

The role of secondary outcomes in multivariate meta-analysis.

Univariate meta-analysis concerns a single outcome of interest measured across a number of independent studies. However, many research studies will have also measured secondary outcomes. Multivariate meta-analysis allows us to take these secondary outcomes into account, and can also include studies where the primary outcome is missing. We define the efficiency (E) as the variance of the overall estimate from a multivariate meta-analysis relative to the variance of the overall estimate from a univariate meta-analysis. The extra information gained from a multivariate meta-analysis of n studies is then similar to the extra information gained if a univariate meta-analysis of the primary effect had a further n(1-E)/E studies. The variance contribution of a study's secondary outcomes (its borrowing of strength) can be thought of as a contrast between the variance matrix of the outcomes in that study and the set of variance matrices of all the studies in the meta-analysis. In the bivariate case this is given a simple graphical interpretation as the borrowing of strength plot. We discuss how these findings can also be used in the context of random effects meta-analysis. Our discussion is motivated by a published meta-analysis of ten anti-hypertension clinical trials.

Publication bias affected the estimate of postoperative nausea in an acupoint stimulation systematic review.

BACKGROUND AND OBJECTIVE: To assess the effect of publication bias and country effect on the results and conclusion of a systematic review of wrist P6 acupoint stimulation for the prevention of postoperative nausea and vomiting. METHODS: Reanalysis of a systematic review of 26 randomized trials comparing P6 acupoint stimulation with sham published in the Cochrane Database of Systematic Reviews using the Copas' sensitivity approach. RESULTS: If it is assumed that all studies that have ever been carried out are included, or that those selected for review are truly representative of all such studies, then the estimated relative risk (RR) for nausea was 0.71 (95% CI: 0.58 to 0.88, P<.01) and for vomiting was 0.70 (95% CI: 0.56 to 0.88, P<.01) after adjusting for country effect. For nausea, adjustment for publication bias suggests that the risk has been overestimated. If around 33% of studies have been unpublished, the RR of nausea (0.92, 95% CI: 0.80 to 1.06, P=.25) is no longer significant. For vomiting, however, there is no strong evidence of publication bias. The number of unpublished studies required to substantially overturn the above significant result is implausibly large. CONCLUSION: Publication bias affects the published estimate of postoperative nausea, not vomiting.

A robust P-value for treatment effect in meta-analysis with publication bias.

Publication bias is a major and intractable problem in meta-analysis. There have been several attempts in the literature to adapt methods to allow for such bias, but these are only possible if we are prepared to make strong assumptions about the underlying selection mechanism. We discuss the assumption that the probability that a paper is published may depend in some unspecified way on the P-value being claimed by that study. We suggest a new robust P-value for the overall treatment effect, which turns out to be closely related to the correlation of the associated radial plot. Properties of the method are discussed and illustrated in two examples.

Confidence intervals and P-values for meta-analysis with publication bias.

We study publication bias in meta-analysis by supposing there is a population (y, sigma) of studies which give treatment effect estimates y approximately N(theta, sigma(2)). A selection function describes the probability that each study is selected for review. The overall estimate of theta depends on the studies selected, and hence on the (unknown) selection function. Our previous paper, Copas and Jackson (2004, Biometrics 60, 146-153), studied the maximum bias over all possible selection functions which satisfy the weak condition that large studies (small sigma) are as likely, or more likely, to be selected than small studies (large sigma). This led to a worst-case sensitivity analysis, controlling for the overall fraction of studies selected. However, no account was taken of the effect of selection on the uncertainty in estimation. This article extends the previous work by finding corresponding confidence intervals and P-values, and hence a new sensitivity analysis for publication bias. Two examples are discussed.