logitOu)=k + P/2, logit(^2k) = ak - P/2 , k =1,..., K

With this model, exp(^) = n1k/n2k is a ratio of success rates, analogous to a relative risk in each centre. (Here, we use notation ^ rather than ft to reflect the effect having a different meaning than in model (1); likewise, the intercept also refers to a different scale, but we use common ak notation for simplicity since this parameter is not the main focus of interest.)

Model (2) has the structural disadvantage of constraining ak±§/2 to be negative, so that %ik falls between 0 and 1. Iterative methods for fitting the model may either ignore this, perhaps yielding estimates of some nik above the permissible [0,1] range, or may fail to converge if estimates at some stage violate this restriction; normally this does not happen when [%ik} are not near 1. This model approximates the logit model when [%ik} are close to 0, but it has interpretations for ratios of probabilities rather than ratios of odds. Model (2) refers to a ratio of success rates, and unlike other models considered in this subsection, when it holds it no longer applies if one interchanges the labelling of 'success' and 'failure' categories.

Simple interpretations also occur with the identity link, by which

For this model, the probability of success is ^1k — n2k = 8 higher for drug than control in each centre. This model has the severe constraint that ak ± 8/2 must fall in [0,1]. Iterative methods often fail for it. It is unlikely to fit well when any %ik are near 0 or 1 as well as somewhat removed from those boundary values, since smaller values of ^1k — %2k typically occur near the parameter space boundary. Thus, the model has less scope than the ones with logit and log links. Even so, unless the model fits very poorly, an advantage of summarizing the effect by 8 is its ease of interpretation by non-statisticians.

In summarizing association for a set of centres by a single measure such as the odds ratio or relative risk, it is preferable to use the measure that is more nearly constant across those centres. In practice, however, for sparse data it is usually difficult to establish superiority of one link function over others, especially when all [%ik} are close to 0. This article discusses all three of these link functions but pays greatest attention to the logit, which is the most popular one in practice.

The standard ML approach for fitting models such as (1) treats jak} as fixed effects. In many applications, such as multi-centre clinical trials and meta analyses, the strata are themselves a sample. When this is true and one would like inferences to apply more generally than to the strata sampled, a random effects approach may be more natural. In practice, the sample of strata are rarely randomly selected. However, Grizzle [16] expressed the belief of many statisticians when he argued that 'Although the clinics are not randomly chosen, the assumption of random clinic effect will result in tests and confidence intervals that better capture the variability inherent in the system more realistically than when clinic effects are considered fixed'. This approach seems reasonable to us for many applications of this type.

For the logit link, a logit-normal random effects model [17] with the same form as (1)

assumes that {ak} are independent from a N(a, a) distribution. The parameter a, itself unknown, summarizes centre heterogeneity in the success probabilities. This model also makes the strong assumption that the treatment effect ft is constant over strata.

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