We included only double blind, parallel design studies with patients (all adults) randomised to either paroxetine or placebo. Altogether 16 studies met these criteria (references 79 to 93 and 95 in the Expert Report), containing respectively 916 and 550 paroxetine and placebo treated patients. The study period was in most instances 6 weeks. One important exception was a study (reference 91) with a preponderance of paroxetine use over placebo and lasting for 17 weeks. Patients were excluded from the studies after a suicide-related event. Taking this censoring into account, paroxetine treatment made up 190.7 patient years altogether and placebo 73.3 patient years. Suicide-related events could be found in tables in the Expert Report, in the adverse reactions section in the individual study reports, and in the individual patient descriptions.
We let θp be the intensity per year of a suicide attempt in the placebo group and θd the intensity per year in the drug group, for a random patient in the 16 studies; correspondingly, Xp and Xd represent the total numbers of suicide attempts. We can have at most one suicide attempt for each patient. Taking this censoring into account, we denoted the corresponding patient years in the 16 studies combined by mp and md. In addition, patients in both the placebo and drug groups are supposed to behave in a similar manner. It then follows that the likelihood of the experiment corresponds to Xp and Xd having Poisson distributions respectively with parameters (mpθp) and (md θd). In addition, we assume that the two variables were conditionally independent given the parameters. The corresponding observed data are (xp, mp) and (xd, md), and the prior information is denoted by (xop, mop) and (xod, mod).
The Bayesian approach is based on the construction of probability distributions for θp
. This does not mean that these parameters are to be interpreted as random variables, but our knowledge of the parameters is uncertain and we describe this uncertainty with the help of probability distributions. Probability distributions describing our initial uncertainty are called prior distributions (that is, before real data are collected). When the real data are taken into account, the prior distributions are updated by Bayes' formula to posterior distributions. An excellent introduction to Bayesian methods in medicine is given by Spiegelhalter et al. [6
We assume that the prior distribution for θp is gamma, with parameters xop and mop, while correspondingly θd has the parameters xod and mod and is assumed to be independent of the prior distribution for θp. Hence, standard Bayesian theory gives the posterior distribution of θp as gamma, with parameters xop + xp and mop + mp, while θd will have the parameters xod + xd and mod + md. We performed simulations by making 80000 random draws of θd and θp from their independent gamma posterior distributions, computed the logarithms of the ratios θd/θp, and constructed diagrams by applying a standard density estimation technique to these logarithms. (The logarithm was introduced to avoid an unwelcome feature of the density estimation method.) Note that the logarithm of the ratio θd/θp is greater than zero whenever θd is greater than θp. Hence, we calculated the probabilities that medication with paroxetine is associated with an increased intensity of a suicide attempt per year as the proportions of logarithmic ratios greater than zero in the samples. This corresponds to areas below the densities to the right of zero in the diagrams.
The grounds for a pessimistic prior have been given by Healy and Whitaker [7
] who, relating the occurrence of suicidal activities to the use of antidepressant drugs, estimated an odds ratio of 2.4 from evidence given in clinical trials, epidemiological observations and case histories. The clinical trial data they used included, but were not restricted to, studies with the active drugs randomised against placebo. Mathematically, we chose to express this view as equivalent to observing two (xod
) events with paroxetine during 50 (mod
) patient years and one (xop
) with placebo during 50 (mop
) patient years, adding up to 3 attempts per 100 patient years, which is similar to our observed average value for paroxetine and placebo taken together. We based the calculations on a total of only 100 (mod
) patient years in the prior, compared to 264 (md
) patient years in the real data, in order to increase the importance of the real data over the prior information. The slightly optimistic and slightly pessimistic priors represent respectively a paper by Lapierre [8
] (appearing in tandem with Healy and Whitaker) and the article that reported suicidal ideation in children medicated with paroxetine [2
]. The former author took the attitude that, if anything, there were slight signs of reduced suicidal activity connected with antidepressants, whereas the latter authors left the reader with the assumption that the observed increased suicidal ideation in children must somehow be reflected in adults. We assigned the numbers of suicidal patients on paroxetine and placebo per 50 patient years to be respectively 1.35 and 1.65 and vice versa