Objective.-This exploratory trial evaluated the safety and efficacy of multiple treatments of botulinum toxin type A (BoNTA; BOTOX , Allergan, Inc., Irvine, CA, USA) as prophylactic treatment of episodic migraine headaches.Design and Methods.-This was an 11-month randomized, double-blind, placebo-controlled, exploratory study. Patients were screened during a 30-day baseline period, and eligible patients with 4 or more migraine episodes and ≤15 headache days entered a single-blind 30-day placebo run-in period. Patients were classified as placebo nonresponders (PNR) if they had at least 4 moderate-to-severe migraine episodes and did not experience at least a 50% decrease from baseline in the frequency of migraine episodes following their placebo treatment. All other subjects were classified as placebo responders (PR). Patients were randomized within each stratum (PNR, PR) to 3 treatments with BoNTA (110 to 260 U of BoNTA per treatment cycle) or placebo at 90-day intervals using a modified follow-the-pain treatment paradigm. The primary efficacy outcome measure was the mean change from baseline in the frequency of migraine episodes for the 30 days prior to day 180 in the PNR group. Secondary efficacy measures included the proportion of patients with a decrease from baseline of 50% or more migraine episodes per 30-day period. Patients were allowed to take concomitant acute and prophylactic headache medications. Adverse events were reported.Results.-A total of 809 patients were screened and 369 patients (89.2% female; mean age, 45 years; range, 20 to 65 years) entered the placebo run-in period and were subsequently randomized to BoNTA or placebo. The mean total dose of BoNTA was 190.5 units (U) (range, 110 U to 260 U). The predetermined primary efficacy endpoint was not met. Substantial mean improvements of 2.4 and 2.2 fewer migraine episodes per month at day 180 in the PNR stratum treated with BoNTA and placebo, respectively, were observed (P > .999). From day 180 through the end of the study (day 270) at least 50% of all patients in each treatment group had a decrease from baseline of 50% or more migraine episodes per 30-day period. However, in the group of patients with ≥12 headache days at baseline (and ≤15 headache days), BoNTA patients experienced a mean change from baseline of −4.0 headache episodes at day 180 compared with −1.9 headache episodes in the placebo group (P= .048). The majority of treatment-related adverse events were transient and mild to moderate in severity. Only 7 patients (1.9%) discontinued the study due to adverse events (6 BoNTA, 1 placebo).

The method of Laplace is used to approximate posterior probabilities for a collection of polynomial regression models when the errors follow a process with a noninvertible moving average component. These results are useful in the problem of period-change analysis of variable stars and in assessing the posterior probability that a time series with trend has been overdifferenced. The nonstandard covariance structure induced by a noninvertible moving average process can invalidate the standard Laplace method. A number of analytical tools is used to produce corrected Laplace approximations. These tools include viewing the covariance matrix of the observations as tending to a differential operator. The use of such an operator and its Green's function provides a convenient and systematic method of asymptotically inverting the covariance matrix.In certain cases there are two different Laplace approximations, and the appropriate one to use depends upon unknown parameters. This problem is dealt with by using a weighted geometric mean of the candidate approximations, where the weights are completely data-based and such that, asymptotically, the correct approximation is used. The new methodology is applied to an analysis of the prototypical long-period variable star known as Mira.