Placebo Effects in Asthma Randomized Clinical Trials: Their Magnitudes, Causes and Potential for Improving Care

Sample Size and Power Level for the Placebo EffectThe respective sample sizes on the placebo arm in the three GSK RCTs are as follows: (1) GSK-MEA115588 (n=191), (2) GSK-200862 (n=277) and (3) GSK-MEA115575 (n=66). These sample sizes were selected originally to detect material differences in the primary outcome between the treatment and placebo arms. The same power profile therefore is provided by the RCT for detecting the placebo effect. To paraphrase Ortega et al (2014), for example, the authors estimated that with 180 patients in each group, the study would have a power of 90% to detect a 40% decrease in the exacerbation rate, from 2.40 per year in the placebo group to 1.44 per year in each of the mepolizumab groups, at a two-sided significance level of 0.05. In performing this calculation, the authors assumed that the number of exacerbations would follow a negative binomial distribution with a dispersion parameter of 0.8. Univariate Statistical Model for Asthma Exacerbations: Negative Binomial Distribution It is common in the statistical analysis of asthma RCTs to assume that the frequency distribution of acute exacerbations in the study population follows a negative binomial distribution (NB). An NB distribution is assumed in the leading articles published for these three RCTs (Ortega et al, 2014; Chupp et al, 2017; and Bel et al, 2014). It is also assumed in many other RCT studies that use asthma exacerbation counts as an outcome measure, including the case demonstration mentioned in the previous section which uses summary data published in Hanania et al (2016). The genesis of the negative binomial distribution is the following. It is taken as plausible that individual subjects in an RCT experience a Poisson number of exacerbations in any study period. The supporting argument is that exacerbations experienced by a subject are rare events that occur over time in a random pattern. Clinical experience tends to support the assumption. The Poisson distribution is assumed to have a mean rate that is particular to the individual subject. The exacerbation rate is assumed to vary across subjects according to a gamma distribution with a mean equal to the population mean for the study population, which we will denote here by m. This Poisson-gamma mixture produces a negative binomial (NB) probability distribution for the population-level frequency distribution of exacerbations. We use a parameterization for the Poisson-gamma mixture that is the default option in Stata statistical software. In this parametrization, the negative binomial distribution is defined by the mean of the study population m and a dispersion parameter, say d. The dispersion parameter d represents the excess variability produced in the population distribution because subjects do not share a common exacerbation rate. If the dispersion parameter d equals 0 then all subjects have the same exacerbation rate, in which case the negative binomial reduces to a Poisson distribution. The larger is the dispersion parameter d the greater the spread of individual subject rates and the greater the variance of the frequency distribution of exacerbations for the study population.Bivariate Model for Exacerbations in Two Periods: Bivariate Negative Binomial Distribution We are interested in comparing exacerbation rates over two time periods for subjects assigned to the placebo arm, namely, the pre-study and on-trial periods. We therefore extend the preceding univariate negative binomial model to a bivariate negative binomial (BNB) model for two periods. In this bivariate model we assume that a given subject experiences probabilistically independent Poisson counts of exacerbations in the two periods. Exacerbation rates in the two periods can differ for a subject but we assume that these two exacerbation rates share the same random effect (a common multiplier) for the individual subject. This multiplier stays the same for each subject across the two periods but varies among subjects according to a gamma distribution, as described earlier. The two-period Poisson-gamma mixture yields a bivariate negative binomial distribution. The bivariate distribution has marginal distributions that are each of the univariate NB form described earlier. Regression to the Mean The negative binomial model for asthma exacerbations exhibits an implicit ‘regression to the mean' with respect to the relationship of the mean exacerbation rate and the observed number of exacerbations. This statistical artifact is seen in the following way. Consider a subject from the study population whose exacerbations are observed over two independent time periods of the same length under the Poisson-gamma mixture process described above. We index these time periods by 0 and 1. Among subjects that have k0 observed exacerbations during period 0, we can ask: What is the expected number of exacerbations k1 that will be observed for the same subject in the follow-up period 1? The answer is that the expected number of exacerbations is a weighted average of the observed number of exacerbations k0 and the study population mean m. Because the weight is a fractional value, the weighting causes `shrinkage' of k0 toward the population mean m, producing what is referred to as 'regression to the mean'. In other words, the number of exacerbations in the follow-up period 1 will tend to be closer to the population mean than the observed number of exacerbations in period 0. For instance, in the Hanania et al case analysis mentioned earlier, the RCT has one subgroup of subjects who had 2 or more exacerbations in a pre-study 12-month period. The negative binomial model in this case shows that the average exacerbations that will be experienced by these subjects in the following 12-month trial period would be 1.68 exacerbations under the same pre-study conditions. It is also found that the estimated mean annual exacerbation rate (the estimate of m) for the whole study population under pre-study conditions is 1.33. This drop from 2 or more exacerbations to an average of 1.68 illustrates the regression to the mean that is inherent to the negative binomial model. The implication of the regression-to-the-mean artifact is that a shrinkage adjustment must be applied to the observed exacerbation count in order to obtain an unbiased estimate of the placebo effect.Conceptual Layout of an RCTTable 1 below presents a conceptual layout of an RCT for our analysis of placebo effects. The layout identifies three sub-contexts: the pre-study or baseline condition (B), the study placebo arm (P) and the study treatment arm (T). This layout considers only one active treatment arm although it could be extended as needed to multiple arms. The layout also shows the set up for the analysis of a subgroup variable with G groups. Only one such variable is used for this presentation but an extension to multiple subgroup variables is straightforward. The table gives notation for the overall mean exacerbation rate for each sub-context as well as for the sub-groups in each context. For example, mP2 denotes the mean exacerbation rate on the placebo arm for subgroup 2. These means are parameters of negative binomial distributions, as described earlier. Estimation methods for these means are taken up shortly. Table 1: Conceptual layout of an RCT and notation for an analysis of placebo effects        Study Arms---------------------------------------------------------------------Subgroup   Pre-Study (Baseline)   Placebo (P)   Treatment (T)--------------------------------------------------------------------- 1    mB1    mP1 mT1 2    mB2    mP2    mT2 …    …    …    … G    mBG    mPG    mTG---------------------------------------------------------------------Study population mB    mP    mT---------------------------------------------------------------We find it useful to define the overall placebo effect observed in an RCT in two mathematical ways. First, we consider the arithmetic difference in annual mean exacerbation rates between the pre-study and on-trial periods, that is, the arithmetic difference mB-mP in the notation of Table 1. Second, we consider the log-ratio ln(mB/mP) as an alternative definition of the placebo effect. The log-ratio is useful for a regression analysis of subgroup and covariate effects, as we show below. Exclusion/Inclusion Criteria Related to the Outcome MeasureStudy populations frequently use inclusion/exclusion criteria that restrict the permissible range for the outcome measure in subject selection. For example, citing Ortega et al (2014, p.1199) for our first RCT, GSK-MEA115588: “All patients had to have had at least two asthma exacerbations in the previous year that were treated with systemic glucocorticoids while they were receiving treatment with at least 880 μg of fluticasone propionate or the equivalent by inhalation per day and at least 3 months of treatment with an additional controller.” Thus, subjects required at least two asthma exacerbations in the year prior to enrolment. The effect of such an inclusion/exclusion criterion is to force an explicit adjustment for bias from regression to the mean into the statistical analysis, as we described earlier.Estimation Methods and Statistical SoftwareMaximum likelihood estimation will be used to estimate the parameters of the negative binomial distribution from observed exacerbation counts for any subgroup of the study population during the pre-study period. Using the length of the pre-study observation period, the mean of the negative binomial distribution can be converted to, say, an average annual exacerbation rate under pre-study conditions. The use of maximum likelihood estimation allows us to report asymptotic inferences about estimates such as asymptotic standard errors of estimates, confidence limits and Wald tests with p-values.Numerical gradient optimization routines will be custom-made for the maximum likelihood estimation involved in this research program. The statistical analysis will be done using Stata software. Some of the requisite routines have already been developed and used for the case demonstration mentioned earlier. Selective simulation routines will also be developed and used to verify numerically more complex statistical formulas to ensure that mathematical developments are sound. Regression AnalysisThe BNB model allows us to model exacerbation counts of individual subjects during the pre-study and on-trial periods and also to attribute the outcome change, if any, to placebo effects adjusted for subgroup and covariate effects of subjects using multivariate regression methods. Maximum likelihood estimation will be used to estimate regression functions taking account of subgroup and covariate effects. To elaborate on the regression setup, we assume that the log-ratio ln(mB/mP)) is a linear combination of covariates or subgroup indicator variables xj, j=1,…,h, as follows:ln(mB/mP)= b0+b1*x1+b2*x2+…+bh*xhThis regression function will estimate the percentage or relative impact of covariates and subgroup indicator variables on the placebo effect as defined by the ratio mB/mP.Covariates such as age, sex, BMI, smoking status, duration of asthma, and FEV1 reversibility will be used in the regression analysis. Model FitHaving subject-level data from the three GSK RCTs will provide an opportunity to check the goodness of fit of the negative binomial model for asthma exacerbation counts using standard tests, such as chi-square tests. Any significant heterogeneity among subgroups would be taken into account in these tests. Separate tests will be performed for exacerbation distributions from the pre-study period, the placebo arm and the active treatment arm, as well as testing of the bivariate negative binomial distribution for the pre-study and on-trial periods. Adjustments for Multiple InferencesAlthough we will be making multiple inferences, we do not intend to make adjustments for multiple inferences such as the Bonferroni correction. Readers will be left to subjectively account for the influence of multiple comparisons, multiple tests, and the like.Association (if any) of Treatment Effects with Placebo EffectsIn Table 1, we included a sub-context for asthma exacerbation rates that occur under an active treatment arm of the RCT. The original purpose of an RCT, of course, concerns the difference mP-mT , the difference in mean exacerbation rates between the treatment arm (for whatever therapeutic agent is under study) and the placebo arm. We have already mentioned that we may gain a better understanding of treatment effects detected in an RCT if we study their association with placebo effects across subgroups. In other words, we can ask, To what extent and in what way are the differences mPg-mTg correlated with the differences mBg-mPg across subgroups g=1,…,G? For example, do treatment effects tend to be larger in subgroups where placebo effects are larger? We will be examining these associations in the three GSK RCTs selected for this project in relation to the impact of mepolizumab in cases of eosinophilic asthma. Placebo Effects in Other Study OutcomesIn addition to exacerbation rates, other study outcomes, such as lung measures (FEV1) and questionnaire scores (ACQ and SGRQ scores), will be considered as secondary indicators of placebo effects. These other outcomes will not be the main focus of our project but may be studied. These outcome measures will have their own statistical models and methods that we would develop if we are led to consider them.