Mortality Rates Among Hospitalized Patients With COVID-19 Infection Treated With Tocilizumab and Corticosteroids

This bayesian reanalysis of a previous meta-analysis of 15 randomized clinical trials evaluates whether treatment with tocilizumab was associated with meaningful mortality benefit among hospitalized patients with COVID-19.


Bayesian meta-analysis
Our random-effect model is defined as: where is the observed mean log odds ratio of tocilizumab versus control and 2 is the known sampling variance in study . Because this is a random-effect model, each study has its own distribution, where represents its mean effect. All s are drawn from normal distribution where the mean effect is and the variance 2 , which represents the between-study heterogeneity. is predicted by a no-intercept linear regression, where each subgroup (simple oxygen only; noninvasive ventilation; invasive mechanical ventilation) has its own parameter , representing the effect of each respective subgroup.
In this case, we can assess tocilizumab's effect in each subgroup while assuming a common between-study heterogeneity: • Simple oxygen only = • Noninvasive ventilation = • Invasive mechanical ventilation =

Weakly informative priors
Because we applied the Bayesian framework, we assigned a prior distribution to each parameter. In our main model, we implemented priors that cover plausible values for all parameters, assigning limited density to impossible values, and thus employed little influence in the results (hereafter, known as weakly informative priors). 1,2 These are our weakly informative priors: Now, we will explain the rationale underlying these distributions.
We find highly unlikely that a pharmacological treatment, such as tocilizumab, will yield a 80% odds reduction in 28-days all-cause mortality regardless of the subgroup of patients, as suggested by empirical evidence. 3 Thus, for , we set a prior distribution of (0, 0.82) in the log odds ratio scale. Another way to assess the plausibility of the priors is to perform a prior predictive check 4 , which can be visualized below: Point estimate depicts the median and interval bar depicts the 95% credible (quantile) interval.
As expected, the distribution approximately ranges from 0.2 to 5.0.
Lastly, we will now discuss the weakly informative prior distribution for . Because we wanted to perform unconditional inferences beyond the included studies, we fitted a random-effect meta-analysis. In this model, one assumes there is within-study heterogeneity (represented by 2 , the known sampling variance in study ) and the between-study heterogeneity (represented by ).
Although the definition of small or large between-study heterogeneity is arbitrary, previous work suggests cutoff values ("reasonable" heterogeneity between 0.1 and 0.5, "fairly high" between 0.5 and 1.0, and "fairly extreme" for values larger than 1.0 log odds ratio). 2,5 We added a category for low heterogeneity (between 0 and 0.1).
The Half-Normal(0.5) distribution yields plausible probabilities in each of these ranges: Here are the corresponding probabilities within each of the heterogeneity ranges:

Alternative priors
To check whether the choice of weakly informative priors meaningfully impacted our results or our conclusions, we also fitted models using vague or informative priors.
Vague priors: Informative priors: 6 Here are graphical representations of these normal distributions (along with the weakly informative mentioned before): Log scale Linear scale Here are graphical representations of distributions for the between-study standard deviation ( ) (along with the weakly informative mentioned before):

Deriving risk difference from odds ratio
We used the odds ratio as our primary estimand. 11,14 We derived the risk in the tocilizumab group using the following formula: 11 where is the mortality risk in the tocilizumab group, is the mortality risk in the control group and is the odds ratio.
We then calculated the risk difference (RD) with the following formula: 11

= −
We assumed different mortality risks in each subgroup. For the simple oxygen only and noninvasive ventilation subgroups, we used the average mortality risk in each subgroup based on the data included in this reanalysis. In contrast, regarding the invasive mechanical ventilation (IVM) subgroup, we found a striking discrepancy between the control mortality risk in the data included in this reanalysis (52%) in comparison to another previously published meta-analysis (34% in patients on IVM and using corticosteroids). 10 Thus, we have decided to use 43% (arithmetic mean between 34 and 52) as our reference risk in the IVM subgroup. Recognizing the potential variability of the subgroup baseline risks, we estimated the risk differences with twenty different plausible baseline risks for each subgroup (spanning +-10% change from the reference risks mentioned above).

Predictive analysis to confirm tocilizumab's association with mortality benefit
In brief, we will update our current evidence (as modeled in our main model) with generated randomized clinical trials (RCTs) of different sample sizes comparing tocilizumab to control on patients on invasive mechanical ventilation.
We will use the estimated marginal posterior mean and standard error on this subgroup to create a prior distribution. Then, we will use normal conjugate analyses to update this prior with new data (likelihood) and form updated posterior distributions.

Prior
As described before, we fitted a Bayesian meta-analysis model, from which we estimated marginal posterior distributions on different subgroups. Once again using a Bayesian approach, we updated our current belief, as expressed by the results of our current Bayesian meta-analysis, and which has now become our new prior, with the results of these new generated RCTs to arrive at revised posterior distributions.
Of note, the only subgroup of interest now is the invasive mechanical ventilation: Marginal posterior distribution of the invasive mechanical ventilation subgroup (also depicted in Figure 1A). The interval bar depicts the mean and 95% credible (quantile) interval.
In the linear scale, the mean of this marginal posterior distribution is 0.89. Because we will use normal conjugate analysis, it is of greater interest to evaluate this distribution on the log scale, which is approximately normally distributed. In this case, the mean is -0.12 and the standard error is 0.17.
Thus, in the following normal conjugate analyses, we will use the following distribution as our prior: Normal(−0.12, 0.17 2 )

Likelihood
We created six different RCTs and update the prior distribution mentioned above six separate times.
Assuming the prior is normally distributed and so is the data (likelihood).

(̂,̂2)
The mean and variance of the posterior distributions can be estimated by the following formulas: 5 In summary, we can update a normally distributed prior distribution (shown in the Figure above) with normally distributed data to generate a normally distributed posterior distribution. Based on the posterior's mean and variance, we generated 100,000 random samples (seed number of 123). Now, we must decide the mean and standard deviation of the likelihood. All RCTs will have a mean of -0.26 (log scale). This value is the equal to 0.77 in the linear scale, which was chosen based on WHO's meta-analysis (page 14 in their Supplement 2). 7 This is the mean odds ratio of tocilizumab vs. control in patients using corticosteroids (overall results). We decided to use this value to reflect a skeptical view to heterogeneity of treatment effect across subgroups, 8 and thus the "real" effect in this subgroup would be equal to the largest body of evidence available for tocilizumab in all hospitalized COVID-19 patients on corticosteroids.
Given that all six generated RCTs were set to find the same effect size, the only difference between them was the total number of included patients: 200, 500, 1000, 1500, 2000, or 4000. To calculate the standard deviation of each corresponding prior based on the number of total patients included, one must also assume the proportion of patients included in each treatment arm and the mortality risk in the control arm:

1.
We assumed equal allocation in both treatment arms 2.
Adapting from the suggestions in the GRADE guidelines, 9(p12) we found a striking discrepancy between the control mortality risk in the data included in this reanalysis (52%) in comparison to another previously published meta-analysis(34% in patients on IVM and using corticosteroids). 10 Thus, we have decided to use 43% (arithmetic mean between 34 and 52) as our reference risk in the IVM subgroup 3.
The mortality risk in the tocilizumab was calculated using the following formula: 11 where is the mortality risk in the tocilizumab group, is the mortality risk in the control group and is the odds ratio mentioned above. Thus, the tocilizumab risk is equal to 37%.
In summary, we are generating RCTs with mean OR equal to 0.77, control risk mortality of 43%, and tocilizumab risk of 37%.
Based on these values, we can estimate the standard deviation (SD) with the following formula: 12 = √ 1 + As previous work has shown, 13 we can estimate these values as: where and are the sample sizes in the tocilizumab and control arms, respectively. As mentioned above, we assume equal allocation in both treatment arms, thus = .
Finally, the based on the 6 different sample sizes mentioned above are: