OBJECTIVES: Evidence synthesis using methods including meta- and network meta-analysis rely on reported sample size, mean and variance indicators. However, studies commonly report other statistics, such as the median, inter-quartile range (IQR), range, confidence interval, percentiles, or various combinations thereof. Several methods have been developed to estimate mean and standard deviation based on reported median + IQR or median + range, but these are limited to those specific cases, and work best in situations where the underlying distribution approximates normal, which is typically not the case where other statistics are reported. We developed a flexible, likelihood-based approach to estimating underlying distributions from reported quantile information.

METHODS: The procedure is implemented in R and combines interval regression with the density function for observation i in a ranked order of N observations. Let φ and Φ denote the probability and cumulative density function for the distribution in question, respectively (not necessarily limited to the normal). Let q₁, q₂, ... q_m denote the reported quantiles, N be the number of observations, and n = N-m. We get the general expression for the loglikelihood to be maximized: p(q)=Φ(q_1 )^(nq_1 ) ϕ(Q_1 ) (Φ(q_2 )-Φ(q_1 ))^n(q_2-q_1 ) ϕ(Q_2 )... ϕ(Q_m ) (1-Φ(q_m ))^(n(1-q_m)) . The method was compared to the quantile estimation and box-cox approaches developed by McGrath 2020 using simulated data as in McGrath 2020, drawn randomly nine different distributions, with between 25 and 1000 observations.

RESULTS: The likelihood-based approach performed as well or better than the best available alternative methods in all the simulated cases, is applicable to other situations.

CONCLUSIONS: The likelihood-based distribution estimation procedure outperforms methods currently used in meta and network-meta analyses, and allows for estimation of underlying distributions in cases not previously supported. Application of this method will allow inclusion of previously excluded studies in evidence synthesis. Real world evidence cases will be presented graphically.