So here is a quick overview :
A. differences in priors : IHDA potentially integrate strong prior information when the user doesn’t select 20% in all control categories. In that case both tools will lead to different results, potentially very different. Expostats will allow informative prior at some point, but probably not before 2023/24, and maybe not in the same form as IHDA.
Even when the user selects 20% in each category in IHDA, there will be small differences between the 2 tools, but this time we expect them to be negligible in most situations. These differences come from the following factors :
B1: the uninformative IHDA prior is different from the Expostat “weakly informative prior” : IHDA has a bonded uniform prior for the log(GM) and log(GSD) parameters. In expostats, log(GM) is also uniform, but bonded wider, and, more importantly, the GSDs have a prior derived from a historical database of GSD values in the workplace. As a consequence, with very small sample size, atypical GSDs in expostats will be pulled towards more typical values. This strategy is the same as adopted by the authors of the Advanced REACH tool.
B2: Treatment of non detects : Bayesian analysis allows the seamless integration of non detects, similar in spirit to the frequentist method called MLE. This is what is used in Expostats. In IHDA, MLE is not always available as an option depending on the version of IHDA.
B3: statistics ( point estimates and confidence intervals for 95th percentile, GSD… ) : Expostats is a fully Bayesian tool, all inferences are made from the posterior samples, i.e. the 25 000 plausible values for the 95th percentile given the prior and our data. As an illustration, the point estimate is the median of the posterior sample, and 95% UCL is the 95th percentile of the posterior sample. When we start implementing informative priors, they will influence the estimation of the statistics.
IHDA on the other hand, only uses Bayesian calculation for the BDA charts, the other statistics are calculated using trafitional frequentist methods. As a consequence, when the user doesn’t select a flat prior, the statistics in IHDA are not consistent with the BDA charts anymore. I believe there is a difference of opinion between Paul and myself about this choice.
B4. Lastly, some slight differences might be observed in the point estimates even when the probabilities for the control categories are the same: This is due to the fact that if you consider the final uncertainty distribution for the 95th percentile for example, even if they yield the same distribution, Expostats and IHDA do not use the same summary of this distribution as a point estimate : Expostats uses the median and IHDA (implied by the MLE frequentist estimates) uses the mode.
Short conclusion : When IHDA is used with a flat prior, the conclusions will be the same in both tools despite possible minor numerical differences.