On probability-based inference under data missing by design

Whether a statistician wants to complement a probability model for observed data with a prior distribution and carry out fully probabilistic inference, or base the inference only on the likelihood function, may be a fundamental question in theory, but in practice it may well be of less importance if the likelihood contains much more information than the prior. Maximum likelihood inference can be justified as a Gaussian approximation at the posterior mode, using flat priors. However, in situations where parametric assumptions in standard statistical models would be too rigid, more flexible model formulation, combined with fully probabilistic inference, can be achieved using hierarchical Bayesian parametrization. This work includes five articles, all of which apply probability modeling under various problems involving incomplete observation. Three of the papers apply maximum likelihood estimation and two of them hierarchical Bayesian modeling. Because maximum likelihood may be presented as a special case of Bayesian inference, but not the other way round, in the introductory part of this work we present a framework for probability-based inference using only Bayesian concepts. We also re-derive some results presented in the original articles using the toolbox equipped herein, to show that they are also justifiable under this more general framework. Here the assumption of exchangeability and de Finetti's representation theorem are applied repeatedly for justifying the use of standard parametric probability models with conditionally independent likelihood contributions. It is argued that this same reasoning can be applied also under sampling from a finite population. The main emphasis here is in probability-based inference under incomplete observation due to study design. This is illustrated using a generic two-phase cohort sampling design as an example. The alternative approaches presented for analysis of such a design are full likelihood, which utilizes all observed information, and conditional likelihood, which is restricted to a completely observed set, conditioning on the rule that generated that set. Conditional likelihood inference is also applied for a joint analysis of prevalence and incidence data, a situation subject to both left censoring and left truncation. Other topics covered are model uncertainty and causal inference using posterior predictive distributions. We formulate a non-parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure, and apply the model in the context of optimal sequential treatment regimes, demonstrating that inference based on posterior predictive distributions is feasible also in this case.

Simplified Treatment of Childhood Acute Bone and Joint Infections

Acute childhood osteomyelitis (OM), septic arthritis (SA), and their combination osteomyelitis with adjacent septic arthritis (OM+SA), are treated with long courses of antimicrobials and immediate surgery. We conducted a prospective multi-center randomized trial among Finnish children at age 3 months to 15 years in 1983-2005. According to the two-by-two factorial study design, children with OM or OM+SA received 20 or 30 days of antimicrobials, whereas those with SA were treated for 10 or 30 days. In addition, the whole series was randomized to be treated with clindamycin or a first-generation cephalosporin. Cases were included only if the causative agent was isolated. The treatment was instituted intravenously, but only for the first 2-4 days. Percutaneous aspiration was done to obtain a representative sample for bacteriology, but all other surgical intervention was kept at a minimum. A total of 265 patients fulfilled our strict inclusion criteria and were analyzed; 106 children had OM, 134 SA, and 25 OM+SA. In the OM group, one child in the long and one child in the short-term treatment group developed sequelae. One child with SA twice developed a late re-infection of the same joint, but the causative agents differed. Regarding surgery, diagnostic arthrocentesis or corticotomy was the only surgical procedure performed in most cases. Routine arthrotomy was not required even in hip arthritis. Serum C-reactive protein (CRP) proved to be a reliable laboratory index in the diagnosis and monitoring of osteoarticular infections. The recovery rate was similar regardless of whether clindamycin or a first-generation cephalosporin was used. We conclude that a course of 20 days of these well-absorbing antimicrobials is sufficient for OM or OM+SA, and 10 days for SA in most cases beyond the neonatal age. A short intravenous phase of only 2-5 days often suffices. CRP gives valuable information in monitoring the course of illness. Besides diagnostic aspiration, surgery should be reserved for selected cases.