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.

Microbiological characterisation of soils : Evaluation of some critical steps in data collection and experimental design

The study of soil microbiota and their activities is central to the understanding of many ecosystem processes such as decomposition and nutrient cycling. The collection of microbiological data from soils generally involves several sequential steps of sampling, pretreatment and laboratory measurements. The reliability of results is dependent on reliable methods in every step. The aim of this thesis was to critically evaluate some central methods and procedures used in soil microbiological studies in order to increase our understanding of the factors that affect the measurement results and to provide guidance and new approaches for the design of experiments. The thesis focuses on four major themes: 1) soil microbiological heterogeneity and sampling, 2) storage of soil samples, 3) DNA extraction from soil, and 4) quantification of specific microbial groups by the most-probable-number (MPN) procedure. Soil heterogeneity and sampling are discussed as a single theme because knowledge on spatial (horizontal and vertical) and temporal variation is crucial when designing sampling procedures. Comparison of adjacent forest, meadow and cropped field plots showed that land use has a strong impact on the degree of horizontal variation of soil enzyme activities and bacterial community structure. However, regardless of the land use, the variation of microbiological characteristics appeared not to have predictable spatial structure at 0.5-10 m. Temporal and soil depth-related patterns were studied in relation to plant growth in cropped soil. The results showed that most enzyme activities and microbial biomass have a clear decreasing trend in the top 40 cm soil profile and a temporal pattern during the growing season. A new procedure for sampling of soil microbiological characteristics based on stratified sampling and pre-characterisation of samples was developed. A practical example demonstrated the potential of the new procedure to reduce the analysis efforts involved in laborious microbiological measurements without loss of precision. The investigation of storage of soil samples revealed that freezing (-20 °C) of small sample aliquots retains the activity of hydrolytic enzymes and the structure of the bacterial community in different soil matrices relatively well whereas air-drying cannot be recommended as a storage method for soil microbiological properties due to large reductions in activity. Freezing below -70 °C was the preferred method of storage for samples with high organic matter content. Comparison of different direct DNA extraction methods showed that the cell lysis treatment has a strong impact on the molecular size of DNA obtained and on the bacterial community structure detected. An improved MPN method for the enumeration of soil naphthalene degraders was introduced as an alternative to more complex MPN protocols or the DNA-based quantification approach. The main advantage of the new method is the simple protocol and the possibility to analyse a large number of samples and replicates simultaneously.