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The concept of missing at random is central in the literature on statistical analysis with missing data. In general, inference using incomplete data should be based not only on observed data values but should also take account of the pattern of missing values. However, it is often said that if data are missing at random, valid inference using likelihood approaches (including Bayesian) can be obtained ignoring the missingness mechanism. Unfortunately, the term \"missing at random\" has been used inconsistently and not always clearly; there has also been a lack of clarity around the meaning of \"valid inference using likelihood\". These issues have created potential for confusion about the exact conditions under which the missingness mechanism can be ignored, and perhaps fed confusion around the meaning of \"analysis ignoring the missingness mechanism\". Here we provide standardised precise definitions of \"missing at random\" and \"missing completely at random\", in order to promote unification of the theory. Using these definitions we clarify the conditions that suffice for \"valid inference\" to be obtained under a variety of inferential paradigms.

The effects of a job training program, Job Corps, on both employment and wages are evaluated using data from a randomized study. Principal stratification is used to address, simultaneously, the complications of noncompliance, wages that are only partially defined because of nonemployment, and unintended missing outcomes. The first two complications are of substantive interest, whereas the third is a nuisance. The objective is to find a parsimonious model that can be used to inform public policy. We conduct a likelihood-based analysis using finite mixture models estimated by the expectation-maximization (EM) algorithm. We maintain an exclusion restriction assumption for the effect of assignment on employment and wages for noncompliers, but not on missingness. We provide estimates under the \"missing at random\" assumption, and assess the robustness of our results to deviations from it. The plausibility of meaningful restrictions is investigated by means of scaled log-likelihood ratio statistics. Substantive conclusions include the following. For compliers, the effect on employment is negative in the short term; it becomes positive in the long term, but these effects are small at best. For always employed compliers, that is, compliers who are employed whether trained or not trained, positive effects on wages are found at all time periods. Our analysis reveals that background characteristics of individuals differ markedly across the principal strata. We found evidence that the program should have been better targeted, in the sense of being designed differently for different groups of people, and specific suggestions are offered. Previous analyses of this dataset, which did not address all complications in a principled manner, led to less nuanced conclusions about Job Corps.

In studying fluctuations in the size of a blackgrouse (Tetrao tetrix) population, an autoregressive model using climatic conditions appears to follow the changes quite well. However, the deviance of the model is considerably larger than its number of degrees of freedom. A widely used statistical rule of thumb holds that overdispersion is present in such situations, but model selection based on a direct likelihood approach can produce opposing results. Two further examples, of binomial and of Poisson data, have models with deviances that are almost twice the degrees of freedom and yet various overdispersion models do not fit better than the standard model for independent data. This can arise because the rule of thumb only considers a point estimate of dispersion, without regard for any measure of its precision. A reasonable criterion for detecting overdispersion is that the deviance be at least twice the number of degrees of freedom, the familiar Akaike information criterion, but the actual presence of overdispersion should then be checked by some appropriate modelling procedure.

For study of the human sex ratio, one of the most important data sets was collected in Saxony in the 19th century by Geissler. The data contain the sizes of families, with the sex of all children, at the time of registration of the birth of a child. These data are reanalysed to determine how the probability for each sex changes with family size. Three models for overdispersion are fitted: the beta-binomial model of Skellam, the 'multiplicative' binomial model of Altham and the double-binomial model of Efron. For each distribution, both the probability and the dispersion parameters are allowed to vary simultaneously with family size according to two separate regression equations. A finite mixture model is also fitted. The model are fitted using non-linear position regression. They are compared using direct likelihood methods based on the Akaike information criterion. The multiplicative and beta-binomial models provide similar fits, substantially better than that of the double-binomial model. All models show that both the probability that the child is a boy and the dispersion are greater in larger families. There is also some indication that a point probability mass is needed for families containing children uniquely of one sex.

In multidimensional models, simultaneous confidence or credibility regions for continuous parameters hold the overall frequentist long run or Bayesian posterior probability constant at some level such as 95%. This means that, as the dimensionality of the problem increases, the precision or information required about each individual parameter also rapidly increases so parameters have progressively less chance of being set to 0 in such a model selection procedure. These methods do not appropriately answer most of the inference questions that are generally encountered in applied statistics. Thus, sample size, model selection criteria and the estimation of parameter precision are intimately related. In contrast with frequentist and Bayesian procedures, direct likelihood inference, calibrating acceptable likelihood regions by criteria derived from model selection, such as the Akaike information criterion, holds the precision requirements per parameter constant as the dimensionality grows, thus allowing the series of inferences to remain compatible. Other model selection criteria, such as the Bayes information criterion, that depend on the sample size, maintain compatibility but decrease the precision per parameter as the sample increases, so, in the limit, the null model tends to be chosen (Lindley's paradox).

A general function to quantify the weight of evidence in a sample of data for one hypothesis over another is derived from the law of likelihood and from a statistical formalization of inference to the best explanation. For a fixed parameter of interest, the resulting weight of evidence that favors one composite hypothesis over another is the likelihood ratio using the parameter value consistent with each hypothesis that maximizes the likelihood function over the parameter of interest. Since the weight of evidence is generally only known up to a nuisance parameter, it is approximated by replacing the likelihood function with a reduced likelihood function on the interest parameter space. The resulting weight of evidence has both the interpretability of the Bayes factor and the objectivity of the p-value. In addition, the weight of evidence is coherent in the sense that it cannot support a hypothesis over any hypothesis that it entails. Further, when comparing the hypothesis that the parameter lies outside a non-trivial interval to the hypothesis that it lies within the interval, the proposed method of weighing evidence almost always asymptotically favors the correct hypothesis under mild regularity conditions. Even at small sample sizes, replacing a simple hypothesis with an interval hypothesis substantially reduces the probability of observing misleading evidence. Sensitivity of the weight of evidence to hypotheses' specification is mitigated by making them imprecise. The methodology is illustrated in the multiple comparisons setting of gene expression microarray data, and issues with simultaneous inference and multiplicity are addressed.

A general procedure for computing Bayes factors for the comparison of arbitrary models is described, based on the use of the posterior mean of the likelihood under each model rather than the usual prior mean. The use of the posterior mean has several advantages, including reduced sensitivity to variations in the prior and the avoidance of the Lindley paradox in testing point null hypotheses. The frequency properties of the new procedure are evaluated in standard examples, and a non-standard example is analysed to show the considerable differences possible between prior and posterior means of the likelihood. Several different justifications of the procedure are given, and a non-Bayesian direct likelihood interpretation is described.