Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.

/pdf/bayesian-measures-of-model-complexity-and-fit-1ouzmtfu7x.pdf

Bayesian measures of model complexity and fit

Propensity score matching (PSM) has become a popular approach to estimate causal treatment effects. It is widely applied when evaluating labour market policies, but empirical examples can be found in very diverse fields of study. Once the researcher has decided to use PSM, he is confronted with a lot of questions regarding its implementation. To begin with, a first decision has to be made concerning the estimation of the propensity score. Following that one has to decide which matching algorithm to choose and determine the region of common support. Subsequently, the matching quality has to be assessed and treatment effects and their standard errors have to be estimated. Furthermore, questions like 'what to do if there is choice-based sampling?' or 'when to measure effects?' can be important in empirical studies. Finally, one might also want to test the sensitivity of estimated treatment effects with respect to unobserved heterogeneity or failure of the common support condition. Each implementation step involves a lot of decisions and different approaches can be thought of. The aim of this paper is to discuss these implementation issues and give some guidance to researchers who want to use PSM for evaluation purposes.

/pdf/some-practical-guidance-for-the-implementation-of-propensity-2qsp4y0xke.pdf

Some practical guidance for the implementation of propensity score matching

Problems involving causal inference have dogged at the heels of statistics since its earliest days. Correlation does not imply causation, and yet causal conclusions drawn from a carefully designed experiment are often valid. What can a statistical model say about causation? This question is addressed by using a particular model for causal inference (Holland and Rubin 1983; Rubin 1974) to critique the discussions of other writers on causation and causal inference. These include selected philosophers, medical researchers, statisticians, econometricians, and proponents of causal modeling.

/pdf/statistics-and-causal-inference-254v25vueh.pdf

Statistics and Causal Inference

We outline a framework for causal inference in settings where assignment to a binary treatment is ignorable, but compliance with the assignment is not perfect so that the receipt of treatment is nonignorable. To address the problems associated with comparing subjects by the ignorable assignment—an “intention-to-treat analysis”—we make use of instrumental variables, which have long been used by economists in the context of regression models with constant treatment effects. We show that the instrumental variables (IV) estimand can be embedded within the Rubin Causal Model (RCM) and that under some simple and easily interpretable assumptions, the IV estimand is the average causal effect for a subgroup of units, the compliers. Without these assumptions, the IV estimand is simply the ratio of intention-to-treat causal estimands with no interpretation as an average causal effect. The advantages of embedding the IV approach in the RCM are that it clarifies the nature of critical assumptions needed for a...

/pdf/identification-of-causal-effects-using-instrumental-16wp7tkypl.pdf

Identification of Causal Effects Using Instrumental Variables

Identification of Causal effects Using Instrumental Variables

R.A. Fisher's classic text on the design of experiments is the principal source of inspiration for a mode of data interpretation that is usually characterized as randomization analysis. In Chapter III of this text, Fisher briefly commented on how to make a randomization test on some data generated by a Darwin experiment. Two variants of this randomization test are discussed in this article. The variant that is discussed in Section 4 may be regarded as the forerunner of all nonparametric tests. The original variant of the test is discussed in Section 6. The author concludes that the Fisher randomization test is not logically viable.

/pdf/randomization-analysis-of-experimental-data-the-fisher-hsyfyhph2h.pdf

Randomization Analysis of Experimental Data: The Fisher Randomization Test

If {Ρ θ}, θЄΩ, be a family of probability measures on an abstract sample space \(\mathcal {G}\) and Τ be a sufficient statistic for θ then for a statistic T 1 to be stochastically independent of Τ it is necessary that the probability distribution of T 1 be independent of θ. The condition is also sufficient if Τ be a boundedly complete sufficient statistic. Certain well-known results of distribution theory follow immediately from the above considerations. For instance, if x 1, x 2,. . . , x n , are independent Ν(μ, σ)’s then the sample mean \(\bar x\) and the sample variance s 2 are mutually independent and are jointly independent of any statistic f (real or vector valued) that is independent of change of scale and origin. It is also deduced that if x 1, x 2, . . ., x n are independent random variables such that their joint distribution involves an unknown location parameter θ then there can exist a linear boundedly complete sufficient statistic for θ only if the x’s are all normal. Similar characterizations for the Gamma distribution also are indicated.

/pdf/on-statistics-independent-of-a-complete-sufficient-statistic-2iexheimt6.pdf

On Statistics Independent of a Complete Sufficient Statistic

In part one of this essay the notion of ‘statistical information generated by a data’ is formulated in terms of some intuitively appealing principles of data analysis. The author comes out very strongly in favour of the unrestricted likelihood principle after demonstrating (to his own satisfaction) the reasonableness of the Bayes-Fisher postulate that, within the framework of a particular statistical model, the ‘whole of the relevant information in the data’ must be supposed to be summarised in the likelihood function generated by the data.

/pdf/statistical-information-and-likelihood-512rhj9nyb.pdf

Statistical Information and Likelihood

Though the marginal distributions of the ancillary statistics are independent of the parameter they are not useless or informationless. A set of ancillaries may sometimes summarise the whole of the information contained in the sample. A classification of the ancillaries in terms of the partial order of their information content is attempted here. In general there are many maximal ancillaries, Among the minimal ancillaries there exists a unique largest one. When there exists a complete sufficient statistic, the problem of tracking down the maximal and minimal ancillaries becomes greatly simplified.

/pdf/the-family-of-ancillary-statistics-22tti2kh3v.pdf

The Family of Ancillary Statistics

Current Issues in Statistical Inference: Essays in Honor of D. Basu

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