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

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In the screening group, 82% of men accepted at least one offer of screening. During a median follow-up of 9 years, the cumulative incidence of prostate cancer was 8.2% in the screening group and 4.8% in the control group. The rate ratio for death from prostate cancer in the screening group, as compared with the control group, was 0.80 (95% confidence interval [CI], 0.65 to 0.98; adjusted P = 0.04). The absolute risk difference was 0.71 death per 1000 men. This means that 1410 men would need to be screened and 48 additional cases of prostate cancer would need to be treated to prevent one death from prostate cancer. The analysis of men who were actually screened during the first round (excluding subjects with noncompliance) provided a rate ratio for death from prostate cancer of 0.73 (95% CI, 0.56 to 0.90). Conclusions PSA-based screening reduced the rate of death from prostate cancer by 20% but was associated with a high risk of overdiagnosis. (Current Controlled Trials number, ISRCTN49127736.)

/pdf/screening-and-prostate-cancer-mortality-in-a-randomized-4rctg8nndu.pdf

Screening and Prostate-Cancer Mortality in a Randomized European Study

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Microbiota Modulate Behavioral and Physiological Abnormalities Associated with Neurodevelopmental Disorders

Autism is a complex neuropsychiatric syndrome with a largely unknown etiology. Inflammation during pregnancy may represent a common pathway by which infections and other insults increase risk for the disorder. Hence, we investigated the association between early gestational C-reactive protein (CRP), an established inflammatory biomarker, prospectively assayed in maternal sera, and childhood autism in a large national birth cohort with an extensive serum biobank. Other strengths of the cohort included nearly complete ascertainment of pregnancies in Finland (N=1.2 million) over the study period and national psychiatric registries consisting of virtually all treated autism cases in the population. Increasing maternal CRP levels, classified as a continuous variable, were significantly associated with autism in offspring. For maternal CRP levels in the highest quintile, compared with the lowest quintile, there was a significant, 43% elevated risk. This finding suggests that maternal inflammation may have a significant role in autism, with possible implications for identifying preventive strategies and pathogenic mechanisms in autism and other neurodevelopmental disorders.

https://www.safeminds.org/wp-content/uploads/2014/09/Research-abstracts-flu-vax-and-inflammatory-response-and-autism.pdf

Elevated maternal C-reactive protein and autism in a national birth cohort

SUMMARY Aalen's additive risk model allows the influence of each covariate to vary separately over time. Although allowing greater flexibility of temporal structure than a Cox model, Aalen's model is more limited in the number of covariates it can handle. We introduce a partly parametric version of Aalen's model in which the influence of only a few covariates varies nonparametrically over time, and that of the remaining covariates is constant. Efficient procedures for fitting this new model are developed and studied. The approach is applied to data from the Medical Research Council's myelomatosis trials. it is the first step of a Taylor series expansion of a general hazard function about the zero of the covariate vector. However, in estimating the unknown functions in such a general model there is a variance-bias trade-off that may be critical in small and medium samples. Also, after fitting the model one does not have parameters or formulae that are easily reported. We propose a model that takes the additive structure of Aalen's model and imposes parametric constraints to obtain a semiparametric submodel, which may be more appropriate in some applications. The model will be illustrated with data from clinical trials on myelomatosis. Covariates include treatment, sex and four age strata, which will be treated parametrically, together with serum levels of haemoglobin and f32-microglobulin, whose effects will be investigated nonparametrically. The additive form can be interpreted loosely in terms of unobserved competing risks since the hazard function for the minimum of independent random vari- ables is the sum of the hazard functions for the individual variables. Microglobulin levels are related to kidney function and tumour mass, whereas haemoglobin is unaffected by kidney function. Hence one might anticipate that the hazard function associated with each covariate represents a different cause of death.

https://www.jstor.org/stable/pdfplus/2337122.pdf

A partly parametric additive risk model

This article extends the scope of empirical likelihood methodology ill three directions: to allow for plug-in estimates Of nuisance parameters in estimating equations, slower than root n-rates of convergence, and settings in which there are a relatively large number of estimating equations compared to the sample size. Calibrating empirical likelihood confidence regions with plug-in is sometimes intractable due to the complexity of the asymptotics, so we introduce a bootstrap approximation that call be used in such situations. We provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results.

/pdf/extending-the-scope-of-empirical-likelihood-4dm6fjaut3.pdf

Extending the Scope of Empirical Likelihood

other supplements, but no folic acid (n=2480 [6.6%], with severe language delay in 22 children [0.9%]; OR, 1.04; 95% CI, 0.62-1.74); (2) folic acid only (n=7127 [18.9%], with severe language delay in 28 children [0.4%]; OR, 0.55; 95% CI, 0.35-0.86); and (3) folic acid in combination with other supplements (n=19 005 [50.5%], with severe language delay in 73 children [0.4%]; OR, 0.55; 95% CI, 0.39-0.78). Conclusion Among this Norwegian cohort of mothers and children, maternal use of folic acid supplements in early pregnancy was associated with a reduced risk of severe language delay in children at age 3 years.

/pdf/folic-acid-supplements-in-pregnancy-and-severe-language-4oa0ldlw87.pdf

Folic acid supplements in pregnancy and severe language delay in children

The theory of wavelets is a developing branch of mathematics with a wide range of potential applications. Compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties. They are useful for the detection of edges and singularities in image and sound analysis and for data compression. But most of the wavelet-based procedures currently available do not explicitly account for the presence of noise in the data. A discussion of how this can be done in the setting of some simple nonparametric curve estimation problems is given. Wavelet analogies of some familiar kernel and orthogonal series estimators are introduced, and their finite sample and asymptotic properties are studied. We discover that there is a fundamental instability in the asymptotic variance of wavelet estimators caused by the lack of translation invariance of the wavelet transform. This is related to the properties of certain lacunary sequences. The practi...

Ian W. McKeague

Papers

Elevated maternal C-reactive protein and autism in a national birth cohort

A partly parametric additive risk model

Extending the Scope of Empirical Likelihood

Folic acid supplements in pregnancy and severe language delay in children

Wavelet Methods for Curve Estimation