Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates

Imputation, weighting, direct likelihood, and direct Bayesian inference (Rubin, 1976) are important approaches for missing data regression. Many useful semiparametric estimators have been developed for regression analysis of data with missing covariates or outcomes. It has been established that some semiparametric estimators are asymptotically equivalent, but it has not been shown that many are numerically the same. We applied some existing methods to a bladder cancer case-control study and noted that they were the same numerically when the observed covariates and outcomes are categorical. To understand the analytical background of this finding, we further show that when observed covariates and outcomes are categorical, some estimators are not only asymptotically equivalent but also actually numerically identical. That is, although their estimating equations are different, they lead numerically to exactly the same root. This includes a simple weighted estimator, an augmented weighted estimator, and a mean-score estimator. The numerical equivalence may elucidate the relationship between imputing scores and weighted estimation procedures.     

Estimation of competing risks with general missing pattern in failure types

In competing risks data, missing failure types (causes) is a very common phenomenon. In this work, we consider a general missing pattern in which, if a failure type is not observed, one observes a set of possible types containing the true type, along with the failure time. We first consider maximum likelihood estimation with missing-at-random assumption via the expectation maximization (EM) algorithm. We then propose a Nelson-Aalen type estimator for situations when certain information on the conditional probability of the true type given a set of possible failure types is available from the experimentalists. This is based on a least-squares type method using the relationships between hazards for different types and hazards for different combinations of missing types. We conduct a simulation study to investigate the performance of this method, which indicates that bias may be small, even for high proportion of missing data, for sufficiently large number of observations. The estimates are somewhat sensitive to misspecification of the conditional probabilities of the true types when the missing proportion is high. We also consider an example from an animal experiment to illustrate our methodology.     

Estimation of Growth Norms from Incomplete Longitudinal Data

Longitudinal studies are rarely complete due to attrition, mistimed visits and observations missing at random. When the data are missing at random it is possible to estimate the primary location parameters of interest by constructing a modification of Zellner's (1962) seemingly unrelated regression estimator. Such a procedure is developed in this paper and is applied to a longitudinal study of coronary risk factors in children. The method consists of two stages in which the covariance matrix is estimated at the first stage. Using the estimated covariance matrix a generalized least squares estimator of the regression parameter vector is then determined at the second stage. Limitations of the procedure are also discussed.     

Estimation of the correlation between nutrient intake measures under restricted sampling

Food frequency questionnaires (FFQs) are commonly used to assess dietary intake in epidemiologic research. To evaluate the FFQ reliability, the commonly used approach is to estimate the correlation coefficient between the data given in FFQ and those in food records (for example, 4-day food records [4DFR]) for nutrients of interest. However, in a dietary intervention study, a criterion for eligibility may be to select participants who have baseline FFQ-measured dietary intake of percent energy from fat above a prespecified quantity. Other instruments, such as the 4DFR, may be subsequently administrated only to eligible participants. Under these circumstances, analysis without adjusting for the restricted population will usually lead to biased estimation of correlation coefficients and other parameters of interest. In this paper, we apply likelihood-based and multiple imputation (MI) methods to accommodate such incomplete data obtained as a result of the study design. A simulation study is conducted to examine finite sample performance of various estimators. We note that both the MI estimate and the maximum likelihood (ML) estimate based on a bivariate-normal model are not sensitive to departures from this normality assumption. This led us to investigate robustness properties of the ML estimator analytically. We present some data analyses from a dietary assessment study from the Women's Health Initiative to illustrate the methods.     

A semiparametric empirical likelihood method for data from an outcome-dependent sampling scheme with a continuous outcome

Outcome-dependent sampling (ODS) schemes can be a cost effective way to enhance study efficiency. The case-control design has been widely used in epidemiologic studies. However, when the outcome is measured on a continuous scale, dichotomizing the outcome could lead to a loss of efficiency. Recent epidemiologic studies have used ODS sampling schemes where, in addition to an overall random sample, there are also a number of supplemental samples that are collected based on a continuous outcome variable. We consider a semiparametric empirical likelihood inference procedure in which the underlying distribution of covariates is treated as a nuisance parameter and is left unspecified. The proposed estimator has asymptotic normality properties. The likelihood ratio statistic using the semiparametric empirical likelihood function has Wilks-type properties in that, under the null, it follows a chi-square distribution asymptotically and is independent of the nuisance parameters. Our simulation results indicate that, for data obtained using an ODS design, the semiparametric empirical likelihood estimator is more efficient than conditional likelihood and probability weighted pseudolikelihood estimators and that ODS designs (along with the proposed estimator) can produce more efficient estimates than simple random sample designs of the same size. We apply the proposed method to analyze a data set from the Collaborative Perinatal Project (CPP), an ongoing environmental epidemiologic study, to assess the relationship between maternal polychlorinated biphenyl (PCB) level and children's IQ test performance.     

The tooth of perfection: functional and spatial constraints on mammalian tooth shape

This paper addresses the question of how close mammalian teeth are to ideal functional forms. An 'ideal' form is a morphology predicted to be the best functional shape according to information of the relationships between shape and function. Deviations from an ideal form are likely to indicate the presence of developmental or genetic constraints on form. Model tools were constructed to conform to functional principles from engineering and dental studies. The final model shapes are very similar to several mammalian tooth forms (carnassial teeth and tribosphenic-like cusps), suggesting that these tooth forms very closely approach ideal functional forms. Further evidence that these tooth forms are close to ideal comes from the conservation over 140 million years, the independent derivation and/or the occurrence over a size range of several orders of magnitude of these basic tooth forms. One of the main functional shapes derived here is the 'protoconoid', a fundamental design for double-bladed tools that fits a large number of functional parameters. This shape occurs in tooth forms such as tribosphenic, dilambdodont and zalambdodont. This study extends our understanding of constraints on tooth shape in terms of geometry (how space influences tooth shape) and function (how teeth divide food).  © 2003 The Linnean Society of London . Biological Journal of the Linnean Society , 2003, 78 , 173–191.     

Crown diameters of the deciduous teeth in Australian Aboriginals

Mesiodistal and buccolingual crown diameters were measured from dental casts representing the deciduous dentitions of 197 Aboriginal children from the Northern Territory of Australia. Double determination analysis indicated that the semi-automatic recording procedure used was reliable leading to observer errors of no practical significance. Tooth-size was greater in the male subjects but the sexual dimorphism was less marked than in the permanent teeth of the same subjects. The mandibular teeth were more uniform than maxillary with respect to buccolingual size relative to mesiodistal. Extremes of general tooth-size were more marked in the deciduous dentition than in the permanent as a consequence of the relatively large deciduous second molar which in Aboriginals approximates in size the permanent first molar of many other ethnic groups.