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Canonical correlation analysis investigates linear relationships between two sets of variables, but often works poorly on modern data sets due to high-dimensionality and mixed data types such as continuous, binary and zero-inflated. To overcome these challenges, we propose a semiparametric approach for sparse canonical correlation analysis based on Gaussian copula. Our main contribution is a truncated latent Gaussian copula model for data with excess zeros, which allows us to derive a rank-based estimator of the latent correlation matrix for mixed variable types without the estimation of marginal transformation functions. The resulting canonical correlation analysis method works well in high-dimensional settings as demonstrated via numerical studies, as well as in application to the analysis of association between gene expression and micro RNA data of breast cancer patients., Accepted to Biometrika. Main text: 19 pages and 3 figures. Supplementary material: 28 pages and 9 figures

Many methods have recently been proposed for efficient analysis of case-control studies of gene-environment interactions using a retrospective likelihood framework that exploits the natural assumption of gene-environment independence in the underlying population. However, for polygenic modelling of gene-environment interactions, which is a topic of increasing scientific interest, applications of retrospective methods have been limited due to a requirement in the literature for parametric modelling of the distribution of the genetic factors. We propose a general, computationally simple, semiparametric method for analysis of case-control studies that allows exploitation of the assumption of gene-environment independence without any further parametric modelling assumptions about the marginal distributions of any of the two sets of factors. The method relies on the key observation that an underlying efficient profile likelihood depends on the distribution of genetic factors only through certain expectation terms that can be evaluated empirically. We develop asymptotic inferential theory for the estimator and evaluate its numerical performance via simulation studies. An application of the method is presented.

We consider a problem of data integration. Consider determining which genes affect a disease. The genes, which we call predictor objects, can be measured in different experiments on the same individual. We address the question of finding which genes are predictors of disease by any of the experiments. Our formulation is more general. In a given data set, there are a fixed number of responses for each individual, which may include a mix of discrete, binary and continuous variables. There is also a class of predictor objects, which may differ within a subject depending on how the predictor object is measured, i.e., depend on the experiment. The goal is to select which predictor objects affect any of the responses, where the number of such informative predictor objects or features tends to infinity as sample size increases. There are marginal likelihoods for each way the predictor object is measured, i.e., for each experiment. We specify a pseudolikelihood combining the marginal likelihoods, and propose a pseudolikelihood information criterion. Under regularity conditions, we establish selection consistency for the pseudolikelihood information criterion with unbounded true model size, which includes a Bayesian information criterion with appropriate penalty term as a special case. Simulations indicate that data integration improves upon, sometimes dramatically, using only one of the data sources.

Covariate measurement error and missing responses are typical features in longitudinal data analysis. There has been extensive research on either covariate measurement error or missing responses, but relatively little work has been done to address both simultaneously. In this paper, we propose a simple method for the marginal analysis of longitudinal data with time-varying covariates, some of which are measured with error, while the response is subject to missingness. Our method has a number of appealing properties: assumptions on the model are minimal, with none needed about the distribution of the mismeasured covariate; implementation is straightforward and its applicability is broad. We provide both theoretical justification and numerical results.

We propose a multiple imputation estimator for parameter estimation in a quantile regression model when some covariates are missing at random. The estimation procedure fully utilizes the entire dataset to achieve increased efficiency, and the resulting coefficient estimators are root-n consistent and asymptotically normal. To protect against possible model misspecification, we further propose a shrinkage estimator, which automatically adjusts for possible bias. The finite sample performance of our estimator is investigated in a simulation study. Finally, we apply our methodology to part of the Eating at American's Table Study data, investigating the association between two measures of dietary intake. 2012 Biometrika Trust2012 © 2012 Biometrika Trust.

We develop an easily computed smooth backfitting algorithm for additive model fitting in repeated measures problems. Our methodology easily copes with various settings, such as when some covariates are the same over repeated response measurements. We allow for a working covariance matrix for the regression errors, showing that our method is most efficient when the correct covariance matrix is used. The component functions achieve the known asymptotic variance lower bound for the scalar argument case. Smooth backfitting also leads directly to design-independent biases in the local linear case. Simulations show our estimator has smaller variance than the usual kernel estimator. This is also illustrated by an example from nutritional epidemiology. Copyright 2009, Oxford University Press.

SUMMARY Hjort & Claeskens (2003) developed an asymptotic theory for model selection, model averaging and subsequent inference using likelihood methods in parametric models, along with associated confidence statements. In this article, we consider a semiparametric version of this problem, wherein the likelihood depends on parameters and an unknown function, and model selection/averaging is to be applied to the parametric parts of the model. We show that all the results of Hjort & Claeskens hold in the semiparametric context, if the Fisher information matrix for parametric models is replaced by the semiparametric information bound for semiparametric models, and if maximum likelihood estimators for parametric models are replaced by semiparametric efficient profile estimators. Our methods of proof employ Le Cam’s contiguity lemmas, leading to transparent results. The results also describe the behaviour of semiparametric model estimators when the parametric component is misspecified, and also have implications for pointwise-consistent model selectors. Somekeywords: Akaike information criterion; Bayes information criterion; Efficient semiparametric estimation; Frequentist model averaging; Model averaging; Model selection; Profile likelihood; Semiparametric model.

With the advent of modern genomic methods to adjust for population stratification, the use of external or publicly available controls has become an attractive option for reducing the cost of large-scale case-control genetic association studies. In this article, we study the estimation of joint effects of genetic and environmental exposures from a case-control study where data on genome-wide markers are available on the cases and a set of external controls while data on environmental exposures are available on the cases and a set of internal controls. We show that under such a design, one can exploit an assumption of gene-environment independence in the underlying population to estimate the gene-environment joint effects, after adjustment for population stratification. We develop a semiparametric profile likelihood method and related pseudolikelihood and working likelihood methods that are easy to implement in practice. We propose variance estimators for the methods based on asymptotic theory. Simulation is used to study the performance of the methods, and data from a multi-centre genome-wide association study of bladder cancer is further used to illustrate their application. Copyright 2013, Oxford University Press.

We consider the problem of maximum-likelihood estimation in case-control studies of gene-environment associations with disease when genetic and environmental exposures can be assumed to be independent in the underlying population. Traditional logistic regression analysis may not be efficient in this setting. We study the semiparametric maximum likelihood estimates of logistic regression parameters that exploit the gene-environment independence assumption and leave the distribution of the environmental exposures to be nonparametric. We use a profile-likelihood technique to derive a simple algorithm for obtaining the estimator and we study the asymptotic theory. The results are extended to situations where genetic and environmental factors are independent conditional on some other factors. Simulation studies investigate small-sample properties. The method is illustrated using data from a case-control study designed to investigate the interplay of BRCA1/2 mutations and oral contraceptive use in the aetiology of ovarian cancer.

SUMMARY We study the profile-kernel and backfitting methods in partially linear models for clustered/longitudinal data. For independent data, despite the potential root-n inconsist ency of the backfitting estimator noted by Rice (1986), the two estimators have the same asymptotic variance matrix, as shown by Opsomer & Ruppert (1999). In this paper, theoretical comparisons of the two estimators for multivariate responses are investigated. We show that, for correlated data, backfitting often produces a larger asymptotic variance than the profile-kernel method; that is, for clustered data, in addition to its bias problem, the backfitting estimator does not have the same asymptotic efficiency as the profile-kernel estimator. Consequently, the common practice of using the backfitting method to compute profile-kernel estimates is no longer advised. We illustrate this in detail by following Zeger & Diggle (1994) and Lin & Carroll (2001) with a working independence covariance structure for nonparametric estimation and a correlated covariance structure for para metric estimation. Numerical performance of the two estimators is investigated through a simulation study. Their application to an ophthalmology dataset is also described.

correlation. We show that a smoothing spline estimator is asymptotically equivalent to a recently proposed seemingly unrelated kernel estimator of Wang (2003) for any working covariance matrix. We show that both estimators can be obtained iteratively by applying conventional kernel or spline smoothing to pseudo-observations. This result allows us to study the asymptotic properties of the smoothing spline estimator by deriving its asymptotic bias and variance. We show that smoothing splines are consistent for an arbitrary working covariance and have the smallest variance when assuming the true covariance. We further show that both the seemingly unrelated kernel estimator and the smoothing spline estimator are nonlocal unless working independence is assumed but have asymptotically negligible bias. Their finite sample performance is compared through simulations. Our results justify the use of efficient, non-local estimators such as smoothing splines for clustered/longitudinal data.

SUMMARY We consider the problem of matched studies with a binary outcome that are analysed using conditional logistic regression, and for which data on some covariates are missing for some study participants. Methods for this problem involve either modelling the distribution of missing covariates or modelling the probability of data being missing. For this second approach, the previously proposed method did not make use of data for those persons with missing covariate data except in the model for the missingness. We propose a new class of estimators that use outcome and available covariate data for all study participants, and show that a particular member of this class always has better efficiency than the previously proposed estimator. We illustrate the efficiency gains that are possible with our approach using simulated data.

In many situations in survival analysis, it may happen that a fraction of individuals will never experience the event of interest: they are considered to be cured. The promotion time cure model takes this into account. We consider the case where one or more explanatory variables in the model are subject to measurement error, which should be taken into account to avoid biased estimators. A general approach is the simulation-extrapolation algorithm, a method based on simulations which allows one to estimate the effect of measurement error on the bias of the estimators and to reduce this bias. We extend this approach to the promotion time cure model. We explain how the algorithm works, we show that the proposed estimator is approximately consistent and asymptotically normally distributed, and that it performs well in finite samples. Finally, we analyze a database in cardiology: among the explanatory variables of interest is the ejection fraction, which is known to be measured with error. ispartof: Biometrika vol:104 issue:1 pages:31-50 ispartof: location:United States status: published

We consider estimation in a semiparametric partially generalised linear model for clustered data using estimating equations. A marginal model is assumed where the mean of the outcome variable depends on some covariates parametrically and a cluster-level covariate nonparametrically. A profile-kernel method allowing for working correlation matrices is developed. We show that the nonparametric part of the model can be estimated using standard nonparametric methods, including smoothing-parameter estimation, and the parametric part of the model can be estimated in a profile fashion. The asymptotic distributions of the parameter estimators are derived, and the optimal estimators of both the nonparametric and parametric parts are shown to be obtained when the working correlation matrix equals the actual correlation matrix. The asymptotic covariance matrix of the parameter estimator is consistently estimated by the sandwich estimator. We show that the semiparametric efficient score takes on a simple form and our profile-kernel method is semiparametric efficient. The results for the case where the nonparametric part of the model is an observation-level covariate are noted to be dramatically different.

In this paper we discuss the relationship between prospective and retrospective sampling problems. Estimates of the parameter of interest may be obtained by solving suitable estimating equations under both sampling schemes. Most common examples of such estimates include the maximum likelihood estimates. Some classical results and more recent development of the first-order asymptotic relationship between the estimators are reviewed. High-order expansions are given for the distributions of the retrospective estimators. Expansions for the marginal distributions of interest are obtained for both prospective and retrospective data. Furthermore, it is shown that the two expansions are asymptotically equal, at least up to order O(n -1 ). This implies that readily available prospective saddlepoint methods may be applied to the analysis of retrospective data without loss of high-order accuracy. The results are also briefly illustrated numerically.

Summary In many regression applications the independent variable is measured with error. When this happens, conventional parametric and nonparametric regression techniques are no longer valid. We consider two diVerent approaches to nonparametric regression. The first uses the simex, simulation-extrapolation, method and makes no assumption about the distribution of the unobserved error-prone predictor. For this approach we derive an asymptotic theory for kernel regression which has some surprising implications. Penalised regression splines are also considered for fixed number of known knots. The second approach assumes that the error-prone predictor has a distribution of a mixture of normals with an unknown number of components, and uses regression splines. Simulations illustrate the results.

SUMMARY In a generalised linear model with a single, normally distributed covariate, for the most part the effect of normally distributed additive measurement error is attenuation, i.e. asymptotic bias towards the null. Prentice & Sheppard (1995) suggested a marginalised random effects approach to combining the results of different studies on binary outcomes. We show that, in probit regression, when the number of observations per study is large, under the stated normality assumptions attenuation never occurs. In fact, the asymptotic bias is away from the null. This appears to be the first known case under reasonable distributional assumptions that the effect of measurement error is reverse-attenuation.

Consider a logistic regression model under case-control sampling. Prentice & Pyke (1979) showed that the logistic slope estimates with case-control sampling may be estimated from a standard prospective logistic regression program and that the resulting standard errors are asymptotically correct. Since logistic regression estimates are nonrobust, we propose and analyze robust estimates of the slope parameters. We focus specifically on estimates which downweight observations on one of three factors: (i) leverage, (ii) extreme fitted values, and (iii) likelihood of misclassification

Efficient estimation of parameters is a major objective in analyzing longitudinal data. We propose two generalized empirical likelihood-based methods that take into consideration within-subject correlations. A nonparametric version of the Wilks theorem for the limiting distributions of the empirical likelihood ratios is derived. It is shown that one of the proposed methods is locally efficient among a class of within-subject variance-covariance matrices. A simulation study is conducted to investigate the finite sample properties of the proposed methods and compares them with the block empirical likelihood method by You et al. (2006) and the normal approximation with a correctly estimated variance-covariance. The results suggest that the proposed methods are generally more efficient than existing methods that ignore the correlation structure, and are better in coverage compared to the normal approximation with correctly specified within-subject correlation. An application illustrating our methods and supporting the simulation study results is presented.

Assay data are often fit by a nonlinear regression model incorporating heterogeneity of variance, as ion radioimmunoassay, for example. Typically, the standard deviation of the response is taken to be proportional to a power theta of the mean. There is considerable empirical evidence suggesting that for assays of a reseasonable size, how one estimates the parameter theta does not greatly affect how well one estimates the the mean regression function. An additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration, for which many definitions exist; we focus on one such definition. The minimum detectable concentration depends both on theta and the mean regression function. We compare three standard method of estimating the parameter theta due to Rodbard (1978), Raab (1981a) and Carroll and Ruppert (1982b). When duplicate counts are taken at each concentration, the first method is only 20% efficient asymptotically in comparison to the third, and the resulting estimate of the minimum detectable concentration is asymptotically 3.3 times more variable for first than the third. Less dramatic results obtain for the second estimator compared to the third; this estimator is still not efficient, however. Simulation results and an example are supportive of the asymptotic theory. Keywords: Least squares method.

SUMMARY The power transformation family is often used for transforming to a normal linear model. The variance of the regression parameter estimators can be much larger when the transformation parameter is unknown and must be estimated, compared to when the transformation parameter is known. We consider prediction of future untransformed observations when the data can be transformed to a linear model. When the transformation must be estimated, the prediction error is not much larger than when the parameter is known.

SUMMARY We consider binary regression models when some of the predictors are measured with error. For normal measurement errors, structural maximum likelihood estimates are considered. We show that if the measurement error is large, the usual estimate of the probability of the event in question can be substantially in error, especially for high risk groups. In the situation of large measurement error, we investigate a conditional maximum likelihood estimator and its properties.

This paper studies estimation of the parameters of generalized linear models in canonical form when the explanatory vector is measured with independent normal error. For the functional case, i.e., when the explanatory vectors are fixed constants, unbiased score functions are obtained by conditioning on certain sufficient statistics. This work generalizes results obtained for logistic regression. In the case that the explanatory vectors are independent and identically distributed with unknown distribtuion, efficient score functions are obtained using the theory developed in Begun et al. (1983). Keywords: Conditional score function; Efficient score function; Functional model; Generalized linear model; Measurement error; Structural model.

SUMMARY We consider a heteroscedastic linear regression model with replication. To estimate the variances, one can use the sample variances or the sample average squared errors from a regression fit. We study the large-sample properties of these weighted least-squares estimates with estimated weights when the number of replicates is small. The estimates are generally inconsistent for asymmetrically distributed data. If sample variances are used based on m replicates, the weighted least-squares estimates are inconsistent for m =2 replicates even when the data are normally distributed. With between 3 and 5 replicates, the rates of convergence are slower than the usual square root of N. With m - 6 replicates, the effect of estimating the weights is to increase variances by (m - 5)/(m -3), relative to weighted least-squares estimates with known weights.

SUMMARY We study optimally bounded score functions for estimating regression parameters in a generalized linear model. Our work extends results obtained by Krasker & Welsch (1982) for the linear model and provides a simple proof of Krasker & Welsch's first-order condition for strong optimality. The application of these results to logistic regression is studied in some detail with an example given comparing the bounded-influence estimator with maximum likelihood. where h( . ), q( . ) and c( . ) are known functions and 0 is a vector of regression parameters. Models of this type include logistic and probit regression, Poisson regression, linear regression with known variance, and certain models for lifetime data. Our motivation for seeking robust estimators is the same as that in the linear model; maximum likelihood estimation is sometimes sensitive to outlying data. For logistic regression, Pregibon (1981,,1982) has documented the nonrobustness of the maximum likelihood estimator and expounded the benefits of diagnostics as well as robust or resistant fitting procedures; see also Johnson (1985). Much of the work on robust estimation concerns finding estimators which sacrifice little efficiency at the assumed model while providing protection against outliers and model violations. We follow this course finding bounded-influence estimators minimizing certain functionals of the asymptotic covariance matrix. Related work includes that of Hampel (1978), Krasker (1980) and Krasker & Welsch (1982). When fitting models to data, two important issues are identification of outliers and influential cases and accommodation of these observations. Frequently when influential cases are present, the fitted model is not representative of the bulk of the data. To rectify this, one can simply delete influential cases and refit via standard methods, but this