Using Hierarchical Likelihood for Missing Data Problems

Most statistical solutions to the problem of statistical inferencewith missing data involve integration or expectation. This canbe done in many ways: directly or indirectly, analytically ornumerically, deterministically or stochastically. Missing-dataproblems can be formulated in terms of latent random variables,so that hierarchical likelihood methods of Lee & Nelder(1996) can be applied to missing-value problems to provide onesolution to the problem of integration of the likelihood. Theresulting methods effectively use a Laplace approximation tothe marginal likelihood with an additional adjustment to themeasures of precision to accommodate the estimation of the fixedeffects parameters. We first consider missing at random caseswhere problems are simpler to handle because the integrationdoes not need to involve the missing-value mechanism and thenconsider missing not at random cases. We also study tobit regressionand refit the missing not at random selection model to the antidepressanttrial data analyzed in Diggle & Kenward (1994).     

Estimation of Spatial Variation in Risk Using Matched Case‐control Data

A common problem in environmental epidemiology is to estimate spatial variation in disease risk after accounting for known risk factors. In this paper we consider this problem in the context of matched case‐control studies. We extend the generalised additive model approach of Kelsall and Diggle (1998) to studies in which each case has been individually matched to a set of controls. We discuss a method for fitting this model to data, apply the method to a matched study on perinatal death in the North West Thames region of England and explain why, if spatial variation is of particular scientific interest, matching is undesirable.     

The Milk Protein Trial: Influence Analysis of the Dropout Process

Diggle and Kenward (1994) proposed a selection model for continuous longitudinal data subject to possible non‐random dropout. Their method in general, and the milk protein example in particular, has provoked a large debate about the role of such models. The original enthusiasm was followed by skepticism about the strong but untestable assumption upon which this type of models invariably rests. Concern was raised about the very nature of incompleteness which is arguably more due to design reasons (the experiment was stopped due to insufficient feed supply), than to genuine dropout. In the meantime, the view has emerged that these models should ideally be made part of a sensitivity analysis. This paper presents a formal and flexible approach to such a sensitivity assessment, based on both global influence (Chatterjee and Hadi , 1988) as well as local influence (Cook , 1986). It will be argued that local influence is more apt to zoom in on a particular source of influence, such as the assumed non‐response mechanism. The method is applied to a set of data on milk protein contents in dairy cattle. The same data were used in the original paper by Diggle and Kenward (1994), who concluded that the dropout process was non‐random.     

Conditional likelihood methods for haplotype-based association analysis using matched case-control data

Genetic epidemiologists routinely assess disease susceptibility in relation to haplotypes, that is, combinations of alleles on a single chromosome. We study statistical methods for inferring haplotype-related disease risk using single nucleotide polymorphism (SNP) genotype data from matched case-control studies, where controls are individually matched to cases on some selected factors. Assuming a logistic regression model for haplotype-disease association, we propose two conditional likelihood approaches that address the issue that haplotypes cannot be inferred with certainty from SNP genotype data (phase ambiguity). One approach is based on the likelihood of disease status conditioned on the total number of cases, genotypes, and other covariates within each matching stratum, and the other is based on the joint likelihood of disease status and genotypes conditioned only on the total number of cases and other covariates. The joint-likelihood approach is generally more efficient, particularly for assessing haplotype-environment interactions. Simulation studies demonstrated that the first approach was more robust to model assumptions on the diplotype distribution conditioned on environmental risk variables and matching factors in the control population. We applied the two methods to analyze a matched case-control study of prostate cancer.     

Comparing the performances of Diggle's tests of spatial randomness for small samples with and without edge-effect correction: application to ecological data

Diggle's tests of spatial randomness based on empirical distributions of interpoint distances can be performed with and without edge-effect correction. We present here numerical results illustrating that tests without the edge-effect correction proposed by Diggle (1979, Biometrics 35, 87-101) have a higher power for small sample sizes than those with correction. Ignoring the correction enables detection of departure from spatial randomness with smaller samples (down to 10 points vs. 30 points for the tests with correction). These results are confirmed by an example with ecological data consisting of maps of two species of trees in a West African savanna. Tree numbers per species per map were often less than 20. For one of the species, for which maps strongly suggest an aggregated pattern, tests without edge-effect correction enabled rejection of the null hypothesis on three plots out of five vs. on only one for the tests with correction.     

Partial‐Likelihood Analysis of Spatio‐Temporal Point‐Process Data

Summary We investigate the use of a partial likelihood for estimation of the parameters of interest in spatio‐temporal point‐process models. We identify an important distinction between spatially discrete and spatially continuous models. We focus our attention on the spatially continuous case, which has not previously been considered. We use an inhomogeneous Poisson process and an infectious disease process, for which maximum‐likelihood estimation is tractable, to assess the relative efficiency of partial versus full likelihood, and to illustrate the relative ease of implementation of the former. We apply the partial‐likelihood method to a study of the nesting pattern of common terns in the Ebro Delta Natural Park, Spain.     

Sensitivity analysis for nonrandom dropout: a local influence approach

Diggle and Kenward (1994, Applied Statistics 43, 49-93) proposed a selection model for continuous longitudinal data subject to nonrandom dropout. It has provoked a large debate about the role for such models. The original enthusiasm was followed by skepticism about the strong but untestable assumptions on which this type of model invariably rests. Since then, the view has emerged that these models should ideally be made part of a sensitivity analysis. This paper presents a formal and flexible approach to such a sensitivity assessment based on local influence (Cook, 1986, Journal of the Royal Statistical Society, Series B 48, 133-169). The influence of perturbing a missing-at-random dropout model in the direction of nonrandom dropout is explored. The method is applied to data from a randomized experiment on the inhibition of testosterone production in rats.     

Use of Generalised Linear Mixed Models for the Analysis of Clustered Survival Data

Data from a litter matched tumorigenesis experiment are analysed using a generalised linear mixed model (GLMM) approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same litter. Maximum likelihood (ML) and residual maximum likelihood (REML) estimates of risk variable parameters, variance component parameters and the prediction of random effects are given. Estimation of treatment effect parameter (carcinogen effect) has good agreement with previous analyses obtained in the literature though the dependence structure within a litter is modelled in different ways. The variance component estimation provides the estimated dispersion of the random effects. The prediction of random effects, is useful, for instance, in identifying high risk litters and individuals. The present analysis illustrates its wider application to detecting increased risk of occurrence of disease in particular families of a study population.     

An assessment of non-randomized medical treatment of long-term schizophrenia relapse using bivariate binary-response transition models

The analyses of observational longitudinal studies involving concurrent changes in treatment and medical conditions present difficulties because of the multitude of directions of potential relationships: past medication influences current symptoms; past symptoms influence current medication; and current medication is associated with current symptoms. In the context of a long-term study of non-randomized pharmacological treatment of schizophrenic relapse, we present an analysis of bivariate discrete-time transitional data with binary responses in an attempt to understand the transitional and concurrent relationships between schizophrenia relapse and medication use. A naive analysis does not show any association between previous medication and current relapse. However, we provide evidence suggesting that current treatment may impact current relapse for those who have previously taken medication, but not for those who haven't taken medication in the past. When univariate models are specified to assess these associations, the bivariate nature of the problem requires a choice of which response, relapse or medication, should be the dependent variable. In this case, the choice of relapse or medication as a dependent variable does matter. Hence, our results derive from models where both relapse and medication are treated as dependent variables. Specifically, we specify a bivariate log odds ratio for current relapse and current medication use and a separate univariate logit component for each of these outcomes. Each of these components contains transitional associations with previous relapse and medication. Such models represent extensions of univariate transitional association models (e.g. Diggle et al. (1994)) and correspond to bivariate transitional models (e.g. Zeger and Liang (1991)). We incorporate changes in transitional associations into the full-data parametric model for final inference, and investigate if these temporal changes are due to learning effects or the impact of drop-out. We also perform residual analyses and sensitivity analyses in the context of missing data patterns.     

Regional Ecological Risk Assessment of a Near Shore Marine Environment: Cherry Point,WA

We conducted a regional ecological risk assessment for a near shore marine environment in northwestern Washington State using the Relative Risk Model. The objectives of this study were threefold: (1) to analyze cumulative impacts from multiple sources of chemical and non-chemical stressors in the near shore region and upland watersheds of Cherry Point (2) to determine the utility of Monte Carlo type uncertainty analysis in a rank-based regional risk assessment and (3) to investigate the effects of model habitat characterization on risk estimates. We used geographic information systems to compile and compare spatial data to determine ranks for sub-regions within the study area. By quantitatively combining ranks with exposure and effects filters, we estimated total relative risk between sub-regions and relative contributions of stressors. Finally, we used Monte Carlo analysis and an alternative ranking scheme to evaluate the effects of model and parameter uncertainty on risk predictions. The regional risk assessment results suggest the major contributors of risk are vessel traffic, upland urban and agricultural land use and shoreline recreational activities. This assessment demonstrated the applicability of regional risk assessment to marine near shore regions and the benefit of Monte Carlo analysis in describing uncertainty in a Relative Risk Model regional risk assessment.     

Bias-reduced estimators of conditional odds ratios in matched case-control studies with unmatched confounding

We study bias-reduced estimators of exponentially transformed parameters in general linear models (GLMs) and show how they can be used to obtain bias-reduced conditional (or unconditional) odds ratios in matched case-control studies. Two options are considered and compared: the explicit approach and the implicit approach. The implicit approach is based on the modified score function where bias-reduced estimates are obtained by using iterative procedures to solve the modified score equations. The explicit approach is shown to be a one-step approximation of this iterative procedure. To apply these approaches for the conditional analysis of matched case-control studies, with potentially unmatched confounding and with several exposures, we utilize the relation between the conditional likelihood and the likelihood of the unconditional logit binomial GLM for matched pairs and Cox partial likelihood for matched sets with appropriately setup data. The properties of the estimators are evaluated by using a large Monte Carlo simulation study and an illustration of a real dataset is shown. Researchers reporting the results on the exponentiated scale should use bias-reduced estimators since otherwise the effects can be under or overestimated, where the magnitude of the bias is especially large in studies with smaller sample sizes.     

Maximum likelihood estimation for Cox's regression model under nested case-control sampling

Nested case-control sampling is designed to reduce the costs of large cohort studies. It is important to estimate the parameters of interest as efficiently as possible. We present a new maximum likelihood estimator (MLE) for nested case-control sampling in the context of Cox's proportional hazards model. The MLE is computed by the EM-algorithm, which is easy to implement in the proportional hazards setting. Standard errors are estimated by a numerical profile likelihood approach based on EM aided differentiation. The work was motivated by a nested case-control study that hypothesized that insulin-like growth factor I was associated with ischemic heart disease. The study was based on a population of 3784 Danes and 231 cases of ischemic heart disease where controls were matched on age and gender. We illustrate the use of the MLE for these data and show how the maximum likelihood framework can be used to obtain information additional to the relative risk estimates of covariates.     

The interpretation of process from pattern using two-dimensional spectral analysis: modelling single species patterns in vegetation

Two-dimensional spectral analyses of spatial patterns are made using (i) the autocorrelation function, (ii) the periodogram, and its polar summary (iii) the R-spectrum and (iv) the -spectrum. Together these give a sensitive analysis of both the complete range of scales of pattern and directional components which exist in data sets and we illustrate how the significance of observed spectral features can be assessed.We investigate the spatial pattern of Calluna vulgaris in a regenerating woodland and of Epilobium angustifolium spreading in a woodland following the thinning of trees. Evidence in the spectra is found for directional, clumping and inhibition patterns, and is discussed in relation to spectra obtained from simulations of known pattern generating processes. Hypotheses about the important biological, environmental and management influences on the structure of the communities are examined.We gratefully acknowledge the Swedish Coniferous Biome Project and Dr P. J. Diggle for access to the heather data.     

Estimation with correlated censored survival data with missing covariates

Incomplete covariate data are a common occurrence in studies in which the outcome is survival time. Further, studies in the health sciences often give rise to correlated, possibly censored, survival data. With no missing covariate data, if the marginal distributions of the correlated survival times follow a given parametric model, then the estimates using the maximum likelihood estimating equations, naively treating the correlated survival times as independent, give consistent estimates of the relative risk parameters Lipsitz et al. 1994 50, 842-846. Now, suppose that some observations within a cluster have some missing covariates. We show in this paper that if one naively treats observations within a cluster as independent, that one can still use the maximum likelihood estimating equations to obtain consistent estimates of the relative risk parameters. This method requires the estimation of the parameters of the distribution of the covariates. We present results from a clinical trial Lipsitz and Ibrahim (1996b) 2, 5-14 with five covariates, four of which have some missing values. In the trial, the clusters are the hospitals in which the patients were treated.     

Cholesteryl ester-transfer protein (CETP) polymorphism and the association of acute coronary syndromes by obesity status in Greek subjects: the CARDIO2000-GENE study

OBJECTIVE: Cholesterol ester transfer protein (CETP) regulates plasma lipid distribution. The present study aimed to investigate whether the CETP gene (Taq1B) polymorphism predisposes to Acute Coronary Syndromes (ACS) depending on obesity status. METHODS: We studied demographic, lifestyle and clinical information in 237 hospitalized patients (185 males) with a first event of an ACS and 237 controls matched by age and sex. CETP Taq1B genotyping was performed by PCR-RFLP analysis. RESULTS: Overall, the CETP genotype frequencies were, in patients: 14% (n = 33), 35% (n = 83) and 51% (n = 121) and in controls: 17% (n = 39), 33% (n = 78) and 50% (n = 120) for B2B2, B1B1 and B1B2 respectively (p = 0.72). A significant interaction (p for interaction <0.001) was observed between obesity status and CETP concerning the likelihood of having ACS. Therefore, we stratified our analysis by obesity status and observed that B2B2 was associated with a 0.27 times lower likelihood of having ACS among normal-weight people (OR = 0.27, p = 0.02). No significant relationships were observed among overweight or obese participants. CONCLUSIONS: These findings provide evidence of a protective effect of the B2B2 genotype of the CETP Taq1B polymorphism on the likelihood of having a first event of ACS in normal-weight persons.     

Dose finding using the biased coin up-and-down design and isotonic regression

We are interested in finding a dose that has a prespecified toxicity rate in the target population. In this article, we investigate five estimators of the target dose to be used with the up-and-down biased coin design (BCD) introduced by Durham and Flournoy (1994, Statistical Decision Theory and Related Topics). These estimators are derived using maximum likelihood, weighted least squares, sample averages, and isotonic regression. A linearly interpolated isotonic regression estimate is shown to be simple to derive and to perform as well as or better than the other target dose estimators in terms of mean square error and average number of subjects needed for convergence in most scenarios studied.     

The interaction of birth order and parental age on sexual orientation: an examination in two samples

A birth order and sexual orientation relationship has been demonstrated numerous times in men, but a related variable, parental age (i.e. age of parents when the participant was born), has been less studied and has demonstrated contradictory results. In this research, the relations among birth order, parental age and sexual orientation were examined in a national probability sample of the US (Kessler, 1994; Kessler et al., 1994) and in a Canadian sample of homosexual and heterosexual men closely matched on demographic characteristics (Blanchard & Bogaert, 1996a). In both studies, an interaction between birth order and parental age was observed in men, such that there was positive association between number of older siblings and the likelihood of homosexuality, but this association weakened with increasing parental age. No significant effects were observed for women. The results are discussed in relation to recent theories of the birth order/sexual orientation relationship.     

Conditional and unconditional categorical regression models with missing covariates

We consider methods for analyzing categorical regression models when some covariates (Z) are completely observed but other covariates (X) are missing for some subjects. When data on X are missing at random (i.e., when the probability that X is observed does not depend on the value of X itself), we present a likelihood approach for the observed data that allows the same nuisance parameters to be eliminated in a conditional analysis as when data are complete. An example of a matched case-control study is used to demonstrate our approach.     

超声下腔内激光消融联合点式剥脱治疗大隐静脉曲张的效果和安全性观察

摘要 目的：探讨超声下腔内激光消融联合点式剥脱治疗大隐静脉曲张的效果和安全性观察。方法：选取我院血管外科在2019年1月到2021年12月收治的140例大隐静脉曲张患者作为研究对象。将所有患者依照住院号单双号进行分组,分为联合组与对照组,每组70例。对照组采用常规高位结扎联合抽剥术进行治疗,联合组采用超声下腔内激光消融联合点式剥脱进行治疗。对比两组手术指标、临床疗效、术前术后下肢深静脉瓣膜功能变化以及术后并发症的发生率。结果：联合组的手术耗时较对照组高,术中出血量、切口个数及住院时间均较对照组低（P<0.05）;两组临床疗效分布对比有明显差异,且联合组显效率高于对照组（P<0.05）;术后3个月两组VCT均有缩短,Vmax均有提升,且联合组术后3个月VCT较对照组短,Vmax较对照组高（P<0.05）;联合组术后并发症较对照组低（P<0.05）。结论：对大隐静脉曲张患者采用超声下腔内激光消融联合点式剥脱方法治疗,与常规手术相比疗效更佳,且能够明显改善患者下肢静脉瓣膜功能,安全性高。     

Nonparametric One‐way Analysis of Variance of Replicated Bivariate Spatial Point Patterns

A common problem in neuropathological studies is to assess the spatial patterning of cells on tissue sections and to compare spatial patterning between disorder groups. For a single cell type, the cell positions constitute a univariate point process and interest focuses on the degree of spatial aggregation. For two different cell types, the cell positions constitute a bivariate point process and the degree of spatial interaction between the cell types is of interest. We discuss the problem of analysing univariate and bivariate spatial point patterns in the one‐way design where cell patterns have been obtained for groups of subjects. A bootstrapping procedure to perform a nonparametric one‐way analysis of variance of the spatial aggregation of a univariate point process has been suggested by Diggle, Lange and Bene? (1991). We extend their replication‐based approach to allow the comparison of the spatial interaction of two cell types between groups, to include planned comparisons (contrasts) and to assess whole groups against complete spatial randomness and spatial independence. We also accommodate several replicate tissue sections per subject. An advantage of our approach is that it can be applied when processes are not stationary, a common problem in brain tissue sections since neurons are arranged in cortical layers. We illustrate our methods by applying them to a neuropathological study to investigate abnormalities in the functional relationship between neurons and astrocytes in HIV associated dementia. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)