Nicholas T. Longford

Factor Analysis for Clustered Observations

Classical factor analysis assumes a random sample of vectors of observations. For clustered vectors of observations, such as data for students from colleges, or individuals within households, it may be necessary to consider different within-group and between-group factor structures. Such a two-level model for factor analysis is defined, and formulas for a scoring algorithm for estimation with this model are derived. A simple noniterative method based on a decomposition of the total sums of squares and crossproducts is discussed. This method provides a suitable starting solution for the iterative algorithm, but it is also a very good approximation to the maximum likelihood solution. Extensions for higher levels of nesting are indicated. With judicious application of quasi-Newton methods, the amount of computation involved in the scoring algorithm is moderate even for complex problems; in particular, no inversion of matrices with large dimensions is involved. The methods are illustrated on two examples.

Selection bias and treatment heterogeneity in clinical trials.

A common perception about many commercially available medical treatments is that they are effective for every patient having the relevant indication and that developers have provided the regulatory authorities with evidence of such a property. We show that the standard of evidence is much lower and that the standard is appropriate only when the treatment effects are almost constant. We discuss the implications on the design and analysis of clinical trials if the standards were made to correspond with the common perception. We conclude that the evidence of positive mean treatment effect should be accompanied by evidence of limited dispersion of the effects and by a sensitivity analysis that explores the impact of the selection bias in recruitment.

Logistic regression with random coefficients

Abstract An approximation to the likelihood for the generalized linear models with random coefficients is derived and is the basis for an approximate Fisher scoring algorithm. The method is illustrated on the logistic regression with one-way classification, but it has an extension to the class of generalized linear models and to more complex data structures, such as nested two-way classification. Both full and restricted maximum likelihood versions of this algorithm are defined. The estimators of the regression parameters coincide with the generalized estimating equations of Zeger and Liang (Biometrics, 42, 1986, p 121–130) but an essentially different class of estimators for the covariance structure parameters is obtained.

Multivariate shrinkage estimation of small area means and proportions

The familiar (univariate) shrinkage estimator of a small area mean or proportion combines information from the small area and a national survey. We define a multivariate shrinkage estimator which combines information also across subpopulations and outcome variables. The superiority of the multivariate shrinkage over univariate shrinkage, and of the univariate shrinkage over the unbiased (sample) means, is illustrated on examples of estimating the local area rates of economic activity in the subpopulations defined by ethnicity, age and sex. The examples use the sample of anonymized records of individuals from the 1991 UK census. The method requires no distributional assumptions but relies on the appropriateness of the quadratic loss function. The implementation of the method involves minimum outlay of computing. Multivariate shrinkage is particularly effective when the area level means are highly correlated and the sample means of one or a few components have small sampling and between‐area variances. Estimations for subpopulations based on small samples can be greatly improved by incorporating information from subpopulations with larger sample sizes.

Factors associated with the prevalence of stereotypic behaviour amongst Thoroughbred horses passing through auctioneer sales

The objective of this study was to evaluate whether sex, age and/or coat colour were associated with the occurrence of stereotypic behaviour in the horse and to assess whether the occurrence of one type of stereotypy in an individual was associated with the occurrence of another specific type of stereotypy. The incidence of stereotypic boxwalking, weaving (both locomotor stereotypies) and oral stereotypy in 4061 Thoroughbred horses passing through five bloodstock auctions were recorded from sale declarations and information on returns. An overall prevalence of 5.1% was recorded, and varied with sex (P<0.001) and age (P<0.001) but not coat colour (P=0.495). Prevalence was higher in females, geldings, and 2-year-olds. Examination of the assumption that stereotypies are acquired independently suggested a higher than expected prevalence of animals with more than one stereotypy. The interaction was not the same for all forms of stereotypy recorded. The effect was greatest between boxwalking and weaving, (odds ratio 13.6) whilst combinations involving oral and locomotor stereotypies had lower odds ratios (between 2.9 and 4.9).

Simulation-based diagnostics in random-coefficient models

Commonly applied diagnostic procedures in random-coefficient (multilevel) analysis are based on an inspection of the residuals, motivated by established procedures for ordinary regression. The deficiencies of such procedures are discussed and an alternative based on simulation from the fitted model (parametric bootstrap) is proposed. Although computationally intensive, the method proposed requires little programming effort additional to implementing the model fitting procedure. It can be tailored for specific kinds of outliers. Some computationally less demanding alternatives are described.

Reliability of Essay Rating and Score Adjustment.

A model-based approach to rater reliability for essays read by multiple readers is presented. The approach is motivated by the generalizability theory. Variation of rater severity (between-rater variation) and rater inconsistency (within-rater variation) is considered in the presence of between-examinee variation. An additive variance component model is posited and the method of moments for its estimation described. The models involve no distributional assumptions other than variance homogeneity and independence of certain random variables. Minimum mean squared error estimators of examinees’ true scores and readers’ severities are derived. Model diagnostic procedures are an integral component of the approach. The methods are illustrated on data from standardized educational tests.

Handling missing data in diaries of alcohol consumption

Missing data can rarely be avoided in large scale studies in which subjects are requested to complete questionnaires with many items. Analyses of such surveys are often based on the records with no missing items, resulting in a loss of efficiency and, when data are missing not at random, in bias. This paper applies the method of multiple imputation to handle missing data in an analysis of alcohol consumption of the subjects in the Medical Research Council National Survey of Health and Development. The outcomes studied are derived from the entries in diaries of food and drink intake over seven designated days. Background variables and other responses related to alcohol consumption and associated problems are used as collateral information. In conventional analyses, subpopulation means of quantities of alcohol consumed are compared. Since we are interested in the harmful effects of alcohol, we make inferences about the percentages of those who consume more than a given quantity of net alcohol. We assess the contribution to the analyses made by the incomplete records and outline a more integrated way of applying multiple imputation in large scale longitudinal surveys.

Poverty and inequality in European regions

The European Union Statistics on Income and Living Conditions (EU-SILC) is the main source of information about poverty and economic inequality in the member states of the European Union. The sample sizes of its annual national surveys are sufficient for reliable estimation at the national level but not for inferences at the sub-national level, failing to respond to a rising demand from policy-makers and local authorities. We provide a comprehensive map of median income, inequality (Gini coefficient and Lorenz curve) and poverty (poverty rates) based on the equivalised household income in the countries in which the EU-SILC is conducted. We study the distribution of income of households (pro-rated to its members), not merely its median (or mean), because we regard its dispersion and frequency of lower extremes (relative poverty) as important characteristics. The estimation for the regions with small sample sizes is improved by the small-area methods. The uncertainty of complex nonlinear statistics is assessed by bootstrap. Household-level sampling weights are taken into account in both the estimates and the associated bootstrap standard errors.

Incidence and enteral feed antecedents of severe neonatal necrotising enterocolitis across neonatal networks in England, 2012-13: a whole-population surveillance study

BACKGROUND Necrotising enterocolitis is a neonatal gastrointestinal inflammatory disease with high mortality and severe morbidity. This disorder is growing in global relevance as birth rates and survival of babies with low gestational age improve. Population data are scant and pathogenesis is incompletely understood, but enteral feed exposures are believed to affect risk. We aimed to quantify the national incidence of severe necrotising enterocolitis, describe variation across neonatal networks, and investigate enteral feeding-related antecedents of severe necrotising enterocolitis. METHODS We undertook a 2-year national surveillance study (the UK Neonatal Collaborative Necrotising Enterocolitis [UKNC-NEC] Study) of babies born in England to quantify the burden of severe or fatal necrotising enterocolitis confirmed by laparotomy, leading to death, or both. Data on all liveborn babies admitted to neonatal units between Jan 1, 2012, and Dec 31, 2013, were obtained from the National Neonatal Research Database. In the subgroup of babies born before a gestational age of 32 weeks, we did a propensity score analysis of the effect of feeding in the first 14 postnatal days with own mothers milk, with or without human donor milk and avoidance of bovine-origin formula, or milk fortifier, on the risk of developing necrotising enterocolitis. FINDINGS During the study period, 118 073 babies were admitted to 163 neonatal units across 23 networks, of whom 14 678 were born before a gestational age of 32 weeks. Overall, 531 (0·4%) babies developed severe necrotising enterocolitis, of whom 247 (46·5%) died (139 after laparotomy). 462 (3·2%) of 14 678 babies born before a gestational age of 32 weeks developed severe necrotising enterocolitis, of whom 222 (48·1%) died. Among babies born before a gestational age of 32 weeks, the adjusted network incidence of necrotising enterocolitis ranged from 2·51% (95% CI 1·13-3·60) to 3·85% (2·37-5·33), with no unusual variation from the adjusted national incidence of 3·13% (2·85-3·42), despite variation in feeding practices. The absolute risk difference for babies born before a gestational age of 32 weeks who received their own mothers milk within 7 days of birth was -0·88% (95% CI -1·15 to -0·61; relative risk 0·69, 95% CI 0·60 to 0·78; number needed to treat to prevent one case of necrotising enterocolitis 114, 95% CI 87 to 136). For babies who received no compared with any bovine-origin products within 14 days of birth, the absolute risk difference was -0·65% (-1·01 to -0·29; relative risk 0·61, 0·39 to 0·83; number needed to treat 154, 99 to 345). We were unable to assess the effect of human donor milk as use was low. INTERPRETATION Early feeding of babies with their own mothers milk and avoidance of bovine-origin products might reduce the risk of necrotising enterocolitis, but the absolute reduction is small. Owing to the rarity of severe necrotising enterocolitis, international collaborations are needed for adequately powered preventive trials. FUNDING National Institute for Health Research.