Yves G. Berger

Body mass index and common mental disorders: exploring the shape of the association and its moderation by age, gender and education

Background:Obesity is known to be associated with increased prevalence of common mental disorders (for example, depression and anxiety), and there is evidence of age and gender differences in this relationship. However, categorisation of body mass index (BMI) and age has limited our ability to understand the nature of these differences. This study used continuous values of BMI and age to explore the shape of the association between common mental disorders and BMI and whether it varied with age, gender and education.Method:The analysis used cross-sectional data on 7043 adults from the English 2007 Adult Psychiatric Morbidity Survey. Common mental disorders were assessed using the revised Clinical Interview Schedule (CIS-R). Cubic splines allowed BMI and age to have non-linear effects in the logistic regression analysis.Results:BMI was strongly associated with the presence of common mental disorders, and there was clear evidence that this association varied with gender and age. In young women the probability of having a disorder increased as BMI increased, whereas in young men the relationship was U-shaped—probabilities were higher for both underweight and obese men. These associations diminished in older age groups, particularly when potential confounders such as physical health were taken into account. There was no evidence that the relationship varied with education.Conclusions:Age and gender differences must be taken into account when investigating the link between BMI or obesity and common mental disorders. Furthermore, results of studies that categorise BMI may be highly sensitive to the width of the ‘normal weight’ reference category.

Variance estimation for a low income proportion

Proportions below a given fraction of a quantile of an income distribution are often estimated from survey data in comparisons of poverty. We consider the estimation of the variance of such a proportion, estimated from Family Expenditure Survey data. We show how a linearization method of variance estimation may be applied to this proportion, allowing for the effects of both a complex sampling design and weighting by a raking method to population controls. We show that, for data for 1998-1999, the estimated variances are always increased when allowance is made for the design and raking weights, the principal effect arising from the design. We also study the properties of a simplified variance estimator and discuss extensions to a wider class of poverty measures. Copyright 2003 Royal Statistical Society.

Rate of convergence to normal distribution for the Horvitz-Thompson estimator

Sampling distinct units from a population with unequal probabilities without replacement is a problem often considered in the literature, e.g. Hanif and Brewer (1980). If we implement such a sampling design, we can estimate the total of an unknown characteristic by the Horvitz-Thompson estimator (1951). One of the aims of statistical inference in a sample survey is to have an asymptotic normal distribution for this estimator. Stenlund and Westlund (1975) examine this problem from an empirical point of view. In this paper, we give a theoretical framework where we show that this problem can be solved by maximizing entropy. Hajek (1981, p. 33) conjectured one of this fact but without a formal expression. Hajek (1964) gives a necessary and sufficient condition for the asymptotic normality of the Horvitz-Thompson estimator, if the rejective sampling is performed. In this work, we give a rate of convergence for any kind of sampling. We apply our results to the Rao (1965) and Sampford (1967) sampling and to the successive sampling (Hajek, 1964).

Rate of convergence for asymptotic variance of the Horvitz–Thompson estimator

Abstract Drawing distinct units without replacement and with unequal probabilities from a population is a problem often considered in the literature (e.g. Hanif and Brewer, 1980 , Int. Statist. Rev. 48, 317–355). In such a case, the sample mean is a biased estimator of the population mean. For this reason, we use the unbiased Horvitz–Thompson estimator (1951). In this work, we focus our interest on the variance of this estimator. The variance is cumbersome to compute because it requires the calculation of a large number of second-order inclusion probabilities. It would be helpful to use an approximation that does not need heavy calculations. The Hajek (1964) variance approximation provides this advantage as it is free of second-order inclusion probabilities. Hajek (1964) proved that this approximation is valid under restrictive conditions that are usually not fulfilled in practice. In this paper, we give more general conditions and we show that this approximation remains acceptable for most practical problems.

Variance estimation for measures of change in probability sampling

We propose to estimate the design variance of absolute changes between two cross-sectional estimators under rotating sampling schemes. We show that the variance estimator proposed is generally positive. We also propose possible extensions for stratified samples, with dynamic stratification; that is, when units move between strata and new strata are created at the second waves.

A Simple Variance Estimator for Unequal Probability Sampling without Replacement

Survey sampling textbooks often refer to the Sen–Yates–Grundy variance estimator for use with without-replacement unequal probability designs. This estimator is rarely implemented because of the complexity of determining joint inclusion probabilities. In practice, the variance is usually estimated by simpler variance estimators such as the Hansen–Hurwitz with replacement variance estimator; which often leads to overestimation of the variance for large sampling fractions that are common in business surveys. We will consider an alternative estimator: the Hájek (1964) variance estimator that depends on the first-order inclusion probabilities only and is usually more accurate than the Hansen–Hurwitz estimator. We review this estimator and show its practical value. We propose a simple alternative expression; which is as simple as the Hansen–Hurwitz estimator. We also show how the Hájek estimator can be easily implemented with standard statistical packages.

Towards optimal regression estimation in sample surveys

The Montanari (1987) regression estimator is optimal when the population regression coefficients are known. When the coefficients are estimated, the Montanari estimator is not optimal and can be extremely volatile. Using design-based arguments, this paper proposes a simpler and better alternative to the Montanari estimator that is also optimal when the population regression coefficients are known. Moreover, it can be easily implemented as it involves standard weighted least squares. The estimator is applicable under single stage stratified sampling with unequal probabilities within each stratum.

Sampling with unequal probabilities

Publisher Summary Since the mid-1950s, there has been a well-developed theory of sample survey design inference embracing complex designs with stratification and unequal probabilities. Unequal probability sampling was first suggested by Hansen and Hurwitz in the context of sampling with replacement. Narain, Horvitz, and Thompson developed the corresponding theory for sampling without replacement. A large part of survey-sampling literature is devoted to unequal probability sampling, and more than 50 sampling algorithms have been proposed. Multistage sampling is one of the applications of unequal probability sampling design where the selection of primary units within strata may be done with unequal probability. For example, self-weighted two-stage sampling is often used to select primary sampling units with probabilities that are proportional to the number of secondary sampling units within the primary units; a simple random sample is selected within each primary unit.Since the mid 1950s, there has been a well-developed theory of sample survey design inference embracing complex designs with stratification and unequal probabilities (Smith, 2001). Unequal probability sampling was first suggested by Hansen and Hurwitz (1943) in the context of sampling with replacement. Narain (1951), Horvitz and Thompson (1952) developed the corresponding theory for sampling without replacement. A large part of survey sampling literature is devoted to unequal probabilities sampling, and more than 50 sampling algorithms have been proposed.

Variance estimation of survey estimates calibrated on estimated control totals-An application to the extended regression estimator and the regression composite estimator

Calibration on control totals is commonly used for survey weighting. It is usually assumed that these totals are values known without sampling errors. However, they can be estimated from other sources. A variance estimator that takes into account the randomness of control totals is derived. Several situations such as calibration on external sources and calibration with sampling on two occasions are investigated. The methodology proposed is general and can be implemented in various situations when control totals are estimated.

Empirical likelihood confidence intervals for adaptive cluster sampling

Adaptive cluster sampling (ACS) is an efficient sampling design for estimating parameters of rare and clustered populations. It is widely used in ecological research. The modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators based on small samples under ACS have often highly skewed distributions. In such situations, confidence intervals based on traditional normal approximation can lead to unsatisfactory results, with poor coverage properties. Christman and Pontius (Biometrics 56:503–510, 2000) showed that bootstrap percentile methods are appropriate for constructing confidence intervals from the HH estimator. But Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) showed that bootstrap confidence intervals from the HT estimator are even worse than the normal approximation confidence intervals. In this article, we consider two pseudo empirical likelihood functions under the ACS design. One leads to the HH estimator and the other leads to a HT type estimator known as the Hájek estimator. Based on these two empirical likelihood functions, we derive confidence intervals for the population mean. Using a simulation study, we show that the confidence intervals obtained from the first EL function perform as good as the bootstrap confidence intervals from the HH estimator but the confidence intervals obtained from the second EL function perform much better than the bootstrap confidence intervals from the HT estimator, in terms of coverage rate.