Maximum likelihood estimation of reference centiles

A two-part model for reference curve estimation subject to a limit of detection

Reference curves are commonly used to identify individuals with extreme values of clinically relevant variables or stages of progression which depend naturally on age or maturation. Estimation of reference curves can be complicated by a technical limit of detection (LOD) that censors the measurement from the left, as is the case in our study of reproductive hormone levels in boys around the time of the onset of puberty. We discuss issues with common approaches to the LOD problem in the context of our pubertal hormone study, and propose a two-part model that addresses these issues. One part of the proposed model specifies the probability of a measurement exceeding the LOD as a function of age. The other part of the model specifies the conditional distribution of a measurement given that it exceeds the LOD, again as a function of age. Information from the two parts can be combined to estimate the identifiable portion (i.e. above the LOD) of a reference curve and to calculate the relative standing of a given measurement above the LOD. Unlike some common approaches to LOD problems, the two-part model is free of untestable assumptions involving unobservable quantities, flexible for modeling the observable data, and easy to implement with existing software. The method is illustrated with hormone data from the Third National Health and Nutrition Examination Survey.

Modelling the association of blood pressure during pregnancy with gestational age and body mass index

Improved understanding of the determinants of blood pressure (BP) changes during pregnancy is essential for decreasing the morbidity and mortality borne by women and their families worldwide. While most epidemiological studies consider associations based on categorical risk factor classifications, using measurements on a continuous scale has been advocated as a means of gaining richer insights into biological processes. We modelled the relationship during pregnancy of continuous systolic (S) and diastolic (D) BP distributions with gestational age and pre-pregnancy body mass index (BMI) using fractional polynomials. We used information, including antenatal BP values abstracted from medical records, from a prospective cohort of 1733 women recruited before 20 weeks' gestation. The percentiles for SBP and DBP changed over pregnancy, with DBP percentiles decreasing initially, followed by an increase starting about mid-second trimester. Modelling the joint impact of BMI and gestational age on mean BP indicated an increase in mean BP with increasing BMI that was attenuated at higher BMI levels, later in pregnancy. This attenuation persisted in a variety of sub-analyses which explored the possibility that it was caused by confounding or by influential groupings of subjects. Estimated longitudinal percentiles that characterise the BP distribution across gestation may facilitate evaluation of BP during pregnancy. BP patterns observed over pregnancy and, in particular, the attenuation of BP increases at high BMI, late in pregnancy, can provide insights towards elucidating the mechanisms that drive BP changes during pregnancy.

Construction of the World Health Organization child growth standards: selection of methods for attained growth curves

The World Health Organization (WHO), in collaboration with a number of research institutions worldwide, is developing new child growth standards. As part of a broad consultative process for selecting the best statistical methods, WHO convened a group of statisticians and child growth experts to review available methods, develop a strategy for assessing their strengths and weaknesses, and discuss methodological issues likely to be faced in the process of constructing the new growth curves. To select the method(s) to be used, the group proposed a two-stage decision-making process. First, to select a few relevant methods based on a list of set criteria and, second, to compare the methods using available tests or other established procedures. The group reviewed 30 methods for attained growth curves. Using the pre-defined criteria, a few were selected combining five distributions and two smoothing techniques. Because the number of selected methods was considered too large to be fully tested, a preliminary study was recommended to evaluate goodness of fit of the five distributions. Methods based on distributions with poor performance will be eliminated and the remaining methods fully tested and compared.

Diagnostics for the exponential normal growth curve model

The exponential normal regression model provides a flexible parametric approach to constructing growth charts or centile curves. We develop diagnostics for this model, with a special emphasis towards assessing the impact of individual cases on estimated centile curves. We also propose a standardized score residual that assesses the fit of individual cases to the location, scale, and skewness functions, respectively. An application of the proposed diagnostics is presented.

Worm plot: a simple diagnostic device for modelling growth reference curves

The worm plot visualizes differences between two distributions, conditional on the values of a covariate. Though the worm plot is a general diagnostic tool for the analysis of residuals, this paper focuses on an application in constructing growth reference curves, where the covariate of interest is age. The LMS model of Cole and Green is used to construct reference curves in the Fourth Dutch Growth Study 1997. If the model fits, the measurements in the reference sample follow a standard normal distribution on all ages after a suitably chosen Box-Cox transformation. The coefficients of this transformation are modelled as smooth age-dependent parameter curves for the median, variation and skewness, respectively. The major modelling task is to choose the appropriate amount of smoothness of each parameter curve. The worm plot assesses the age-conditional normality of the transformed data under a variety of LMS models. The fit of each parameter curve is closely related to particular features in the worm plot, namely its offset, slope and curvature. Application of the worm plot to the Dutch growth data resulted in satisfactory reference curves for a variety of anthropometric measures. It was found that the LMS method generally models the age-conditional mean and skewness better than the age-related deviation and kurtosis.

Community-based surveillance of cardiovascular risk factors in Geneva: methods, resulting distributions, and comparisons with other populations

OBJECTIVES

This paper presents the results of an ongoing community-based surveillance program of cardiovascular risk factors in Geneva, Switzerland, using percentiles with their associated 95% confidence intervals and compares the Genevan results with published data from international surveys reporting percentiles.

METHODS

A random survey of adults ages 35 to 74 years was conducted from 1993 to 1994. Confidence intervals for percentiles based on parametric and non-parametric methods are given.

RESULTS

The distribution of total cholesterol was shifted upward with increasing age. The median reached a maximum at 55-64 years among men (5.7 mmol/L) and at 65-74 years among women (5.9 mmol/L), and remained relatively stable thereafter. In both genders, systolic and, less so, diastolic blood pressure increased progressively with advancing age. The median daily energy intake among men declined from 2,390 kcal at age 35-44 years to 2,169 kcal at age 65-74 years, while among women it remained stable at about 1,900 kcal. In both males and females, the relative intake of saturated fat was stable throughout life (14 to 13%). The median body mass index (BMI) was about 25 kg/m2 across all age groups among men, but increased with age among women, with a peak of 23.6 kg/m2 occurring at ages 65-74 years. Compared with U.S. and western European surveys, Genevan men and women had lower total plasma cholesterol and Genevan women tended to have lower BMIs.

CONCLUSIONS

Percentiles with their associated precision appear particularly well suited for international comparison of surveillance data. They could be used in the future to monitor shifts in distributions resulting from mass prevention strategies.

Age-related standards for ordinal data: modelling the changes in visual acuity from 2 to 9 years of age

The visual acuities of 1140 children between the ages of 2 and 9 years were measured using the Sonksen-Silver Acuity System (SSAS) from a distance of 3 metres. Ordered logistic regression was used to construct smoothly changing age-related reference ranges for the proportions of children at different ages expected to have acuities in each of three ordinal categories. Exponential functions of age fitted the data better than polynomials. The resulting reference ranges can be used to derive the ages at which a given percentage of children achieve each of the acuity cut-offs. However, since it is unlikely that all children will be available for testing at the exact appropriate age, two possible alternatives are introduced and discussed. First, a new ordinal category that would result in a chosen outcome at a pre-selected age is calculated. Secondly, because it may be more feasible to use the current SSAS system whilst varying the distance of measurement, age-related standards showing the distances required to give constant pass-rates with the usual letter-sizes are also presented.

Age-related reference ranges: significance tests for models and confidence intervals for centiles

A method of fitting age-related reference ranges using maximized likelihood is presented. The method is illustrated using CD4 counts of uninfected children born to HIV-1 infected women. The likelihood incorporates exponential functions to describe the Box-Cox transformation parameter that induces Normality and the mean and standard deviation of the transformed data, at all ages in the range. This guarantees that the centiles are asymptotic to notional adult values and avoids edge effects. Likelihood ratio tests were used to assess the significance of the model parameters, and profile likelihood confidence intervals were constructed around the fitted centiles. The asymptotes of the resulting CD4 reference range for the childhood dataset were compared with CD4 counts available from a group of uninfected adults. The proposed approach limits the need for arbitrary decisions about grouping of the data, degree of smoothing or choice of model.

Controlling for a confounder monotonically related to exposure by means of isotonic quantile regression

We present a new approach to evaluating the effect of a continuous exposure factor when that factor increases with a covariate (for example, age) which is regarded as a potential confounder. The basic idea is to estimate, as functions of the covariate, some selected quantiles of the exposure distribution, under the assumption that the dependence of each quantile on the covariate is monotonic. The resulting estimates are then used to divide the data into different exposure categories. This method of categorizing the data implies that the covariate distribution will be almost the same in each exposure group. We illustrate the approach with a study of blood pressure and cardiovascular mortality.

Smoothing reference centile curves: the LMS method and penalized likelihood

Refence centile curves show the distribution of a measurement as it changes according to some covariate, often age. The LMS method summarizes the changing distribution by three curves representing the median, coefficient of variation and skewness, the latter expressed as a Box-Cox power. Using penalized likelihood the three curves can be fitted as cubic splines by non-linear regression, and the extent of smoothing required can be expressed in terms of smoothing parameters or equivalent degrees of freedom. The method is illustrated with data on triceps skinfold in Gambian girls and women, and body weight in U.S.A. girls.

Constructing time-specific reference ranges

Reference ranges which take time (such as age) into account are often required in medicine, but simple, systematic and efficient statistical methods for constructing them are lacking. A method is described which is based on low order polynomial curves (linear, quadratic or occasionally cubic), together with guidelines for when and how to apply a logarithmic transformation to the variable analysed, testing for departures from normality, and assessment of the adequacy of the reference range which is constructed from the regression line plus or minus a multiple of the standard deviation. Standard statistical packages may be used to carry out the calculations. The question of comparing two or more groups of patients is addressed. Three examples are discussed in detail.