Power and Sample Size Calculations for Generalized Estimating Equations via Local Asymptotics

We consider the problem of calculating power and sample size for tests based on generalized estimating equations (GEE), that arise in studies involving clustered or correlated data (e.g., longitudinal studies and sibling studies). Previous approaches approximate the power of such tests using the asymptotic behavior of the test statistics under fixed alternatives. We develop a more accurate approach in which the asymptotic behavior is studied under a sequence of local alternatives that converge to the null hypothesis at root-m rate, where m is the number of clusters. Based on this approach, explicit sample size formulae are derived for Wald and quasi-score test statistics in a variety of GEE settings. Simulation results show that in the important special case of logistic regression with exchangeable correlation structure, previous approaches can inflate the projected sample size (to obtain nominal 90% power using the Wald statistic) by over 10%, whereas the proposed approach provides an accuracy of around 2%.

Low maternal retinol as a risk factor for schizophrenia in adult offspring

BACKGROUND

Prenatal micronutrient deficiency has been linked to later development of schizophrenia among offspring; however, no study has specifically investigated the association between vitamin A and this disorder. Vitamin A is an essential nutrient which is required by the early embryo and fetus for gene expression and regulation, cell differentiation, proliferation and migration. Previous work suggests that vitamin A deficiency in the second trimester may be particularly relevant to the etiopathogenesis of neurobehavioral phenotypes some of which are observed in schizophrenia.

METHODS

We examined whether low maternal vitamin A levels in the second trimester are associated with the risk of schizophrenia and other schizophrenia spectrum disorders (SSD) in the Prenatal Determinants of Schizophrenia study; third trimester vitamin A levels were also examined in relation to SSD. The cases were derived from a population-based birth cohort; all cohort members belonged to a prepaid health plan. Archived maternal serum samples were assayed for vitamin A in cases (N=55) and up to 2 controls per case (N=106) matched on length of membership in the health plan, date of birth (±28 days), sex, and gestational timing and availability of archived maternal sera.

RESULTS

For the second trimester, low maternal vitamin A, defined as values in the lowest tertile of the distribution among controls, was associated with a greater than threefold increased risk of SSD, adjusting for maternal education and age (OR=3.04, 95% CI=1.06, 8.79, p=.039). No association between third trimester maternal vitamin A and SSD was observed.

CONCLUSIONS

Although further investigations are warranted, this is the first birth cohort study to our knowledge to report an association between low maternal vitamin A levels and SSD among offspring.

Visualizing Multiple Quantile Plots

Multiple-quantile plots provide a powerful graphical method for comparing the distributions of two or more populations. This article develops a method of visualizing triple-quantile plots and their associated confidence tubes, thus extending the notion of a quantile-quantile (QQ) plot to three dimensions. More specifically, we consider three independent one-dimensional random samples with corresponding quantile functions Q₁, Q₂, and Q₃. The triple-quantile (QQQ) plot is then defined as the three-dimensional curve Q(p) = (Q₁(p), Q₂(p), Q₃(p)), where 0 < p < 1. The empirical likelihood method is used to derive simultaneous distribution-free confidence tubes for Q. We apply our method to an economic case study of strike durations and to an epidemiological study involving the comparison of cholesterol levels among three populations. These data as well as the Mathematica code for computation of the tubes are available in the online supplementary materials.

Folic acid supplements in pregnancy and severe language delay in children

CONTEXT

Prenatal folic acid supplements reduce the risk of neural tube defects and may have beneficial effects on other aspects of neurodevelopment.

OBJECTIVE

To examine associations between mothers' use of prenatal folic acid supplements and risk of severe language delay in their children at age 3 years.

DESIGN, SETTING, AND PATIENTS

The prospective observational Norwegian Mother and Child Cohort Study recruited pregnant women between 1999 and December 2008. Data on children born before 2008 whose mothers returned the 3-year follow-up questionnaire by June 16, 2010, were used. Maternal use of folic acid supplements within the interval from 4 weeks before to 8 weeks after conception was the exposure. Relative risks were approximated by estimating odds ratios (ORs) with 95% CIs in a logistic regression analysis.

MAIN OUTCOME MEASURE

Children's language competency at age 3 years measured by maternal report on a 6-point ordinal language grammar scale. Children with minimal expressive language (only 1-word or unintelligible utterances) were rated as having severe language delay.

RESULTS

Among 38,954 children, 204 (0.5%) had severe language delay. Children whose mothers took no dietary supplements in the specified exposure interval were the reference group (n = 9052 [24.0%], with severe language delay in 81 children [0.9%]). Adjusted ORs for 3 patterns of exposure to maternal dietary supplements were (1) other supplements, but no folic acid (n = 2480 [6.6%], with severe language delay in 22 children [0.9%]; OR, 1.04; 95% CI, 0.62-1.74); (2) folic acid only (n = 7127 [18.9%], with severe language delay in 28 children [0.4%]; OR, 0.55; 95% CI, 0.35-0.86); and (3) folic acid in combination with other supplements (n = 19,005 [50.5%], with severe language delay in 73 children [0.4%]; OR, 0.55; 95% CI, 0.39-0.78).

CONCLUSION

Among this Norwegian cohort of mothers and children, maternal use of folic acid supplements in early pregnancy was associated with a reduced risk of severe language delay in children at age 3 years.

Association of maternal genital and reproductive infections with verbal memory and motor deficits in adult schizophrenia

Maternal exposure to genital and reproductive infections has been associated with schizophrenia in previous studies. Impairments in several neuropsychological functions, including verbal memory, working memory, executive function, and fine-motor coordination occur prominently in patients with schizophrenia. The etiologies of these deficits, however, remain largely unknown. We aimed to assess whether prospectively documented maternal exposure to genital/reproductive (G/R) infections was related to these neuropsychological deficits in offspring with schizophrenia and other schizophrenia spectrum disorders. The cases were derived from a population-based birth cohort; all cohort members belonged to a prepaid health plan. Cases were assessed for verbal memory, working memory, executive function, and fine-motor coordination. Compared to unexposed cases, patients exposed to maternal genital/reproductive infection performed more poorly on verbal memory, fine-motor coordination, and working memory. Stratification by race revealed associations between maternal G/R infection and verbal memory and fine-motor coordination for case offspring of African-American mothers, but not for case offspring of White mothers. Significant infection-by-race interactions were also observed. Although independent replications are warranted, maternal G/R infections were associated with verbal memory and motor function deficits in African-American patients with schizophrenia.

FRACTALS WITH POINT IMPACT IN FUNCTIONAL LINEAR REGRESSION

This paper develops a point impact linear regression model in which the trajectory of a continuous stochastic process, when evaluated at a "sensitive time point", is associated with a scalar response. The proposed model complements and is more interpretable than the functional linear regression approach that has become popular in recent years. The trajectories are assumed to have fractal (self-similar) properties in common with a fractional Brownian motion with an unknown Hurst exponent. Bootstrap confidence intervals based on the least-squares estimator of the sensitive time point are developed. Misspecification of the point impact model by a functional linear model is also investigated. Non-Gaussian limit distributions and rates of convergence determined by the Hurst exponent play an important role.

Prenatal exposure to maternal infection and executive dysfunction in adult schizophrenia

OBJECTIVE

Executive dysfunction is one of the most prominent and functionally important cognitive deficits in schizophrenia. Although strong associations have been identified between executive impairments and structural and functional prefrontal cortical deficits, the etiological factors that contribute to disruption of this important cognitive domain remain unclear. Increasing evidence suggests that schizophrenia has a neurodevelopmental etiology, and several prenatal infections have been associated with risk of this disorder. The authors examined whether prenatal infection is associated with executive dysfunction in patients with schizophrenia.

METHOD

The authors assessed the relationship between serologically documented prenatal exposure to influenza and toxoplasmosis and performance on the Wisconsin Card Sorting Test and the Trail Making Test, part B (Trails B), as well as other measures of executive function, in 26 patients with schizophrenia from a large and well-characterized birth cohort.

RESULTS

Patients who were exposed to infection in utero committed significantly more total errors on the Wisconsin Card Sorting Test and took significantly more time to complete the Trails B than unexposed patients. Exposed patients also exhibited deficits on figural fluency, letter-number sequencing, and backward digit span.

CONCLUSIONS

Prenatal infections previously associated with schizophrenia are related to impaired performance on the Wisconsin Card Sorting Test and Trails B. The pattern of results suggests that cognitive set-shifting ability may be particularly vulnerable to this gestational exposure. Further work is needed to elucidate the specificity of prenatal infection to these executive function measures and to examine correlates with neuroanatomic and neurophysiologic anomalies.

PROPORTIONAL HAZARDS MODELS WITH CONTINUOUS MARKS

For time-to-event data with finitely many competing risks, the proportional hazards model has been a popular tool for relating the cause-specific outcomes to covariates [Prentice et al. Biometrics34 (1978) 541-554]. This article studies an extension of this approach to allow a continuum of competing risks, in which the cause of failure is replaced by a continuous mark only observed at the failure time. We develop inference for the proportional hazards model in which the regression parameters depend nonparametrically on the mark and the baseline hazard depends nonparametrically on both time and mark. This work is motivated by the need to assess HIV vaccine efficacy, while taking into account the genetic divergence of infecting HIV viruses in trial participants from the HIV strain that is contained in the vaccine, and adjusting for covariate effects. Mark-specific vaccine efficacy is expressed in terms of one of the regression functions in the mark-specific proportional hazards model. The new approach is evaluated in simulations and applied to the first HIV vaccine efficacy trial.

Maternal iron deficiency and the risk of schizophrenia in offspring

CONTEXT

Iron is essential for brain development and functioning. Emerging evidence suggests that iron deficiency in early life leads to long-lasting neural and behavioral deficits in infants and children. Adopting a life course perspective, we examined the effects of early iron deficiency on the risk of schizophrenia in adulthood.

OBJECTIVE

To determine whether maternal iron deficiency, assessed by maternal hemoglobin concentration during pregnancy, increases the susceptibility to schizophrenia spectrum disorders (SSDs) among offspring.

DESIGN

Data were drawn from a population-based cohort born from 1959 through 1967 and followed up for development of SSD from 1981 through 1997.

PARTICIPANTS

Of 6872 offspring for whom maternal hemoglobin concentration was available, 57 had SSDs (0.8%) and 6815 did not (99.2%).

MAIN OUTCOME MEASURE

Prospectively assayed, the mean value of maternal hemoglobin concentration was the primary exposure. Hemoglobin concentration was analyzed as a continuous and a categorical variable.

RESULTS

A mean maternal hemoglobin concentration of 10.0 g/dL or less was associated with a nearly 4-fold statistically significant increased rate of SSDs (adjusted rate ratio, 3.73; 95% confidence interval, 1.41-9.81; P = .008) compared with a mean maternal hemoglobin concentration of 12.0 g/dL or higher, adjusting for maternal education and ethnicity. For every 1-g/dL increase in mean maternal hemoglobin concentration, a 27% decrease in the rate of SSDs was observed (95% confidence interval, 0.55-0.96; P = .02).

CONCLUSIONS

The findings suggest that maternal iron deficiency may be a risk factor for SSDs among offspring. Given that this hypothesis offers the potential for reducing the risk for SSDs, further investigation in independent samples is warranted.

A fingerprint characteristic associated with the early prenatal environment

Fingerprints and fingertip ridge counts (RCs) have a significant genetic component. However, they also reflect the nongenetic environment of early pregnancy, an important time window for tissue differentiation and organogenesis. Fingerprints are permanently configured before the 20th week of gestation, and each fingertip's RC is related to the growth and regression of its early fetal volar pads. Rostral and caudal aspects of the embryonic limb bud have different relations to somite segments and to morphogen-activator functions. We hypothesized, therefore, that early fetal circumstances would be associated with a contrast in RCs between the thumb (digit 1) and little finger (digit 5). We obtained RCs from the fingerprints of a sample of 658 Dutch adults identified through prenatal and delivery records of Dutch urban births occurring during 1943-1947, an historical era that included months of wartime disruption with a winter famine. We calculated the mean of left- and right-hand RC differences between digits 1 and 5 (Md15). The Md15 fluctuated in relation to the calendar season of the mother's last menstrual period, but only if the gestation occurred outside of the wartime disruption interval. If the gestation occurred during the disruption interval, the Md15 seasonal fluctuation was not evident. This finding suggests that parental environmental factors may influence the fingerprints of the offspring. Fingerprint RC differences observed in postnatal life may be useful in the study of metabolic or anatomic programming related to the early prenatal environment.

The 2-sample problem for failure rates depending on a continuous mark: an application to vaccine efficacy

The efficacy of an HIV vaccine to prevent infection is likely to depend on the genetic variation of the exposing virus. This paper addresses the problem of using data on the HIV sequences that infect vaccine efficacy trial participants to (1) test for vaccine efficacy more powerfully than procedures that ignore the sequence data and (2) evaluate the dependence of vaccine efficacy on the divergence of infecting HIV strains from the HIV strain that is contained in the vaccine. Because hundreds of amino acid sites in each HIV genome are sequenced, it is natural to treat the genetic divergence as a continuous mark variable that accompanies each failure (infection) time. Problems (1) and (2) can then be approached by testing whether the ratio of the mark-specific hazard functions for the vaccine and placebo groups is unity or independent of the mark. We develop nonparametric and semiparametric tests for these null hypotheses and nonparametric techniques for estimating the mark-specific relative risks. The asymptotic properties of the procedures are established. In addition, the methods are studied in simulations and are applied to HIV genetic sequence data collected in the first HIV vaccine efficacy trial.

Tests for comparing mark-specific hazards and cumulative incidence functions

It is of interest in some applications to determine whether there is a relationship between a hazard rate function (or a cumulative incidence function) and a mark variable which is only observed at uncensored failure times. We develop nonparametric tests for this problem when the mark variable is continuous. Tests are developed for the null hypothesis that the mark-specific hazard rate is independent of the mark versus ordered and two-sided alternatives expressed in terms of mark-specific hazard functions and mark-specific cumulative incidence functions. The test statistics are based on functionals of a bivariate test process equal to a weighted average of differences between a Nelson-Aalen-type estimator of the mark-specific cumulative hazard function and a nonparametric estimator of this function under the null hypothesis. The weight function in the test process can be chosen so that the test statistics are asymptotically distribution-free. Asymptotically correct critical values are obtained through a simple simulation procedure. The testing procedures are shown to perform well in numerical studies, and are illustrated with an AIDS clinical trial example. Specifically, the tests are used to assess if the instantaneous or absolute risk of treatment failure depends on the amount of accumulation of drug resistance mutations in a subject's HIV virus. This assessment helps guide development of anti-HIV therapies that surmount the problem of drug resistance.

Comparison of HIV-1 and HIV-2 infectivity from a prospective cohort study in Senegal

From a prospective cohort study of 1948 initially human immunodeficiency virus (HIV) uninfected female commercial sex workers followed between 1985 and 1999 in Dakar, Senegal, the authors compared the male to female per infectious sexual exposure transmission probability of HIV types one (HIV-1) and two (HIV-2). New non-parametric competing risks failure time methods were used, which minimized modelling assumptions and controlled for risk factors for HIV infection. The HIV-1 versus HIV-2 infectivity ratio over time was estimated by the ratio of smoothed non-parametric kernel estimates of the HIV-1 and HIV-2 infection hazard functions in sex workers, adjusted by an estimate of the relative HIV-1 versus HIV-2 prevalence in the partner population. HIV-1 was found to be significantly more infectious than HIV-2 throughout the follow-up period (P < 0.001). The HIV-1/HIV-2 infectivity ratio was inferred to be approximately constant over time, with estimated common value 3.55. The finding of greater HIV-1 infectivity persisted in sensitivity analyses and in covariate-adjusted analyses, with adjusted infectivity ratio estimates ranging between 3.40 and 3.86. Understanding the mechanisms by which HIV-1 infects more efficiently than HIV-2 may be useful in the development of HIV-1 vaccines. Additionally, the methodology developed here may be useful for analysing other data sets.