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1.  Sample size requirements to detect an intervention by time interaction in longitudinal cluster randomized clinical trials with random slopes 
In longitudinal cluster randomized clinical trials (cluster-RCT), subjects are nested within a higher level unit such as clinics and are evaluated for outcome repeatedly over the study period. This study design results in a three level hierarchical data structure. When the primary goal is to test the hypothesis that an intervention has an effect on the rate of change in the outcome over time and the between-subject variation in slopes is substantial, the subject-specific slopes are often modeled as random coefficients in a mixed-effects linear model. In this paper, we propose approaches for determining the samples size for each level of a 3-level hierarchical trial design based on ordinary least squares (OLS) estimates for detecting a difference in mean slopes between two intervention groups when the slopes are modeled as random. Notably, the sample size is not a function of the variances of either the second or the third level random intercepts and depends on the number of second and third level data units only through their product. Simulation results indicate that the OLS-based power and sample sizes are virtually identical to the empirical maximum likelihood based estimates even with varying cluster sizes. Sample sizes for random versus fixed slope models are also compared. The effects of the variance of the random slope on the sample size determinations are shown to be enormous. Therefore, when between-subject variations in outcome trends are anticipated to be significant, sample size determinations based on a fixed slope model can result in a seriously underpowered study.
doi:10.1016/j.csda.2012.11.016
PMCID: PMC3580878  PMID: 23459110
longitudinal cluster RCT; three level data; power; sample size; random slope; effect size
2.  Multivariate Longitudinal Modeling of Cognitive Aging 
GeroPsych  2012;25(1):15-24.
We illustrate the use of the parallel latent growth curve model using data from OCTO-Twin. We found a significant intercept-intercept and slope-slope association between processing speed and visuospatial ability. Within-person correlations among the occasion-specific residuals were significant, suggesting that the occasion-specific fluctuations around individual’s trajectories, after controlling for intraindividual change, are related between both outcomes. Random and fixed effects for visuospatial ability are reduced when we include structural parameters (directional growth curve model) providing information about changes in visuospatial abilities after controlling for processing speed. We recommend this model to researchers interested in the analysis of multivariate longitudinal change, as it permits decomposition and directly interpretable estimates of association among initial levels, rates of change, and occasion-specific variation.
doi:10.1024/1662-9647/a000051
PMCID: PMC3625423  PMID: 23589712
cognitive aging; longitudinal analysis; growth curve modeling; multivariate analysis
3.  Differences in HIV Natural History among African and Non-African Seroconverters in Europe and Seroconverters in Sub-Saharan Africa 
PLoS ONE  2012;7(3):e32369.
Introduction
It is unknown whether HIV treatment guidelines, based on resource-rich country cohorts, are applicable to African populations.
Methods
We estimated CD4 cell loss in ART-naïve, AIDS-free individuals using mixed models allowing for random intercept and slope, and time from seroconversion to clinical AIDS, death and antiretroviral therapy (ART) initiation by survival methods. Using CASCADE data from 20 European and 3 sub-Saharan African (SSA) cohorts of heterosexually-infected individuals, aged ≥15 years, infected ≥2000, we compared estimates between non-African Europeans, Africans in Europe, and Africans in SSA.
Results
Of 1,959 (913 non-Africans, 302 Europeans - African origin, 744 SSA), two-thirds were female; median age at seroconversion was 31 years. Individuals in SSA progressed faster to clinical AIDS but not to death or non-TB AIDS. They also initiated ART later than Europeans and at lower CD4 cell counts. In adjusted models, Africans (especially from Europe) had lower CD4 counts at seroconversion and slower CD4 decline than non-African Europeans. Median (95% CI) CD4 count at seroconversion for a 15–29 year old woman was 607 (588–627) (non-African European), 469 (442–497) (European - African origin) and 570 (551–589) (SSA) cells/µL with respective CD4 decline during the first 4 years of 259 (228–289), 155 (110–200), and 199 (174–224) cells/µL (p<0.01).
Discussion
Despite differences in CD4 cell count evolution, death and non-TB AIDS rates were similar across study groups. It is therefore prudent to apply current ART guidelines from resource-rich countries to African populations.
doi:10.1371/journal.pone.0032369
PMCID: PMC3295758  PMID: 22412867
4.  Segregation and linkage analysis for longitudinal measurements of a quantitative trait 
BMC Genetics  2003;4(Suppl 1):S21.
We present a method for using slopes and intercepts from a linear regression of a quantitative trait as outcomes in segregation and linkage analyses. We apply the method to the analysis of longitudinal systolic blood pressure (SBP) data from the Framingham Heart Study. A first-stage linear model was fit to each subject's SBP measurements to estimate both their slope over time and an intercept, the latter scaled to represent the mean SBP at the average observed age (53.7 years). The subject-specific intercepts and slopes were then analyzed using segregation and linkage analysis. We describe a method for using the standard errors of the first-stage intercepts and slopes as weights in the genetic analyses. For the intercepts, we found significant evidence of a Mendelian gene in segregation analysis and suggestive linkage results (with LOD scores ≥ 1.5) for specific markers on chromosomes 1, 3, 5, 9, 10, and 17. For the slopes, however, the data did not support a Mendelian model, and thus no formal linkage analyses were conducted.
doi:10.1186/1471-2156-4-S1-S21
PMCID: PMC1866456  PMID: 14975089
5.  A new approach to analyse longitudinal epidemiological data with an excess of zeros 
Background
Within longitudinal epidemiological research, ‘count’ outcome variables with an excess of zeros frequently occur. Although these outcomes are frequently analysed with a linear mixed model, or a Poisson mixed model, a two-part mixed model would be better in analysing outcome variables with an excess of zeros. Therefore, objective of this paper was to introduce the relatively ‘new’ method of two-part joint regression modelling in longitudinal data analysis for outcome variables with an excess of zeros, and to compare the performance of this method to current approaches.
Methods
Within an observational longitudinal dataset, we compared three techniques; two ‘standard’ approaches (a linear mixed model, and a Poisson mixed model), and a two-part joint mixed model (a binomial/Poisson mixed distribution model), including random intercepts and random slopes. Model fit indicators, and differences between predicted and observed values were used for comparisons. The analyses were performed with STATA using the GLLAMM procedure.
Results
Regarding the random intercept models, the two-part joint mixed model (binomial/Poisson) performed best. Adding random slopes for time to the models changed the sign of the regression coefficient for both the Poisson mixed model and the two-part joint mixed model (binomial/Poisson) and resulted into a much better fit.
Conclusion
This paper showed that a two-part joint mixed model is a more appropriate method to analyse longitudinal data with an excess of zeros compared to a linear mixed model and a Poisson mixed model. However, in a model with random slopes for time a Poisson mixed model also performed remarkably well.
doi:10.1186/1471-2288-13-27
PMCID: PMC3599839  PMID: 23425202
Two-part joint model; Excess of zeros; Count; Mixed modelling; Longitudinal; Statistical methods
6.  Neighborhood influences on the association between maternal age and birth weight: A multilevel investigation of age-related disparities in health 
Social science & medicine (1982)  2008;66(9):2048-2060.
It was hypothesized that the relationship between maternal age and infant birthweight varies significantly across neighborhoods and that such variation can be predicted by neighborhood characteristics. We analyzed 229,613 singleton births of mothers aged 20–45 from Chicago, USA in 1997–2002. Random coefficient models were used to estimate the between-neighborhood variation in age-birthweight slopes, and both intercepts- and-slopes-as-outcomes models were used to evaluate area-level predictors of such variation.
The crude maternal age-birthweight slopes for neighborhoods ranged from a decrease of 17 grams to an increase of 10 grams per year of maternal age. Adjustment for individual-level covariates reduced but did not eliminate this between-neighborhood variation. Concentrated poverty was a significant neighborhood-level predictor of the age-birthweight slope, explaining 44.4 percent of the between-neighborhood variation in slopes. Neighborhoods of higher economic disadvantage showed a more negative age-birthweight slope. The findings support the hypothesis that the relationship between maternal age and birthweight varies between neighborhoods. Indicators of neighborhood disadvantage help to explain such differences.
doi:10.1016/j.socscimed.2008.01.027
PMCID: PMC2794800  PMID: 18313187
birth weight; maternal age; poverty; social environment; socioeconomic factors; multi-level modeling
7.  Individual (N-of-1) trials can be combined to give population comparative treatment effect estimates: Methodologic considerations 
Journal of clinical epidemiology  2010;63(12):1312-1323.
Abstract/Summary
Objective
To compare different statistical models for combining N-of-1 trials to estimate a population treatment effect.
Study Design and Setting
Data from a published series of N-of-1 trials comparing amitriptyline therapy and combination treatment (amitriptyline + fluoxetine ) were analyzed to compare summary and individual participant data meta-analysis, repeated measures models, Bayesian hierarchical models, single-period, single-pair and averaged outcome crossover models.
Results
The best fitting model included a random intercept (response on amitriptyline) and fixed treatment effect (added fluoxetine). Results supported a common, uncorrelated within-patient covariance structure that is equal between-treatments and across patients. Assuming unequal within-patient variances, a random effects model was favored. Bayesian hierarchical models improved precision and were highly sensitive to within-patient variance priors.
Conclusion
Optimal models for combining N-of-1 trials need to consider goals, data sources, and relative within and between patient variances. Without sufficient patients, between-patient variation will be hard to explain with covariates. N-of-1 data with few observations per patients may not support models with heterogeneous within-patient variation. With common variances, models appear robust. Bayesian models may improve parameter estimation but are sensitive to prior assumptions about variance components. With limited resources, improving within-patient precision must be balanced by increased participants to explain population variation.
doi:10.1016/j.jclinepi.2010.04.020
PMCID: PMC2963698  PMID: 20863658
N-of-1 trials; methodology; comparisons; population estimate; meta-analysis; comparative effectiveness
8.  Replication of genetic associations as pseudoreplication due to shared genealogy 
Genetic epidemiology  2009;33(6):479-487.
The genotypes of individuals in replicate genetic association studies have some level of correlation due to shared descent in the complete pedigree of all living humans. As a result of this genealogical sharing, replicate studies that search for genotype-phenotype associations using linkage disequilibrium between marker loci and disease-susceptibility loci can be considered “pseudoreplicates” rather than true replicates. We examine the size of the pseudoreplication effect in association studies simulated from evolutionary models of the history of a population, evaluating the excess probability that both of a pair of studies detect a disease association compared to the probability expected under the assumption that the two studies are independent. Each of nine combinations of a demographic model and a penetrance model leads to a detectable pseudoreplication effect, suggesting that the degree of support that can be attributed to a replicated genetic association result is less than that which can be attributed to a replicated result in a context of true independence.
doi:10.1002/gepi.20400
PMCID: PMC2766236  PMID: 19191270
9.  Marginal and Random Intercepts Models for Longitudinal Binary Data With Examples From Criminology 
Two models for the analysis of longitudinal binary data are discussed: the marginal model and the random intercepts model. In contrast to the linear mixed model (LMM), the two models for binary data are not subsumed under a single hierarchical model. The marginal model provides group-level information whereas the random intercepts model provides individual-level information including information about heterogeneity of growth. It is shown how a type of numerical averaging can be used with the random intercepts model to obtain group-level information, thus approximating individual and marginal aspects of the LMM. The types of inferences associated with each model are illustrated with longitudinal criminal offending data based on N = 506 males followed over a 22-year period. Violent offending indexed by official records and self-report were analyzed, with the marginal model estimated using generalized estimating equations and the random intercepts model estimated using maximum likelihood. The results show that the numerical averaging based on the random intercepts can produce prediction curves almost identical to those obtained directly from the marginal model parameter estimates. The results provide a basis for contrasting the models and the estimation procedures and key features are discussed to aid in selecting a method for empirical analysis.
PMCID: PMC2893373  PMID: 20592941
10.  Random regression models for detection of gene by environment interaction 
Two random regression models, where the effect of a putative QTL was regressed on an environmental gradient, are described. The first model estimates the correlation between intercept and slope of the random regression, while the other model restricts this correlation to 1 or -1, which is expected under a bi-allelic QTL model. The random regression models were compared to a model assuming no gene by environment interactions. The comparison was done with regards to the models ability to detect QTL, to position them accurately and to detect possible QTL by environment interactions. A simulation study based on a granddaughter design was conducted, and QTL were assumed, either by assigning an effect independent of the environment or as a linear function of a simulated environmental gradient. It was concluded that the random regression models were suitable for detection of QTL effects, in the presence and absence of interactions with environmental gradients. Fixing the correlation between intercept and slope of the random regression had a positive effect on power when the QTL effects re-ranked between environments.
doi:10.1186/1297-9686-39-2-105
PMCID: PMC2682832  PMID: 17306196
gene by environment interaction; QTL detection; random regression; reaction norms
11.  NONINDEPENDENT AND NONSTATIONARY RESPONSE TIMES IN STOPPING AND STEPPING SACCADE TASKS 
Saccade stop-signal and target-step tasks are used to investigate the mechanisms of cognitive control. Performance of these tasks can be explained as the outcome of a race between stochastic GO and STOP processes. The race-model analyses assume that response times (RTs) measured throughout an experimental session are independent samples from stationary stochastic processes. This article demonstrates that RTs are neither independent nor stationary for humans and monkeys performing saccade stopping and target-step tasks. We investigate the consequences this has on analyses of these data. Nonindependent and nonstationary RTs artificially flatten inhibition functions and account for some of the systematic differences in RTs following different types of trials. However, nonindependent and nonstationary RTs do not bias the estimation of the stop-signal RT. These results demonstrate the robustness of the race model to some aspects of nonindependence and nonstationarity, and point to useful extensions of the model.
doi:10.3758/APP.72.7.1913
PMCID: PMC3237060  PMID: 20952788
12.  A meta-analysis of the relation between cumulative exposure to asbestos and relative risk of lung cancer. 
OBJECTIVES: To obtain summary measures of the relation between cumulative exposure to asbestos and relative risk of lung cancer from published studies of exposed cohorts, and to explore the sources of heterogeneity in the dose-response coefficient with data available in these publications. METHODS: 15 cohorts in which the dose-response relation between cumulative exposure to asbestos and relative risk of lung cancer has been reported were identified. Linear dose-response models were applied, with intercepts either specific to the cohort or constrained by a random effects model; and with slopes specific to the cohort, constrained to be identical between cohorts (fixed effect), or constrained by a random effects model. Maximum likelihood techniques were used for the fitting procedures and to investigate sources of heterogeneity in the cohort specific dose-response relations. RESULTS: Estimates of the study specific dose-response coefficient (kappa 1.i) ranged from zero to 42 x 10(-3) ml/fibre-year (ml/f-y). Under the fixed effect model, a maximum likelihood estimate of the summary measure of the coefficient (k1) equal to 0.42 x 10(-3) (95% confidence interval (95% CI) 0.22 to 0.69 x 10(-3)) ml/f-y was obtained. Under the random effects model, implemented because there was substantial heterogeneity in the estimates of kappa 1.i and the zero dose intercepts (Ai), a maximum likelihood estimate of k1 equal to 2.6 x 10(-3) (95% CI 0.65 to 7.4 x 10(-3)) ml/f-y, and a maximum likelihood estimate of A equal to 1.36 (95% CI 1.05 to 1.76) were found. Industry category, dose measurements, tobacco habits, and standardisation procedures were identified as sources of heterogeneity. CONCLUSIONS: The appropriate summary measure of the relation between cumulative exposure to asbestos and relative risk of lung cancer depends on the context in which the measure will be applied and the prior beliefs of those applying the measure. In most situations, the summary measure of effect obtained under the random effects model is recommended. Under this model, potency, k1, is fourfold lower than that calculated by the United States Occupational Safety and Health Administration.
PMCID: PMC1128699  PMID: 9166131
13.  Longitudinal age-and cohort trends in body mass index in Sweden – a 24-year follow-up study 
BMC Public Health  2013;13:893.
Background
The aim of this longitudinal study was to analyze whether mean Body Mass Index (BMI), assessed at four occasions, changed within different age groups and birth cohorts over time, i.e., between 1980/81 and 2004/05, after adjustment for possible confounders.
Methods
A sample of 2728 men and 2770 women aged 16–71 years at study start were randomly drawn from the Swedish Total Population Register and followed from 1980/81 to 2004/05. The same sample was assessed on four occasions during the 24-year study period (i.e., every eighth year). The outcome variable, BMI, was based on self-reported height and weight. A mixed model, with random intercept and random slope, was used to estimate annual changes in BMI within the different age groups and birth cohorts.
Results
Mean BMI increased from 24.1 to 25.5 for men and from 23.1 to 24.3 for women during the 24-year study period. The annual change by age group was highest in the ages of 32–39, 40–47 and 48–55 years among men, and in the ages of 24–31, 32–39, and 40–47 years among women. The highest annual changes were found in the youngest birth cohorts for both men and women, i.e., those born 1958–65, 1966–73, and 1974–81. For each birth cohort, the annual change in BMI increased compared to the previous, i.e., older, birth cohort. In addition, age-by-cohort interaction tests revealed that the increase in BMI by increasing age was higher in the younger birth cohorts (1966–1989) than in the older ones.
Conclusions
Public health policies should target those age groups and birth cohorts with the highest increases in BMI. For example, younger birth cohorts had higher annual increases in BMI than older birth cohorts, which means that younger cohorts increased their BMI more than older ones during the study period.
doi:10.1186/1471-2458-13-893
PMCID: PMC3849274  PMID: 24074433
Age; Birth cohort; Body mass index; Longitudinal data; Mixed models
14.  Selection on plasticity of seasonal life-history traits using random regression mixed model analysis 
Ecology and Evolution  2012;2(4):695-704.
Theory considers the covariation of seasonal life-history traits as an optimal reaction norm, implying that deviating from this reaction norm reduces fitness. However, the estimation of reaction-norm properties (i.e., elevation, linear slope, and higher order slope terms) and the selection on these is statistically challenging. We here advocate the use of random regression mixed models to estimate reaction-norm properties and the use of bivariate random regression to estimate selection on these properties within a single model. We illustrate the approach by random regression mixed models on 1115 observations of clutch sizes and laying dates of 361 female Ural owl Strix uralensis collected over 31 years to show that (1) there is variation across individuals in the slope of their clutch size–laying date relationship, and that (2) there is selection on the slope of the reaction norm between these two traits. Hence, natural selection potentially drives the negative covariance in clutch size and laying date in this species. The random-regression approach is hampered by inability to estimate nonlinear selection, but avoids a number of disadvantages (stats-on-stats, connecting reaction-norm properties to fitness). The approach is of value in describing and studying selection on behavioral reaction norms (behavioral syndromes) or life-history reaction norms. The approach can also be extended to consider the genetic underpinning of reaction-norm properties.
doi:10.1002/ece3.60
PMCID: PMC3399192  PMID: 22837818
Bird; clutch size; natural selection; phenotypic plasticity; reaction norm
15.  Calibrating disease progression models using population data: a critical precursor to policy development in cancer control 
Biostatistics (Oxford, England)  2010;11(4):707-719.
There are many more strategies for early detection of cancer than can be evaluated with randomized trials. Consequently, model-projected outcomes under different strategies can be useful for developing cancer control policy provided that the projections are representative of the population. To project population-representative disease progression outcomes and to demonstrate their value in assessing competing early detection strategies, we implement a model linking prostate-specific antigen (PSA) levels and prostate cancer progression and calibrate it to disease incidence in the US population. PSA growth is linear on the logarithmic scale with a higher slope after disease onset and with random effects on intercepts and slopes; parameters are estimated using data from the Prostate Cancer Prevention Trial. Disease onset, metastatic spread, and clinical detection are governed by hazard functions that depend on age or PSA levels; parameters are estimated by comparing projected incidence under observed screening and biopsy patterns with incidence observed in the Surveillance, Epidemiology, and End Results registries. We demonstrate implications of the model for policy development by projecting early detections, overdiagnoses, and mean lead times for PSA cutoffs 4.0 and 2.5 ng/mL and for screening ages 50–74 or 50–84. The calibrated model validates well, quantifies the tradeoffs involved across policies, and indicates that PSA screening with cutoff 4.0 ng/mL and screening ages 50–74 performs best in terms of overdiagnoses per early detection. The model produces representative outcomes for selected PSA screening policies and is shown to be useful for informing the development of sound cancer control policy.
doi:10.1093/biostatistics/kxq036
PMCID: PMC2950791  PMID: 20530126
Decision analysis; Population health; Prostatic neoplasm; Screening
16.  A generalized linear mixed model for longitudinal binary data with a marginal logit link function 
The annals of applied statistics  2011;5(1):449-467.
Summary
Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis (2003) proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper, we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall’s τ. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.
doi:10.1214/10-AOAS390
PMCID: PMC3082943  PMID: 21532998
Correlated binary data; multivariate normal distribution; probability integral transformation
17.  The use of statistical tools in field testing of putative effects of genetically modified plants on nontarget organisms 
Ecology and Evolution  2013;3(8):2739-2750.
Abstract
To fulfill existing guidelines, applicants that aim to place their genetically modified (GM) insect-resistant crop plants on the market are required to provide data from field experiments that address the potential impacts of the GM plants on nontarget organisms (NTO's). Such data may be based on varied experimental designs. The recent EFSA guidance document for environmental risk assessment (2010) does not provide clear and structured suggestions that address the statistics of field trials on effects on NTO's. This review examines existing practices in GM plant field testing such as the way of randomization, replication, and pseudoreplication. Emphasis is placed on the importance of design features used for the field trials in which effects on NTO's are assessed. The importance of statistical power and the positive and negative aspects of various statistical models are discussed. Equivalence and difference testing are compared, and the importance of checking the distribution of experimental data is stressed to decide on the selection of the proper statistical model. While for continuous data (e.g., pH and temperature) classical statistical approaches – for example, analysis of variance (ANOVA) – are appropriate, for discontinuous data (counts) only generalized linear models (GLM) are shown to be efficient. There is no golden rule as to which statistical test is the most appropriate for any experimental situation. In particular, in experiments in which block designs are used and covariates play a role GLMs should be used. Generic advice is offered that will help in both the setting up of field testing and the interpretation and data analysis of the data obtained in this testing. The combination of decision trees and a checklist for field trials, which are provided, will help in the interpretation of the statistical analyses of field trials and to assess whether such analyses were correctly applied.
We offer generic advice to risk assessors and applicants that will help in both the setting up of field testing and the interpretation and data analysis of the data obtained in field testing.
doi:10.1002/ece3.640
PMCID: PMC3930044  PMID: 24567836
Environmental risk assessment; experimental design; field trials; generalized linear models
18.  Statistical Analysis of the Recovery of Coliform Organisms on Gelman and Millipore Membrane Filters 
Applied Microbiology  1974;28(4):605-607.
The recovery of coliform organisms on Gelman and Millipore membranes was analyzed by using both a model I (which assumes no error in the x variable) and model II (which allows errors in both the variables) regression analysis. The two models afford estimates of the slope which agree within their 95% confidence limits. Using equations derived in this paper, the model II confidence limits on the intercept are obtained. This range does not include the model I intercept limits, thereby demonstrating the differences between results from an incorrect (model I) and correct (model II) approach. In addition, fecal coliform show no differences in response to the two membranes, whereas total coliform exhibit higher recoveries on Gelman membranes.
PMCID: PMC186781  PMID: 4608420
19.  Sample Sizes Required to Detect Interactions between Two Binary Fixed-Effects in a Mixed-Effects Linear Regression Model 
Summary
Mixed-effects linear regression models have become more widely used for analysis of repeatedly measured outcomes in clinical trials over the past decade. There are formulae and tables for estimating sample sizes required to detect the main effects of treatment and the treatment by time interactions for those models. A formula is proposed to estimate the sample size required to detect an interaction between two binary variables in a factorial design with repeated measures of a continuous outcome. The formula is based, in part, on the fact that the variance of an interaction is fourfold that of the main effect. A simulation study examines the statistical power associated with the resulting sample sizes in a mixed-effects linear regression model with a random intercept. The simulation varies the magnitude (Δ) of the standardized main effects and interactions, the intraclass correlation coefficient (ρ ), and the number (k) of repeated measures within-subject. The results of the simulation study verify that the sample size required to detect a 2 × 2 interaction in a mixed-effects linear regression model is fourfold that to detect a main effect of the same magnitude.
doi:10.1016/j.csda.2008.06.010
PMCID: PMC2678722  PMID: 20084090
interaction; mixed-effects linear regression; statistical power; sample size
20.  Reduced Intellectual Development in Children with Prenatal Lead Exposure 
Environmental Health Perspectives  2005;114(5):791-797.
Objective
Low-level postnatal lead exposure is associated with poor intellectual development in children, although effects of prenatal exposure are less well studied. We hypothesized that prenatal lead exposure would have a more powerful and lasting impact on child development than postnatal exposure.
Design
We used generalized linear mixed models with random intercept and slope to analyze the pattern of lead effect of the cohort from pregnancy through 10 years of age on child IQ from 6 to 10 years. We statistically evaluated dose–response nonlinearity.
Participants
A cohort of 175 children, 150 of whom had complete data for all included covariates, attended the National Institute of Perinatology in Mexico City from 1987 through 2002.
Evaluations/Measurements
We used the Wechsler Intelligence Scale for Children–Revised, Spanish version, to measure IQ. Blood lead (BPb) was measured by a reference laboratory of the Centers for Disease Control and Prevention (CDC) quality assurance program for BPb.
Results
Geometric mean BPb during pregnancy was 8.0 μg/dL (range, 1–33 μg/dL), from 1 through 5 years was 9.8 μg/dL (2.8–36.4 μg/dL), and from 6 through 10 years was 6.2 μg/dL (2.2–18.6 μg/dL). IQ at 6–10 years decreased significantly only with increasing natural-log third-trimester BPb (β = −3.90; 95% confidence interval, −6.45 to −1.36), controlling for other BPb and covariates. The dose–response BPb–IQ function was log-linear, not linear–linear.
Conclusions
Lead exposure around 28 weeks gestation is a critical period for later child intellectual development, with lasting and possibly permanent effects. There was no evidence of a threshold; the strongest lead effects on IQ occurred within the first few micrograms of BPb.
Relevance to Clinical Practice
Current CDC action limits for children applied to pregnant women permit most lead-associated child IQ decreases measured over the studied BPb range.
doi:10.1289/ehp.8552
PMCID: PMC1459938  PMID: 16675439
child development; intelligence; lead; prenatal exposure delayed effects
21.  Recreational Drug Use and T Lymphocyte Subpopulations in HIV-uninfected and HIV-infected Men 
Drug and alcohol dependence  2008;94(1-3):165-171.
The effects of recreational drugs on CD4 and CD8 T cells in humans are not well understood. We conducted a longitudinal analysis of men who have sex with men (MSM) enrolled in the Multicenter AIDS Cohort Study to define associations between self-reported use of marijuana, cocaine, poppers and amphetamines, and CD4 and CD8 T cell parameters in both HIV-uninfected and HIV-infected MSM. For the HIV-infected MSM, we used clinical and laboratory data collected semiannually before 1996 to avoid potential effects of antiretroviral treatment. A regression model that allowed random intercepts and slopes as well as autoregressive covariance structure for within subject errors was used. Potential confounders adjusted for included length of follow-up, demographics, tobacco smoking, alcohol use, risky sexual behaviors, history of sexually transmitted infections, and antiviral therapy. We found no clinically meaningful associations between use of marijuana, cocaine, poppers, or amphetamines and CD4 and CD8 T cell counts, percentages, or rates of change in either HIV-uninfected or -infected men. The regression coefficients were of minimum magnitude despite some reaching statistical significance. No threshold effect was detected for frequent (at least weekly) or continuous substance use in the previous year. These results indicate that use of these substances does not adversely affect the numbers and percentages of circulating CD4 or CD8 T cells in either HIV-uninfected or -infected MSM.
doi:10.1016/j.drugalcdep.2007.11.010
PMCID: PMC2691391  PMID: 18180115
marijuana; cocaine; poppers; recreational drug use; T cells; HIV infection
22.  Prediction models for clustered data: comparison of a random intercept and standard regression model 
Background
When study data are clustered, standard regression analysis is considered inappropriate and analytical techniques for clustered data need to be used. For prediction research in which the interest of predictor effects is on the patient level, random effect regression models are probably preferred over standard regression analysis. It is well known that the random effect parameter estimates and the standard logistic regression parameter estimates are different. Here, we compared random effect and standard logistic regression models for their ability to provide accurate predictions.
Methods
Using an empirical study on 1642 surgical patients at risk of postoperative nausea and vomiting, who were treated by one of 19 anesthesiologists (clusters), we developed prognostic models either with standard or random intercept logistic regression. External validity of these models was assessed in new patients from other anesthesiologists. We supported our results with simulation studies using intra-class correlation coefficients (ICC) of 5%, 15%, or 30%. Standard performance measures and measures adapted for the clustered data structure were estimated.
Results
The model developed with random effect analysis showed better discrimination than the standard approach, if the cluster effects were used for risk prediction (standard c-index of 0.69 versus 0.66). In the external validation set, both models showed similar discrimination (standard c-index 0.68 versus 0.67). The simulation study confirmed these results. For datasets with a high ICC (≥15%), model calibration was only adequate in external subjects, if the used performance measure assumed the same data structure as the model development method: standard calibration measures showed good calibration for the standard developed model, calibration measures adapting the clustered data structure showed good calibration for the prediction model with random intercept.
Conclusion
The models with random intercept discriminate better than the standard model only if the cluster effect is used for predictions. The prediction model with random intercept had good calibration within clusters.
doi:10.1186/1471-2288-13-19
PMCID: PMC3658967  PMID: 23414436
Logistic regression analysis; Prediction model with random intercept; Validation
23.  Regression dilution bias: Tools for correction methods and sample size calculation 
Upsala Journal of Medical Sciences  2012;117(3):279-283.
Background
Random errors in measurement of a risk factor will introduce downward bias of an estimated association to a disease or a disease marker. This phenomenon is called regression dilution bias. A bias correction may be made with data from a validity study or a reliability study.
Aims and methods
In this article we give a non-technical description of designs of reliability studies with emphasis on selection of individuals for a repeated measurement, assumptions of measurement error models, and correction methods for the slope in a simple linear regression model where the dependent variable is a continuous variable. Also, we describe situations where correction for regression dilution bias is not appropriate.
Results
The methods are illustrated with the association between insulin sensitivity measured with the euglycaemic insulin clamp technique and fasting insulin, where measurement of the latter variable carries noticeable random error. We provide software tools for estimation of a corrected slope in a simple linear regression model assuming data for a continuous dependent variable and a continuous risk factor from a main study and an additional measurement of the risk factor in a reliability study. Also, we supply programs for estimation of the number of individuals needed in the reliability study and for choice of its design.
Conclusions
Our conclusion is that correction for regression dilution bias is seldom applied in epidemiological studies. This may cause important effects of risk factors with large measurement errors to be neglected.
doi:10.3109/03009734.2012.668143
PMCID: PMC3410287  PMID: 22401135
Correction methods; measurement errors; regression dilution bias; SAS and R programs
24.  Longitudinal variance components models for systolic blood pressure, fitted using Gibbs sampling 
BMC Genetics  2003;4(Suppl 1):S25.
This paper describes an analysis of systolic blood pressure (SBP) in the Genetic Analysis Workshop 13 (GAW13) simulated data. The main aim was to assess evidence for both general and specific genetic effects on the baseline blood pressure and on the rate of change (slope) of blood pressure with time. Generalized linear mixed models were fitted using Gibbs sampling in WinBUGS, and the additive polygenic random effects estimated using these models were then used as continuous phenotypes in a variance components linkage analysis. The first-stage analysis provided evidence for general genetic effects on both the baseline and slope of blood pressure, and the linkage analysis found evidence of several genes, again for both baseline and slope.
doi:10.1186/1471-2156-4-S1-S25
PMCID: PMC1866460  PMID: 14975093
25.  The effect of reinforcer delays on the form of the forgetting function. 
Pigeons were trained in a matching-to-sample procedure with retention intervals of 0, 2, 4, 6, and 8 s mixed within each session. In different conditions, reinforcement was delayed by 0, 1, 2, 4, 6, or 8 s from correct choice responses. Discriminability decreased with increasing retention-interval duration and with increasing reinforcer delay. Exponential forgetting functions were fitted to discriminability measures plotted as a function of retention interval. Initial discriminability (intercept of the fitted functions) decreased with increasing reinforcer delay. Rate of forgetting (slope of the fitted functions) increased with reinforcer delay, suggesting an interaction between the effects of reinforcer delay and retention interval. The data were well described by multiplying an exponential function describing the effects of retention interval by a hyperbolic function describing the effect of reinforcer delay. This description included an interaction term that allowed for a greater effect of reinforcer delay at longer retention intervals.
doi:10.1901/jeab.2003.80-77
PMCID: PMC1284948  PMID: 13677610

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