Intentionally incomplete longitudinal designs: I. Methodology and comparison of some full span designs

A power approximation for the Kenward and Roger Wald test in the linear mixed model

We derive a noncentral F power approximation for the Kenward and Roger test. We use a method of moments approach to form an approximate distribution for the Kenward and Roger scaled Wald statistic, under the alternative. The result depends on the approximate moments of the unscaled Wald statistic. Via Monte Carlo simulation, we demonstrate that the new power approximation is accurate for cluster randomized trials and longitudinal study designs. The method retains accuracy for small sample sizes, even in the presence of missing data. We illustrate the method with a power calculation for an unbalanced group-randomized trial in oral cancer prevention.

Applying planned missingness designs to longitudinal panel studies in developmental science: An overview

Longitudinal panel studies are widely used in developmental science to address important research questions on human development across the lifespan. These studies, however, are often challenging to implement. They can be costly, time-consuming, and vulnerable to test-retest effects or high attrition over time. Planned missingness designs (PMDs), in which partial data are intentionally collected from all or some of the participants, are viable solutions to these challenges. This article provides an overview of several PMDs with potential utilities in longitudinal studies, including the multi-form designs, multi-method designs, varying lag designs, accelerated longitudinal designs, and efficient designs for analysis of change. For each of the designs, the basic rationale, design considerations, data analysis, advantages, and limitations are discussed. The article is concluded with some general recommendations to developmental researchers and promising directions for future research.

Optimal design for longitudinal studies to estimate pubertal height growth in individuals

BACKGROUND

The SITAR model expresses individual pubertal height growth in terms of mean size, peak height velocity (PHV) and age at PHV.

AIM

To use SITAR to identify the optimal time interval between measurements to summarise individual pubertal height growth.

SUBJECTS AND METHODS

Heights in 3172 boys aged 9-19 years from Christ's Hospital School measured on 128 679 occasions (a median of 42 heights per boy) were analysed using the SITAR (SuperImposition by Translation And Rotation) mixed effects growth curve model, which estimates a mean curve and three subject-specific random effects. Separate models were fitted to sub-sets of the data with measurement intervals of 2, 3, 4, 6, 12 and 24 months, and the different models were compared.

RESULTS

The models for intervals 2-12 months gave effectively identical results for the residual standard deviation (0.8 cm), mean spline curve (6 degrees of freedom) and random effects (correlations >0.9), showing there is no benefit in measuring height more often than annually. The model for 2-year intervals fitted slightly less well, but needed just four-to-five measurements per individual.

CONCLUSIONS

Height during puberty needs to be measured only annually and, with slightly lower precision, just four biennial measurements can be sufficient.

Sample size determination for multilevel hierarchical designs using generalized linear mixed models

A unified statistical methodology of sample size determination is developed for hierarchical designs that are frequently used in many areas, particularly in medical and health research studies. The solid foundation of the proposed methodology opens a new horizon for power analysis in presence of various conditions. Important features such as joint significance testing, unequal allocations of clusters across intervention groups, and differential attrition rates over follow up time points are integrated to address some useful questions that investigators often encounter while conducting such studies. Proposed methodology is shown to perform well in terms of maintaining type I error rates and achieving the target power under various conditions. Proposed method is also shown to be robust with respect to violation of distributional assumptions of random-effects.

Accelerated longitudinal designs: An overview of modelling, power, costs and handling missing data

Longitudinal studies are often used to investigate age-related developmental change. Whereas a single cohort design takes a group of individuals at the same initial age and follows them over time, an accelerated longitudinal design takes multiple single cohorts, each one starting at a different age. The main advantage of an accelerated longitudinal design is its ability to span the age range of interest in a shorter period of time than would be possible with a single cohort longitudinal design. This paper considers design issues for accelerated longitudinal studies. A linear mixed effect model is considered to describe the responses over age with random effects for intercept and slope parameters. Random and fixed cohort effects are used to cope with the potential bias accelerated longitudinal designs have due to multiple cohorts. The impact of other factors such as costs and the impact of dropouts on the power of testing or the precision of estimating parameters are examined. As duration-related costs increase relative to recruitment costs the best designs shift towards shorter duration and eventually cross-sectional design being best. For designs with the same duration but differing interval between measurements, we found there was a cutoff point for measurement costs relative to recruitment costs relating to frequency of measurements. Under our model of 30% dropout there was a maximum power loss of 7%.

The Effects of Tai Chi on Cardiovascular Risk in Women

PURPOSE

This study examined the effects of tai chi (TC) on biobehavioral factors associated with cardiovascular disease (CVD) risk in women.

DESIGN

A randomized trial used a wait-list control group, pretest-posttest design. Data were collected immediately before, immediately after, and 2 months following the intervention.

SETTING

The study was community based in central Virginia.

SUBJECTS

Women aged 35 to 50 years at increased risk for CVD.

INTERVENTION

The 8-week intervention built on prior work and was designed to impact biobehavioral factors associated with CVD risk in women.

MEASURES

Biological measures included fasting glucose, insulin, and lipids as well as C-reactive protein and cytokines. Behavioral measures included fatigue, perceived stress, depressive symptoms, social support, mindfulness, self-compassion, and spiritual thoughts and behaviors.

ANALYSIS

A mixed effects linear model was used to test for differences between groups across time.

RESULTS

In 63 women, TC was shown to decrease fatigue (∂ [difference in group means] = 9.38, p = .001) and granulocyte colony stimulating factor (∂ = 12.61, p = .052). Consistent with the study model and intervention design, significant changes observed 2 months post intervention indicated that TC may help down-regulate proinflammatory cytokines associated with underlying CVD risk, including interferon gamma (∂ = 149.90, p = .002), tumor necrosis factor (∂ = 16.78, p = .002), interleukin (IL) 8 (∂ = 6.47, p = .026), and IL-4 (∂ = 2.13, p = .001), and may increase mindfulness (∂ = .54, p = .021), spiritual thoughts and behaviors (∂ = 8.30, p = .009), and self-compassion (∂ = .44, p = .045).

CONCLUSION

This study contributes important insights into the potential benefits and mechanisms of TC and, with further research, may ultimately lead to effective strategies for reducing CVD risk in women earlier in the CVD trajectory.

LADES: a software for constructing and analyzing longitudinal designs in biomedical research

One of the most important steps in biomedical longitudinal studies is choosing a good experimental design that can provide high accuracy in the analysis of results with a minimum sample size. Several methods for constructing efficient longitudinal designs have been developed based on power analysis and the statistical model used for analyzing the final results. However, development of this technology is not available to practitioners through user-friendly software. In this paper we introduce LADES (Longitudinal Analysis and Design of Experiments Software) as an alternative and easy-to-use tool for conducting longitudinal analysis and constructing efficient longitudinal designs. LADES incorporates methods for creating cost-efficient longitudinal designs, unequal longitudinal designs, and simple longitudinal designs. In addition, LADES includes different methods for analyzing longitudinal data such as linear mixed models, generalized estimating equations, among others. A study of European eels is reanalyzed in order to show LADES capabilities. Three treatments contained in three aquariums with five eels each were analyzed. Data were collected from 0 up to the 12th week post treatment for all the eels (complete design). The response under evaluation is sperm volume. A linear mixed model was fitted to the results using LADES. The complete design had a power of 88.7% using 15 eels. With LADES we propose the use of an unequal design with only 14 eels and 89.5% efficiency. LADES was developed as a powerful and simple tool to promote the use of statistical methods for analyzing and creating longitudinal experiments in biomedical research.

Statistical analysis of longitudinal neuroimage data with Linear Mixed Effects models

Longitudinal neuroimaging (LNI) studies are rapidly becoming more prevalent and growing in size. Today, no standardized computational tools exist for the analysis of LNI data and widely used methods are sub-optimal for the types of data encountered in real-life studies. Linear Mixed Effects (LME) modeling, a mature approach well known in the statistics community, offers a powerful and versatile framework for analyzing real-life LNI data. This article presents the theory behind LME models, contrasts it with other popular approaches in the context of LNI, and is accompanied with an array of computational tools that will be made freely available through FreeSurfer - a popular Magnetic Resonance Image (MRI) analysis software package. Our core contribution is to provide a quantitative empirical evaluation of the performance of LME and competing alternatives popularly used in prior longitudinal structural MRI studies, namely repeated measures ANOVA and the analysis of annualized longitudinal change measures (e.g. atrophy rate). In our experiments, we analyzed MRI-derived longitudinal hippocampal volume and entorhinal cortex thickness measurements from a public dataset consisting of Alzheimer's patients, subjects with mild cognitive impairment and healthy controls. Our results suggest that the LME approach offers superior statistical power in detecting longitudinal group differences.

Analytic methods in Project HeartBeat!

Project HeartBeat! (1991-1995) was an observational study of the development of cardiovascular disease (CVD) risk factors in childhood and adolescence using an accelerated longitudinal design. The purpose of this paper is to explain the analytic methods used in the study, particularly multilevel statistical models. Measurements of hemodynamic, lipid, anthropometric, and other variables were obtained in 678 children who were enrolled in three cohorts (baseline ages 8, 11, and 14 years) and followed for 4 years, resulting in data for children aged 8-18 years. Patterns of change of blood pressure, serum lipid concentration, and obesity with age, race, and gender were of particular interest. The design specified 12 measurements of each outcome variable per child. Multilevel models were used to account for correlations resulting from repeated measurements on individuals and to allow use of data from incomplete cases. Data quality-control measures are described, and an example of multilevel analysis in Project HeartBeat! is presented. Multilevel models were also used to show that there were no differences attributable to the cohorts, and combining data from the three age cohorts was judged to be reasonable. Anthropometric data were compared with national norms and shown to have similar patterns; thus, the patterns seen in the CVD risk factors may be generalized, with some caveats, to the U.S. population of children.

Statistical Power Calculations for Clustered Continuous Data

To calculate the sample size for a research study it is important to take into account several aspects of the study design. In particular, one needs to take into account the hypotheses being tested, the study design, the sampling design, and the method to be used for the analysis. In this paper we propose a simple method to calculate sample size for clustered continuous data under various scenarios of study design.

A comparison of methods for determining HIV viral set point

During a course of human immunodeficiency virus (HIV-1) infection, the viral load usually increases sharply to a peak following infection and then drops rapidly to a steady state, where it remains until progression to AIDS. This steady state is often referred to as the viral set point. It is believed that the HIV viral set point results from an equilibrium between the HIV virus and immune response and is an important indicator of AIDS disease progression. In this paper, we analyze a real data set of viral loads measured before antiretroviral therapy is initiated, and propose two-phase regression models to utilize all available data to estimate the viral set point. The advantages of the proposed methods are illustrated by comparing them with two empirical methods, and the reason behind the improvement is also studied. Our results illustrate that for our data set, the viral load data are highly correlated and it is cost effective to estimate the viral set point based on one or two measurements obtained between 5 and 12 months after HIV infection. The utility and limitations of this recommendation will be discussed.

Sample size determination for hierarchical longitudinal designs with differential attrition rates

We consider the problem of sample size determination for three-level mixed-effects linear regression models for the analysis of clustered longitudinal data. Three-level designs are used in many areas, but in particular, multicenter randomized longitudinal clinical trials in medical or health-related research. In this case, level 1 represents measurement occasion, level 2 represents subject, and level 3 represents center. The model we consider involves random effects of the time trends at both the subject level and the center level. In the most common case, we have two random effects (constant and a single trend), at both subject and center levels. The approach presented here is general with respect to sampling proportions, number of groups, and attrition rates over time. In addition, we also develop a cost model, as an aid in selecting the most parsimonious of several possible competing models (i.e., different combinations of centers, subjects within centers, and measurement occasions). We derive sample size requirements (i.e., power characteristics) for a test of treatment-by-time interaction(s) for designs based on either subject-level or cluster-level randomization. The general methodology is illustrated using two characteristic examples.

Number of days, number of subjects, and sources of variation in longitudinal intervention or crossover feeding trials with multiple days of measurement

Dietary studies are often conducted as longitudinal intervention or crossover trials using multiple days of measurement on each subject during each of several measurement periods, and determining the required numbers of days and subjects is important in designing these studies. Linear mixed statistical models were used to derive equations for precision, statistical power and sample size (number of days and number of subjects) and to obtain estimates of between-subject, period-to-period, and day-to-day variation needed to apply the equations. Two cohorts of an on-going exercise intervention study, and a crossover study of Olestra, each with 14 d of measurement/subject per period, were used to obtain estimates of variability for energy and macronutrient intake. Numerical examples illustrate how the equations for calculating the number of days or number of subjects are applied in typical situations, and sample SAS code is given. It was found that between-subject, period-to-period, and day-to-day variation all contributed significantly to the variation in energy and macronutrient intake. The ratio of period-to-period and day-to-day standard deviations controls the trade-off between the number of days and the number of subjects, and this remained relatively stable across studies and energy and macronutrient intake variables. The greatest gains in precision were seen over the first few measurement days. Greater precision and fewer required days were noted in the study (Olestra) that exerted greater control over the subjects and diets during the feeding protocol.

Properties of internal pilots with the univariate approach to repeated measures

Uncertainty surrounding the error covariance matrix often presents the biggest barrier to achieving accurate power analysis in the 'univariate' approach to repeated measures analysis of variance (UNIREP). A poor choice gives either an overpowered study which wastes resources, or an underpowered study with little chance of success. Internal pilot designs were introduced to resolve such uncertainty about error variance for t-tests. In earlier papers, we extended the use of internal pilots to any univariate linear model with fixed predictors and independent Gaussian errors. Here we further extend our exact and approximate results to UNIREP analysis. For a fixed treatment effect, the inaccuracy in a power calculation depends only on the ratio of the true variance to the value used for planning. The greater complexity of repeated measures requires generalizing misspecification of error variance to the misspecification of the eigenvalues of the error covariance. We recommend approximating the misspecification in terms of the first and second moments of the eigenvalues, for both fixed sample and internal pilot designs. We also describe an unadjusted approach for internal pilots with repeated measures. Simulations illustrate the fact that both positive and negative properties in the univariate setting extend to repeated measures analysis. In particular, internal pilots allow maintaining power or reducing expected sample size when the covariance matrix used for planning differs from the true value. However, an unadjusted approach can inflate test size, at least with small to moderate sample sizes. Hence new, adjusted methods must be developed for small samples. At this time, we caution against using an internal pilot design with repeated measures without first conducting simulations to document the amount of test size inflation possible for the conditions of interest.

Guidelines for the design of clinical trials with longitudinal outcomes

A common objective of longitudinal clinical trials is to compare rates of change in a continuous response variable between two groups. The power realized for such a study is a function of both the number of people recruited and the planned number of measurements for each participant. By varying these two quantities in opposite directions, power can be kept at the desired level. We consider the problem of how best to choose the sample size and frequency of measurement, with a view to minimizing either the total number of measurements or the cost of a study. Some general guidelines are first developed for the situation in which all participants have complete observations. In practice, however, longitudinal studies often suffer from dropout, where a participant leaves the study permanently so that no further observations are possible. We therefore consider the impact of unanticipated dropout on power and also ways of allowing for dropout at the design stage. Based on our results, we propose some general design guidelines for longitudinal trials comparing rates of change when dropout is present.

Modern statistical techniques for the analysis of longitudinal data in biomedical research

Longitudinal study designs in biomedical research are motivated by the need or desire of a researcher to assess the change over time of an outcome and what risk factors may be associated with the outcome. The outcome is measured repeatedly over time for every individual in the study, and risk factors may be measured repeatedly over time or they may be static. For example, many clinical studies involving chronic obstructive pulmonary disease (COPD) use pulmonary function as a primary outcome and measure it repeatedly over time for each individual. There are many issues, both practical and theoretical, which make the analysis of longitudinal data complicated. Fortunately, advances in statistical theory and computer technology over the past two decades have made techniques for the analysis of longitudinal data more readily available for data analysts. The aim of this paper is to provide a discussion of the important features of longitudinal data and review two popular modern statistical techniques used in biomedical research for the analysis of longitudinal data: the general linear mixed model, and generalized estimating equations. Examples are provided, using the study of pulmonary function in cystic fibrosis research.