Biostatistics in Psychiatry（4) Analysis of repeated outcome measures from longitudinal studies

Yuanjia WANG，Naihua DUAN

Biostatistics in Psychiatry（4) Analysis of repeated outcome measures from longitudinal studies

Yuanjia WANG*，Naihua DUAN

In many clinical studies repeated measurements of an outcome are collected over time.For example，in an 8-week study of treatment for obsessive compulsive disorder，the severity of the disorder may be measured weekly using the Yale-Brown-Obsessive-Compulsive-Disorder-Scale (YBOCS).For each study participant who completes the study，there will be nine repeated measures of YBOCS(a baseline assessment plus eight assessments during the course of treatment).Such a study in which participants are followed and measured repeatedly over time is called a longitudinal study and the resulting data are called longitudinal data［1-3］.

A naïve way to analyze longitudinal data is to perform a univariate analysis(for example，a twosample t-test)at each time point.However，this strategy does not provide an overall summary of the treatment effects over time nor does it allow one to examine how disease symptoms change over time，so the results are difficult to interpret.In addition，performing multiple two-sample t-tests(one at each time point)increases the likelihood of incorrectly reporting statistically significant differences—the problem of multiple comparisons.

A variety of statistical models are available to examine how symptoms change over time.To introduce these models，we will start by defining necessary notation.Let Yij denote the j-th outcome measurement(such as the symptom severity score) collected on the i-th participant in the sample.Let Tij denote the time of assessing the j-th outcome on the i-th participant.And let Gi denote the treatment group(Gi=1 for experimental group，and Gi=0 for control group).

The simplest longitudinal model is a linear model，which assumes that each participant's expected outcome follows a linear trajectory，therefore Yij，Tij and Gi are related as follows：

In this model，the expected trajectory for participants in the control group is

that is，a straight line with intercept b0 and slope b1.For the study about obsessive compulsive disorder，based on Model(2)the expected YBOCS score(i.e.，‘outcome')at baseline is b0 points，and the YBOCS score is expected to change over time at the rate of b1 points per unit time(such as per week，if the study time is measured in weeks). The slope coefficient b1 can be interpreted as the rate of‘natural recovery'，that is，the rate of recovery under the control condition;in this example it is the mean weekly decrease in the YBOCS score in the control group.

Participants in the experimental group follow an alternative expected trajectory，

that is，a straight line with intercept b0+b2 and slope b1+b3.The expected difference between the experimental group and the control group is

where b2 denotes the expected difference between the two groups at the baseline and b3 denotes the expected difference in the rate of change over time between two groups(i.e.，the difference in the slopes).In randomized trials，the coefficient b2 is expected to be zero because the randomly assigned groups are expected to have the same baseline score.Therefore Model(4)is simplified to

The coefficient b3(the difference in the rate of change or the difference in the slope)can be interpreted as the incremental effect of the experimental treatment.Using our study of obsessive compulsive disorder as an example，if the estimated b3= -0.5 YBOCS points，the interpretation is that the rate of symptom decrease in the experimental group is 0.5 YBOCS points per week faster than in the control group.The expected difference between the two groups at the end of an 8-week trial is therefore 0.5*8=4 YBOCS points according to Model(5).

The b3 coefficient is usually called the"time by group interaction"coefficient，because this effect is represented by the time x group interaction term，Tij*Gi，in Model(1).For randomized trials the significance of the treatment effect is tested by testing the coefficient b3 for time x group interaction，that is，testing the following null hypothesis：

As part of testing this hypothesis one must take intoconsiderationthelikelihoodthatrepeated measures on a particular subject are usually correlated，so the correlation of the random errors associated with each measurement in Model(1)is not zero：

Failure to account for this correlation can lead to erroneous estimation of the standard error of the treatment effect and，thus，to incorrect conclusions about the statistical significance of the results.To address this problem when analyzing longitudinal data the researcher must specify the correlation structure(i.e.，the form for the correlation in Equation 6).There are a variety of correlation structures for repeated measures that are commonly used for longitudinal models.One popular correlation structureisthefirst-orderautoregressivestructure (‘AR1'，page 87-89 in reference 1)，which assumes the correlation decreases exponentially with increasing time between two measurements［i.e.，corr(eij，eij')=ρ［j-j'［］］.Another popular correlation structure isthecompoundsymmetrystructure，which assumes that the correlation between two distinct measures is a constant regardless of how far apart they are in time［i.e.，corr(eij，eij')=ρ，for any j≠j'］.There are various model selection procedures(Chapter 7 in reference 2)that can be used to select the appropriate correlation structure for a specific longitudinal dataset—such as using goodness of fit summaries AIC(an information criterion) and BIC(Bayesian information criterion).

Once the appropriate correlation structure is specified，longitudinal models such as(1)can be fitted and tested using PROC MIXED or PROC GENMOD in SAS or the MIXED procedure in SPSS. The correlation(covariance)structure is specified in the subcommand TYPE in SAS or the subcommand COVTYPE in SPSS.The difference in slopes is tested for the regression coefficient b3 in Model (1).

The linear model(1)can be extended in a number of ways.Firstly，if the outcome trajectories do not follow the linear model，we can add nonlinear terms，such as quadratic terms，to the model：

The treatment effect is represented in this model by the two time x group interaction terms，Tij*Gi and Tij2*Gi，and can be tested by testing the following null hypothesis：

Here we test both the difference in the slopes(b4) and the difference in the quadratic coefficient(b5). The same model selection procedures such as AIC and BIC can also be used to select the correlation structure for Model(7).

Secondly，for dichotomous outcomes(Y=0 or 1)，the linear model(1)can be extended into a logistic regression model：

where Pij denotes the conditional probability for the dichotomous outcome Yij to assume the value 1 given Tij and Gi.As an example，a longitudinal study of patients with major depression examines whether the patient is in remission(Y=0)or relapse(Y=1) at each time point.Model(8)can be used to represent the changes in the rate of remission over time，and how the rate of change differs across treatment groups.The hypothesis of primary interest is

that is，whether the coefficient b3，the difference in the rate of change across groups，is null.Logistic regression will be discussed in further details in a future column in this series.

10.3969/j.issn.1002-0829.2011.04.015

Department of Biostatistics，Mailman School of Public Health，Columbia University and Division of Biostatistics，Department of Psychiatry，Columbia University Medical Center，New York，NY，USA.

*Correspondence：yw2016@columbia.edu

1. Diggles P，Heagerty P，Liang KY and Zeger S.Analysis of longitudinal data.New York：Oxford University Press，2001.

2. Fitzmaurice G，Laird N，Ware B.Applied Longitudinal Analysis. New York：Wiley-Interscience，2004.