Epidemiology of Mental Disorders, New York State Psychiatric Institute, New York, NY, USA

Department of Psychiatry, College of Physicians and Surgeons, Columbia University, New York, NY, USA

Department of Epidemiology, Mailman School of Public Health, Columbia University, New York, NY, USA

Abstract

Background

The individual growth model is a relatively new statistical technique now widely used to examine the unique trajectories of individuals and groups in repeated measures data. This technique is increasingly used to analyze the changes over time in quality of life (QOL) data. This study examines the change from adolescence to adulthood in physical health as an aspect of QOL as an illustration of the use of this analytic method.

Methods

Employing data from the Children in the Community (CIC) study, a prospective longitudinal investigation, physical health was assessed at mean ages 16, 22, and 33 in 752 persons born between 1965 and 1975.

Results

The analyses using individual growth models show a linear decline in average physical health from age 10 to age 40. Males reported better physical health and declined less per year on average. Time-varying psychiatric disorders accounted for 8.6% of the explained variation in mean physical health, and 6.7% of the explained variation in linear change in physical health. Those with such a disorder reported lower mean physical health and a more rapid decline with age than those without a current psychiatric disorder. The use of SAS PROC MIXED, including syntax and interpretation of output are provided. Applications of these models including statistical assumptions, centering issues and cohort effects are discussed.

Conclusion

This paper highlights the usefulness of the individual growth model in modeling longitudinal change in QOL variables.

Background

Quality of life (QOL) has now become firmly established as an important and broad set of concerns in patient care and clinical research

The individual growth model

In this paper, we show how to use SAS PROC MIXED

Methods

Participants and study procedure

This study examined longitudinal data from the now-grown youths in the Children in the Community (CIC) study, an ongoing investigation of childhood behavior and development based on a sample of families randomly selected on the basis of residence in two upstate New York counties (21, 22). Approximately 800 mothers and one randomly sampled child from each family (mean age 5.5, SD = 2.8, in 1975) have been re-interviewed in their homes by extensively trained and supervised lay interviewers in 1985–1986 (n = 752), 1991–1994 (n = 751) and 2002–2004 (n = 641). These families were generally representative of the northeastern United States in terms of demographic characteristics and socioeconomic status (22). The sample also reflects the relatively high proportion of Catholic (54%) and Caucasian (91%) residents living in the sampled region. Detail of sampling, comparison to population, and retention rates are provided in the study website

Measures

Quality of life

Participating youth in 1985–86, 1991–94, and 2001–04 interviews completed the Quality of Life Instrument for Young Adults (YAQOL)

Individual Physical Health Change (raw data, n = 20)

Individual Physical Health Change (raw data, n = 20).

Psychiatric disorders

The parent and youth versions of the Diagnostic Interview Schedule for Children (DISC-I)

Individual growth models

In longitudinal QOL data we have measures of QOL at multiple time points for each individual. Individual growth models allow us to use the trajectories of individuals across time or age as the basic unit of analysis. Trajectory aspects include mean over time or age: is an individual's average QOL score higher or lower than that of others? Does it rise or fall with age? Is change non-linear, such as declining gradually but then later plunging? In individual growth models, those questions represent the individual intercept, slope and quadratic slope. Individual growth models may estimate change trajectories over time measured as age at each assessment. In clinical samples time since illness or treatment onset is a common alternative. In the current illustration we "centered" age at 23 years, the age closest to the mean over the entire data set, by subtracting 23 from each participant's age at each assessment. Linear, quadratic, cubic or other models can be fit, as a function of age or time.

Setting up the data file

Virtually all programs that analyze growth or time-changing variables of individuals require that the basic file to be analyzed be set up such that each row represents a specific measurement time for a specific individual and each column a different variable. In this file some variables will be repeated unchanged for each participant, including that persons ID and gender. Other variables may change in each assessment, including the dependent variable, age, and possible time-varying predictors. There may be different numbers of assessments for different participants.

Unconditional growth model

For the unconditional linear growth model, the level-1 model is:

QOL_{it }= α_{i }+ β_{i }+ r_{it}

The level-1 model indicates each individual's standing on QOL as a function of his or her level of QOL at age 23 (α_{i}), his or her linear growth trajectory (β_{i}), plus his or her random error as it varies by age (r_{it}). Level-1 models thus directly represent individuals' change trajectories.

The level-2 model is:

α_{i }= G_{00 }+ U_{0i }and β_{i }= G_{10 }+ U_{1i}

The level-2 model provides intercept and linear growth (slope over time) terms as the sample average, measured with some error. In addition to the average of the intercept and slope (fixed effects), the variances of the intercept and slope (random effects) are also obtained. It is important to note that even if the average slope is not significantly different from zero, significant variability in slope associated with the time variable in the level-1 model indicates that individuals are changing in QOL, although in different directions.

Conditional growth model

Once the unconditional linear growth model was selected for our QOL data, we may further determine whether the intercepts and linear slopes vary as a function of differences between the participants. The level-2 model may be expanded to become a "conditional" model. As in ordinary linear regression, additional predictors may be included in subsequent models. If those measures are constant across the time/age points they are considered "fixed" predictors (e.g., gender). If they also may change over the multiple assessments they are considered "time-varying predictors (e.g., psychiatric disorder). In either case such variables are added to the level 2 model to determine their association with QOL and the extent to which they may account for a fraction of the sample mean or linear trajectory. For example, with gender in the level-2 model:

α_{i }= G_{11 }+ G_{12 }(gender) + U_{1i}and β_{i }= G_{21 }+ G_{22 }(gender) + U_{2i}

We coded female 0 and male 1 in our data. In the conditional level-2 model, G_{11 }and G_{21 }represent the average intercept at age 23 and linear slope for female. G_{12 }and G_{22 }represent the mean difference between men and women for the average intercept at age 23 and linear slope.

Fitting individual growth models using SAS

Unconditional growth model (basic growth model)

We can fit the unconditional growth model in SAS PROC MIXED (12) quite easily using the following syntax:

The PROC MIXED statement calls the procedure. NOCLPRINT prevents printing the CLASS level information. COVEST tests the variance and covariance components (random effects). NOITPRINT statement tells SAS not to print the iteration history. The CLASS variable specifies that ID is a classification variable to indicate that the data represents multiple observations over time for individuals. MODEL statement is an equation whose left-side contains the name of the dependent variable, in this case HEALTH. The right-hand side contains a list of the fixed-effect variables (predictors). The intercept is contained in all models. This unconditional model tests only the intercept and slope without any predictors. DDFM = BW asks SAS to use the "Between/Within" method for computing the denominator degrees of freedom for tests of the fixed effects. NOTEST prevents the printing results of type 3 tests of fixed effects. RANDOM statement contains a list of the random effects, in this case intercept and age.

Conditional growth model for gender

Based on the unconditional growth model, we can add gender into the model and test the mean and slope differences in physical health by gender. The SAS syntax is:

The only change in this model is adding gender and gender*age in the right-hand side of the MODEL statement as predictors.

Conditional growth model for psychiatric disorders

Based on the conditional growth model for gender, we add a time-varying variable reflecting the presence of a psychiatric disorder and its product with age into the model and test the mean and slope differences in physical health associated with psychiatric disorder in a model that includes gender and age-gender product. The SAS syntax is:

Results

Unconditional linear growth model

Table

Individual growth models for longitudinal changes in physical health^{a}

**Unconditional Linear Model**

**Unconditional Non-linear Model**

**Gender**

**Psychiatric Disorder**

Estimate (SE)

Estimate (SE)

Estimate (SE)

Estimate (SE)

Random Variance

Intercept

101.53 (8.14) ***

101.68 (8.11) ***

87.34 (7.39) ***

79.86 (7.09) ***

Linear Slope

0.30 (0.08) ***

0.31 (0.08) ***

0.30 (0.08) ***

0.28 (0.08) ***

Residual

130.45 (7.17) ***

129.36 (7.12) ***

128.50 (6.96) ***

128.71 (7.04) ***

Fixed Effects

Intercept

74.71 (0.44) ***

75.19 (0.52) ***

70.95 (0.59) ***

72.26 (0.60) ***

Age

-0.63 (0.04) ***

-0.59 (0.05) ***

-0.73 (0.06) ***

-0.67 (0.06) ***

Age^{2}

-0.01 (0.01)

--

--

Gender

7.61 (0.84) ***

7.24 (0.81) ***

Gender × Age

0.25 (0.08) **

0.22 (0.08) **

Psychiatric Disorder

-5.95 (0.87) ***

Psychiatric Disorder × Age

-0.23 (0.11) *

Goodness of Fit^{b}

Parameters

5

6

7

9

Raw Likelihood (-2LL)

17624.0

17627.0

17538.3

17485.8

X^{2}

3.0

85.7 ***

138.2***

Degrees of Freedom

1

2

4

Note. SE = standard error; LL = log likelihood.

^{a}All parameter entries are maximum likelihood estimates fitted using SAS PROC MIXED.

Age was centered at 23 years, Gender was coded 0 = Female, 1 = Male.

Psychiatric disorder was coded 0 = no disorder, 1= disorder.

^{b}Models for non-linear, gender and psychiatric disorder are compared with the unconditional linear growth model.

* p < 0.05; ** p < 0.01; *** p < 0.001

Unconditional non-linear growth model

We add age*age (quadratic age) in the unconditional linear growth model to test the non-linear change in physical health. There was a non-significant negative quadratic age change in physical health (p = 0.08). The unconditional non-linear growth model was not significantly improved compared to the unconditional linear growth model (X^{2 }= 3.0, df = 1, p > 0.05). Therefore, we used the unconditional linear growth model as our basic growth model.

Conditional growth model for gender

Gender was powerful predictor of level of physical health

Physical Health Change by Gender

Physical Health Change by Gender.

Conditional growth model for psychiatric disorders

Psychiatric disorders have been associated with a lower level of physical health

Physical Health Change by Psychiatric Disorder

Physical Health Change by Psychiatric Disorder.

Discussion

Individual growth models are increasingly used to analyze the change in QOL data over time as more clinical trials include patients' longitudinal QOL data now

As noted in a basic regression text

The statistical maximum likelihood model used to generate these estimated effects assumes multivariate normality of the model residuals, linear relationships, and homoscedasticity. When the dependent variable distribution is seriously non-normal this assumption may be violated and a transform of the original dependent variable to more nearly normal distribution is likely to be necessary

We fit a growth model for our QOL data in which both intercepts and slopes vary across persons. We did not explore the within-person error covariance structure because these data consisted of only three longitudinal time points. With additional observations per person, additional structures for the within-person error covariance are possible. Three of the most commonly used structures are compound symmetry, unstructured, and autoregressive order one. The structure of the within-person error covariance matrix is specified using a REPEATED statement in SAS. The interested reader is referred to the SAS PROC MIXED (12), the helpful paper by Wolfinger (32) and the book by Singer and Willett (33).

Conclusion

This paper highlights the utility of growth model analyses in modeling longitudinal change in QOL variables.

Authors' contributions

Patricia Cohen and Henian Chen were responsible for conceptualization and design of the study and quality of life data collection. Henian Chen analyzed the data, interpreted the findings, and drafted the article. Patricia Cohen supervised the data analysis and assisted with the interpretation of findings and the critical revision of the article. Both read and approved the final manuscript.

Acknowledgements

This study was supported by National Institute of Mental Health Grant MH-36971, MH-38916, MH-49191 and MH-60911