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Brain Trauma Stroke
Application of Latent Curve Models in Medical Research – A Review
Jun-Hao Pan,
1
Xin-Yuan Song
2
and Timothy Kwok
3
1. Lecturer, Department of Psychology, Sun Yat-sen University, Guangzhou; 2. Associate Professor, Department of Statistics, Chinese University of Hong Kong;
3. Professor, Department of Medicine and Therapeutics, Chinese University of Hong Kong
Abstract
Recently, latent curve modelling (LCM) has received increasing attention in the analysis of longitudinal data. It is a method to model individual
change and to assess the effects of co-variates and the relationship among multiple outcomes. It provides an integrated and flexible approach
in modelling developmental processes from both inter- and intra-individual perspectives. Similar to conventional longitudinal analysis, the main
objectives of this model are to characterise changes in the response of interest over time and to examine the selected covariates that contribute
to those changes. In this article the fundamental principle of LCM is briefly introduced. Several important kinds of LCM, including linear LCM,
non-linear LCM, multilevel LCM and mixture LCM, together with their applications in medical research, are reviewed. We believe that this
statistical technique should become more popular in medical applications, and that the medical field would benefit from increased use of this
powerful and flexible statistical method.
Keywords
Longitudinal data, medical application, latent curve models
Disclosure: The research described herein was fully supported by a research grant (GRF 450607) from the Research Grants Council of the Hong Kong Special Administration
Region, a grant (grant no. 931012) from the Health and Health Services Research Fund in Hong Kong and a research grant (grant no. 16000-3126133) from the Hundred Talent
Program of Sun Yat-sen University. The authors have no conflicts of interest to declare.
Received: 21 January 2009 Accepted: 7 October 2009
Correspondence: Timothy Kwok, Professor, Department of Medicine and Therapeutics, Prince of Wales Hospital, the Chinese University of Hong Kong, Shatin,
New Territories, Hong Kong. E:
tkwok@cuhk.edu.hk
Longitudinal data comprising repeated measurements of the same individual across time are fitted through a regression-type curve,
individuals on a number of occasions arise frequently in a wide range of which is either linear or non-linear. In the second stage, the focus
fields: medicine, public health, psychology, biology and more. The main of the analysis is on the latent growth factors, which are used to
objectives of a longitudinal study are to characterise changes in the identify the individual’s growth curve. The interest is no longer
response of interest over time and to examine the selected co-variates specifically on the original repeated measures observed, but on the
that contribute to those changes. Traditional methods used to analyse unobserved latent growth factors that lead to the repeated
longitudinal data are varied. Examples of these methods include measures. The LCM not only takes into account the mean of latent
autoregressive models, repeated measures multivariate analysis of growth factors, which represent the group-level change, but also
variance, mixed-effects models, multiple regressions and so on. considers the variances that measure the degree of individual
differences. This combination of group- and individual-level analyses
Recently, there has been growing interest in models that have the ability is synthesised in the LCM procedure.
to incorporate information concerning not only the group or population,
but also changes in the individual. Latent (growth) curve modelling The interpretation of the model parameters in the LCM is illustrated
allows for the testing of complex models regarding developmental with a simple two-factor linear model. The first-stage regression type
trends at both inter- and intra-individual levels. It has received increasing equation, also called trajectory equation, is:
attention in medical research recently and has been well recognised as
a useful longitudinal technique in the analysis of patterns of change.
1–6
y
it
= α
i
+ t
i
β
i
+ ε
it
(1)
Latent Curve Model where y is the measurement for individual i at time t, and α
it i
and β
i
The latent curve model (LCM) is a method to model individual are, respectively, the random intercept and slope for individual i, t
i
change, assess the effects of co-variates and assess the relationship represents the sequential value of each time record and ε
it
is the
among multiple outcomes, and take the measurement error into measurement error for individual i at time t.
account. It provides a means of modelling developmental processes
from both the inter- and intra-individual perspective. Generally In the second stage, the random intercept and slope are of interest,
speaking, LCM consists of two stages in modelling the patterns leading to to the following two equations, called the intercept
of change. In the first stage, the repeated measures of each equation and slope equation, respectively:
52 © TOUCH BRIEFINGS 2009
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