➢ In orthopaedic research, investigators often encounter a variety of unbalanced data structures such as serial data sets with different follow-up intervals and points, longitudinal data sets with missing values, data sets with multiple measurements for individual subjects (for example, measurements from left and right sides), and so on.
➢ If investigators analyze these data sets with a traditional method such as multiple linear regressions and separate one-way analysis of variance at each time point, then statistical assumptions would be violated and the level of significance could be inflated.
➢ The linear mixed model provides a useful approach for analyzing unbalanced data structures. Because of recent advances in methods, the linear mixed model is now readily available for data analysis.
➢ The purposes of the present report are to introduce the use of the linear mixed model in orthopaedic research, to discuss estimation and inference, and to summarize data-analysis procedures for clinical investigators facing this problem.
A longitudinal study is among the most popular methods used to test hypotheses in orthopaedic research. However, clinical data analyses can present challenges in several situations.
The following orthopaedic scenario will be used throughout this article; in this scenario, the data structure is longitudinal. A researcher evaluated the long-term outcomes of the Dega osteotomy1 in fifty patients (sixty hips) with symptomatic developmental dysplasia of the hip. The mean age at the time of the operation was thirty months, and the mean duration of follow-up was sixty months (range, twelve to 144 months). The clinical outcome was assessed with use of preoperative and postoperative radiographic evaluations of the acetabular index. The preoperative and postoperative radiographs were repeatedly assessed during the follow-up period. However, some patients may not have completed the required follow-up, and therefore some of the necessary data were unavailable. Furthermore, for some patients, the survey may have been partially completed or not completed at all. These discrepancies could be interpreted as different follow-up durations, intervals, and visit numbers for each patient.
Traditionally, the repeated-measures analysis of variance (ANOVA) test is used to analyze an equal number of repeated measurements at regular time values for individual subjects. This type of analysis does not allow for the inclusion of any patient with even a small amount of missing data. Therefore, data loss in prospective studies could be substantial as the duration of follow-up becomes longer and longer. Furthermore, it could cause selection bias when subjects with unfavorable outcomes are lost to follow-up. In a prospective study, it is somewhat difficult to obtain the data at exact time points, and time effects can be frequently ignored. Similarly, in a retrospective study, the repeated data are usually obtained at irregular time points, which inevitably causes the problems of missing data and irregular time effects2.
In the above example involving the Dega osteotomy1, the researcher studied fifty patients (sixty hips), including forty women (forty-five hips) and ten men (fifteen hips). Thirty-four patients had an osteotomy on the right side and twenty-six had an osteotomy on the left side, with ten patients having the procedure bilaterally. For the patients undergoing bilateral procedures, little consideration was given to the statistical bias regarding the inclusion of only one side or even the inclusion of both sides as independent cases (assuming that the right and left sides in a single person were treated as two cases from two different persons). However, to achieve a true data analysis, no data should be discarded, and the unusual data correlations or irregular external effects should be appropriately reflected in the statistical analysis.
Linear mixed model analysis, which seeks to overcome the above-described problems, recently has become available to analysts. One major advantage of linear mixed model analysis is that it accommodates the complexities of typical data sets. The use of linear mixed model methodology for the analysis of repeated measurements is becoming increasingly common because the linear mixed model can be used to analyze data with correlations between or within cases, longitudinal data with different time effects, and data sets with missing data.
The purposes of the present review are to introduce the use of the linear mixed model, to discuss estimations and inferences, and to summarize data-analysis procedures in the context of orthopaedic research.
Issues Related to Longitudinal Studies in Orthopaedic Research
In clinical research, longitudinal studies are often designed to study changes in a particular parameter that is measured repeatedly over time. Variability among measurements for the same subject could be homogeneous but alternatively may be expected to change over time. In a repeated-measurements design, data are analyzed for subjects who have undergone measurements repeatedly over time. The repeated measurements refer to data in which the response of each subject is observed on multiple occasions or under multiple conditions (Fig. 1). The data require statistical methods that accept many repeated measurements as well as the time span between these measurements, which vary between subjects. Furthermore, the correct method is determined by considering the correlation between repeated measurements for individual subjects. Longitudinal studies also are often used to describe repeated observations of the same variables over the follow-up period. With longitudinal data, the response from each subject is observed on two or more occasions. As noted by Diggle et al.3, the benefit of longitudinal data is that such data can discriminate between changes over time within subjects and can allow for observation of changes from the baseline values for each subject.
In the example cited above, the radiographic values were measured for fifty subjects (ten male and forty female) who underwent Dega osteotomy1. During a follow-up period of sixty months, >300 radiographic values were measured repeatedly. The data structure is longitudinal. Within the groups of male and female patients, we observed considerable between-subject variation, with some subjects exhibiting relatively high values for their group and others exhibiting relatively low values. Within-subject variability indicates variation between repeated measurements for a single subject. Between-subject variability indicates variation between the measurements for multiple subjects. The data set should be modeled to address within and between-subject variations by assuming different random intercepts for each subject (Fig. 2).
The goals of longitudinal data research are (1) to identify patterns within the same subjects over time and (2) to inspect the effects of patterns such as the variation of covariates between subjects and within subjects.
Missing Data: Unequal Numbers of Repeated Measurements
The problem of missing values is common throughout statistical research and is almost always present in the analysis of longitudinal data. When longitudinal data have missing values, there are three important implications. First, the data are unavoidably unbalanced over time because of the different numbers of repeated measurements for the same subjects. Second, the loss of information will necessarily occur. As missing data for individual subjects are spread over time, the precision of statistical analysis will be reduced. Third, the missing data could cause bias, resulting in misleading inferences about changes of response. For example, if the missing data set were to be analyzed with use of the classic ANOVA method, then the data set would be discarded from the analysis, which could cause bias when the amount of missing data is considerable.
Rubin4 developed a useful method for describing the assumptions about the process that causes missing data. Generally, the types of missing data are randomly missing, randomly missing completely, and not randomly missing. The last-observation-carried-forward method is simple to use, but it makes the strong assumption that the value of the outcome remains unchanged after dropout, which seems highly unrealistic in many settings. However, the distortion of correlations and covariance could be a serious drawback5. For cases with missing values, the expectation-maximization algorithm is the optimum technique for estimating the model parameters. The expectation-maximization algorithm is an iterative algorithm that identifies parameters that maximize the log likelihood. Furthermore, in combination with powerful computing methods, it can provide a solution from the point of view of the problem.
Retrospective Studies: Repeated Measurements at Irregular Time Points
In retrospective studies, all of the data types are recorded with reference to fixed and predetermined time points. If data are measured in a continuous fashion, then the number and sequence of events and the interval between them can be calculated. Data recorded in a continuous fashion are often gathered retrospectively by means of a patient history and chart review. The main benefit of a retrospective study is in the greater detail and accuracy of information6. Longitudinal studies are restricted to a prospective design; however, retrospective studies have been described as having a quasi-longitudinal design7.
Data sets in orthopaedic research often include cases of bilateral involvement in which the data on both the left and right sides of the same subject are included in the analysis8. However, this issue can be avoided by analyzing only one side per subject. In cases of bilateral involvement, the value of the best or worst side could be selected or either side could be selected at random. However, in either scenario, the problem is that the values for both sides are not used. Separate analyses could be performed on each side to address this problem, but spreading the data across two analyses could cause loss of statistical power––that is, it reduces the chances of the study detecting significant results when an association exists9. These problems occur frequently in orthopaedic research. Park et al.8 wrote about the assumption of statistical independence and the frequent violations of this assumption (Fig. 3). There is also the issue of interpretation in cases in which the results for the two sides differ.
In the earlier example involving the Dega osteotomy1, the patients had different follow-up intervals, different follow-up durations, and different numbers of visits (Fig. 4). Furthermore, the data set included cases of bilateral involvement. If the goal of such a study is to estimate the annual changes in acetabular indices after the operation, what kind of statistical methods should the researchers consider?
A classic ANOVA might ignore an unequal number of repeated measurements at irregular time points and the correlation between the right and left sides. The linear mixed model should be used rather than analyzing the data with classic statistical frameworks.
Traditional methods assume that data exhibit normality and homogeneous variances. However, data sets that include cases of bilateral involvement often depart from the view of classic statistical methods. Whereas classic statistical methods depend on normally distributed data and are used to identify the effects of each predictor variable, data sets that include bilateral cases often involve random factors, and the aim of statistical analysis is to identify the variation among subjects. In the ANOVA model, all factors are handled as fixed factors. Fixed factors include qualitative covariates, such as sex, region, and age group. A random factor is a classification variable with levels that can be thought of as being randomly sampled from a population of levels. Traditional methods may violate statistical assumptions such as heterogeneity of variance by group. Furthermore, they restrict the limitation of inference; for example, the estimation of fixed effects cannot be extrapolated. Moreover, when follow-up data for a subject are missing, all of the data on the subject are discarded from the data analysis.
Modern methods such as the linear mixed model help to overcome these problems. The linear mixed model is a useful method for the analysis of longitudinal or repeated measurements that identifies the relationship between a continuous dependent variable and predictor variables10,11. The linear mixed model is an extension that includes additional random-effect terms in a linear model (see Appendix). The linear mixed model involves fixed effects and random effects12. Fixed effects describe the relationships between the dependent variable and the predictor variables for an entire population of subjects. Fixed effects are assumed to be unknown fixed quantities in the linear mixed model. Random effects are random values associated with the levels of a random factor in the linear mixed model. These values represent random deviations from the relationship described by fixed effects. In addition, random effects are factors that could be indicated according to within-subject and between-subject variations. The within-subject variation is the deviation between different observations in the same individual. The between-subject variation is the variation between the intercepts13. The linear mixed model is useful in situations in which repeated measurements are taken from the same subjects as those used in the longitudinal study. Furthermore, given its ability to deal with missing values, the linear mixed model is often preferred over more traditional approaches such as repeated-measures ANOVA.
The maximum likelihood method is used to estimate the parameters in many modern statistical tools. When the response variables are normal and the designs are balanced, the maximum likelihood method can be used to estimate the parameters in a classic ANOVA. However, the maximum likelihood method does not estimate the parameter of random effects, which is one way of estimating variance components in a linear mixed model. The restricted maximum likelihood method is an alternative to the maximum likelihood method, which estimates all effects as fixed effects. The restricted maximum likelihood method is the simplest and most widely used approximation of the linear mixed model14.
Inference and Model Selection
The restricted maximum likelihood method is used to estimate parameters for the linear mixed model, and then the proper inference types should be applied. The proper inferences select the best statistical model and draw the estimates and confidence intervals, which eventually test the hypothesis in the best way. The types of inference include hypothesis testing and model comparison according to the likelihood ratio test. In general, the likelihood ratio test is appropriate for inference of the linear mixed model, including random effects. The useful model-selection tools that provide a measure of relative goodness of fit among models are the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Smaller AIC or BIC values indicate a better model15.
The statistical software used for linear mixed model analysis included (1) the R version 2.15.3 (R Foundation for Statistical Computing, Vienna, Austria; ISBN 3-900051-07-0; http://www.r-project.org) using the NLME (Nonlinear and Linear Mixed Effects model) package and (2) the SAS (http://www.sas.com) system PROC MIXED procedure to fit the linear mixed model.
With use of data from the hypothetical case example, the linear mixed model application was used as follows. For each measured angle, the rate of improvement was adjusted according to multiple factors with use of the linear mixed model, with sex as a fixed effect and individual subject, age, and laterality as random effects. When applying the linear mixed model, the covariance structure enables the model to accept different types of data. The restricted maximum likelihood method was used to provide unbiased estimators. The linear mixed model was constructed to estimate the rate of the measured angle by including the linear age, sex, and laterality effects as covariates. A model with a random slope and a random intercept was proposed along with a test of the individual pattern of the angular correction rate and the duration of follow-up. The linear age, sex, and laterality effects were integrated to estimate the acetabular index. The best model was selected with use of the AIC or the BIC. In terms of model selection, a smaller AIC or BIC value is chosen. Therefore, the model was considered reasonable for estimating the acetabular index16.
Longitudinal data are used in statistical studies that accept many repeated measurements as well as the different time spans of the measurements between or within subjects. Furthermore, correct inferences can particularly be obtained by considering the correlation between repeated measurements within subjects. The linear mixed model can analyze data with correlations between or within cases and longitudinal data with different time effects. In this review, we suggest an appropriate method for longitudinal studies and a practical guide for application.
The findings of some orthopaedic research studies involving patients with bilateral involvement are statistically wrong and most likely result in wrong inferences and conclusions. The analysis of longitudinal data also is an important statistical approach that is generally used in orthopaedic research. The linear mixed model has been widely used for the analysis of longitudinal data. The advantages of the linear mixed model are useful in situations in which repeated measurements are taken from the same subjects and situations in which there are missing values.
In this review, we introduced the linear mixed model and its practical applications for orthopaedic data. It is now widely adopted by orthopaedic researchers who have the potential to choose appropriate tools for linear mixed model analysis.
The linear model
has one random effect, the error term . The parameters of the model are the regression coefficients, , and the error variance, . We rewrite the linear model in matrix form,
where is the response vector; is the model matrix, with typical row ; is the vector of regression coefficients; is the vector of errors; represents the n-variable multivariate-normal distribution; is an vector of zeroes; and is the order-n identity matrix.
So-called mixed models include additional random-effect terms and are often appropriate for representing clustered, and therefore dependent, data arising, for example, when data are collected hierarchically, when observations are taken on related individual, or when data are gathered over time on the same individuals.
Let be the vector of response from subject for . The general linear mixed model for longitudinal data is , , where is the model matrix for the fixed effects for observations in group i, is the vector of fixed-effect coefficients, is the vector of random-effect coefficients for group i, is the model matrix for the random effects for observations in group i, and is the vector of errors for observations in group i. The and vectors are assumed to be independent and . In addition, is independent , when .
These matrices and are subject-specific. This model is very general because subjects can have varying numbers of observation times as well as the various measurement values. The within-subject covariance matrix is assumed to depend on i only through its dimension ; unknown parameters in do not depend on i. A wide variety of covariance structures for and can be considered.
Laired and Ware12 considered the linear mixed model as two random effects model. In the first stage, they assumed that the model for the ith subject is . The vectors are assumed to be independently distributed as . The vector regression coefficient and the subject specific vectors are considered to be fixed. In second stage are assumed to be independent variates and and are assumed independent. Therefore, .
Source of Funding: There was no external source of funding for this investigation.
Investigation performed at the Department of Orthopaedic Surgery, Seoul National University Bundang Hospital, Kyungki, South Korea
Disclosure: None of the authors received payments or services, either directly or indirectly (i.e., via his or her institution), from a third party in support of any aspect of this work. None of the authors, or their institution(s), have had any financial relationship, in the thirty-six months prior to submission of this work, with any entity in the biomedical arena that could be perceived to influence or have the potential to influence what is written in this work. Also, no author has had any other relationships, or has engaged in any other activities, that could be perceived to influence or have the potential to influence what is written in this work. The complete Disclosures of Potential Conflicts of Interest submitted by authors are always provided with the online version of the article.
- Copyright © 2014 by The Journal of Bone and Joint Surgery, Incorporated