Latent Class Growth Modelling: A Tutorial

The present work is an introduction to Latent Class Growth Modelling (LCGM). LCGM is a semi ‐ parametric statistical technique used to analyze longitudinal data. It is used when the data follows a pattern of change in which both the strength and the direction of the relationship between the independent and dependent variables differ across cases. The analysis identifies distinct subgroups of individuals following a distinct pattern of change over age or time on a variable of interest. The aim of the present tutorial is to introduce readers to LCGM and provide a concrete example of how the analysis can be performed using a real ‐ world data set and the SAS software package with accompanying PROC TRAJ application. The advantages and limitations of this technique are also discussed.

One approach is to study raw change scores.By this method, change is computed as the difference between the Time 1 and the Time 2 scores and the resulting raw change values are analyzed as a function of individual or group characteristics (Curran & Muthén, 1999).Raw change scores are typically analyzed using t-tests, analysis of variance (ANOVA), or multiple regression.An alternative approach is to study residualized change scores.By this method, change is computed as the residual between the observed Time 2 score and the expected Time 2 score as predicted by the Time 1 score (Curran & Muthén, 1999).Residualized change scores are typically analyzed using multiple regression or analysis of covariance (ANCOVA).
Although both raw and residualized change scores can be useful for analyzing longitudinal data under some circumstances, one limitation is that they tend to consider change between only two discrete time points and are thus more useful for prospective research designs (Curran & Muthén, 1999).Frequently, however, researchers are interested in modelling developmental trajectories, or patterns of change in an outcome across multiple (i.e., at least three) time points (Nagin, 2005).For instance, psychologists try to identify the course of various psychopathologies (e.g., Maughan, 2005), criminologists study the progression of 12 criminality over life stages (e.g., Sampson & Laub, 2005), and medical researchers test the impact of various treatments on the progression of disease (e.g., Llabre, Spitzer, Siegel, Saab, & Schneiderman, 2004).
A common approach to studying developmental trajectories is to use standard growth analyses such as repeated measures multivariate analysis of variance (MANOVA) or structural equation modelling (SEM; Jung & Wickrama, 2008).Standard growth analyses estimate a single trajectory that averages the individual trajectories of all participants in a given sample.Time or age is used as an independent variable to delineate the strength and direction of an average pattern of change (i.e., linear, quadratic, or cubic) across time for an entire sample.This average trajectory contains an averaged intercept (i.e., the expected value of the dependent variable when the value of the independent variable(s) is/are equal to zero) and an averaged slope (i.e., a line representing the predicted strength and direction of the growth trend) for the entire sample.This approach captures individual differences by estimating a random coefficient that represents the variability surrounding this intercept and slope.By this method, researchers can use categorical or continuous independent variables, representing potential risk or protective factors, to predict individual differences in the intercept and/or slope values.By centering the age or time variable, a researcher may set the intercept to any predetermined value of interest.For instance, a researcher could use self-esteem to predict individual differences in the intercept of depression at the start, middle, or end of a semester, depending on the research question.Results could indicate that people with higher self-esteem report lower levels of depression at the start of the semester.Similarly, researchers could use self-esteem to predict individual differences in the linear slope of depression.Results could indicate that people with higher self-esteem at baseline experience a slower increase in depressive symptoms over the course of the semester, indicating that self-esteem is a possible protective factor against a more severe linear increase in depression.
Standard growth models are useful for studying research questions for which all individuals in a given sample are expected to change in the same direction across time with only the degree of change varying between people (Raudenbush, 2001).Nagin (2002) offers time spent with peers as an example of this monotonic heterogeneity of change.With few exceptions, children tend to spend more time with their peers as they move from childhood to adolescence.In this case, it is useful to frame a research question in terms of an average trajectory of time spent with peers.However, some psychological phenomena may follow a multinomial pattern in which both the strength and the direction of change are varying between people (Nagin, 2002).Raudenbush (2001) uses depression as an example by arguing that it is incorrect to assume that all people in a given sample would be experiencing either increasing or decreasing levels of depression.In a normative sample, he states, many people will never be high on depression, others will always be high, others will become increasingly depressed, while others may fluctuate between high and low levels of depression.In such instances, a single averaged growth trajectory could mask important individual differences and lead to the erroneous conclusion that people are not changing on a given variable.Such conclusions could be drawn if 50% of the sample increased by the same amount on a particular variable whereas 50% of the sample decreased by the same amount on that variable.Here, a single growth trajectory would average to zero, thus prompting researchers to conclude an absence of change despite the presence of substantial yet opposing patterns of change for two distinct subgroups in the sample (Roberts, Walton, & Viechtbauer, 2006).For this class of problems, alternative modelling strategies are available that consider multinomial heterogeneity in change.One such approach is a group-based statistical technique known as Latent Class Growth Modelling (LCGM).

Theoretical basis of LCGM
Given the substantial contribution of Nagin (1999;2005) to both the theory and methodology of LCGM, the following explanations of the technique draw primarily from his work and from recent extensions proposed by his collaborators.LCGM is a semi-parametric technique used to identify distinct subgroups of individuals following a similar pattern of change over time on a given variable.Although each individual has a unique developmental course, the heterogeneity or the distribution of individual differences in change within the data is summarized by a finite set of unique polynomial functions each corresponding to a discrete trajectory (Nagin, 2005).Given that the magnitude and direction of change can vary freely across trajectories, a set of model parameters (i.e., intercept and slope) is estimated for each trajectory (e.g., Nagin, 2005).Unlike standard latent growth modelling techniques in which individual differences in both the slope and intercept are estimated using random coefficients, LCGM fixes the slope and the intercept to equality across individuals within a trajectory.Such an approach is acceptable given that individual differences are captured by the multiple trajectories included in the model.Given that both the slope and intercept are fixed, a degree of freedom remains available to estimate quadratic trajectories of a variable measured at three time points or cubic trajectories with data ) * it available at four time points.
Although the model is widely applicable, the rating scale of the instrument used to measure the variable of interest dictates the specific probability distribution used to estimate the parameters.Psychometric scale data necessitate the use of the censored normal model distribution, dichotomous data require the use of the binary logit distribution, and frequency data dictate the use of the Poisson distribution.For example, in the censored normal model, each trajectory is described as a latent variable ( y that represents the predicted score on a given dependent variable of interest for a given trajectory ( ) Υ ( ) j at a specific time ( ) t and is defined by the following function: In this equation, , , and represent the independent variable (i.e., Time or Age) entered in a regular, squared, or cubed term, respectively.Further, parameter and a linear trend may either steadily increase or decrease at varying magnitudes or remain stable.A quadratic pattern of change is defined by the parameter and a quadratic trend may increase, decrease, or remain stable up to a certain time point before changing in either magnitude or direction.Furthermore, a cubic trajectory is defined by the ( parameter and a cubic trend will have two changes in either the magnitude or direction across time points. ) Using LCGM, researchers must specify the number of distinct trajectories to be extracted from the data and select the model with the number of trajectories that best fits the data.It is preferable to have a priori knowledge concerning the number and the shape of trajectories whenever theory and literature exists in the area of study.Researchers evaluate which model provides the best fit to the data by interpreting and comparing both the fit statistics and the posterior probabilities for each model tested.The Bayesian Information Criterion (BIC) value is obtained for each model tested and is a fit index used to compare competing models that include different numbers of trajectories or trajectories of various shapes (e.g., linear versus quadratic).More specifically, nested models testing the inclusion of a different number of trajectories can be compared using an estimate of the log Bayes Factor defined by the following formula (Jones, Nagin, & Roeder, 2001): The estimate is approximately equal to two times the difference in the BIC values for the two models being compared.Here, the difference is calculated by subtracting the BIC value of the simpler model (i.e., the model with the smaller number of trajectories) from the more complex model (i.e., the model with the larger number of trajectories).A set of guidelines has been adopted for interpreting the estimate of the log Bayes Factor in order to measure the extent of evidence surrounding the more complex model thereby ensuring model parsimony.According to these guidelines, values ranging from 0 to 2 are interpreted as weak evidence for the more complex model, values ranging from 2 to 6 are interpreted as moderate evidence, values ranging from 6 to 10 are interpreted as strong evidence, and values greater than 10 are interpreted as very strong evidence (Jones et al., 2001).Initially, for each model, the linear, quadratic, and cubic functions of each trajectory can be tested, depending on the number of time points.To ensure parsimony, consistent with the recommendations of Helgeson, Snyder, and Seltman (2004), non-significant cubic and quadratic terms are removed from trajectories in a given model, but linear parameters are retained irrespective of significance (as cited in Louvet, Gaudreau, Menaut, Genty, & Deneuve, 2009).Once nonsignificant terms have been removed, each model is retested yielding a new BIC value.The fit of each nested model is then compared using the estimate of the log Bayes factor.This process of comparing the fit of each subsequent, more complex model, to the fit of the previously tested, simpler model, continues until there is no substantial evidence for improvement in model fit.In addition, both the posterior probabilities and the averaged group membership probabilities for each trajectory are examined to evaluate the tenability of each model.
The parameter coefficients estimated in LCGM provide direct information regarding group membership probabilities.A group membership probability is calculated for each trajectory and corresponds to the aggregate size of each trajectory or the number of participants belonging to a given trajectory.Ideally, each trajectory should hold an approximate group membership probability of at least five percent.However, in clinical samples, some trajectories may model the profile of change of only a fraction of the sample.
Posterior probabilities can be calculated post hoc to estimate the probability that each case, with its associated profile of change, is a member of each modelled trajectory.The obtained posterior probabilities can be used to assign each individual membership to the trajectory that best terior probabilities o erior probabilities of p matches his or her profile of change.A maximum-probability assignment rule is then used to assign each individual membership to the trajectory to which he or she holds the highest posterior membership probability.An example demonstrating the use of the maximum probability assignment rule is presented below.Table 1 displays a hypothetical data set with six individuals whose developmental course on a variable are modelled by three trajectories, thus resulting in three posterior probability values for each individual.Using the maximum probability assignment rule, participants 1 and 5 would be assigned group membership to Trajectory 3, participants 2 and 3 would be considered as members of Trajectory 2, and participants 4 and 6 would be regrouped into Trajectory 1.
The average posterior probability of group membership is calculated for each trajectory identified in the data.The average posterior probability of group membership for a trajectory is an approximation of the internal reliability for each trajectory.This can be calculated by averaging the posterior probabilities of individuals having been assigned group membership to a trajectory using the maximum probability assignment rule.Average posterior probabilities of group membership greater than .& Figueredo, 2007).Given this, it is recommended that any missing data be treated

Performing LCGM using SAS
For illustration purposes, a data set obtained with permission from Louvet, Gaudreau, Menaut, Genty, and Deneuve (2007) will be used to demonstrate how to perform the LCGM analyses (see Appendix A).This data set, labelled DISEN, includes a measure of disengagement coping at three time points for 107 participants.Typically, a data set of at least 300 to 500 cases is preferable for running LCGM, although the analysis can be applied to data sets of at least 100 cases (Nagin, 2005).It should be noted that performing LCGM with smaller sample sizes limits the power of the analysis as well as the number of identifiable trajectories (Nagin, 2005).In such instances, as in the example presented this tutorial, the researcher may adopt a more liberal significance criterion (e.g., p < .10;Tabachnick & Fidell, 2007) which is then applied consistently throughout the analysis.
In order to perform the analysis using SAS, the user has to install the PROC TRAJ application (Jones et al., 2001) which is freely available at the following w p://www.andrew.cmu.edu/user/bjones/.Complete instructions for downloading and installing the PROC TRAJ application are available on the website (Jones, 2005).
Before running the analysis using the PROC TRAJ application in SAS, the data set to be analyzed will need to be imported from the computer program on which it was prepared (e.g., Excel).The variables from the data set should be labelled before being imported into SAS.As seen in Figure 1, the first column in our data set is labelled "ID" and identifies each case in the data set.Each row should contain the data of one individual, or case.The next columns are the dependent variable scores.In this data set, there are three measures of disengagement coping, one at each time point.Thus, in this example, the second column is disengagement coping at Time 1, the third column is disengagement coping at Time 2, and the fourth column is disengagement coping at Time 3. The next columns in the data set are variables representing the time points at which the dependent variable was measured; in this example, columns 5, 6, and 7 represent Time 1, Time 2, and Time 3, respectively.In this data set, each measurement point is separated by the same amount of time; therefore, Time 1 is coded as "0", Time 2 is coded as "1", and Time 3 is coded as "2".However, LCGM can be performed even when the measurement points are separated by different time intervals that are relevant to the data analysis (e.g., Time 1 = baseline, Time 2 = 1 month, Time 3 = 6 months).In these instances, the user can code the time variable to represent the age or time of each measurement point (e.g., 5, 7, 13 [years old] or 1, 6, 18 [months since baseline]).The specific age/time can also vary across individuals to account for the fact that it is often impossible to measure each case at the exact same time.For this reason, the time variable is entered individually for each case in the database.Also, it is important to note that SAS uses an imputation procedure to assign values for missing data which may not be suitable for every data set To import the data into SAS, four steps must be followed.First, the user selects Import Data from the File drop-down menu.A window o licks on the syntax window and enters nes of syntax (please note, the colour of the text will change in any wa Third, once the syntax has been entered, copied and pasted into the syntax window from a program like Excel.Finally, on the next line after the data set, the syntax "RUN;" is entered.Figure 1 provides an example of the syntax window at this point. After adding the data set to the syntax window, the user enters the syntax to run the analysis.In this case, given that there are three time poin q yntax as NGROUPS 1) using the following syntax.For a ore complete description of each of the syntax items, lease refer to Appendix B. the second BIC value is interpreted as the index of model for one linear trajectory is run to determine the BIC t significant, the r B i r of the qua can be run (syntax "ORDER 1 2" or "ORDER 2 1").e the second BIC value is interpreted as the index of model for one linear trajectory is run to determine the BIC t significant, the r B i r of the qua can be run (syntax "ORDER 1 2" or "ORDER 2 1").e read from the Output for each model.The Group column labels the number of trajectories tested and the Parameter column labels the estimated parameters.The test statistic, standard error, and significance for each parameter are displayed in the Estimate column, the Standard Error column, the Prob > |T| column, respectively.The T for H0: Parameter = 0 column provides a value for the test of the null hypothesis that determines whether the parameter is significant or not.The value of the t-test has to be higher than 1.96 (p < .05)or 2.58 (p < .01).Two BIC values are also provided in each output and hypothesis that determines whether the parameter is significant or not.The value of the t-test has to be higher than 1.96 (p < .05)or 2.58 (p < .01).Two BIC values are also provided in each output and fit for the model.In this analysis modelled with one quadratic trajectory, the results can be summarized as follows: β0 = 2.09, p = .000;β1=.028, p = 0.86; and β2 = .01,p = .92;Sigma = .62,p = .000;BIC = -309.79(see Figure 3).The user can also scroll down in this window to view the predicted values of the dependent variable at different values of the independent variable for each point on the graph.
As a general rule, for data sets with three time points, a single quadratic trajectory model is tested first.If the quadratic component of this model is not significant, the fit for the model.In this analysis modelled with one quadratic trajectory, the results can be summarized as follows: β0 = 2.09, p = .000;β1=.028, p = 0.86; and β2 = .01,p = .92;Sigma = .62,p = .000;BIC = -309.79(see Figure 3).The user can also scroll down in this window to view the predicted values of the dependent variable at different values of the independent variable for each point on the graph.
As a general rule, for data sets with three time points, a single quadratic trajectory model is tested first.If the quadratic component of this model is not significant, the best fit is obtained, as determined by comparing the BIC values.
Coming back to our example, since the quadratic component of the one trajectory model was no best fit is obtained, as determined by comparing the BIC values.
Coming back to our example, since the quadratic component of the one trajectory model was no model for one linear trajectory is run by changing the order of the model from 2 (for quadratic) to 1 (for linear) in the syntax window.The rest of the syntax remains the same.After running this analysis, a new graph and a new output with the equired IC value is also generated.
To run the analysis for the quadratic model for two trajectories the user changes the syntax from "NGROUPS 1" to "NGROUPS 2" to estimate two trajectories.The syntax for the order is also changed to "ORDER 2 2".When using the ORDER syntax, the f rst number represents the first trajectory and the second number represents the second trajectory.Further, a "2" indicates the trajectory should be modelled on a quadratic trend whereas a "1" indicates a linear trend.This analysis can now be run.If neithe model for one linear trajectory is run by changing the order of the model from 2 (for quadratic) to 1 (for linear) in the syntax window.The rest of the syntax remains the same.After running this analysis, a new graph and a new output with the equired IC value is also generated.
To run the analysis for the quadratic model for two trajectories the user changes the syntax from "NGROUPS 1" to "NGROUPS 2" to estimate two trajectories.The syntax for the order is also changed to "ORDER 2 2".When using the ORDER syntax, the f rst number represents the first trajectory and the second number represents the second trajectory.Further, a "2" indicates the trajectory should be modelled on a quadratic trend whereas a "1" indicates a linear trend.This analysis can now be run.If neithe dratic components of the two-trajectory model is significant, the linear model can be run (syntax "ORDER 1 1").Likewise, if only one component is significant, a model with one quadratic component and one linear component dratic components of the two-trajectory model is significant, the linear model can be run (syntax "ORDER 1 1").Likewise, if only one component is significant, a model with one quadratic component and one linear component value for this model.If the quadratic component of the model for one trajectory is significant, the analysis for the quadratic model for two trajectories is performed.Following these analyses, the BIC value of the appropriate twotrajectory model will be compared to the BIC value of the appropriate one-trajectory model.This process is repeated with an increasing number of trajectories until the model of In our example, given that both quadratic components of the two-trajectory model are significant, the BIC value obtained from this analysis is compared to the BIC valu from the previous analysis to test for improvement of fit.Using this data set as an example, the model with two quadratic trajectories (BIC = -289.05) is compared to the model with one linear trajectory (BIC = -307.79)using the value for this model.If the quadratic component of the model for one trajectory is significant, the analysis for the quadratic model for two trajectories is performed.Following these analyses, the BIC value of the appropriate twotrajectory model will be compared to the BIC value of the appropriate one-trajectory model.This process is repeated with an increasing number of trajectories until the model of In our example, given that both quadratic components of the two-trajectory model are significant, the BIC value obtained from this analysis is compared to the BIC valu from the previous analysis to test for improvement of fit.Using this data set as an example, the model with two quadratic trajectories (BIC = -289.05) is compared to the model with one linear trajectory (BIC = -307.79)using the   Using the guidelines for interpreting the estimate of the log Bayes factor (Jones et al., 2001), these results provide very strong evidence for the model with two quadratic trajectories compared to a model with one trajectory.However, it is necessary to continue testing more complex models with an increasing number of trajectories in order to det Next, a model with three quadratic trajectories is run by changing the syntax from "NGROUPS 2" to "NGROUPS 3" and by changing the order to read "ORDER 2 2 2".The output of this model, displayed in Figure 4, shows that the quadratic terms of Trajectory 1 and Trajectory 3 are significant whereas the quadratic parameter of Trajectory 2 is not significant.Given this, the user can test a model with two quadratic trajectories and one linear trajectory with the following sy tax c des indicating that Trajec ory is linear: "NGROUPS 3; ORDER 2 1 2".As shown in Figure 5, the results of this analysis indicate that all modelled components of the trajectories can be considered significant as the small sample size warrants a more liberal significance idell, 2007).Comparing this three-trajectory model to the tw  it for the four-trajectory model.As a result, the three-trajectory model is retained as the final and most parsimonious model.The output for this final model is displayed in Figure 5 and the parameters presented in the output will be interpreted.Figure 6 displays the graph for this final model with Trajectory 1 shown on the bottom, Trajectory 2 in the middle, and Trajectory 3 on top.Trajectory 1 follows a quadratic trend in which disengagement coping decreases from Time 1 to Time 2 and returns to its initial level by Time 3. The participants whose profiles of change are best represented by Trajectory 1 tended to report low levels of disengagement coping which decreased from Time 1 to Time 2 and then returned to the initial level by Time 3 (β0 = 1.78, p < 0.001; β1 = -.032,p = 0.07; β2 = 0.16, p = 0.06).Trajectory 2, which follows a linear trend, represents the profile of change of the participants who reported moderate levels of disengagement at Time 1 followed by a marginally significant linear increase across time (β0 = 2.33, p < 0.001; β1 = 0.09, p = 0.07).Trajectory 3 follows a quadratic trend in which disengagement coping increases from Time 1 to Time 2 and then decreases by Time 3. The participants whose profiles of change are best represented by Trajectory 3 tended to report high levels of disengagement copi After the selection of the most suitable model, the posterior probabilities (i.e., the likelihood of each case belonging to each trajectory) are calculated.This is done by running the syntax "PROC PRINT DATA=OF" which will generate a tabular output that identifies each case and the likelihood of each case belonging to each trajectory.Participants are then assigned group membership to a trajectory based on the maximum probability assignment rule.Finally, the posterior probabilities of the individuals assigned membership to a given trajectory are averaged to obtain the average posterior probability of group membership for each trajectory and can be examined to assess the reliability of the trajectory.For the final model, the average posterior probabilities for Trajectory 1, Trajectory 2, and Trajectory 3 were .88,.93,and .88,respectively.In addition, the group membership probabilities are provided in an output for each model (provided by the syntax OUT=OF), but only the group membership probabilities for the final model are interpreted in this example.Based on these probabilities, it is estimated that 47% of the sample is categorized within the first trajectory, 50% is categorized within the second trajectory and 3% of the sample is categorized within the third trajectory (see Figure 5).

Recent Extensions of LCGM
A recent extension to the PROC TRAJ application allows researchers to calculate the 95% confidence interval surrounding each trajectory to determine if the trajectories are overlapping at any of the measurement points (Jones & Nagin, 2007).This is done by adding "CI95M" to the first line of syntax and "%TRAJPLOTNEW" after the very last line of syntax.Below is an example of the syntax used to calculate the 95% confidence intervals surrounding each trajectory in the sample data set: This syntax yields a new graph of the trajectories that includes the upper and lower confidence interval for each trajectory.It also provides an output with the numerical values of the confidence intervals for each trajectory at each time point, as shown in Table 2.For a more detailed description of this extension and several others, including performing LCGM with covariates, please refer to Jones and Nagin (2007) and Gaudreau, Amiot, and Vallerand (in press).
Another extension allows researchers to test whether the intercept and the slopes are significantly different across the trajectories using equality constraints, also known as invariance testing (Jones & Nagin, 2007).The Wald test is the statistical test used to compare the intercepts ( ) 0 β as well as the linear ( ) β and quadratic ( ) 2 β growth com- ponents of the trajectories to determine whether they are significantly different across trajectories.Using the sample data set, it is possible to compare if there is a significant difference between the intercepts of any two trajectories.As an example, to compare the intercepts of Trajectories 1 and 3 the following syntax is added to the line following the ORDER command: This analysis revealed that the intercepts of these two trajectories were significantly different ( χ 2 (1) =27.68, p < .0001),thus indicating that Trajectory 3 had a greater disengagement score at the time point on which the independent variable was centered (i.e., Time 1 was coded as 0; see Figure 1) .To compare the slopes of Trajectories 1 and 3, the "%TRAJTEST" syntax is changed to read: %TRAJTEST(ʹlinear1=linear3,quadra3=quadra1ʹ) /*linear&quadratic equality test*/ The rest of the syntax remains the same and this analysis reveals that the slopes of these two quadratic trajectories are also significantly different, χ 2 (2) =7.64 p < .05,which indicates that the quadratic slope of Trajectory 3 is steeper than that of Trajectory 1.

Conclusion
LCGM is a useful technique for analyzing outcome variables that change over age or time.This method provides a number of advantages over standard growth modelling procedures.First, rather than assuming the existence of a particular type or number of trajectories a priori, this method uses a formal statistical procedure to test whether the hypothesized trajectories actually emerge from the data (Nagin, 2005).As such, the method permits the discovery of unexpected yet potentially meaningful trajectories that may have otherwise been overlooked (Nagin, 2005).LCGM also bypasses a host of other challenges (e.g., over-or under-fitting the data, trajectories reflecting only random variation) associated with assignment rules sometimes used in conjunction with standard growth modelling approaches (Nagin, 2002).Although not presented in detail here, extensions of the basic LCGM model also allow the researcher to estimate the probability that an individual will belong to a particular trajectory based on their score on a covariate.Other extensions allow the researcher to obtain estimates of whether a turning point event (such as an intervention or important life transition) can alter a developmental trajectory (see Nagin, 2005;Jones & Nagin, 2007).In addition, LCGM serves as a steppingstone to growth mixture modelling analyses in which the precise number and shape of each trajectory must be known a priori in order for the researcher to impute the requisite start values for the model to converge in software packages such as Mplus (Jung & Wickrama, 2008).Finally, the method lends itself well to the presentation of results in graphical and tabular format, which can facilitate the dissemination of the findings to wide-ranging audiences (Nagin, 2005).Notwithstanding the numerous advantages of LCGM, one limitation concerns the number of assessments needed to run the analysis.As with all growth models, a minimum of three time points is required for proper estimation and four or five time points are preferable in order to estimate more complex models involving trajectories following cubic or quadratic trends (Curran & Muthén, 1999).Given the need for numerous repeated assessments, greater attrition rates are expected (Raudenbush, 2001).Attrition can weaken statistical precision and potentially introduce bias if the data are not missing at random (MAR) or missing completely at random (MCAR;McKnight et al., 2007).
In sum, the aim of this tutorial was to introduce readers to LCGM and to provide a concrete example of how the analysis can be performed using the SAS software package and accompanying PROC TRAJ application.With the aforementioned advantages and limitations in mind, readers are encouraged to consider LCGM as an alternative to raw and residualized change scores as well as to standard growth approaches whenever multiple and somewhat contradictory patterns of change are part of a research question.This introduction should serve as a helpful guide to researchers and graduate students wishing to use this technique to explore multinomial patterns of change in longitudinal data.

β
disturbance term assumed to be normally distributed with a mean of zero and a constant standard deviation.Finally, are the parameters defining the intercept and slopes (i.e., linear, quadratic, cubic) of the trajectory for a specific subgroup ( ) j .As demonstrated in the above polynomial function, the trajectories are most often modelled using either a linear ( ) number of time points measured.A linear pattern of change is defined by the ( )

Figure 1 .
Figure 1.Syntax window with the associated data set.
70 to .80 are taken to indicate that the modelled trajectories group individuals with similar patterns of change and discriminate between individuals with dissimilar patterns of change.Returning to the hypothetical data set displayed in

Figure 2 .
Figure 2. Syntax for the analysis of a model with one quadratic trajectory.
The syntax "PROC PRINT DATA=OP" provides a table of the predict asurement point for each trajectory.These values are used to create a figure depicting the shape of each trajectory.SAS provides a figure with low resolution that may need to be recreated using an alternative statistical program such as hs are to be used in publication or modified gure 2 illustrates the syntax window at this nalysis, the user clicks the icon of the person running in the top right hand corner of the syntax windo .The analysis ill run and the graph of the trajectory wi pen in a new windTo view the actual output of the analysis, the user clicks on the Output window located in the bar at the bottom of the screen.The output for this one quadratic trajectory model is displayed in Figure3.In each output, statistics are provided fo each e timate parameter including for each trajectory.The intercept ( ) 0 β corresponds to the value of the dependent variable when the value of the independent variable is equal to zero.A t-test for the intercept provi in the output e value of the dependent variable from zero when the independent ro.The linear slope parameter on the dependent variable for each unit of increase on the squared independent variable.The amount of variance in the data accounted for by the model and its significance is given by Sigma.The above information is ic trajectory.Figure3.Output from the analysis of a model with one quadrat

Figure 4 .
Figure 4. Output from the analysis of a model with three quadratic trajectories.

Figure 5 .
Figure 5. Output from the analysis of a quadratic model for Trajectories 1 and 3 and linear model for Trajectory 2.
model with two trajectories provides the best fit to the data.
evidence for the three-trajectory model.significant improvement in model fit compared to the previously tested model.For this example, an analysis for four quadratic trajectories is run, followed by a model deleting the non-significant components of each of the four trajecto the three-trajectory modelThe addition of trajectories , p < 0.001; β1 = 1.47, p = 0.02; β2 = -0.70,p = 0.02).
Figure 6.Graph of the final model.

Table 1 .
Hypothetical data set with six participants and three trajectories

Table 2 .
Upper and lower limits of the 95% confidence intervals