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STATISTICAL CORNER |
From the Behavioral Medicine Research Center, Department of Psychology, University of Miami, Coral Gables, Florida.
Address reprint requests to: Maria M. Llabre, PhD, Department of Psychology, University of Miami, P.O. Box 248185, Coral Gables, FL 33124. Email: mllabre{at}miami.edu
ABSTRACT
OBJECTIVE: This paper provides an introduction to latent growth curve (LGC) modeling, a modern method for analyzing data resulting from change processes such as cardiovascular recovery from stress. LGC models are superior to traditional approaches such as repeated measures analysis of variance and simple change scores.
METHODS: The basic principles of LGC modeling are introduced and applied to data from 167 men and women whose systolic blood pressure was assessed before, during, and after the cold pressor and evaluated speech stressors and who had completed the Cook-Medley Hostility Inventory.
RESULTS: The LGC models revealed that systolic blood pressure recovery follows a different nonlinear trajectory after speech relative to the cold pressor. The difference resulted not from the initial decline at the completion of the stressor, but from higher levels at the end of the stressor and slower rate of change in decline for the speech. Hostility predicted the trajectory for speech but not for cold pressor. This relationship did not differ as a function of gender, although men had larger systolic blood pressure responses than women to both stressors.
CONCLUSIONS: LGC modeling yields an understanding of the processes and predictors of change that is not attainable through traditional statistical methods. Although our application concerns cardiovascular recovery from stress, LGC modeling has many other potential applications in psychosomatic research.
Key Words: latent growth, • cardiovascular, • blood pressure, • recovery.
Abbreviations: LGC = latent growth curve;; BP = blood pressure;; SBP = systolic blood pressure;; DBP = diastolic blood pressure;; CFI = comparative fit index;; RMSEA = root mean squared error of approximation;; CHD = coronary heart disease;; FIML = full information maximum likelihood;; SEM = structural equation modeling;; RM-ANOVA = repeated measures analysis of variance.
INTRODUCTION
Latent growth curve (LGC) modeling is a versatile tool for analyzing data in repeated measures designs (1–4). Under the names of growth models (1, 5), hierarchical linear models (6, 7), random regression (8), or mixed models (9, 10), the popularity of growth modeling is the result of theoretical developments and the availability of software to conduct the analyses. LGC modeling considers change over time as an underlying latent process. In analyzing this process, a trajectory of change over time is established for each individual in a sample, and therefore, characteristics of the trajectory (eg, slope) may vary across individuals and are treated as latent variables. These latent variables describe parameters of change and may be treated as independent, dependent, control, or mediating variables. In this manner LGC provides flexibility to address questions about change.
Modeling the process of change requires certain prior knowledge about the general functional form of the trajectory that describes the change for a sample. When change occurs over brief periods, it is typical to model a linear trajectory, which presumes that the rate of change, or slope, is constant over time. This rate of change together with an intercept, become the variables of interest in LGC modeling. The average slope and intercept may then be estimated for the group. These are fixed effects. But because individuals within the group may deviate from group means in individual slopes and intercepts, the variability around the mean slope and mean intercept are also estimated. These are random effects. It is the combination of fixed and random effects that renders these models "mixed" models. Because each individual’s contribution has its own slope and intercept, these models qualify as random regression. The models are also considered a special case of hierarchical linear models with two levels in the hierarchy: level 1 is the individual observation repeated over time; level 2 is the individual within which the observations are nested. But it is the assumption of a latent process of growth that places these models in the context of latent models and allows them to take full advantage of structural equation modeling (SEM) methodology.
There are several advantages to using LGC modeling in repeated measures designs over more conventional techniques such as repeated measures analysis of variance (RM-ANOVA). When the restrictive assumptions and conditions of RM-ANOVA hold, both approaches yield the same parameter estimates.1 Thus RM-ANOVA may be viewed as a restricted special case of LGC modeling. But the flexibility of LGC models is worth the investment in their application. LGC models allow the study of a wide range of parameters of change including linear and nonlinear effects along with their variances and covariances, providing a more complete picture of the change process. Also, the parameters in LGC models are estimated separately from measurement error and therefore, are not affected by the attenuation associated with such error. We know from test theory that measurement error attenuates the magnitude of parameter estimates. Another advantage of LGC modeling is the flexibility in specification of the variances and covariances of the repeated measurements. Error variances and covariances may be estimated freely or specified to conform to a predetermined pattern. In the traditional RM-ANOVA, an assumption of compound symmetry is made that the variances are equal across time and that all covariances are equal. Even the more relaxed assumption of sphericity holds that the variances of the differences between times are equal. Violations of these assumptions lead to bias in the estimation of the error terms. These assumptions are frequently unrealistic because observations that are closer in time are likely to be more strongly correlated than those with greater separation.
Perhaps one of the key advantages of LGC models for researchers working with longitudinal data is the ability to use all available data. Traditional RM-ANOVA programs use listwise deletion of missing data, often drastically reducing sample size and power and biasing parameter estimates. LGC models may be analyzed with programs that employ an approach known as full information maximum likelihood (FIML), useful for handling missing data. In FIML, all the available data for each person are used in obtaining a likelihood function for that person that is then summed across persons.
The use of individual information in obtaining the likelihood function in newer versions of SEM programs allows LGC models to incorporate missing observations and/or unequally spaced observations across persons. These features make LGC modeling an attractive alternative to the RM-ANOVA approach favored by most in the field.
Any research question in which change over time is a phenomenon of interest may benefit from using this methodology. In psychosomatic research, there has been a long-standing interest in understanding how the cardiovascular system reacts to and recovers from stressful stimuli. Under the rubric "cardiovascular reactivity," individual differences in physiological responses to stress have been linked, either via direct or mediating mechanisms (11), to hypertension and coronary heart disease (CHD) (12–15). The early literature demonstrated that hypertensive people show not only greater cardiovascular reactivity but also delayed recovery compared with those who are normotensive (16, 17). Arguably, recovery may be more relevant than reactivity to the understanding of hypertension (13, 18) and CHD (19) given the potential for delays in recovery to continue long after even brief stressors (20). That is, impaired and delayed cardiovascular recovery may lead to long-term adjustments to elevated BP. A number of individual difference factors have been related to recovery (21), including gender (22), race (23), responder type (24), and hostility (25–28). Nevertheless, cardiovascular recovery is vastly understudied relative to the study of cardiovascular reactivity (29). Measurement concerns may represent one reason why cardiovascular recovery has not been well-studied: there is no standard method of quantification and the most commonly employed methods have only modest reliability (30). By describing the trajectory of change separate from error and using the individual time-dependent observations during the recovery period, LGC modeling can improve reliability. In fact, Christenfeld, Glynn, and Gerin (30) demonstrated that a curve-fitting approach similar to LGC proved to have superior reliability relative to more common methods.
A recent in-depth analysis of the status of reactivity and recovery research (31) leaves the reader with the impression that stress reactivity is an important concept, that stress recovery is as or more important, but that the validity of each has not been fully established. Furthermore, the review recommends that future reactivity and recovery research should include individual differences in personality, situational independent variables, and outcome measures from multiple physiological systems, in a psychometrically sound design. But the scope of the review did not include the statistical methodology that will give reactivity and recovery their "best chance" at demonstrating their utility by 1) ruling out measurement error and 2) providing a quantification method amenable to study in the context of complex designs. In fact, the recommendations resulting from the review cannot be fully realized without capitalizing on newer statistical methods, in particular LGC models.
In a previous paper (32), we applied LGC modeling to quantify reactivity and recovery simultaneously using a piecewise LGC approach. The piecewise approach, useful when studying different processes, breaks up the trajectory of change into pieces, each described by a separate function. We modeled reactivity as one piece and recovery as a second piece. The piecewise illustration showed the ability of LGC modeling to demonstrate the generalizibility of blood pressure (BP) reactivity and recovery in the laboratory to ambulatory BP in the work setting. However, it was limited to a single stressor and a single sample. This earlier paper presented these processes in a mediational context and complements the models described here.
The main purpose of the present paper is to illustrate how LGC modeling can address questions about change in cardiovascular recovery research. We use cardiovascular recovery as the focus of this illustration because its reliability and generalizibility are still in question (20). Limiting the focus to one process permitted an in-depth analysis while keeping the models relatively simple. This application begins with an examination and comparison of the trajectory of recovery for two stressors and systematically builds more complex models by including a personality characteristic, a grouping variable, and their interaction as predictors. Each section will address a specific research question about recovery and will illustrate how important aspects of the question may be understood by using LGC modeling. Our target audience is psychosomatic researchers interested in the change process who have or would like to develop some general familiarity with linear models and structural equations.
The following research questions will guide our application of LGC modeling: How can we characterize BP recovery from a stressor as a process of change over time? With knowledge that patterns of BP reactivity vary across stressors (33), do BP recovery patterns also differ as a function of stressor type? Given the evidence linking personality characteristics such as hostility to hypertension (34) and CHD (35), does hostility predict systolic blood pressure (SBP) recovery? Similarly, are there gender differences in recovery patterns? Is the hostility-recovery prediction moderated by gender?
Participants
Participants were 167 adults, 25 to 54 years of age (M = 41, SD = 8). The sample included 25 African-American women, 30 African-American men, 49 White American women, and 63 white American men. Their resting BP levels were M= 118 mm Hg SBP/72 mm Hg diastolic blood pressure (DBP) (SD = 17.3/10.5). Twenty-four were hypertensive (SBP 
140 mm Hg or DBP 90 mm Hg), but otherwise healthy. No participant was on antihypertensive medication.
Procedures
The present study utilized a 6-day protocol described in (36). For this study, we used the Cook-Medley Hostility Inventory (37). The Cook-Medley is a well-established measure of hostility and distrust of others. We also used data from the speech presentation and cold pressor tasks. BP was recorded while engaged in the tasks and during recovery periods that lasted 27 and 21 minutes after the speech and cold pressor, respectively. Baseline BP measurements were obtained at 2-minute intervals during the 6-minute period preceding each task.
Speech
The evaluated speech task (38) is a social-evaluative stressor with demonstrable effects on state anxiety (39). It involves responding to a hypothetical scenario about being wrongfully accused of shoplifting. Participants were instructed to first mentally prepare their speeches for 3 minutes and then to deliver their speeches for 3 minutes. BP was assessed at 2 minutes into the talking phase of the task. Three recovery readings were assessed at approximately 2 minutes intervals after the last task reading.
Cold Pressor
The cold pressor is often considered a physical task, but it also involves the psychological dimension of pain (40). Participants placed their left foot into a bucket of ice water for 90 seconds. BP was assessed approximately at 1.5 minutes into the task. Three recovery readings were assessed at approximately 2 minutes intervals after the last task reading.
Simple Linear Model
The first step in the analysis was to graph the data (not shown). Creating separate graphs of the response variable over time for each individual in the data set informs about the functional form of the data. Although the graphs indicated a nonlinear functional form, we started with the simplest model by expressing the SBP for each individual as a function of the recovery time period. For now let us assume that the model would take the most parsimonious functional form, that of a line: there would be a steady decrease in SBP from the end of the task and across the first 6 minutes of the recovery period. The corresponding linear model may be specified as:equation
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where Yij represents the SBP measure for person j at time i. 
0j is the intercept for the SBP trajectory for participant j or the predicted SBP value at the end of the task, where time = 0 represents the last task reading. 1j is the slope of the trajectory for participant j, or the true decline from task back to basal level per minute change in time. tij is the time, i, corresponding to each measurement for person j, and rij represents the random error or unexplained deviations from the line for person j. This stochastic or error term contains measurement error combined with any time specific error and is assumed to be normally distributed with zero mean and nonzero variance. This error variance is typically assumed to be constant across time; however, this assumption may be relaxed in LGC models, as discussed later. Importantly, the intercept and slope parameters are estimated separately from the measurement error and are free from such error. Having one such equation for each person, we can then treat the true intercept (0j) and slope (1j) parameters as random dependent variables that may be modeled. An initial simple model may be 
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where ß0 represents the mean intercept for the sample and ß1 represents the mean slope. These are fixed parameters in the model. u0j and u1j are random and represent the residual difference between the mean of all participants and each person’s intercept or slope, respectively. Estimates of the variance associated with each random variable are obtained as well as their covariance. Examination of the magnitude of the variance components in this simple model can serve as a guide in determining whether there is sufficient individual variability to warrant questions about predictors of change. If there is variability around the means, then additional predictors may be included in each model equation. LGC modeling permits the estimation of model fit, an evaluation of whether the proposed functional form is consistent with the data.
Fitting a linear function to the recovery data for both the cold pressor and the speech stressor indicated poor fit to the data. The fit indices were 
2 (5) = 113.06, p < .001, CFI = 0.86, root mean squared error of approximation (RMSEA) = 0.36 for the cold pressor and 2 (5) = 145.71, p < .001, CFI = 0.83, RMSEA = 0.41 for the speech. In a commonly cited study, Hu and Bentler (41) provide criteria to determine fit beyond a nonsignificant 2. These include the CFI > 0.95 and RMSEA < 0.06.2 Our initial results show that recovery, specifically over the first 6 minutes of the recovery period does not occur in a linear fashion. With evidence of poor fit, interpretation of the resulting parameters was not appropriate. Rather, modification of the model was the appropriate next step.
Nonlinear Change
Although linear change is parsimonious when describing short intervals, as we just saw, it is not always appropriate. For many processes and when time intervals are long, nonlinear functional forms may be necessary to describe the trajectory. In this situation, the alternatives may be to use a polynomial to describe the function or to specify a nonlinear function such as a logistic curve (42). Working with true nonlinear functions such as growth curves provides additional flexibility in that the parameters can be altered and the curves can take on a variety of shapes, but complicates the estimation process (43, 44), and has received limited attention in the context of structural modeling (42, 45, 46) . In our application, we will use a polynomial to describe the recovery trajectory.
Quadratic Polynomial
The advantage of the polynomial is that, although it describes a nonlinear pattern, it is linear in the parameters, thus facilitating the ease of estimation. Although the polynomial function may not be the most correct model, it often provides an approximation close enough to capture the parameters of interest, within a specific period of time.
Again, SBP for each individual is expressed as a function of time; however, we expect the initial decline to stabilize over the recovery period. In this fashion, we will model a quadratic polynomial as shown below.
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By coding the last time during the task as time = 0, 0j represents the intercept for the SBP trajectory for participant j, or the SBP value at the last task reading. 1j is the linear coefficient of the trajectory for participant j, or the instantaneous slope at time = 0. The change in meaning for the slope parameter results from fitting a quadratic function, where the slope changes as a function of time. 2j is the quadratic coefficient, reflecting the change in the rate of change, and contributing to the conditional slope as described in the results below. tij is the time i corresponding to each measurement for person j. And rij contains random measurement error or the residual deviation from the curve for person j with the assumption of normality and zero mean.
There are now three latent variables that may be modeled. Specifically, equation
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Each latent variable may be modeled by a fixed component expressing the mean across all participants, and a random component representing individuals’ deviations from the mean. For the random variables, uij, variance and covariance components are estimated.
Figure 1 shows the path diagram for cold pressor recovery, which includes three latent variables describing the change process. Path diagram symbols include ovals for latents, rectangles for observed, direct arrows for effect coefficients, curved arrows for covariances or variances, and triangles for the constants necessary to estimate means and intercepts. Notice the set of coefficients for the loadings of the latent variables. These coefficients map the time feature of the repeated measures design onto the latent process. Unlike a conventional latent variable analysis (factor analysis) where the loadings are estimated and the latent variables assumed to have a mean of 0, here the loadings are fixed and the mean of the latent variables (indicated by the paths from the constant 1 in the triangles) are estimated, along with their variances and covariances. Speech recovery would be similarly diagrammed.
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The loadings for the quadratic latent variable are the squares of the times in the design. By specifying these loadings in LGC modeling, the latent variables are made to represent the latent process, which results in the repeated measurements. The time structure of the repeated measurements, again, is conveyed by the loadings.
Table 1 shows the results of the parameter estimates for SBP recovery for the cold pressor and the speech. The model fit the data well for both stressors: for the cold pressor, 2 (1) = 0.645, p = .42, CFI = 1.0, RMSEA = 0.001; for the speech 2 (1) = 0.015, p = .902, CFI = 1.0, RMSEA = 0.001. As shown in Table 1, the average SBP at the end of the cold pressor test, the intercept, was 138 mm Hg. On completion of the task, SBP instantaneously declined by an average of 7.63 mm Hg (linear parameter), and this decline was reduced by an average of 1.53 mm Hg per minute (2 · 0.765, or the constant in the first derivative). The first derivative of the quadratic polynomial of SBP as a function of time gives us an equation of the conditional slope (conditional on time). The equation is -7.63 + 1.53 · time. Thus, estimation of the slope in the context of a quadratic function takes into account both the linear and quadratic parameters, as well as the specific time at which the changing slope is calculated. Setting the conditional slope equation equal to 0 yields the average time associated with the end of the decline. For the cold pressor, this time is 4.98 minutes into the recovery period. For the speech stressor, the average SBP at the end of the task was 145 mm Hg and the initial decline was 8.11 mm Hg on average with a stabilization of 1.36 (2 · 0.682) mm Hg per minute. The end of the decline occurred at about 5.96 minutes.
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To determine whether these SBP levels associated with the lowest point in the recovery trajectory were equivalent to the baseline levels before each stressor, two baseline latent variables were incorporated into the model, each defined by the three SBP readings obtained during the baseline period preceding each task. The mean baseline SBP for the cold pressor was 119 mm Hg and for the speech 117 mm Hg. The difference in baseline means was statistically significant and could have been the result of presenting the cold pressor later in the sequence. Comparing the SBP baseline levels to the levels observed at the end of the decline resulted in no difference for the cold pressor (119 vs. 119 mm Hg) but a significant difference for the speech (117 vs. 121 mm Hg). Thus, it appears that full recovery occurred for the cold pressor after about 5 minutes; however, for the speech, full recovery had not occurred after about 6 minutes post task.
There was considerable variability in the intercepts and baselines for both tasks and less variability in the linear and quadratic terms, although all variances were significantly different from 0. The results of these random effects suggest that seeking predictors of the individual differences is warranted, particularly in the intercept. The results of the fixed effects suggest that a more detailed comparison of the recovery trajectories of these two stressors may be interesting, given their differential time to recovery. Correlations among the intercept, linear, and quadratic latent variables were comparable between stressors, moderate and negative between intercept and linear, negative and large, as expected, between the linear and quadratic terms, and moderate and positive between the intercept and quadratic (see Table 1), all significantly different from 0. The correlations between the intercept on one hand and the linear and the quadratic terms on the other suggest that greater reactivity at the end of the stressor is associated with a greater initial decline and a quicker change in slope. Because the quadratic coefficients are derived from the linear coefficients, the high correlation between these terms is an arithmetic artifact and should not be given a substantive interpretation. Baselines were highly correlated with each other (0.96) and with the intercepts (range, 0.77–0.85), indicating that people with higher SBP levels at baseline were likely those with higher values at the end of the task.
Comparing LGC Models Between Stressors
Comparisons between stressors were done using two approaches. The two stressors were first examined in terms of their similarity in the characteristics of the growth parameters for the sample. In addition, corresponding parameters were correlated between stressors to address similarity in the ordering of individual differences. Taken together, these two approaches provide a comprehensive view of these two stressors by examining levels, variability, and covariation.
To compare characteristics of the growth parameters between stressors, a series of nested models were tested, each constraining a different parameter or set of parameters to be equal between the two stressors. An advantage of LGC modeling is the ability to directly test differences between models with a 2 difference test; that is, to determine whether one model shows improvement or deterioration in fit relative to another model. The validity of this 2 difference test depends on the two models being nested, meaning that the parameters of one are a subset of the parameters in the other. This occurs when the parameters are estimated freely in one model but constrained to 0 or equal to other parameters in a second model (47). Typically, models where parameters are estimated freely display better fit relative to models in which some parameters are constrained.
As a point of reference for comparing models, the polynomial models for both stressors were fit together, but the fixed and random effects were estimated freely for each stressor. (To simplify the presentation and because of the high correlations between baselines and intercepts, subsequent models did not include the baseline latent variables.) The results of the test of this model, shown as model 2.1 in Table 2, showed adequate fit according to some indices, although the chi-squared test was significant [2 (9) = 17.2, p = .046, CFI = 0.99, RMSEA = 0.074]. When the means of the linear terms (instantaneous decline) were constrained equal between stressors in model 2.2, the change was not significant. These results showed the more restrictive model with equal linear terms was an appropriate model to characterize the data. This is important because it indicates that, regardless of the type of stressor, the mean instantaneous recovery decline is similar. Subsequent tests then retained the constraint of equal means of the linear terms between stressors and used model 2.2 as the reference model. Constraining either the means of the intercepts (model 2.3) or the quadratic terms (model 2.4) equal between stressors resulted in deterioration in model fit, as shown in Table 2. Thus, in terms of fixed effects, the means of the linear terms were comparable between stressors, but the intercept, indicative of the task level at the end of the stressor, and the quadratic terms were not. The discrepancy in end task levels (ie, intercept) was expected given that these two stressors have been differentially characterized in the literature, and that the speech has consistently been shown to produce strong SBP reactivity effects (48). The difference in quadratic terms, responsible for the previous demonstration of differential recovery times, provokes speculation about what may be differentially happening during recovery to these stressors and whether the difference is meaningful. Before we turn to an assessment of the differences in means, the variances (model 2.5) and covariances (model 2.6) for corresponding latent variables were constrained equal between stressors; in both cases, the results showed improvement in model fit (see Table 2). (We actually constrained them one at a time, but we only report the change when all the variances were constrained equal and when all covariances were constrained equal.) Thus, in terms of the characteristics of the growth parameters for this sample, we see a pattern of consistency in variability and average initial decline, but differences in starting level and rate of change in decline.
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Hostility as a Predictor of Recovery
We next examined hostility as a predictor of the different components of recovery because hostility is perhaps the most frequently examined psychosocial predictor of CHD (49–56). This involved extending the models of the intercept (0j), linear (1j), and quadratic (2j) terms for each stressor to include hos-tility (hj), a continuous variable, as a predictor as shown below:equation
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In these conditional models, the intercepts ß00, ß10, and ß20 represent the mean parameter estimate for persons with no hostility. ß01, ß11, and ß21 represent the effect of hostility on each recovery parameter. The constraints on the variances and covariances tested in the earlier model were retained in these analyses. The path diagram associated with these models is shown in Figure 2. A test of model fit indicated good fit to the data 2 (17) = 22.98, p =.15, CFI = 0.997, RMSEA = 0.046. All paths from hostility to the intercept, linear, and quadratic components were statistically significant for the speech but not the cold pressor stressor.
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2 (20) = 31.75, p = .04], and a significant change [
2 (3) = 8.77, p < 0.05], suggesting deterioration in model fit. Therefore, it may be concluded that hostility predicts the intercept, linear, and quadratic terms of the cold pressor and the speech differentially. The parameter estimates for the model with the paths allowed to differ between stressors are shown in Table 3. As can be seen, correlations between hostility and the cold pressor parameters were low; however, for the speech stressor, the correlation with the intercept was -0.31; with the linear, 0.39; and with the quadratic, -0.31. These correlations suggest that while hostility does not predict recovery from the cold pressor, more hostile people tend to have lower SBP levels at the end of the speech (intercept), less steep initial declines in SBP on completion of the speech (linear), and a longer period before recovery (quadratic).
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In a multiple group analysis, the models are estimated within each group simultaneously, and the parameters specified to be either estimated freely for each group or constrained to be equal across groups. By comparing a model with freely estimated parameters to a nested model having specific equality constraints, the appropriateness of those constraints can be tested using a 2 difference test. The hypothesis tested is whether equal parameters across groups are consistent with the data. This approach is similar to the approach used to compare the two different stressors.
We first specified the model found to be consistent with the data in the total sample with equality constraints between the stressors (model 2.6), allowing the parameters to be different between men and women. Model 4.1 in Table 4 indicates good fit for the model to the data [2 (32) = 44.0, p =.076, CFI = 0.99, RMSEA = 0.067]. Subsequent tests constrained the linear terms to be equal between groups, the variances for corresponding latent variables equal between groups, and the corresponding covariances equal between groups. As indicated in models 4.2 to 4.4 in Table 4, the 2 difference test showed no significant decrement in model fit resulting from these equality constraints.
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2 (41) = 49.7, p = .165, CFI = 0.997, RMSEA = 0.050]. Lastly, we constrained the means of the intercept for the cold pressor (model 4.6) and the speech (model 4.7) to be equal between groups; however, in each case, the model fit deteriorated. Parameter estimates associated with model 4.5, the most restrictive well-fitting model, are presented in Table 5. The parameter estimates shown in Table 5 indicate that men responded with higher SBP levels than women at the end of both stressors (cold pressor: 142 vs. 134 mm Hg; speech: 148 vs. 140 mm Hg, for men and women, respectively), yet their patterns of recovery were the same for both stressors. For both genders, the similarity in initial decline between stressors and difference in the quadratic component was confirmed.
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2 (40) = 52.49, p = .089, CFI = 0.99, RMSEA = 0.061. The second model constrained corresponding coefficients to be equal between groups; that is, the effect of hostility on each of the six parameters was held equal between men and women. This more restrictive model also showed good fit [2 (46) = 56.20, p = .144, CFI = 0.99, RMSEA = 0.052] with no deterioration [2 (6) = 3.71, p > .05]. Table 6 shows the parameter estimates in common to both gender groups. Also included in the last two columns of the table are the correlations between hostility and the latent recovery variables calculated for each gender.4
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LGC assumes that change over time is a dynamic process characterized by the parameters of a general functional form that describes that change process. The parameters of that change process address meaningful characteristics, and these characteristics may be predicted from other variables or used as predictors of subsequent outcomes.
Traditionally, cardiovascular recovery from stress has been quantified with change scores from either baseline or reactivity. Investigators have typically looked at change over a specified amount of time, but they have also studied time to achieve recovery. With either method, the recovery process is reduced to a single value and potentially important information is lost. If, hypothetically, recovery could best be characterized by a negative linear trajectory, then a simple change score may be tempting, although it would include measurement error. Empirically, however, we know that recovery is not a linear process; indeed, it is dynamic and nonlinear. Furthermore, there may be individual differences in the different parameters used to describe the nonlinear trajectory of change. Because the intercept, linear, and quadratic terms are estimated for each participant, these components can be used as independent, dependent, or mediating variables. There is also evidence to suggest that change scores as measures of recovery do not capture individual differences reliably, whereas curve-fitting techniques do (30). In LGC modeling, the parameters of change are estimated without contamination from measurement error. Recall that the association between recovery and any other variable (eg, hostility) is limited by each measure’s degree of reliability. Thus, to better capture the dynamic processes of recovery and for psychometric reasons, LGC modeling offers several advantages over simple change scores. LGC modeling also has its disadvantages. Perhaps most obvious is its complexity. However, the complexity of the method only matches the complexity of the research question posed. And the recommendations for cardiovascular reactivity research (31) in the new millennium and for understanding psychosocial influences on the development of disease (55) indicate that complex mediational models are essential. Finally, modeling recovery (or any other variable) over time necessitates repeated measurements in meaningful increments. In cardiovascular research, there already exists the technology to sample BP and other cardiovascular variables repeatedly over time. In many cases, investigators may already have the requisite cardiovascular measurements necessary to conduct LGC modeling. With the advent of user-friendly structural equation and hierarchical linear modeling software, LGC modeling has become more accessible.
As mentioned earlier, change scores have been traditionally used to quantify recovery. Several issues in the measurement of change relevant in this context were specified by Kamark and Lovallo (69). One additional consideration is very important in measuring change, and that is the number of readings used to characterize the response. In an influential paper, Rogosa and colleagues (70) demonstrated the limitations of assessing change with only two measurements, even when modeling linear change. Having more than 2 readings allows the separation of the slope parameter from the idiosyncrasies of the individual observations, including their measurement error. Having more than 3 observations allows the exploration of nonlinear trajectories. This is particularly important in examining recovery patterns that tend to be nonlinear. With nonlinear functions, both the spacing and the number of the observations warrant consideration because observations should be sampled at the times when the most rapid change is taking place. Theoretical knowledge or a preliminary study of a small number of subjects should lend information regarding the appropriate spacing of observations.
We used Mplus (71) software to run the models. However, any SEM software package that uses FIML and can incorporate a mean structure will do, including LISREL (72), or AMOS (74). This feature, in newer versions of SEM programs, allows measurements at different times across participants, making them comparable to multilevel programs (76). Any multilevel model program, such as HLM (77) or PROC MIXED in SAS (78) will estimate the models up to the point of including growth parameters as predictors of outcome variable. We are not aware of a way to get multilevel model programs to do so at this time; however, since all packages are continually being upgraded and improved, it may be possible to do so in the future. The code used for the model presented in Table 6 is included in the appendix.
What did we learn from this application of LGC modeling about the SBP recovery process? We know that SBP recovery cannot be quantified with a single index because the trajectory is nonlinear, requiring three parameters to describe it. This general nonlinear pattern appears to be similar between stressors, with higher SBP starting levels and greater delay in recovery for the speech relative to the cold pressor. But the difference in recovery time did not result from the initial drop in SBP. On termination of the stressor, one can expect an immediate physiological response to attain homeostasis. At the group level, this immediate SBP response was similar for both stressors. The difference in trajectory resulted from the differential quadratic term, which yielded slower recovery for the speech stressor. This distinction between the initial drop and the change in rate of change is not possible with a single index. These findings are consistent with the idea that psychological aspects of the speech, possibly associated with rumination are responsible for the maintenance of elevated SBP after the stressor has ended. The speech stressor is considered a more psychological task than the cold pressor because of its link to anxiety (39) and emotionality, and it may lend itself to a greater degree of rumination (79), leading to a longer recovery period. Glynn and colleagues (79) suggested that the emotional nature of the stressor may be what is associated with extended response elevations. This differential characteristic of stressors raises the possibility that stressors in which the BP elevations are sustained for longer periods of time may be those that will make the greater contributions to disease.
Additional support for the involvement of a psychological process in recovery comes from the findings that hostility predicted all three components of SBP recovery for the speech but not for the cold pressor. Although the cold pressor has been shown to involve a component of cold and a component of pain (40), and pain has been construed as emotionally laden (80), there did not appear to be lingering effects of these components once the stressor was removed. Hostility was associated with slower SBP recovery from the speech that derived from a slower initial decline and a prolonged turn to plateau. To the extent that recovery from stressors might be useful for understanding the relation between hostility and hypertension (34) or CHD (35), future prospective studies may use LGC to examine hostility and recovery from the speech stressor to predict hypertension and/or CHD morbidity and mortality.
Surprisingly, hostility was also inversely related to the intercept, which reflects an aspect of reactivity. Earlier papers using mental arithmetic (81) or anagrams (82) reported a similar reciprocal relationship, perhaps a sign of less engagement by more hostile individuals or lower levels of apprehension. We observed an inverse relationship (r = -0.19, p < .05) between the Cook-Medley and an item asking "how frustrated were you by the speech task?" The relation between hostility and reactivity depends on the nature of the task because hostility has been associated with increased reactivity when the stressor involves interpersonal conflict (83, 84) and a series of studies by Suarez and colleagues (26–28) showed that hostility was associated with heightened BP reactivity only under conditions of harassment. The hostility effects observed in our analyses were apparent once measurement error was removed. Correlations between hostility and the individual SBP readings post stressor, the average over the interval, the change from baseline, or the change from task level yielded mostly nonsignificant results (r = -0.02 to 0.16), except for the negative correlation between hostility and the last task reading of the speech (-0.30). Clearly, LGC methods provided a means for giving important relations between hostility and SBP recovery a chance to be known.
While gender differences in SBP recovery have been reported in previous studies, our data indicated that the gender differences observed for both stressors were strictly the result of differences apparent in the levels at the end of the tasks, but they did not translate into differences in other aspects of recovery. Men and women were comparable in their initial SBP decline and their eventual recovery. This similarity between the gender groups was established for both stressors. Additionally, while men displayed higher levels of hostility (M = 22, SD = 8.97) than women (M = 18, SD = 6.23) and greater variability, the effect of hostility on the parameters of recovery were the same for men and women. This held true for both stressors and for each of the parameters. Again, had a single value been used to index recovery, this one composite may have given the impression that men and women differed in recovery when, in fact, they only differed in overall level.
The data used in this illustration are limited and did not permit a comprehensive assessment of all aspects of recovery that may be of interest. Because the data set did not include measures of disease outcome, the critical hypothesis of recovery as a mediator of stress on disease was not tested. However, the methodology illustrated lends itself to the performance of such tests with extension of the models; for example, growth parameters may be specified to predict disease outcome. Also, it is now recognized that BP and heart rate provide a limited view of the pathways taken by the stress response and the evidence implicates multiple systems (85, 86) including the immune and neuroendocrine systems. Because LGC models may be embedded in more extensive structural models, the responses of multiple systems may be examined simultaneously and their possible links, including reciprocal relations, further explored.
Appendix A
TITLE: Speech and Cold Pressor Recovery LGC Model with Anger in men and women;
DATA: FILE IS c:Mplusrecoverydata.dat;
TYPE IS INDIVIDUAL;
VARIABLE: NAMES ARE sn sex sbs1 sbs2 sbs3 sps1 sps2 sts1
sts2 srs1-srs3 cbs1 cbs2 cbs3 cts1 cts2 crs1-crs3 cookmd;
MISSING ARE .;
USEVAR = sts2 srs1-srs3 cts2 crs1-crs3 cookmd sex;
GROUPING = sex (0 = female 1 = male);
ANALYSIS: TYPE IS MGROUP MISSING H1;
MODEL:
MODEL female:
OUTPUT: SAMPSTAT STANDARDIZED MODINDICES;
NOTES
This work was supported by Research Grant HL36588 from the National Heart, Lung and Blood Institute of the National Institutes of Health
1 Most conventional RM-ANOVA programs treat time as a categorical variable but allow the specification of polynomial trends (ie, linear, quadratic, etc.) using orthogonal contrasts. Because the interpretation of the parameter estimates depends on the coding of the time variable, it is important to know what codes are assigned to the time variable in a particular program. Typically, the intercept is coded to represent the grand mean or a function of the grand mean and time is coded assuming equal unit intervals. Interpretation of the intercept parameter is typically not useful. Other parameters have to be interpreted relative to the arbitrary time coding. It is possible to substitute the orthogonal contrasts from a RM-ANOVA program in a LGC model and obtain the same results, but using the actual times yields more meaningful parameters. 
2 In this context, the 2 test of fit is based on the difference in the deviance (-2 log likelihood) between a saturated model with 0 degrees of freedom and the hypothesized model.
3 The mean level at any time during the recovery trajectory may be obtained by substituting the desired time into these equations.
4 Note that the correlations calculated for the men appear to be of larger magnitude than those for the women. The difference in correlations between men and women may seem inconsistent with the statement that the effect of hostility on recovery is the same for the two gender groups. This discrepancy is reconciled once the variance of hostility is taken into account. The variability in hostility in the men is 80.45, about double that in the women, which was 38.89. It is this difference in variances that translates into different correlations, although the actual effect coefficients are the same. This finding reminds us of the importance of working with unstandardized coefficients when comparing groups, as opposed to relying on correlations, which may artificially imply differences in effects because of differential variability.
Received for publication September 26, 2003.
Revision received October 31, 2003.
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