Presentation on theme: "Growth Curve Models (being revised)"— Presentation transcript:
1 Growth Curve Models (being revised) Thanks due to Betsy McCoachDavid A. KennyAugust 26, 2011
2 Overview Introduction Estimation of the Basic Model Nonlinear Effects Exogenous VariablesMultivariate Growth Models
3 Not Discussed or Briefly Discussed Modeling NonlinearityLDS ModelTime-varying CovariatesPoint of Minimal Intercept VarianceComplex Nonlinear Models(see extra slides at the end)
4 Two Basic Change Models StochasticI am like how I was, but I change randomly.These random “shocks” are incorporated into who I am.Autoregressive models (last week)Growth Curve ModelsEach of us in a definite track.We may be knocked off that track, but eventually we end up “back on track.”Individuals are on different tracks.
5 Linear Growth Curve Models We have at least three time points for each individual.We fit a straight line for each person:The parameters from these lines describe the person.
6 The Key Parameters Slope: the rate of change Some people are changing more than others and so have larger slopes.Some people are improving or growing (positive slopes).Some are declining (negative slopes).Some are not changing (zero slopes).Intercept: where the person startsError: How far the score is from the line.
7 Latent Growth Models (LGM) For both the slope and intercept there is a mean and a variance.MeanIntercept: Where does the average person start?Slope: What is the average rate of change?VarianceIntercept: How much do individuals differ in where they start?Slope: How much do individuals differ in their rates of change: “Different slopes for different folks.”
8 Measurement Over Time measures taken over time chronological time: 2006, 2007, 2008personal time: 5 years old, 6, and 7missing data not problematicperson fails to show up at age 6unequal spacing of observations not problematicmeasures at 2000, 2001, 2002, and 2006
9 Data Types Raw data Covariance matrix plus means Means become knowns: T(T + 3)/2Should not use CFI and TLI (unless the independence model is recomputed; zero correlations, free variances, means equal)Program reproduces variances, covariances (correlations), and means.
10 Independence Model Default model in Amos is wrong! No correlations, free variances, and equal means.df of T(T + 1)/2 – 1
11 Specification: Two Latent Variables Latent intercept factor and latent slope factorSlope and intercept factors are correlated.Error variances are estimated with a zero intercept.Intercept factorfree mean and varianceall measures have loadings set to one
12 Slope Factor free mean and variance loadings define the meaning of timeStandard specification (given equal spacing)time 1 is given a loading of 0time 2 a loading of 1and so onA one unit difference defines the unit of time. So if days are measured, we could have time be in days (0 for day 1 and 1 for day 2), weeks (1/7 for day 2), months (1/30) or years (1/365).
13 Time Zero Where the slope has a zero loading defines time zero. At time zero, the intercept is defined.Rescaling of time:0 loading at time 1 ─ centered at initial statusstandard approach0 loading at the last wave ─ centered at final statususeful in intervention studies0 loading in the middle wave ─ centered in the middle of data collectionintercept like the mean of observations
14 Different Choices Result In Samemodel fit (c2 or RMSEA)slope mean and varianceerror variancesDifferentmean and variance for the interceptslope-intercept covariance
15 some intercept variance, and slope and intercept being positively correlated no intercept varianceintercept variance, with slope and intercept being negatively correlated
16 Identification Need at least three waves (T = 3) Need more waves for more complicated modelsKnowns = number of variances, covariances, and means or T(T + 3)/2So for 4 times there are 4 variances, 6 covariances, and 4 means = 14Unknowns2 variances, one for slope and one for intercept2 means, one for the slope and one for the interceptT error variances1 slope-intercept covariance
17 Model df Known minus unknowns General formula: T(T + 3)/2 – T – 5 Specific applicationsIf T = 3, df = 9 – 8 = 1If T = 4, df = 14 – 9 = 5If T = 5, df = 20 – 10 = 10
18 i.e., the means have a linear relationship with respect to time. Three-wave ModelHas one df.The over-identifying restriction is:M1 + M3 – 2M2 = 0(where “M” is mean)i.e., the means have a linear relationship with respect to time.
19 Example Data Curran, P. J. (2000) Adolescents, ages 10.5 to 15.5 at Time 13 times, separated by a yearN = 363MeasurePerceived peer alcohol use0 to 7 scale, composite of 4 items
25 Parameter Estimates Estimate SE CR MEANS Intercept 1.304 .091 14.395 SlopeVARIANCESInterceptSlopeErrorErrorErrorCOVARIANCE*Intercept-Slope*Correlation = -.378
26 Interpretation Mean Variance Intercept: The average person starts atSlope: The average rate of change per year is .555 units.VarianceIntercept+1 sd = = 2.86-1 sd = – 1.56 =Slope+1 sd = =1.19-1 sd = .56 – .63 = -0.07% positive slopes P(Z > -.555/.634) = .80
27 Model Fit c2(1) = 4.98, p = .026 RMSEA = .105 CFI = ( – 5 – )/ ( – 5)= .991Conclusion: Good fitting model. (Remember that the RMSEA with small df can be misleading.)
28 Nonlinearity Latent Basis Model: Some Loadings Free Fix the loadings for two waves of data to different nonzero values and free the other loadings.In essence rescales time.SlopeIntercept1?2
29 Results for Alcohol Data Wave 1: 0.00Wave 2: 0.84Wave 3: 2.00Function fairly linear as 0.84 is close to 1.00.
30 Trimming Growth Curve Models Almost never trimSlope-intercept covarianceIntercept varianceNever have the intercept “cause” the slope factor or vice versa.Slope variance: OK to trim, i.e., set to zero.If trimmed set slope-intercept covariance to zero.Do not interpret standardized estimates except the slope-intercept correlation.
31 Using Amos Must tell the Amos to “Estimate means and intercepts.” Growth curve plug-inIt names parameters, sets measures’ intercepts to zero, frees slope and intercept factors’ means and variance, sets error variance equal over time, fixes intercept loadings to 1, and fixes slope loadings from 0 to 1.
32 Second ExampleOrmel, J., & Schaufeli, W. B. (1991). Stability and change in psychological distress and their relationship with self-esteem and locus of control: A dynamic equilibrium model. Journal of Personality and Social Psychology, 60,389 Dutch Adults after College Graduation5 Waves Every Six MonthsDistress Measure
34 Parameter Estimates Estimate SE CR MEANS Intercept 3.276 .156 20.946 SlopeVARIANCESInterceptSlopeAll error variances statistically significantCOVARIANCE*Intercept-Slope*Correlation = -.433
35 Interpretation Large variance in distress level. Average slope is essentially zero.Variance in slope so some are increasing in distress and others are declining.Those beginning at high levels of distress decline over time.
36 Model Fit c2(10) = 110.37, p < .001 RMSEA = .161 CFI = ( – 14 – )/ ( – 14)= .886Conclusion: Poor fitting model.
37 Alternative Options for Error Variances Force error variances to be equal across time.c2(4) = (not helpful)Non-independent errorserrors of adjacent waves correlatedc2(4) = (not much help)autoregressive errors (err1 err2 err3)c2(4) = (not much help)
38 Exogenous Variables Often in this context referred to as “covariates” TypesPerson – e.g., age and genderTime varying: a different measure at each timeSee “extra” slides.Need to center (i.e., remove their mean) these variables.For time-varying use one common mean.
39 Person CovariatesCenter (failing the center makes average slope and intercept difficult to interpret)These variables explain variation in slope and intercept; have an R2.Have them cause slope and intercept factors.Intercept: If you score higher on the covariate, do you start ahead or behind (assuming time 1 is time zero)?Slope: If you score higher on the covariate, do you grow at a faster and slower rate.Slope and intercept now have intercepts not means. Their disturbances are correlated.
40 Three exogenous person variables predict the slope and the intercept (own drinking)
41 Effects of Exogenous Variables Variable Intercept SlopeAge *Gender *COA *Rc2(4) = 4.9Intercept: Older children start out higher.Slope: More change for Boys and Children of Alcoholics.(Trimming ok here.)
42 Extra Slides Relationship to multilevel models Time varying covariates Multivariate growth curve modelPoint of minimal intercept varianceOther ways of modeling nonlinearityEmpirically scaling the effect of timeLatent difference scoresNon-linear dynamic models
43 Relationship to Multilevel Modeling (MLM) Equivalent if ML option is chosenAdvantages of SEMMeasures of absolute fitEasier to respecify; more options for respecificationMore flexibility in the error covariance structureEasier to specify changes in slope loadings over timeAllows latent covariatesAllows missing data in covariatesAdvantages of MLMBetter with time-unstructured dataEasier with many timesBetter with fewer participantsEasier with time-varying covariatesRandom effects of time-varying covariates allowable
44 Time-Varying Covariates A covariate for each time point.Center using time 1 mean (or the mean at time zero.)Do not have the variable cause slope or intercept.Main EffectHave each cause its measurement at its time.Set equal to get the main effect.Interaction: Allow the covariate to have a different effect at each time.
45 Interpretation Main effects of the covariate. Path: .504 (p < .001)c2(3) = 8.44, RMSEA = .071Peer “affects” own drinkingCovariate by Time interactionChi square difference test: c2(2) = 4.24, p = .109No strong evidence that the effect of peer changes over time.
52 Example Basic Model: c2(4) = 8.18 Same Factors: c2(13) = 326.30 CorrelationsIntercepts: .81Slopes: .67Same Factors: c2(13) =One common slope and intercept for both variables.9 less parameters:5 covariances2 means2 covariancesMuch more variance for Own than for Peer
53 Point of Minimal Intercept Variance ConceptThe variance of intercept refers to variance in predicted scores a time zero.If time zero is changed, the variance of the intercept changes.There is some time point that has minimal intercept variance.PossibilitiesPoint is before time zero (negative value)Divergence or fan spreadIncreasing variance over timePoint is after the last point in the studyConvergence of fan closeDecreasing variance over timePoint is somewhere in the studyConvergence and then divergenceMay wish to define time zero as this point
54 ComputationShould be computed only if there is reliable slope variance.Compute: sslope,intercept/sslope2Curran Example-0.458/0.170 = 1.931.93, just before the last waveConvergence and decreasing variabilityPeer perceptions become more homogeneous across time.
55 More Elaborate Nonlinear Growth Models Latent basis modelfix the loadings for two waves of data (typically the first and second waves or the first and last waves) and free the other loadingsBilinear or piecewise modelinflection pointtwo slope factorsStep functionlevel jumps at some point (e.g., treatment effect)two intercept factors
56 Bilinear or Piecewise Model Inflection pointTwo slope factors
57 Bilinear or Piecewise Model OPTION 1: 2 distinct growth ratesOne from T1 to T3The second from T3 to T5OPTION 2: Estimate a baseline growth plus a deflection (change in trajectory)One constant growth rate from T1 to T5Deflection from the trajectory beginning at T3Two options are equivalent in term of model fit.
60 Results Bilinear: c2(6) = 102.91, p < .001 RMSEA = .204Piecewise: c2(6) = , p < .001Conclusion: No real improvement of fit for these two different but equivalent methods
61 Step Function: Change in Intercept Level jumps at some point (e.g., point of intervention)Two intercept factorsSlopeInt1Int21234Note Int2 measures the size of intervention effect for each person.
62 Results Change in intercept Conclusion: No real improvement of fit RMSEA = .199Conclusion: No real improvement of fit
63 Modeling Nonlinearity Quadratic EffectsSeasonal EffectsEmpirically based slopes of any form.
64 Add a Quadratic FactorAdd a second (quadratic) slope factor (0, 1, 4, 9 …)Correlate with the other slope and intercept factor.Adds parameters1 mean1 variance2 covariances (with intercept and the other slope)No real better fit for the Distress Examplec2(6) = ; RMSEA = .199
65 Modeling Seasonal Effects Note the alternating positive and negative coefficients for the slope
66 Resultsc2(6) = , p < .001RMSEA = .120No evidence of Slope Variance (actually estimated as negative!)Conclusion: Fit better, but still poor.
67 Empirically Estimated Scaling of Time Allows for any possible growth model.Fix one slope loading (usually one).No intercept factor.
68 Results Curvilinear Trend Wave 1: 1.00 Wave 2: 0.74 Wave 3: 0.95 Better Fit, But Not Good Fitc2(9) = 62.5, p < .001
69 Latent Difference Score Models Developed by Jack McArdleCreates a difference score of each timeUses SEMTraditional linear growth curve models are a special caseCalled LDS Models
71 Relation to a Linear Growth Curve Model The same if a = 0If a not equal to zero, the model can be viewed as a blend of growth curve and autoregressive models.
72 Nonlinear Growth: Negative Exponential One Unit Moving Through TimeConstant Rate of Change (no error)The Force Pulling the Score to the Mean Is a ConstantThe First Derivative Is a Constant
73 More Complex Nonlinear Growth SinusoidNonzero first and second order derivativePendulumdampening
74 Estimation Using AR(2) Model Negative Exponential1 > a1 > -1 (the rate of change) and a2 = 0Sinusoid2 > a1 > 1 and a2 = -1Cobb formula for period length = p/cos-1√a1Pendulumdampening factor = 1 - a2
75 Go to the next SEM page.Go to the main SEM page.