1. 1 SEM and Longitudinal Data Latent Growth Models UTD 07.04.2006

2. 2 Why Growth models? Aren‘t autoregressive and cross-lagged models enough to test change and relationships over time? 1) In autoregressive models we can see stability over time but not type of development. We might have a stability of 1 – that is the relative placement of people is unchanged, and still everyone increases (or decreases).

3. 3 Number of cigarettes smoked after meal as a function of the day of the course

4. 4 Stability is 1 in an autoregressive model. Higher ones remain higher, and lower ones remain lower. However, there is a development. They all increase the number of cigarettes smoked. We cannot see it in the autoregressive model. We need a developmental model, which takes into account this development, but- also the differences in development across individuals.

5. 5 In the example, each individual had an intercept and a slope. Person1 had a slope 1, and an intercept 1 Person2 had a slope 1, and an intercept 2 Person3 had a slope 1, and an intercept 3 The mean of their slope is 1 The mean of their intercept is 2 The developmental model should take this individual information into account Still, the model should allow us to study development at the group level

6. 6 The Latent Growth Curve Model These criteria are met by the growth curve model. Meredith and Tissak (1990) belonged to the first to develop the growth model mathematically. The model uses an SEM methodology The results are meaningful when there is time gap between the measurements, and not just repeated measures How long the time gap is between the time points- is also meaningful The number of time points and the spacing between time points across individuals should be the same

7. 7 The latent factors in the growth model are interpreted as common factors representing individual differences over time. Remark: Latent growth model was developped from ANOVA, and expanded over time. Basically, with two time points we can have only a linear process of change. However, for deductive purpose, we will start with modeling a growth model for two time points, and then expand it to more points in time.

8. 8 A two-factor LGM for anomia for 2 time points

9. 9 Intercept: The intercept represents the common or mean intercept for all individuals, since it has a factor loading of 1 to all the time points. In the previous example it will have a mean 2. It is the point where the common line for all individuals crosses the y axis. It presents information in the sample about the mean and variance of the collection of intercepts that characterize each individual‘s growth curve.

10. 10 Slope: It represents the slope of the sample. In this case it is the straight line determined by the two repeated measures. It also has a mean and a variance, that can be estimated from the data. Slope and intercept are allowed to covary. In this example with two time points, in order to get the model identified, the coefficients from the slope to the two measures have to be fixed. For ease of interpretation of the time scale, the first coefficient is fixed to zero. With a careful choice of factor loadings, the model parameters have familiar straightforward interpretations.

11. 11 Exercise:is the model identified? How many df? How many parameters are to be estimated? In this example: The intercept factor represents initial status The slope factor represents the difference scores anomia2-anomia1 since: Anomia1=1*Intercept + 0*Slope + e1 Anomia2=1*Intercept + 1*Slope + e2 If errors are the same then Anomia2 – Anomia1 = Slope

12. 12 This model is just identified (if we set the measurement errors to zero). By expanding the model to include error variances, the model parameters can be corrected for measurement error, and this can be done when we have three measurement time points or more. Three or more time points provide an opportunity to check non linear trajectories. For those interested, Duncan et al. Shows the technical details for this model on p. 15-19.

13. 13 A two-factor LGM for anomia for 3 time points

14. 14 Representing the shape of growth With three points in time, the factor loadings carry information about the shape of growth over time. In this example we specify a linear model. We have reasons to believe that anomia is increasing as a linear process, and this way we can test it. If we are not sure, we can test a model where the third factor loading is free

15. 15 A two-factor LGM for anomia. 3rd time point free

16. 16 level time Parallel stabilityLinear stability Strict stability Monotone stability

17. 17 Sometimes there are reasons to believe that the process is not linear. For example, a process might take a quadratic form. In this case, one can model a three-factor polynomial LGM Anomia=intercept +slope 1 *t+slope 2 *t 2 However, this is more rare in sociology and political sciences. It might be reasonable in contexts such as learning, tobacco reduction etc.

18. 18 3-factor polynomial LGM

19. 19 Summary1 In all the examples shown we use LGM when we believe that the process at hand is a function of time. What is the meaning of the covariance between slope and intercept? Intercept represents the initial stage, and slope the change. A negative covariance suggests that people with a lower initial status, change more and people with a higher initial status change less. For positive covariances: people with a higher initial status change more, and people with a lower initial status change less.

20. 20 Summary 2 There is no direct test for cross lagged effects. The means of the latent slope and the latent intercept represent the developmental process over time for the whole group; their variance represents the individual variability of each subject around the group parameters.

21. 21 Single-indicator model vs. multiple- indicator model Instead of using a single-scale score to measure at each time point authoritarianism or anomie for example, we could use latent factors to estimate these constructs, and could therefore be purged from measurement error.

22. 22 Single-indicator model without auto-correlation

23. 23 multiple-indicator model without auto- correlation

24. 24 In a 2nd order LGM The same 1st order variable is chosen as the scale indicator for each first-order factor. Corresponding variables whose loadings are free have those loadings constrained to be equal across time. This ensures a comparable definition of the construct over time (referred to as „stationarity“, Hancock, Kuo & Lawrence 2001, Tisak and Meredith 1990).

25. 25                           t2          =1   =1       t1       Item a Item b Item c Item d Item e Item f Measurement Invariance: Equal factor loadings across groups                           t2          =1   =1       t1       Item a Item b Item c Item d Item e Item f Group AGroup B

26. 26 Configural Invariance Metric Invariance Scalar Invariance Invariance of Factor Variances Invariance of Factor Covariances Invariance of latent Means Invariance of Unique Variances Steps in testing for Measurement Invariance between groups and/or over time

27. 27 Configural Invariance Metric Invariance Equal factor loadings Same scale units in both groups/time points Presumption for the comparison of latent means Scalar Invariance Invariance of Factor Variances Invariance of Factor Covariances Invariance of latent Means Invariance of Unique Variances Steps in testing for Measurement Invariance

28. 28 Concept of ‘partial invariance’ introduced by Byrne, Shavelson & Muthén (1989) Procedure Constrain complete matrix Use modification indices to find non-invariant parameters and then relax the constraint Compare with the unrestricted model Steenkamp & Baumgartner (1998): Two indicators with invariant loadings and intercepts are sufficient for mean comparisons One of them can be the marker + one further invariant item Full vs. Partial Invariance

29. 29 Autocorrelation As in the autoregressive model, we believe that measurement errors of repeated measures are related to one another. Therefore, we correlate them (Hancock, Kuo & Lawrence 2001, Loehlin 1998).

30. 30 Latent Curve Model with Autocorrelations

31. 31 Intercepts In a 2nd order factor LGM, intercepts for corresponding 1st order variables at different time points are constrained to be equal, reflecting the fact that change over time in a given variable should start at the same initial point.

32. 32 MIMIC and LGM, time-invariant covariates in the latent growth modeling Sometimes a model in which longitudinal development is predicted by an intercept and growth curve is too restrictive. Such a model is called unconditional. In such a case we may try to predict the latent slope and intercept by background variables (for example demographic variables), which are time invariant. This would be called a conditional model.

33. 33 Growth MIMIC model – anomia

34. 34 Another complication: the intercept and slope may be not only conditioned on some other variables, they could also cause them. For example, the intercept of anomia could be a cause of a variable named satisfaction in life.

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36. 36 Analyzing growth in multiple populations Sometimes our data contains information on several populations: males and females, different age cohorts, people from former east and west Germany, voters of right and left wing parties, ethnicities, treatment and control groups etc. The SEM methodology to analyze multiple groups can be applied also here. We can compare the means of the slope and intercept latent variables as well as growth parameters, equality of covariance between slope and intercept etc.

37. 37 The coding of time (Biesanz, Deeb- Sossa, Papadakis, Bollen and Curran, 2004) Misinterpretation regarding the relationships among growth parameters (intercepts and slopes) appear frequently. Therefore it is important to pay attention to the coding of time Covariance between intercept and slope, and variance of the intercept and slope are directly determined by the choice of coding When the coefficient between slope and the first time point is set to zero, the covariance and variances are related to the first time point. For example, a negative covariance between the slope and intercept indicates that at the first (0) time point people with a lower starting point change more quickly. It is not necessarily true for later time points.

38. 38 If we are interested at the relation between the intercept and the slope at a later time point, for example the second one, we have to fix at this point the coefficient from the slope to zero. The first coefficient will change from 0 to -1, and the third coefficient will change from 2 to 1. It is useful to code the coefficients from the slope according to the time interval on a yearly basis, if we believe in a linear process.

39. 39 Example: if we have data collected in January, then in July, and then again in July in the following year, a possible coding of the coefficients from the slopes to the measurements could be: 0, 0.5 (since the measurement took place half a year later) and 1.5. Exercise: If we are interested in the relations between the slope and the intercept at the second time point, how could we code the coefficients?

40. 40 Answer: -0.5, 0 and 1. Using a yearly basis, we keep the interpretation simple. If we have a quadratic model, the interpretation of the highest order coefficient (for example its variance) does not change with different codings and placement of time origins. But the interpretation of lower order terms (intercept and linear slope) does. The choice of where to place the origin of time has to be substantially driven. This choice determines that point in time at which individual differences will be examined for the lower order coefficients.

41. 41 time Level of anomia 1.0 1.5 2.0

42. 42 Coding and Mimic The meaning of the variance of the intercept and the slope changes in Mimic models. If the intercept is explained (conditioned) by age for example, the residual variance of the intercept indicates the variability across individuals in the starting point not accounted for by age. This should be taken into account when we interpret our results.

43. 43 The bivariate latent trajectories (growth curve) analysis We can extend the univariate latent trajectory model to consider change in two or more variables over time. The bivariate trajectory model is simply the simultaneous estimation of two univariate latent trajectory models. The relation between the random intercepts and slopes is evaluated for each series. Then it is possible to determine whether development in one behavior covaries with other behaviors.

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45. 45 So far we could demonstrate LGM which allow multiple measures, multiple occasions and multiple behaviors simultanuously over time. We could estimate the extent of covariation in the development of pairs of behaviors. We can go one level higher, and extend the test of dynamic associations of behaviors by describing growth factors in terms of common higher order constructs.

46. 46 Factor-of-curves LGM To test whether a higher order factor could describe the relations among the growth factors of different processes, the models can be parameterized as a factor of curves LGM. The covariances among the factors are hypothesized to be explained by the higher order factors (McArdle 1988). The method is useful in determining the extent to which pairs of behaviors covary over time. Rarely used. The test if the approach is better can be done by comparing the fit measures of alternative models.

47. 47 Factor-of-curves LGM d1 d3 d4 d2

48. 48 Missing values and LGM As in AR models, missing data constitute a problem in LGM. Also here we distinguish between 3 kinds of MD: MCAR, MAR and MNAR. The diagnosis and solutions discussed in the AR apply also for LGM models.

49. 49 Estimating Means and Getting the model identified As Sörbom (1974) has shown, in order to estimate the means, we must introduce some further restrictions: 1) setting the mean of the latent variable in one group-the reference group- to zero. The estimation of the mean of the latent variable in the other group is then the mean difference with respect to the reference group. In the growth model, one could alternatively set all intercepts of the constructs in both groups to zero and intercept of one indicator per construct to zero (constraining the second to be equal across time points), and then compare the means of the latents mean and intercepts in both groups. 2) in case of a one group analysis: setting the measurement models invariant across time, since it makes no sense to compare the means of constructs having a different measurement model over time. At least one intercept (of an indicator) per construct has to be set equal across time.

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56. 56 Additional uses of LGM models- Intervention studies 1) Using the multiple group option to test effects of intervention programs 2) The effect of interventions in experimental settings can be done also as a mimic model See Curran and Muthen, 1999.

57. 57 First questionnair e was sent Second questionnair e was sent Third questionnair e was sent The move 2-3 weeks 6-7 weeks 4 weeks The intervention Figure 3. The development of the experiment before and after the move to Stuttgart

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60. 60 Figure 7a. The latent curve model with a multi group analysis for the low intention group (standardized coefficients).

61. 61 Figure 7b. The latent curve model with a multi group analysis for the high intention group (standardized coefficients).

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63. 63 ALT/Hybrid Modeling Goals Combining features of both autoregressive and latent growth curve models to result in a more comprehensive model for longitudinal data than either the autoregressive or latent trajectory model provide alone.

64. 64 Model specification: unconditional model We incorporate key elements from the latent trajectory and autoregressive models in the development of the univariate ALT model: from the latent trajectory model we include the random intercept and random slope factors to capture the fixed and random effects of the underlying trajectories over time. From the autoregressive model we include the standard fixed autoregressive parameters to capture the time specific influences between the repeated measures themselves. The mean structure enters solely through the latent trajectory factors in the synthesized model.

65. 65 Usually we will treat the first time point measurement as predetermined in the ALT model and it can be expressed simply by an unconditional mean and an individual deviation from the mean. It will correlate with the intercept and the slope. There are some instances where treating the initial measure as endogenous will be required in order to achieve identification (For equations, see Bollen/Curran 2004 page 349-352). we assume the residuals have zero means and are uncorrelated with the exogenous variables.

66. 66 Identifying the ALT model 1) With five or more waves of data, the model is identified while treating the wave one y variable as predetermined without making any further assumptions. 2) With four waves we need a constant autoregressive parameter. 3) If we have only three waves of data, we can have an identified model when we assume an equal autoregressive parameter throughout the past, make the wave one endogenous, and introduce further (nonlinear) constraints for the first wave.

67. 67 Unconditional ALT model- exogenous time 1 construct

68. 68 Conditional ALT model- endogenous time 1 construct

69. 69 Conditional ALT model- exogenous time 1 construct

70. 70 Bivariate unconditional ALT model

71. 71 Bivariate Conditional

72. 72 Third order LGM An example of a third order LGM.

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76. 76 Level of latent variablesContent First orderLatent variables of different aspects of group related enmity, each measured by two indicators: racism (r), enmity towards foreigners (f), anti-Semitism (a), enmity towards homosexuals (h), enmity towards homeless people (ob), Islam-phobia (i)and enmity of the non- established (eta) Measured in 2002, 2003 and 2004 in Germany on a representative sample of the German population Second orderGRE- Second order variable of group related enmity Third orderGrowth variables- slope and intercept

77. 77 Racism: ra 01 r Aussiedler (Russian immigrants with German ancestors) should be better employed than foreigners, since they have a German origin. ra 03 r The white people are justifiably leading in the world. Foreigners Enmity ff04 d1r Too many foreigners live in Germany. ff08 d1r If working places become scarce, one should send foreigners living in Germany back to their home country. Antisemitism as 01r Jews have too much influence in Germany. as 02r Jews are to be blamed due to their behavior for their persecution.

78. 78 Heterophobia 1. Rejection of homosexuals he01hMarriage between two women or two men should be permitted. he02hrIt is disgusting, when homosexuals kiss in public. 2. Rejection of disabled He01br One feels sometimes not comfortable in the presence of disabled people. He02brSometimes on is not sure how to behave with disabled people. 3. Rejection of homeless people he01oHomeless beggars should be removed from pedestrian zones. he02orThe homeless people in towns are unpleasant.

79. 79 Islamphobia he01 m The Muslims in Germany should have the right to live according to their belief. he02 m It is only a matter of Muslims, if they call to pray over loudspeakers. Rights of the established ev03 r One who is new somewhere should be at first satisfied with less. ev04 r Those who have always lived here should have more rights than those who came later. Classical sexism sx03rWomen should take again the role of wives and mothers.

80. 80 Growth of group related enmity (GFE)- slope and intercept 3rd level 1st level 2nd levelGFE 1st time point GFE 2nd time point GFE 3rd time point r, f, a, h, ob, i, eta- 1st time point r, f, a, h, ob, i, eta- 2nd time point r, f, a, h, ob, i, eta- 3rd time point

81. 81 SUMMARY 4.0) Evaluation of the different strategies for analysis of panel data in SEM Each of the two models (AR and LGC) has a distinct approach to modeling longitudinal data. Each has been widely used in many empirical applications. Two key components of the autoregressive and cross lagged models are the assumptions of lagged influences of a variable on itself and that the coefficients of effects are the same for all cases, when we do not conduct a multiple-group analysis.

82. 82 Summary (continuation) In contrast, the latent trajectory model has no influences of the lagged values of a variable on itself. The intercept and the slope parameters governing the trajectories differ over subjects in the analysis. Measurements are modeled alternatively as a function of time. The LGM gives us a description of a process. We do not get it from the AR. However, in bivariate Lgm we have the same problem as in cross section: we have one slope trajectory and one intercept trajectory variables for each process. It is again not clear what is the cause of what… Each of these assumptions about the nature of changes is empirically or theoretically plausible. The hybrid model combines for these reasons both assumptions into one framework.

83. 83 Further SEM applications such as a multiple group comparison, can also be done with the ALT model. In a discussion with Muthen, he criticizes the ALT model. His critique concentrates in the difficulty to interpret the parameters in this model. An alternative is to use continuous time modelling with differential equations(Oud, Singer), but it is not as straight forward to be applied as the AR and Lgm modeling An alternative is to run AR and LGM models separately. Depending on the research question, each model would provide complementary answers.

84. 84 Thank you very much for your attention!!!!