Longitudinal data analysis with structural equations

Longitudinal data analysis with SEM Running Head: Longitudinal data analysis with SEM Longitudinal data analysis with structural equations Jesús Rosel(1) and Ian Plewis(2) (1) Departamento de Psicología Evolutiva, E., S. y Metodología. Universitat Jaume I, Castellón, Spain (2) Centre for Longitudinal Studies, Institute of Education, University of London, United Kingdom Dr. Jesús Rosel Departamento de Psicología E, E, S y Metodología Universitat Jaume I Apdo. 224, 12080 Castellón, SPAIN. Tel: 964 72 93 30. Fax: 964 72 93 49. E-mail: [email protected] Suprimit: ¶ Longitudinal data analysis with SEM Longitudinal data analysis with structural equations Abstract In this paper we review different structural equation models for the analysis of longitudinal data: (a) univariate models of observable variables, (b) multivariate models of observable variables, (c) models with latent variables, (d) models that are unconditioned or conditioned to other variables (depending on the variability of the independent variables: time-varying or time-invariant, and depending on the type of independent variables: of latent variables or of observable variables), (e) models with interaction of variables, (f) models with non-linear variables, (g) models with a constant, (h) with single level and multilevel measurement, and (i) other advances in SEM of longitudinal data (latent growth curve model, latent difference score, etc.). We have paid more attention to the interaction of variables and to non-linear transformations of variables because they are not frequently used in empirical investigation. They do, however, offer interesting possibilities to researchers who wish to verify relations between the variables they obtain. Potential applications are described, with their advantages and disadvantages. Suprimit: We Suprimit: unconditioned Suprimit: and Suprimit: ¶ Longitudinal data analysis with SEM 1 Longitudinal data analysis with structural equations Since Jöreskog (1969), Keesling (1972) and Wiley (1973) first developed the statistical model of structural equations, it has become one of the most widely used techniques for analysing longitudinal data. So much is this the case that: (a) in most handbooks about longitudinal data analysis one or several chapters are devoted to exemplifying structural equation models (Amsel & Renninger, 1997; Bijleveld & van der Kamp, 1998; Collins & Horn, 1991; Collins & Sayer, 2001; Dwyer, Feinlieb & Hoffmeister, 1992; Fitzmaurice, Laird & Ware, 2004; Frees, 2004; Gottman, 1995; Little, Schnabel & Baumert, 2000; Plewis, 1985; Singer & Willet, 2003; von Eye & Clogg, 1994); (b) centres that provide training in longitudinal research offer courses in structural equation modelling (SEM); and (c) there is a steady increase in the number of journal articles in which this methodology is applied (Card & Little, 2007). An example of the importance of SEM applied to longitudinal studies is reflected by the two chapters that Jöreskog (1974, 1977) published on this topic 30 years ago. In this article we will briefly review the different models applied to the analysis of longitudinal data by means of SEM. We keep the statistical theory down to a basic level, but this work is also intended to be a practical guide for researchers who have to analyse longitudinal data. For this reason, we largely avoid dwelling on the basics of SEM and it is therefore assumed that the reader has some prior knowledge about this subject matter. One important point to be taken into account in any SEM model is that the researcher, on conducting his or her research, must ensure a proper integration of the following aspects: (a) a basic theory with the correctly formulated hypotheses so as to be able to check whether the data match the theory; (b) a correct research design, with a thorough study of the variables to be measured, the time between measurements, the number of measurements, the age or ages of the sample, the time the research lasted, and so forth; and (c) the statistical model of data analysis that is to be used as a method of confirming (or, should it be the case, rejecting) the hypotheses that have been posited (Collins, 2006; Embretson, 2007; Little, Bovaird & Slegers, 2006; Little, Preacher, Selig & Card, 2007; Ram & Grimm, 2007). Before beginning any longitudinal study it is important to put forward hypotheses about the stability of the observable and latent variables, as well as the relations between them, in order to check whether: (a) the variances of the latent and observable variables Suprimit: Un Suprimit: importante aspecto a tener en cuenta en cualquier modelo SEM es que el investigador, cuando lleva a cabo su investigación, ha de tener bien articulados entre sí los siguientes aspectos Suprimit: una teoría de base con las correspondientes hipótesis correctamente formuladas con el fin de comprobar si los datos se corresponden con la teoría, Suprimit: un correcto diseño de investigación Suprimit: con las variables a medir, el tiempo entre mediciones, el número de mediciones, la edad o edades de la muestra, la duración de la investigación, etc. bien estudiadas, Suprimit: el modelo estadístico de análisis de datos que sirva de método de comprobación (o, en su caso, de refutación) de las hipótesis planteadas Suprimit: Antes de comenzar cualquier estudio longitudinal es importante plantear hipótesis acerca de la estabilidad de las variables observables y latentes Suprimit: así como de las relaciones entre las mismas comprobando si Suprimit: ¶ Longitudinal data analysis with SEM 2 are stable or differ over time; (b) the same is true of the measurements of both the observable and latent variables; (c) the loads, or coefficients, between the latent variables and their corresponding observable variables are equal at each time of measurement; (d) the internal correlations between latent and/or observable variables remain stable between different times of measurement; and (e) the effects between different times of measurement are equal or changing (Brown, 2006). The different models we will be looking at are as follows: (a) univariate and multivariate models; (b) observable variable models and latent variable models; (c) unconditioned versus conditioned to other variables; (d) with and without interaction terms; (e) with linear and non-linear effects; (f) with and without constants; (g) with measures at one and more than one level; and (h) diverse advances in longitudinal SEM. In fact, there can be as many different models as there are combinations among the preceding conditions, which is what makes SEM a highly versatile research tool. Univariate models of observable variables The theoretical foundation of effect models with observable variables lies in simple regression (Pearson, 1896) and in path analysis (Wright, 1918, 1921). Path analyses are a further development of regression, but with the addition of intermediate variables to predict the response variable of interest. SEM encompasses both these observable variable models and those with latent variables (or factors). Univariate repeated measures models are those in which the same persons are measured on a single variable on several occasions. Thus, if the variable V1 is measured 4 times (e.g. let us suppose we are dealing with a group of children for whom the variable ‘level of knowledge of mathematics’ is measured at 6, 7, 8 and 9 years of age; in panel designs this is represented by 4W1V, that is to say, 4 ‘waves’ or times, and 1 variable), then the representation of the variable at the four times would be: V1,1, V1,2, V1,3 and V1,4, where the first subscript indicates the variable (in this example it is always the same: V1,t) and the second subscript indicates the time of measurement (the first time, at the age of 6, is represented by V1,1; the second time, at 7 years old, by V1,2, and so forth). The simplest model for representing the nature of these data is shown in Figure 1. Insert Figure 1 about here In the graphic representation of the SEM we have followed Bentler’s (1995) system of notation, which uses rectangles to represent the observed variables (V1,1, V1,2, V1,3 and V1,4); the effects of some variables on others are represented by arrows (which show the Suprimit: las varianzas de las variables latentes y observables son estables o difieren a lo largo del tiempo, Suprimit: lo mismo respecto de las medias, tanto de las variables observables como de las latentes, Suprimit: las cargas, o coeficientes, entre las variables latentes y sus correspondientes variables observables son iguales en cada momento de medición, Suprimit: las correlaciones internas entre variables latentes y/o observables se mantienen estables entre diferentes momentos de medición, Suprimit: los efectos entre diferentes momentos de medición son equivalentes o cambiantes Suprimit: , Suprimit: ¶ Longitudinal data analysis with SEM 3 direction of each respective effect), with the corresponding size of the effect (b1, b2 and b3); and the arrows that are labelled with the letter “E” (E1,2, E1,3 and E1,4) are the residuals. The model in Figure 1 is called a first-order autoregressive model (AR(1)), or ‘simplex’ or Markov model. The term ‘autoregressive’ refers to the fact that each value of the variable V1 at time t depends only on the value of that same variable in the previous measurement t-1 (V1,t=f(V1,(t-1))). This model was put forward by Guttman (1954), but has since been developed by others (Anderson, 1960; Heise, 1969; Humphreys, 1960). It should be noted that there is no E1,1 in the model because one of the assumptions of the model is that the independent (or explanatory) variables have been measured without error. This model can be represented algebraically in a compact form: y = Βy + Γx + ε, (1) where y is the p × 1-order vector of dependent (or response) variables: y’ = [V1,2, V1,3, V1,4], x is the q × 1-order vector of independent variables: x=[V1,1], ε is the vector of residuals: ε’ =[E1,2, E1,3, E1,4], while the relations between the dependent variables are represented by matrix Β, of order p × p, and are as follows: Β = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 00 00 000 3 2 b b, (2) Γ is a coefficient matrix of order p × q that relates the independent variables with the dependent ones, which in our case will be: Γ’ = [b1, 0, 0]. In order to estimate the corresponding parameters, matrix Ψ = Cov(ε), where Cov is the covariance operator so that Cov(Y,Y) = Var(Y), and Ψ is a matrix of order p × p, must be added to the previous matrices. In our case, the residual variances (Cov(E1,2, E1,2), Cov(E1,3, E1,3), Cov(E1,4, E1,4)) will be represented on the main diagonals of this matrix. Matrix Φ = Cov(x), of order q × q, which represents the matrix of the covariances of the independent variables, would also have to be added to the model. The model in Figure 1 (which has 3 degrees of freedom) can be reformulated so that each measurement depends on the measurement immediately before it but also on the measurement performed at the last time but one. As a result, the model in Figure 2 could be proposed. Suprimit: ¶ Longitudinal data analysis with SEM 4 Insert Figure 2 about here The model in Figure 2 (which would have 1 degree of freedom) is called second-order autoregressive (AR(2)) because each value is a function of the two measurements immediately preceding it (V1,t=f(V1,(t-1),V1,(t-2))), and if there were evidence that the measurements of the different times were closely related, a third-order autoregressive model could be contemplated, where a new effect (b6) would have to be added to the model in Figure 2. This effect would go from V1,1 to V1,4, and in general for any AR(3) model it would be: V1,t=f(V1,(t-1),V1,(t-2),V1,(t-3)). To be able to formulate a p-order autoregressive model, we need at least p+1 measurements. The AR(3) model above would have 0 degrees of freedom, and therefore it would not be possible to estimate coefficients of overall fit for the model. Different hypotheses about the variables can be tested. The main purpose of these hypotheses (as they are shown in Figures 1 and 2) are the equality of the effect parameters, of the variances and between covariances of the residuals (Brown, 2006): (a) If it is assumed that the effects among variables remain constant over time, they can be constrained so that coefficients continue to be equal; thus, in the AR(1) model in Figure 1, it could be hypothesised that b1 = b2 = b3 and, following on with the same logic, in the AR(2) model in Figure 2, it could be considered that b1 = b2 = b3, and also b4 = b5. These assumptions make more sense if the time span between variables is the same, and if it is assumed that the process does not change over time (which is more likely to occur with adults than with small children). (b) Since the same variable V1,t is measured on several occasions, the residual variances could be assumed to be equal; thus, in the models in Figures 1 and 2, Cov(E1,2, E1,2) = Cov(E1,3, E1,3) = Cov(E1,4, E1,4) can be included in the syntax of the constraints paragraph of the input for the statistical software that is being used (Cov is the covariance operator so that Cov(Y,Y) = Var(Y)). This assumption makes more sense when the variances of the variables have roughly the same value, which is more frequent in research conducted with adults. This is due to the fact that when the same variable is examined in small children, the variance is usually seen to increase with age and the previous constraint will not be fulfilled empirically. (c) Different assumptions can be made about the residuals. It might be reasonable to suppose that, because the same variable is being measured several times, the residuals at time 2 (E1,2) will covary with those from time 3 (E1,3), that is to say, Cov(E1,2,E1,3) will Suprimit: hypothesis Suprimit: hypothesis Suprimit: Suprimit: : Suprimit: : Suprimit: : Suprimit: ¶ Longitudinal data analysis with SEM 5 be left free (i.e. ≠ 0), those from time 3 will covary with those from 4 (Cov(E1,3,E1,4)), which could also be left free, and so forth, if there were more measurement times (Cov(E1,4, E1,5), …). These parameters will therefore be included in the corresponding covariance section as being free, since by default they are assumed to be equal to 0. If this assumption were added to Figure 1, the resulting model would be like the one in Figure 3. From Figure 3 onwards, we will follow the convention of representing covariances by means of curved lines with arrowheads at both ends. Insert Figure 3 about here (d) Bearing in mind that it is the same variable, and that the same covariance processes occurring between errors can be repeated between consecutive measurements, the following constraint can be added: Cov(E1,2, E1,3) = Cov(E1,3, E1,4) = etc. (e) The residuals at each occasion can be interpreted as ‘innovations’ of the variable at the time of measurement (Box, Jenkins and Reinsel, 1994), as they are a part of the same variable that is not explained by the previous variable or variables. Thus, a firstorder moving average model (MA(1)) can be put forward, in which each value of the variable at any given time (V1,t) is a function of the previous error for the same person (E1,t-1), that is V1,t = f(E1,t-1), which results in the model in Figure 4. Insert Figure 4 about here In fact, the model in Figure 4 is a first-order autoregressive and first-order moving average model (ARMA(1,1)). It is possible to add the constraint whereby the effects of the moving averages are invariant over time (as is assumed in Box-Jenkins time series models), by leaving b4 = b5. The constraints mentioned above for the effects, variances and covariances of the residuals can also be applied to the model in Figure 4. The foregoing statistical hypothesis must be based on a theoretical justification for the model, but if used wisely they help to identify the underlying model that generates the data. The models that have been outlined here constitute the basis of SEM, and understanding and being able to apply them is an almost indispensable condition before moving on to more complex models. Multivariate models of observable variables Multivariate models of observable variables, as we have already stated, are an extension of univariate models. For example, let us suppose that (as in Figure 1) the same children are measured at 6, 7, 8 and 9 years of age on the variable ‘level of knowledge of Suprimit: , Suprimit: , Suprimit: ing Suprimit: hypotesis Suprimit: ¶ Longitudinal data analysis with SEM 6 language’ in addition to their ‘level of knowledge of mathematics’: the design would now be represented by 4W2V. As we have seen, the variable ‘level of knowledge of mathematics’ was represented at the four times by V1,1, V1,2, V1,3 and V1,4 and now the ‘level of knowledge of language’ is represented as V2,1, V2,2, V2,3 and V2,4. One advantage of measuring two variables is that it becomes possible to study the directional effect of one variable on the other. That is to say, the researcher can formulate two research questions, namely, do the two variables develop independently of each other, or does the variable ‘level of knowledge of mathematics’ exert a greater influence over that concerning ‘level of knowledge of language’ or vice versa? This gives us the model in Figure 5. Insert Figure 5 about here In Figure 5 the covariance between the variables V1,1, V2,1 is represented by lines with arrowheads at both ends that join the two variables; if the variable ‘maths ability’ (V1,1, V1,2, V1,3 and V1,4) exerted an influence on ‘language ability’ (V2,1, V2,2, V2,3 and V2,4), then the effects b8, b10 and b12 could be expected to be statistically significant; if the opposite were true, then the significant effects would be b9, b11 and b13. In each case the researcher must have some substantive hypotheses that justify the specification; if only a few of the previous effects were significant, the researcher would have to establish an explanation for these findings and their meaning. The mathematical representation of the model in Figure 5 would have the same form as the one in Equation 1, but the y vector of the dependent variables is now of order 6 × 1: y’ = (V1,2, V1,3, V1,4, V2,2, V2,3, V2,4), the x vector is now of order 2 × 1: x’ = (V1,1, V2,1), while ε is the vector of residuals, which in our case is of order 6 × 1: ε’ = (E1,2, E1,3, E1,4, E2,2, E2,3, E2,4), Β being of order 6 × 6: Β = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ 0000 0000 0000 0000 000000 000000 612 133 510 112 bb bb bb bb , and Γ is a coefficient matrix of order 6 × 2 that relates the independent variables with the dependent ones: Suprimit: ; Suprimit: ( ,,, , VV VV 22413121 Suprimit: ¶ Longitudinal data analysis with SEM 7 Γ’ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 0000 0000 49 81 bb bb . Matrix Ψ is of order 6 × 6, and in our case the variances of the residuals (Cov(E1,2, E1,2), Cov(E1,3, E1,3), …, Cov(E2,4, E2,4)) will be represented on its main diagonal; the main diagonal of matrix Φ = Cov(x), of order 2 × 2, will contain the variances of V1,1 and of V2,1, while the other two elements of the matrix contain the covariance between V1,1 and V2,1. Sometimes the crossed effects between variables V1,t and V2,t are contemporaneous (either reciprocal or unidirectional) rather than lagged; thus, it can be supposed that in the model in Figure 5 the variables ‘level of knowledge of mathematics’ and ‘level of knowledge of language’ influence each other at the same time. The model would therefore be non-recursive (two or more variables exert an influence on each other); a feasible representation of such a model is shown in Figure 6. Insert Figure 6 about here If the variable ‘level of knowledge of mathematics’ (V1,1, V1,2, V1,3 and V1,4) influenced ‘level of knowledge of language’ (V2,1, V2,2, V2,3 and V2,4), then only effects b9, b11 and b13 in Figure 6 would be significant. If, in contrast, the variable ‘level of knowledge of language’ were the one that exerted an influence on ‘level of knowledge of mathematics’, then parameters b8, b10 and b12 would be significant. If no crossed effects between the variables (b8, b9, b10, …, b13) were significant, then this would indicate that the variables V1,t and V2,t develop independently, with just the initial covariance (Cov(V1,1, V2,1)) representing the preliminary relation between the two. The model in Figure 6 is non-recursive because there are one or several interdependence ‘loops’ between variables; note that there is a reciprocal influence between V1,2 and V2,2, between V1,3 and V2,3, as well as between V1,4 and V2,4. These models are also called interdependence and simultaneous equation models and can lead to identification difficulties (Bentler & Raykov, 2000; Bollen, 1989; Hayduk, 1996). The constraints outlined in the section about univariate models of observable variables for the effects, variances and covariances can be added to the models in Figures 5 and 6, but in Figures 5 and 6 the covariances of contemporaneous measurements can be taken as being free: Cov(E1,2, E2,2), Cov(E1,3, E2,3), Cov(E1,4, E2,4). In Figures 5 and 6, this could be represented by lines with arrowheads at both ends that join the respective measurement errors. The constraint Cov(E1,2, E2,2) = Cov(E1,3, E2,3) = Cov(E1,4, E2,4) Suprimit: by Suprimit: Suprimit: ¶ Longitudinal data analysis with SEM 14 As in the model in Figure 14, if we propose that X1 and X2 present a permanent effect over all the measurement times, the corresponding effects of X1 and X2 on V1,2, V1,3, V1,4, V2,2, V2,3 and V2,4 can be added. The two time-invariant conditioned models described here, that is, latent variables (see Figure 14) and observable variables (see Figure 15), can be generalised and applied to the models of latent variables mentioned above, for example, from Figure 8 to Figure 13. With respect to the time-varying conditional models, let us suppose the model in Figure 12, in which the cognitive capacity of each child has been measured at each time as the number of correct answers in an intelligence test (X1,t). This exerts an influence at each respective measurement time and, through the respective factors, would affect their capacity for language and mathematics; the result would therefore be the model in Figure 16. Insert Figure 16 about here Note that in Figure 16 we have included all the possible covariances between the independent variables X1,1, X1,2, X1,3 and X1,4: Cov(X1,1, X1,2), Cov(X1,1, X1,3), …, Cov(X1,3, X1,4), assuming that there is a correlation between the children’s scores at each time of measurement. If we assumed that the independent variable X1,t was temporally dependent on itself (instead of covariation), then we would have to remove the covariances between X1,1, X1,2, X1,3 and X1,4 and put in the effects that correspond from X1,1 to X1,2, from X1,2 to X1,3 and from X1,3 to X1,4. Figure 17 shows a development model conditioned to time-varying factors. If we assume that the following are measured in the children: the variable ‘level of knowledge of mathematics’, represented at the four times by V1,1, V1,2, V1,3, and V1,4; the variable ‘level of knowledge of language’, represented as V2,1, V2,2, V2,3, and V2,4; and a general intelligence test (X1,t) and Wechsler’s block design test (X2,t) are used as independent variables (or covariables), and that both intelligence tests form a temporal intelligence factor (F5, F6, F7 and F8) and, furthermore, that there is a time dependence between the factors, then the model would be the one in Figure 17. Insert Figure 17 about here Note that, statistically, the independent variable of the system is factor F5, which means it has been measured with no error; if we were to assume that the temporal intelligence factor (F5, F6, F7 and F8) is covarying over time (instead of having a temporal Suprimit: Suprimit: ¶ Longitudinal data analysis with SEM 15 dependence), then we would have to remove effects b16, b17 and b18, the errors of measurement of the factors D6, D7 and D8, and add the covariances among factors F5, F6, F7 and F8: Cov(F5, F6), Cov(F5, F7), …, Cov(F7, F8). The same considerations as those made in the section about ‘Univariate models of observable variables’ and ‘Multivariate models of observable variables’ can be repeated here with respect to the measurement errors of the variables V1,t and V2,t in order to gain degrees of freedom and facilitate the convergence of the model estimators. As can be seen, including substantive (‘time-invariant’ or ‘time-varying’) independent variables in SEM adds new possibilities to the estimation of effects between variables, and to the search for dependence among them. Models with interaction of variables When two or more variables are being measured, one of them, or a new dependent variable, can be hypothesised as being a function of the interaction of two (or more) independent variables (Aiken & West, 1991; Aiken, West & Pitts, 2003; Jaccard, Turrisi & Wan, 2003). Interaction of variables is understood to mean the product of them. Thus, if we propose that the interaction between consuming alcohol and tobacco favours the appearance of cancer, then we have to measure the rate of alcohol and tobacco consumption for each person, and we must create a third variable that will be the product of both for each person, while at the same time, as a dependent variable, it would serve as an indicator of cancer. In interaction of variables, an overriding principle is that of ‘nesting’: when an interaction of variables is used, we must utilise the variables and corresponding interactions that lie on a lower level. Thus, if the interaction of the variables X, Y and Z were used in a model, the variables and interactions XYZ, XY, XZ, YZ, X, Y, and Z would have to be employed so that the model makes substantive sense and the forecasts established by the model can be performed correctly. There are three interaction procedures in SEM: interaction between independent observable variables, interaction when the observed variables are part of the same factor and interaction between factors, with their corresponding observable variables (Jaccard, Turrisi & Wan, 1999; Kenny & Judd, 1984; Ping, 1995; Schumacker, 2002; Schumacker & Marcoulides, 1998; Schumacker & Lomax, 2004). Suprimit: Suprimit: ¶ Suprimit: rate Suprimit: ¶ Longitudinal data analysis with SEM 16 Let us suppose that in Figure 5 the interaction of V1,t and V2,t exerts an influence on the development of the variables V1,t and V2,t. Thus, if a new variable is generated, it should be the interaction of V1,t and V2,t for each time: V1*2,t = V1,t*V2,t, (1) In this way we would have a new variable for each measurement time and, bearing in mind that it has an influence on V1,t and V2,t, we would propose the model in Figure 18. Insert Figure 18 about here Note that in Figure 18: (a) the interaction between variables V1*2,1, V1*2,2 and V1*2,3 have been included in the model; (b) the covariances between all the independent variables of the model have also been included; (c) the variable V1,2 is a function of V1,1, of V2,1 and of the interaction between both of them (V1*2,1). The same thing happens with the variables V1,3 [V1,3 = f(V1,2, V2,2, V1*2,2)] and V1,4 [V1,4 = f(V1,3, V2,3, V1*2,3)], and the variable V2,t depends on the interaction of V1*2,t-1, that is to say: V2,2 = f(V1,1, V2,1, V1*2,1), V2,3 = f(V1,2, V2,2, V1*2,2)] and V2,4 = f(V1,3, V2,3, V1*2,3). It can also be supposed that the interaction between variables is time dependent and, in that case, we would need to remove Cov(V1*2,1, V1*2,2, V1*2,3) and add both V1*2,2= f(V1*2,1) and V1*2,3= f(V1*2,2). Should there be an interaction between variables within the same factor, so that they affect another factor, we would have to create a new factor resulting from the interaction between the two of them. Thus, if in Figure 12 it is assumed that the interaction of the variables V1,t and V2,t within each factor Ft affects the factor Ft+1, we would have to obtain the product of the variables from equation (6), including the product within a new factor, as shown in Figure 19. Insert Figure 19 about here We can hypothesise that F5, F6 and F7 are time dependent, and therefore we would need to remove Cov(F5, F6, F7) and add F6 = f(F5) and F7 = f(F6). It is easy to apply the interaction of variables between factors and the references cited above can be consulted for further information on this matter. In models of interaction of variables it is also possible to use the same procedure as that proposed in relation to the errors in measuring the variables V1,t and V2,t in the previous sections. Suprimit: in Suprimit: Suprimit: , Suprimit: ¶ Longitudinal data analysis with SEM 17 Models with non-linear variables Strictly speaking, these ought to be called models with transformation to non-linear variables; thus, it becomes possible to propose a model in which a variable (Yt) is a function of the value of the same variable at a previous time (Yt-1) and of the square of the same variable at the previous time (Y2t-1). In other words Yt=f(Yt-1,Y2t-1), which indicates that the relation between Yt and Yt-1 is quadratic. This non-linear relation can be of any other type (and not only quadratic): cubic, quadratic, logarithmic, square rooted, and so forth, or other more complex, exponential-type functions: logistic, Gomperz, and so forth. Even interaction models are a special case of non-linear transformation (Etezadi-Amoli & McDonald, 1983; Jaccard, Turrisi & Wan, 1999; Jöreskog & Yang, 1996; Kenny & Judd, 1984; McArdle & Nesselroade, 2003; Schumacker, 2002; Schumacker & Marcoulides, 1998; Seber & Wild, 1989). If, in the model in Figure 1, it were assumed that Yt=f(Yt-1,Y2t-1), then the graphic representation would be that shown in Figure 20. Insert Figure 20 about here In Figure 20: (a) the squares of the respective independent variables have been included in the model; (b) the covariances between all the independent variables in the model are also included, Cov(V1,1, V21,1, V21,2, V21,3) meaning all the pairwise covariances of the independent variables, that is to say, Cov(V1,1, V21,1, V21,2, V21,3): Cov(V1,1,V21,1), Cov(V1,1,V21,2), …, Cov(V21,2,V21,3); and (c) note that the variable V1,2 is a function of V1,1, and of the square of the same variable [V1,2=f(V1,1,V21,1)]. The same thing happens with the variable V1,3, which is a function of V1,2 and of the square of this same variable (V21,2) (that is: V1,3=f(V1,2,V21,2)), and lastly: V1,4=f(V1,3,V21,3). If the quadratic effects were not significant, it would be necessary to withdraw effects b4, b5 and b6, together with their corresponding independent variables (V21,1, V21,2 and V21,3). By so doing, the model would become the one shown in Figure 1 again. Instead of the quadratic model between observable variables suggested here, any other non-linear transformation of the variables could also be hypothesised. The non-linear model of observed variables can be generalised to latent variables and, thus, a quadratic model of the effect of one factor (F2t) on another factor (Ft+1) could also be proposed and, hence, Ft+1=f(Ft, F2t). The model in Figure 9 would then become the one in Figure 21. Suprimit: : Suprimit: ¶ Longitudinal data analysis with SEM 18 Insert Figure 21 about here Each quadratic factor must contain its respective squared variables and the corresponding interaction of the variables. It is strongly advisable to include orthogonal coefficients in the product and the power of the variables in order to avoid colinearity (Little, Bovaird & Widaman, 2006); this is also valid for interaction between variables, as we have explained in the previous section. The covariances among the pairwise independent variables, Cov(F1, F5, F6, F7) = Cov(F1, F5), Cov(F1, F6), ..., Cov(F6, F7), must be added to Figure 21. Supposing the quadratic factors were not significant, then the corresponding effects b12, b13 and b14 would not be significant either. If the quadratic effect b13 were significant (that is to say, from F6 to F3), but one of the effects of F6 on its indicator variables were not significant, then all the effects and their corresponding indicator variables would have to be left (due to the principle of hierarchy, for the appropriate forecast of F3). We can also assume that F5, F6 and F7 are time dependent and, therefore, we would have to remove Cov(F5, F6, F7) and add F6 = f(F5) and F7 = f(F6). When the time intervals between successive measurements are different and it is also assumed that the passage of time exerts a constant effect on the different measures, the following strategy can be used to include non-linear effects among the variables: (i) each measuring unit is identified by a constant (e.g. a); (ii) each autoregressive coefficient is made equal to the constant a raised to the power of the time interval; additionally, (iii) the value of a must be less than unity, so that the more time has passed, the lower the effect of the passage of time will be. Thus, let us suppose in Figure 21 that one year passes between the first and the second measurement, a year and a half between the second and the third, and 10 months between the third and the fourth measurement. In the constraints paragraph of the syntax of an SEM statistical program it could be indicated (by reducing the time unit to months) that b1= a12, b2= a18, b3= a10, and a<1. The criteria of the variances and covariances of the errors that were proposed earlier can be applied. Both the procedures of interaction between variables and those involving their nonlinear transformation enable the researcher to optimise the information that is available. This is because, although generally speaking data collection tends to be the most Suprimit: In factor F5 (factor of F1 squared), the parameters have to be set in accordance with the following criterion: b15=(b4)2; b16=b4*b5; b17=(b5)2; b18=(b6)2; b19=b6*b7; b20=(b7)2; b21=(b8)2; b22=b8*b9; b23=(b9)2; that is to say, the variable V1,1 squared (V21,1) that receives the effect of F5 must have the square of the coefficient of F1 on V1,1, the same being true for the factors F6 and F7 with respect to their variables. Suprimit: El Suprimit: The system for estimating parameters for models that include interactions of variables or non-linear effects can be implemented in two phasessistema de estimación de parámetros para modelos que incluyan interacción de variables o efectos no-lineales puede ser implementado en dos fases (Ping, 1995), and Suprimit: even possible Suprimit: e incluso pueden incluirse coeficientes ortogonales en el producto de parámetros para evitar la colinealidad Suprimit: over Suprimit: para Suprimit: incluir efectos nolineales entre las Suprimit: ¶ Longitudinal data analysis with SEM 19 demanding part, with the data already available to the researcher it is possible to test more hypotheses concerning the same variables with their corresponding transformation. Models with constant SEM initially relate covariances and variances between the variables. The model differentiates (or centres) the observable and latent variables with respect to the mean, with the resulting loss of information about the constants. Nevertheless, it is sometimes very important to obtain the value of the constants of the model. In longitudinal designs, this is the case when comparing the results in terms of an intervention in different groups or to determine the change of level in a variable. As an example, let us suppose that in the case of Figure 1 we were interested in verifying the constant at each measurement time; this would therefore give us the model in Figure 22. Insert Figure 22 about here In the representation of the constants we followed the system put forward by McArdle and Epstein (1987), in which the constants are distinguished by an arrow with a triangle placed at the origin with a number ‘1’ inside it. It must be borne in mind that the variable V1,1 now admits forecast error (E1,1). The substantive interpretation of the other elements and symbols in Figure 22 is the same as that for Figure 1 (except in the constants that have been added). The model in Figure 22 can easily be extended by including the vector of constants in equation (1): y = αy + Βy + Γx + ε, (6) where additionally: x = αx + Λxξ + δ. Interpretation of the symbols is the same as that explained in Equation 1. In our case, αy’ = [k1 k2 k3 k4], αx, Λx, ξ and δ are null matrices, and Ψ’ = [(Cov(E1,1, E1,1), (Cov(E1,2, E1,2), Cov(E1,3, E1,3), Cov(E1,4, E1,4)]. Similarly, the substantive interpretation of the other elements and symbols in Figure 22 is the same as that for Figure 1 (except in the constants that have been added). One important aspect to be taken into account in Figure 22 is that, as in the other regression models, the values of the constants do not necessarily have to be those of the means (Rosel, Arnau & Jara, 1998; Rosel, Jara & Arnau, 2002). Instead, they are Suprimit: )]; Suprimit: similarly Suprimit: ¶ Longitudinal data analysis with SEM 20 conditioned to the value of the mean of the variable that they have as a dependent variable, and also that of the effects they receive through the independent variables. For example, in Figure 1, let us suppose that the means of each variable were 1,1 V= 50, 1,2 V = 60, 1,3 V= 70 and 1,4 V= 80 and that the results of the coefficients were b1 = 0.60, b2 = 0.70 and b3 = 0.85; the equation of the variable V1,1 would be V1,1 = k1 + E1,1, and its expected values E(V1,1) = E(k1 + E1,1). As a result, 1,1 V= k1, and the value of k1 would therefore be the mean of V1,1, that is, k1 = 50; for V1,2 it would be V1,2 = k2 + b1·V1,1 + E1,2, and its expected values would be E(V1,2) = E(k2 + b1·V1,1 + E1,2), which results in 1,2 V = k2 + b1·1,1 V, k2 = 1,2 Vb1·1,1 V= 60 - 0.60·50 =30. Hence, the value of k2 no longer depends only on the mean of V1,2, but also on the coefficient b1 and on the mean of V1,1. Likewise, we would have k3 = 28, and k4 = 20.50, which proves that the value of the constants diminishes over time, despite the fact that the means of the variables increase. As can be seen with this simple example, there does not necessarily have to be a relation between the means and the constants, and therefore the magnitudes of the constants cannot be taken directly to compare the means. In conclusion, the values of what most authors call ‘structural means’ are not these means at all but instead constants from each respective forecast equation. We recommend comparing the means of the observable variables taking the covariances among variables (with no effects among them) as the reference model, which thus allows the respective means to be compared directly. On the other hand, for the latent variables it is advisable to use the procedure proposed by Little, Slegers and Card (2006), which constrains the mean of the effects of the latent variable on its corresponding observable variables to a value of one. With single level versus multilevel measurement Up to this point the models have been developed under the assumption that the parameters are fixed. In other words, we have not considered the possibility that these parameters (for example: b1, in Figure 1) might vary across units, perhaps from one higher level unit such as a geographical area to another. Suppose we have a response or dependent variable, V, measured on four possibly unequally spaced occasions with any pattern of missing data after the first occasion. Let us also suppose that our sample is structured in such a way that each case is located in one and only one cluster (pupils (i) within schools (j), for example). Our observations are Vtij, t (measurement occasion) = Suprimit: : Suprimit: : Suprimit: : Suprimit: : Suprimit: Como Suprimit: conclusión Suprimit: los valores de lo que la mayoría de autores denomina ‘medias estructurales’, no son tales medias, sino que son constantes de cada respectiva ecuación de pronóstico Suprimit: Nuestra recomendación es comparar las medias de las variables observables estableciendo como modelo de referencia las covarianzas entre las variables (sin ningún efecto entre las mismas), Suprimit: así se comparan directamente las respectivas medias; Suprimit: mientras para las medias de las variables latentes es conveniente usar el procedimiento de Suprimit: que constriñe igual a uno la media de los efectos de la variable latente sobre sus correspondientes variables observables Suprimit: , Suprimit: ¶ Longitudinal data analysis with SEM 21 1, …, Tij (max. Tij =4); i (case) = 1, …, nj; j (cluster) = 1, …, J and we will assume that V ~ MVN (µ, Ω). We can approach the analysis of these data in a number of ways, depending on our underlying research question. The first method is to model Vt as a function of one or more earlier or lagged measures of V, that is Vt-k, k > 0, along with other explanatory variables. This approach is particularly useful if we seek causal conclusions about, for example, the effect of an intervention in the absence of randomisation (see, for example, Plewis, 1985). We can regard each of the three responses Vt, t > 1 as a multivariate set (V1 is assumed to be exogenous here) and so our model is: ijijjij eVbbV 2121202 + + = (2.a) ijijijjij eVbVbbV 3232131303 + + + = (2.b) ijijijijjij eVbVbVbbV 4343242141404 + + + + = (2.c) 3,2,1, 000 = + =kubb kjkjk (2.d) The intercept terms (bk0j , k = 2, 3 ,4) vary randomly from cluster to cluster around an overall mean bk00 as shown in equation (2.d). The regression coefficients bk1 represent the relations between each response and the exogenous V1; bk2, k = 3, 4 link the last two responses to their value on the second occasion and b43 links the final response to its previous or lagged value. Equations (2.a) to (2.c) define a recursive model for the cases, equation (2.d) defines the model at the cluster level and we assume that the case residuals are uncorrelated with the cluster residuals. Maximum likelihood estimation is relatively straightforward, albeit iterative, and can be carried out in most statistical packages as well as with the more specialised multilevel packages such as MLwiN and HLM. Providing any missing data are ‘missing at random’, in other words, any missing Vt (t > 1) depend only on Vt-k, k > 0, then the algorithms are fully efficient. The model generates two covariance matrices, one for the cases and the other for the clusters. The multivariate model allows all the residuals at the case level to be correlated as might happen if there is an omitted explanatory variable that is related to Vt (t > 1) after controlling for Vt-k, k > 0, in other words, to the change in V from occasion to occasion. This part of the model is essentially the same as the model for ‘seemingly unrelated regressions’ as introduced into the econometrics literature by Zellner (1962). The model can be simplified. For example, we might constrain the parameters b31, b41 and b42 to be zero so that any response Vt depends only on its immediately preceding Suprimit: ..., Suprimit: ..., Suprimit: ..., Suprimit: i.e. Suprimit: , Suprimit: exogeneous Suprimit: exogeneous Suprimit: at Suprimit: specialist Suprimit: ; Suprimit: for Suprimit: ¶ Longitudinal data analysis with SEM 22 measure as in Figure 1 for a single level model. In the same vein, we might also constrain some of the elements of the covariance matrix at both levels to be zero so that only the first-order autoregressive elements are non-zero. On the other hand, it will sometimes be appropriate to make the model more complicated. In particular, we might want to allow some or all of the regression coefficients b to vary across clusters so that we replace bpq by bpqj and the covariance matrix at the cluster level is extended to include both the extra variances but also the covariances between the intercepts (bk0j) and the slopes (bpqj). We might also want to introduce explanatory variables measured at the cluster level and these might include cluster variables created from the explanatory variables at the case level, such as the cluster mean. Rather than relating the response at one occasion to lagged measures of the response, a second approach to analysing repeated measures data comes from modelling the evolution of the response over time (or age). In other words, we model Vt as a function of time within a growth curve or unconditional framework. This can be particularly useful in a developmental or ageing context. Our model is now: ∑ = += Q q tij q tijqijtij eaV 0 β (3.a) qjjqqij u+ = 0 β β (3.b) qjqjq v+ = 000 β β (3.c) Here Q is the order of the polynomial at age (or time) a, determined by the data but with the restriction that Q << max Tj, and βq are the model coefficients that show how the response varies with age. These (random) coefficients can vary from case to case and from cluster to cluster. The equations (3.a) to (3.c) are a three-level model with variation between occasions within cases (level one), represented by 2 e σ , variation between cases within clusters (level two) represented by the random effects u for the intercept, slope etc., and variation between clusters (level three) represented by a second set of random effects v. It can be estimated with the same algorithm used for model 3, with equations (3.a) to (3.c). In the structural equations literature, the random effects u and v are sometimes treated as latent variables that are determined by the observed responses over time (Meredith & Tisak, 1984, 1990). The multilevel growth curve model is a flexible tool that can be adapted and extended. Thus, it can accommodate the fact that there is often variation in the age or time of Suprimit: in Suprimit: a Suprimit: three Suprimit: and Suprimit: ¶ Longitudinal data analysis with SEM 23 measurement within any one measurement occasion. It is possible to allow the level-one residuals to be autocorrelated (Goldstein et al., 1994) and to allow the variance at level one to change with age or time. Explanatory variables can be introduced at each level (time-varying variables at level one, case-level variables at level two and cluster variables at level three) and these variables can be used to explain variation in any of the growth parameters βq. The conditional (or regression) and the unconditional (or growth curve) approaches can, with sufficient occasions of measurement, be combined, as shown by Plewis (1996). Then it is possible to condition on the measure at the first occasion (the baseline measure), thereby making it possible to compare growth parameters across groups for cases starting off at the same point. Other extensions are also possible – for example, to binary, unordered and ordered categorical responses (Plewis et al., 2006), to multivariate responses (Plewis, 2005), and to combinations of these. Other advances in SEM of longitudinal data Although we do not intend to go into great depth on the matter, other advances in SEM of longitudinal data have been made in recent years, some of the most important being the following: (a) Latent growth curve models (Bollen & Curran, 2006; Duncan, Duncan & Strycker, 2006; Meredith & Tisak, 1984, 1990), which use latent variables and constants. One of the latent variables would thus represent the starting level and the other latent variable would represent the rate of growth of the group, but now adding indicator constants to each latent variable. New latent variables that represent the quadratic, cubic or some other tendency of the development curve can also be added. In this model the respective variances of the latent variables indicate the random variability of the individuals that make up the sample, and therefore each latent variable becomes a variable with a random (multilevel) coefficient. This latent curve model is thus directly related to the multilevel model. Figure 23 shows an example of a linear latent growth curve model with a random constant and a linear effect. Note that all the coefficients from the latent variables towards the observable variables are fixed effects and that F1 represents the level of the constant (fixing the values: b1 = b2 = b3 = b4 = 1), while F2 is the linear slope of the model (fixing the values: b5 =1, b6 = 2, b7 = 3). Insert Figure 23 about here Suprimit: – Suprimit: – Suprimit: En Suprimit: la Figura Suprimit: se muestra un ejemplo de modelo de Suprimit: con una constante aleatoria y un efecto lineal Suprimit: Obsérvese Suprimit: que todos los coeficientes desde las variables latentes hacia las variables observables son efectos fijos (fixed effects) y que Suprimit: representa Suprimit: el nivel de la constante Suprimit: mientras Suprimit: es Suprimit: la pendiente lineal del modelo Suprimit: ¶ Longitudinal data analysis with SEM 30 Steyer, R. (2005). Analyzing individual and average causal effects via structural equation models. Methodology, 1, 39-54. Steyer, R., Eid, M. & Schwenkmezger, P. (1997). Modeling true intraindividual change: True change as a latent variable. Methods of Psychological Research-Online, 2, 21-33. Steyer, R., Krambeer, S. & Hannöver, W. (2004). Modeling Latent Trait-Change. In: K. Van Montfort, H. Oud & A. Satorra (Eds.), Recent developments on structural equation modeling: theory and applications. Amsterdam: Kluwer. Tukey, J.W. (1977). Exploratory data analysis. Reading, MA: Addison-Wesley. von Eye, A. & Clogg, C.C. (Eds.) (1994). Latent variables analysis: Applications for developmental research. Thousand Oaks, CA: Sage. Wiley, D.E. (1973). The identification problem for structural equation models with unmeasured variables. In: A.S. Goldberger & O.D. Duncan (Eds.), Structural equation models in the social sciences. New York: Seminar Wright, S. (1918). On the nature of size factors. Genetics, 3, 367-374. Wright, S. (1921). Correlation and causation. Journal of Agricultural Research, 10, 557-585. Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests of aggregation bias. Journal of the American Statistical Association, 57, 348-368. Suprimit: (www.mpronline.de). Suprimit: In Suprimit: Suprimit: ¶ Longitudinal data analysis with SEM 1 Figure 1. First-order autoregressive univariate model (AR(1)), 4W1V. V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 Suprimit: 1 Longitudinal data analysis with SEM 2 Figure 2. Second-order autoregressive univariate model (AR(2)), 4W1V. V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 b4 b 5 Suprimit: 1 Longitudinal data analysis with SEM 3 Figure 3. First-order autoregressive model with covariance between the measurement errors (4W1V). V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 Cov(E1,2, E1,3) Cov(E1,3, E1,4) Suprimit: 1 Longitudinal data analysis with SEM 4 Figure 4. First-order moving average and autoregressive model, ARMA(1,1), design 4W1V. V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b3 b4 b 5 Suprimit: 1 Longitudinal data analysis with SEM 5 V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 V2,1 V2,2 V2,3 V2,4 E2,2 E2,3 E2,4 b4 b 5 b 6 Cov(V1,1,V2,1) b8 b9 b10 b11 b12 b13 Figure 5. AR(1) bivariate model with crossed effects between the two variables, corresponding to the design 4W2V. Suprimit: 1 Longitudinal data analysis with SEM 6 V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 V2,1 V2,2 V2,3 V2,4 E2,2 E2,3 E2,4 b4 b 5 b 6 b8 b 9 b 10 b 11 b 12 b 13 Cov(V1,1, V2,1) Figure 6. Recursive model with reciprocal influences between variables at the same moment of measurement. Suprimit: medition Suprimit: 1 Longitudinal data analysis with SEM 7 Figure 7. Simple factor model in repeated measures. V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 b3 E1,1 F1 b4 b1 b2 E1,4 Suprimit: 1 Longitudinal data analysis with SEM 8 Figure 8. Simple factor model in repeated measures, with AR(1) effects. V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 b3 E1,1 F1 b4 b1 b2 E1,4 b5 b6 b7 Suprimit: 1 Longitudinal data analysis with SEM 9 Figure 9. AR(1) univariate model in the factor. V1,1 V1,2 V1,4 E1,2 E1,3 E1,4 E1,1 F1 F2 F3 F4 D2 D3 D4 b1 b 2 b 3 b4 b 5 b 6 b 7 V1,3 Suprimit: 1 Longitudinal data analysis with SEM 16 Figure 16. Time-varying model conditioned to the variable X1,t. Cov(X1,1, X1,2, X1,3, X1,4) indicates all the possible covariances between the variables X1,1, X1,2, X1,3 and X1,4: Cov(X1,1, X1,2), Cov(X1,1, X1,3), … , Cov(X1,3, X1,4). V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4 E2,2 E2,3 E2,4 b4 b5 b6 F1 F2 F3 F4 D2 D3 D4 b1 b 2 b 3 b7 b 8 b 9 b 10 b 11 E1,1 E2,1 E1,2 E1,3 E1,4 D1 X1,1 X1,2 X1,3 X1,4 Cov(X1,1, X1,2, X1,3, X1,4) b12 b 13 b 14 b 15 Suprimit: 1 Longitudinal data analysis with SEM 17 Figure 17. Time-varying model conditioned to the intellectual capacity time factor, indicated from F5 to F8. V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4 E2,2 E2,3 E2,4 b4 b5 b6 F1 F2 F3 F4 D2 D 3 D 4 b1 b 2 b 3 b7 b 8 b 9 b 10 b 11 E1,1 E2,1 E1,2 E1,3 E1,4 D1 b12 b 13 b 14 b 15 F5 F6 F7 F8 b16 b 17 b 18 X1,1 X1,2 X1,3 X1,4 X2,1 X2,2 X2,3 X2,4 b19 b20 b21 b 22 b 23 b 24 b 25 b 26 D6 D 7 D 8 Ex2,2 EX2,3 EX2,4 EX1,1 EX2,1 EX1,2 EX1,3 EX1,4 Suprimit: 1 Longitudinal data analysis with SEM 18 Figure 18. Variable interaction model, with effects on the next time of measurement. Cov(V1,1, V2,1, V1*2,1, V1*2,2, V1*2,3) indicates all the possible (pairwise) covariances between the variables. V1*2,2 V1*2,3 b9 b11 b13 b14 b15 b16 b17 b18 b19 Cov(V 1,1 , V 2,1 , V 1*2,1 , V 1*2,2 , V 1*2,3 ) b10 b 12 V1,1 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 V2,1 V2,2 V2,3 V2,4 E2,2 E2,3 E2,4 b4 b 5 b 6 b8 V1,2 V1*2,1 Suprimit: 1 Longitudinal data analysis with SEM 19 V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4 E2,2 E2,3 E2,4 b4 b5 b6 F1 F2 F3 F4 D2 D3 D4 b1 b 2 b 3 b7 b 8 b 9 b 10 b 11 E1,1 E2,1 E1,2 E1,3 E1,4 Figure 19. Observed variable interaction model within the measurement time factors. Double-headed arrows must be added to the model to represent the covariances between the independent variables: Cov(F1, F5, F6, F7)= Cov(F1, F5), Cov(F1, F6), …, Cov(F6, F7) F5 F6 F7 V1*2,1 V1*2,2 V1*2,3 b12 b 13 b 14 Cov(F1, F5, F6, F7) E 1 * 2,1 E 1*2,2 E1*2,3 b15 b 16 b 17 Suprimit: . Suprimit: 1 Longitudinal data analysis with SEM 20 Figure 20. Lag 1 quadratic effects model. Cov(V1,1, V21,1, V21,2, V21,3) indicates all the (pairwise) covariances of the independent variables: Cov(V1,1,V21,1), Cov(V1,1,V21,2), …, Cov(V21,2,V21,3). Cov(V1,1, V21,1, V21,2, V21,3) b4 b 5 b 6 V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 V21,1 V21,2 V21,3 Suprimit: 1 Longitudinal data analysis with SEM 21 Figure 21. Model of quadratic effects of one factor (F2t) on another factor (Ft+1). Double-headed arrows must be added to the model to represent the covariances between the independent variables: Cov(F1, F5, F6, F7)= Cov(F1, F5), Cov(F1, F6), …, Cov(F6, F7). V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4 E2,2 E2,3 E2,4 b4 b5 b6 F1 F2 F3 F4 D2 D3 D4 b1 b 2 b 3 b7 b 8 b 9 b 10 b 11 E1,1 E2,1 E1,2 E1,3 E1,4 V2 1,1 V2 1,2 V2 1,3 V2 2,1 V2 2,2 V2 2,3 F5 F6 F7 b15 b17 b18 b 20 b21 b23 V1*2,1 V1*2,2 V1*2,3 E4E6 E1 E2 E3 E5 b12 b 13 b 14 b16 b19 b 22 Cov(F1, F5, F6, F7) E1*2,1 E 1*2,2 E 1*2,3 Suprimit: 1 Longitudinal data analysis with SEM 22 Figure 22. Univariate model AR(1) with constants in the variables. V1,1 V1,2 V1,3 V1,4 E1,2 E1,3 E1,4 b1 b 2 b 3 E1,1 1 1 11 k1 k2 k3 k4 Suprimit: 1 Longitudinal data analysis with SEM 23 Figure 23. Linear latent growth curve model, with a random constant and a random slope F1 F2 b1 b2 b3 b4 b5 b6 b7 D1 D2 k1 k2 V1,1 E1,1 V1,2 E1,2 V1,3 E1,3 V1,4 E1,4 1 1 Cov(D1,D2) Suprimit: . Suprimit: 1 Longitudinal data analysis with SEM 24 Figure 24. Latent growth difference score model, with a constant initial level and an additive coefficient. k1 b11 b12 b13 b14 b2 V1 , 1 V1 , 2 V1 , 4 E1,2 E1,3 E1,4 E1,1 F 1 F2 F3 F4 b1 b 3 b4 b 5 b 6 b 7 V1 , 3 Δ F 2 ΔF3 ΔF4 b8 b 9 b 10 F5 F6 1 1 k2 Cov(D5,D6) D5 D6 b15 b 16 b 17 Suprimit: random Suprimit: random Suprimit: 1