6.1 Stage development of drug use: LTA
The idea behind latent transition analysis (LTA) was first applied to drug use by Kandel (1975), who by means of a Guttmann1 scale tested for the first time the gateway hy-pothesis of drug involvement. The results confirmed the existence of gateway drugs that facilitate the subsequent consumption of other, more dangerous substances. She found that tobacco and alcohol are gateway drugs to hard liquors, which subsequently increase the chance to use cannabis, which in turn act as a gateway for harder substances. She also pointed out that being in an earlier stage is necessary, but not sufficient, for advancing to the next, this process being not a simple causal relationship among drugs, but rather the simultaneous action of individual and social factors (Kandel, 1975; Kandel & Faust, 1975). The development of longitudinal techniques and the availability of panel data have recently allowed a more precise empirical test of the gateway hypothesis. Collins and col-leagues (see Graham et al., 1991; Collins & Wugalter, 1992), for instance, have been quite active in the field over the last fifteen years; they developed a particular technique called LTA, which is a special case of LCA2 applied to longitudinal data. This technique has been extensively used to test the existence of sequential stages in the development of drug consumption (see also Graham et al., 1991; Collins, Graham, Rousculp, & Hansen, 1997;
Collins & Flaherty, 2002).
In this chapter I will first introduce the concept of LTA, its methods and some special features that can be applied to drug use behaviors. Thereafter I will present the results of a LTA applied to five waves of the German study CriMoC (Boers et al., 2010).
6.2 LTA - General model
In the previous chapter growth mixture models (GMM) have been used to measure the quantitative development of the frequencies of drug use across time. Although a lot has been learned by means of GMM, these models do not provide much information about qualitative change between qualitatively different stages (which are represented by categorical latent variables). In fact, a growth model cannot be used for the analysis of categorical observed variables, which are widely used to define qualitative states.
1This method stems from the psychological research and was used with cross sectional data. LTA, developed later, is particularly suited for panel data and adds thus more reliability to the results.
2LCA - Latent Class Analysis (see McCutcheon, 1987).
Latent transition analysis (LTA) is a special case of latent class analysis (LCA) that enables the researcher to specify a number of discrete stages using categorical variables, and to calculate the transition probabilities between them. The main purpose of LTA is to statistically describe movement across discrete latent stages.
Mathematically speaking a LTA is based on the combination of two independent models:
a Markov chain and a LCA.
In the simplest case, where there is a single observed categorical variable measured four times, in a Markov chain (see Blalock, 1970) the probability P of being in a particular state over four time points can be expressed as follow3:
Pijkl= δ1iτj|i21τk|j32τl|k43 (6.1) where i, j, k, l are the categories of a single categorical variable measured at four time points (response category i = 1 . . . n at time 1, response category j = 1 . . . n at time 2, etc...), the τ ’s represent the time specific transition probabilities to move from one category to another across time (for four time points there are obviously three transition probabilities), and the δi1 parameter represents the observed proportion of individuals in each category of the observed outcome at time point 1.
However, a simple Markov chain gives us information about movements across discrete stages, but being an observed model does not allow for measurement error; this is, in turn, an important feature when modelling in social science. Furthermore, Markov models cannot be used when the variable of interest is not directly measured but it is captured by means of observed indicators. Latent class analysis can be used to overcome this shortcomings (see Lazarsfeld & Henry, 1968). LCA assumes that an individual belongs to one exclusive class, i.e. the classes are mutually exclusive. “Latent classes are, in essence, categorical factors arising from the pattern of response frequencies to categorical items”, where the response frequencies (i.e. the cross-table of the response patterns) play a similar role to that of the correlation matrix in factor analysis (D. Kaplan, 2008, p. 459).
LCA assumes that the association between the observed variables is totally explained by the latent variable. According to this definition, also known as local independence, the number of classes of the latent variable should be larger than 2, whereas a single latent class would mean total independence among the observed items (McCutcheon, 1987). An LCA for four items can be represented as follows:
Pijkl=
A
X
a=1
δaρi|aρj|aρk|aρl|a (6.2) The conditional probabilities ρ are analogue to factor loadings in factor analysis and estimate the probability of a particular response i = 1 . . . n on a manifest item given membership in the latent class a = 1 . . . A. The parameter δ represents the proportion of individuals in latent class a.
LCA can be employed to represent class membership at a specific point in time, but in its simple form cannot be used to test changes in latent class composition across time.
However, this limitation can be overcome when combining it with a Markov chain into a single model. In this way it is possible to calculate transitions across different latent classes estimated at different points in time; this model is known as latent transition analysis (LTA). In fact, LTA shares the properties of the transition probabilities of a Markov model applied to latent classes, allowing for measurement error in the estimation of the unobserved classes. Expanding the two examples presented above and joining them into a single model - in the case of two items with categories i and j measured at two time points - the probability of having a given combination of responses on the observed items will be:
3For the equations in this chapter the notation of Collins and Wugalter (1992) is used.
6.2. LTA - General model where P represents the expected probabilities of the multivariate contingency table, i.e. all the possible response patterns generated by the combination of the observed items. The parameters are the same as for the above mentioned two models. The most interesting parameters are obviously the transition probabilities τ (in this case only one because of only two time points), and the δ, which represents the estimated proportion of individuals in each category of latent class a at time point 1. For what concerns the second time point the values for δ are not directly estimated but can be derived from δ for time 1, and are presented in the results4. Finally, the ρ parameters represent the measurement part of the model which map the observed items onto the latent classes. In LTA different restrictions can be applied to these parameters according to the researcher’s interest. The most important restrictions are generally applied to the ρ parameters in order to assure the stability of the latent classes at each time point. In fact, although not always necessary, achieving the same number of latent classes with the same substantial meaning is a necessary condition for the interpretation of the transition probability and the results; this can be done using either an explorative or a confirmatory procedure.
In the first case, for instance, the analyses carried out by D. Kaplan (2008), Nylund et al. (2006) and Nylund (2007) involved the use of a LCA to map the indicators on the categories of the latent variables. That means, the ρ values are freely estimated by means of a LCA, and then used as starting values for the ρ’s in the LTA. On the other hand, in the LTA specified by Collins and Wugalter (1992) and Graham et al. (1991) the ρ’s were modelled by means of specific starting values, chosen to correctly map the observed indicator onto the latent statuses, and thus assure the stability of the latent model across the time points. In this confirmatory approach the developmental position of each latent status (in the stages sequence) is defined by means of the probability to endorse a specific value of the observed indicator; the ρ’s are not freely estimated, but rather are fixed by the researcher in order to match the wanted developmental process (see Table 6.3 in the result section). The decision of the number and type of constrained ρ’s is left to the researcher and can be used to decide the best fitting model. In fact, models with different restrictions on the ρ’s are nested, and nested models can be compared with each other to find the more parsimonious ones.
The results of a LTA are interpretated by means of the transition matrix, which contains all the transition probabilities between time point 1 (on the left side) and time point 2 (above, on the upper side of the matrix).
The τ parameters on the diagonal represent the probabilities of staying in a given latent status between time 1 and time 2 (i.e., those subjects that do not change their behavior over time). The τ ’s in the upper corner represent the probabilities of moving upwards across stages (i.e., the percentage of subjects that between time 1 and 2 move
4The software Mplus (Muth´en & Muth´en, 1998-2010) reports the values for δ at each time point in the output. These values can also be calculated by: δ1a∗ T , where T is the transition probabilities matrix and δa1is the value at time point 1 (see Reinecke, 1999).
to another, higher status). The τ ’s in the lower corner, on the other hand, represent the percentage of individuals who move back from one latent status to a previous one.
All these transition probabilities can be restricted to one, to zero, or to other values according to theoretically specified needs.5 For the interpretation of the matrix it should be remembered that P τb|a,g= 1: all the lines of the transition matrix sum up to one.
Similarly to the models presented in the previous section, a LTA can be graphically represented using the general notation of a structural equation model as specified by Muth´en and Muth´en (1998-2010):