 Prvim prezentacijami, da lahko je slavljana izgleda z modačenje, in način tudi, način dražičnih, izgledaj, da so vse modače, že se pričo se izgleda tudi, in nekaj nekaj nagobnih, že se izgleda vzvečenje, in v družje, da se izgleda. Na b sincere veliko je, da prikazujemo tudi svoje kulte, za všechno zpronedi zvarovani, in ste imemo všechno vznažitost, ki bomo tako površati, kar je namočen, denetvene, in tudi, nekaj všechno, nekaj imaju dovolj nekaj, ki za pravidovani idej jaz izgleda, ki si se ohoodenja, nas tu da uvedila, ker je i sebem dobar silhouette, ... in kaj jeimatega je tudi kategorična ... pa se prejze. ... in vse častrof vse vse dobar verjali. V svej prezentacij, ... z tudi, ... ... z bom izgledal, Svisim, da vedimo, kar možemo sajati z njega, katero je značite vspe in izgleda, kot pa je to, da da se možemo, in da je to, ko se je začeli, in da se je začeli, da se je začeli, da se je začeli, da se je začeli, da se je začeli, da se je začeli. I zelo, da se dobro, da se je začeli, da se je začeli, kako drugi pražitev, zelo na odličenje in zelo na dobroj krišenje. Ca danes počnijo, nekaj me vse imaš nekaj vsej modez, vse vsej modez do štešnjih trajektori in nječ nekaj vsej modez. Prepo zopršenjem, nekaj nekaj nekaj nekaj vsej modez nekaj došli očinati trajektori. In nekaj došli očinje na nekaj došli očinji, kaj ga se izgleda, da je tudi čanjev v zelo. Vzelo izgledam je predvaj na ovoj propoziv izgleda na ncrem, tako če bo se vzela, da je našlično. Zelo je vzelo od te, da je propočilo, že bilo vzelo v tega vsega tudi vsega tudi vsega, da je zelo, da je vsega kvalitativ, zelo, v tom sega. Vdovo, zvokovice hraloga v tradijtli v prvnih pogledu, je nebezresivna zapartina, kdaj se tukaj počas in se vratem, na svega tukaj. Na razmih na počasernost in izdožosta, tudi ne pa publika vflir in je tukaj počas in se vrati, na počasernost 5-i malosti. Zvokovice in tradijtli pravse tukaj ne težko težko zevokovico, that happen at each specific measurement occasion. So, trajectory-based group models are more ideal for, how adequate for investigating individual differences in developmental patterns that unfold over a significant period of time or over a significant number of measurement occasions, whereas latent transition analysis is more adequate to investigate stadial developmental models. Models, where individuals are supposed to move across different stages of development. So, in the second presentation, I had provided a more formal description of growth-mixture models and latent class growth analysis, focusing on cases where we deal with ordered categorical variables. However, the models I am presenting can be run with any type of variables from continuous to nominal. And one of the main purposes of these models is to investigate predictors and covariates that may influence the probability of being one trajectory group or another. For example, does family separation affect the probability of adolescence being in a chronic drug-use trajectory group. When we introduce these covariates, because latent classes are nominal variables and, in other words, they are not ordered, we run multinomial logistic regression of the latent classes on the covariates. In the exercises, there are some examples of how to introduce covariates and interpret the results. So, and you are welcome to look at them. However, since growth-mixture models also allows individual variability around the growth parameters within classes, the covariates that we introduce in the models also influence intra-class individuals' variation around the intercepts and the slopes. For example, within adolescence they show a chronic trajectory of drug-use. The specific trajectory of an adolescence within this group may be affected negatively by parental separation, and the increase in drug-use may be more dramatic for those adolescents that within the chronic-use group were exposed to family separation. So, in this case, models we covariates can be more complex and explore more nuances in mechanisms of influence. So, when we are using growth-mixture models, there are also some more nuances that we can model as approach. In the case of latent class growth analysis, the intra-class intercepts and slopes are constant across individuals. So, there is no individual variation around the class-specific growth parameters. And when we introduce covariates, therefore we estimate how covariates influence latent class affiliation, that is the probability of being in one trajectory group or another. Once again, these associations are modeled by multinomial logistic regressions of latent classes on covariates. So, we are often interested in introducing different covariates that can act as predictors of trajectory group affiliation. And we can estimate the influence of these covariates using multinomial logistic regression of the latent classes on the covariates, and one appealing feature of this method is that we can also calculate probabilities for different configurations of covariates. And there are some exercises on this in the material I have prepared. And this really allows to test hypothesis on the role that the community of exposure to some risk factors may have on the probability of being in a specific trajectory group. So, it's possible to look at the effects, the cumulative effect of risk factors on the probability of displaying some pattern of developmental trajectories. In here, I display a fictional example where the vertical axis in the graph represents the probability of the lessens being a chronic antisocial behavior trajectory group. In this example, the probability of being chronically antisocial is relatively low when adolescents are exposed to only one of these risk factors, low social economic status, parental separation, insecure attachment. But if adolescents are exposed to all three factors over time, the risk of being in the chronically antisocial group is significantly higher. So, this is an example of how we could use these approaches to investigate cumulative risk. Any examples and applications, the covariates that are used to predict trajectory group, trajectory-based groups, are invariant. For example, they represent events that happened before the study started. It is, however, possible to include time-vary covariates. And here, if they are to a study by machine aerocodigs and the QR code links to this publication if you are interested. The researchers investigated trajectories of violent behavior in adolescence and tested if school failure was associated with changes in the expected trajectory. In Quebec, where the studies was conducted, school failure led to great retention. Here also report the equation that the authors devised to test these hypothesis where you can see that the latent response white star displays a quadratic trajectory across age. And there were additional parameters for the effect of failing schools for the first time between age 6 and 10, failing schools for the first time between age 11 and 12 or failing schools for the first time between age 13 and 15. And note that parameters for school failure have indexes for different latent classes, k being estimated. But the effects of these covariates varied across classes, or in other words, individual across classes had different trajectories, as well as displaying differences in the effects of school failure. The graph I produced here represents the potential effect of school failure at an early age, say age 9, on that trajectory. So in this example, the effect of school failure is basically to elevate the expected trajectory to a higher level. And this is an example of how time-vary covariates can be included in the model. So I report the QR code so that you can look at this paper if you want to know more. In other key interests of researchers that use growth mixture models of or latent class growth analysis lies in the possibility of testing if trajectory groups differ in the distal outcome or achievements later on. It is easy to include distal outcomes in the models and the exercises I devised for with these presentations also provide some examples. However, when we introduce distal outcomes, we introduce further observed variables that vary together with the other variables that are in the model. This means that the latent classes are not just model in the heterogeneity we observed in the observed variables and their trajectories, but also the heterogeneity of these and the distal outcomes. So consequently, this means that often the latent class models we had estimated without including distal outcomes may change and we will have to recheck the latent class models to see how much it has changed and how. And this is sometimes impractical but also requires some caveats in interpreting the results. And a solution to this problem had been to consider individual latent class affiliation in trajectory groups as if there were variables that could be used as predictors in other analysis for example in ANOVA. But the problem with this approach where the latent class affiliation is used as if it were similar to an observed variable is that latent class affiliations are uncertain and in the first presentation I gave an example of how one individual could have say 50% probability of being in that latent class. So if we fail to take into account this uncertainty the analysis we run are going to be biased. An elegant solution to this problem has been provided by the freestyle approach. The advantage of this approach lies in separating the step where we estimate the latent class affiliations from the step where we impose associations between latent classes and caveats of distal outcomes of both. So the first step in this approach is to estimate the model and decide the number of classes as well as estimate the most likely class of each participant based on the posterior probability. The second step involves calculating parameters that represent the uncertainty in this latent class allocation and these parameters are the goals of being the assigned latent class rather than another one. Once we have estimated the most likely class and the parameter that summarizes uncertainty around this classification we can include the latent class classification as a variable in the analysis making sure that we feed the calculated uncertainty parameter into the model in this way controlling for this uncertainty. In the online material provide more details and examples and also I talked more details about this in my presentations on latent class analysis and latent transition analysis. So this approach can be very helpful when you want to separate the step of developing the model, the latent class model from looking at the associations between the latent classes and these outcomes or covariates of both. And this approach however is not adequate if covariates have react effects on the measurement parameters. Finally I'm going to talk about some more advanced models and for example we are often interested in investigating the associations between trajectories in different groups in mental health and drug use. Using trajectory based models we can model individual differences in the trajectories across the two processes and we can test then the associations between trajectory groups in one for example mental health and trajectory groups in the other drug use. In this case one process for example mental health in the graph you see here happens slightly before drug use and this type of model would provide some cross tabs where we can test if those that show for example chronic mental health problems are significantly more likely to fall in a chronic drug use trajectory group and we can also test hypothesis about these associations between different trajectory groups by restricting some of the probabilities of associations between the latent classes. So it's possible to create different models where we can investigate the associations between individual differences in trajectories across the two processes and test specific hypothesis about those associations. And finally it's possible to extend the models to represent trajectories in different behaviors and here provide an example of a study that I have provided a QR code if you are interested and the researchers identified groups of Chinese elderly people that were taken in a long part in a longitudinal study who differed in their trajectories of physical activity, smoking and social participation. You can see in the pictures representation of these different trajectory groups we identified and for example large group labored isolated active non-smokers because they were consistently physically active across time but showed consistent low levels of smoking but also low levels of social participation. The results indicated that trajectory groups characterised by low levels of social participation displayed significantly higher the patient's scores. So to finalize I have talked about how we can add different type of covariates to growth mix to models or latent class growth analysis models and I have talked about how we can include these outcomes and I mentioned some advanced models so thank you very much for your attention and if you want to know more you can look at the exercises and some of the references I've put together with these presentations. Thank you very much. Bye now.