thanks so much for your comment -- we're glad you enjoy our silliness -- Patrick

thanks for your kind words. You are correct: when you use multiple indicator latent factors in the full SEM, you are able to separate the true factor variance from the item-specific residual variance. As such, the latent factors are assumed to be "error free". I'm careful here in saying "assumed" because this only holds if you meet certain underlying assumptions of the model. But in many situations the latent factor is a marked improvement over simple scale scores. Thanks again for watching -- patrick

the reason is that the level-1 and level-2 distinctions are for heuristic value only -- the actual model is the reduced form that is defined by substituting level-2 back into level-1. So the level-2 is critical to express both the fixed and random effects at the between-persons level.

Thank you SO much for the kind words. Seriously....Dan and I make this stuff and have no idea if anyone finds it useful or not. I really appreciate you sharing your thoughts. If you're a true glutton for punishment, Dan and I have a full 3-day version of intro to SEM that is completely free and comes with PDFs of course notes and detailed demos in lavaan, Mplus, and Stata. See centerstat.org/ for details. Good luck with your work -- Patrick

Good morning -- thanks for your note and your kind words. I think having a foundational knowledge in matrix algebra is hugely helpful when applying a variety of statistical models to real data. It both helps you understand what the models are doing, but also what is happening when things go wrong. Unfortunately, it's challenging to find general introductions because things tend to get very complicated, very quickly. Often you can get a good presentation of this in a classic multivariate statistics class, or maybe a factor analysis class. Also, there are many (many) tutorials on KZbin, some of which are better than others. One of my favorite is at Kahn Academy -- they do great work in everything they do. I hope you find this helpful -- take care -- patrick

Thanks for the positive comments!

There are several ways to allow for non-independence. For panel data, a common approach is to represent each repeated measure as a manifest variable. Then, the correlations among the repeated measures can be directly modeled, for instance, via a latent curve model. For clustered data, there is a multilevel SEM that models both the within- and between-groups covariance matrices. And -- most relevant for you -- for time series data (many observations per unit), there are extensions to the usual SEM to allow for non-independence within the framework known as Dynamic Structural Equation Modeling. You may want to look into this approach for your situation

@@centerstat Thanks a lot

Thanks for the kind words -- the interaction between time and a TVC would test whether the magnitude of the effect of the TVC in the prediction of the outcome varies systematically with the passage of time. In other words, might the magnitude of the TVC systematically strengthen or weaken as time progresses.

Hi -- by building a model that explicitly represents the two separate sources of variability (time within person, and between persons) then the conditional distributions among the residuals at level 1 are independent.

Thanks for your kind words -- I'm glad you have found these videos helpful. When you have TICs predicting the latent growth factors, the intercept terms of the latent factors are interpreted in the same way as a traditional regression -- that is, they are the model-implied means of the factors when all TICs are equal to zero. If that implied value has some meaning (say you mean-center your continuous predictors or have a binary predictor where a value of zero reflects a given group) then these may be meaningful to interpret. If zero values on your TICs are not interpretable (say you have a TIC that ranges from 1 to 5, so zero is not a valid value) then interpreting these can be quite misleading. I hope this helpful -- good luck with your work -- Patrick

@@centerstat Thank you!! I did not center the TIC continuous variables and it led to a very confusing intercept! Thanks so much :)

Hi Denika -- thanks for the kind words. I don't know if you're aware, but Dan Bauer and I teach a completely free 3-day online course in SEM. It's both live streaming (on May 8-10 for 2024) but you also can get indefinite access to recordings and all materials. If interested, see centerstat.org/structural-equation-modeling/ Good luck with your work -- Patrick

@@centerstat Hi Patrick, I just signed up! Thank you for providing all this great material for free, as a current grad student I can say it is greatly appreciated :) All the best!

Thanks for your nice words. With a 2nd time-varying covariate in addition to your time metric, it is still a growth model but it can be interpreted in two ways. First, you can examine growth in the DV net the effects of the TVC (so you are looking at growth curves in the adjusted outcome); or second, you can examine the relation between the TVC and the outcome above and beyond the effect of the underlying growth trajectory. These are precisely the same model and simply give different interpretational priority depending on your theory. The same holds for the plotting of effects. Raudenbush and Bryk have a really nice section on this in their 2002 book on hierarchical linear modeling.

​@@centerstat Sorry, could you please explain this again. Not a native speaker. What is the neaning of "net the effects" and "above abd beyond"?

@@GazallaAltaf of course -- it simply means the unique effect of each predictor while holding all other predictors constant. This is sometimes referred to as "controlling" for other predictors -- statistically, it's as if the only characteristic individuals varied on is the predictor of interest because all others are held constant.

thanks for your nice comments -- the lme4 package and lmer function jointly allows for the estimation of a general class of mixed effects models, including the growth models described here.

@@centerstat Thank you. Leaving aside lmer(), what is the fundamental theoretical difference between the mixed effects models and the growth models described here?

@@OmarRafique-op7bv there is no difference -- they are one in the same. You estimate a growth model using a mixed effects framework. If you're interested, this might help: Curran, P. J., Obeidat, K., & Losardo, D. (2010). Twelve frequently asked questions about growth curve modeling. Journal of cognition and development, 11(2), 121-136.

unless gender varied with time (say self-reported variability in perceived gender) then you could enter this at level 1. However, if you treat a characteristic as immutable to the passage of time (say biological sex at birth) then it would go into level 2 given the values do not vary as a function of time.

Thanks for the comment -- if you don't want a cross-level interaction with time, then you need only include your predictor in the intercept equation and not the slope equation. Then it will be a main effects-only model.

@@centerstat thank you for the reply. But what is an intercept equation and what is a slope equation? And by predictor, do you mean Age or Gender?

@@OmarRafique-op7bv the intercept equation if B0 and the slope equation is B1; age would only enter at level 1 and gender at level 1.

Hello -- yes...you can use MNLFA in nearly any parameterization of the CFA or SEM. It naturally gets more complex because you need to consider covariate effects on the set of thresholds between categories. Bauer & Hussong (2009, Psych Methods) give an example of this. Hope that helps -- patrick

Thanks for the very nice words. New Zealand is one of my very favorite places in the world -- you gotta love a place that has more sheep than people. I'm really glad you found the video helpful. If you're a true glutton for punishment, Dan Bauer and I have a full 3-day workshop on the SEM that is completely free -- see centerstat.org/introduction-to-structural-equation-modeling-async/ Good luck with your work -- Patrick

​@@centerstat Thanks so much! it is very generous of you to off the course for free, I will have a look.... glad you enjoyed your time here! ... we do have the odd sheep, also lots of need for your type of research & LOTS of jobs if you ever wanted to join us! :)

Hi Julie Anne -- thanks for the very kind words. If you're truly a glutton for punishment, Dan and I have a full 3-day workshop on the SEM that is completely free-of-charge (the beauty of both being tenured). See centerstat.org if you're interested. Good luck with your work -- Patrick

@@centerstat yes I already found it on your website! I’m very excited for it haha

​​@@centerstat I also look forward to attending! In the meanwhile I would appreciate a clarification. At about 5:32 you assume that the exogenous variables are correlated. Can you explain how this would happen considering they all point a collider? I thought the path was closed and the variables couldn't be associated...

@@domenicoscarpino3715 Thanks for the comment -- we always allow exogenous variables to freely correlate, either in the SEM or in any form of the GLM, because this allows the regression coefficients to be partialed for all other predictors (that is, the relation between one predictor and the outcome above and beyond all other predictors). Substantively why we do this is to represent the shared causes that might exist from things outside of our model that led to the predictors being correlated in the first place. I hope this helps -- patrick

@@centerstat please correct me if I'm wrong. If we don't include correlated exogenous variables in the model we would also incur in a confounding case scenario.

Hi -- thanks for the kind words. Briefly, there are two issues to consider: do your items have a sufficient number of response options to be considered continuous, but remains non-normal; or do you have so few response options that the linearity assumption no longer holds and you must move to a nonlinear model. If the former, there are many good options using robust maximum likelihood; if the latter, you have to adopt an alternative estimator to the typical normal theory ML. One option is based on polychoric correlations and uses some form of weighted least squares estimation; another option is to use an ML estimator that is based directly on the discrete item responses. As of now, Amos provides neither of these options but uses a Bayesian approach instead. Different packages offer different options (e.g., lavaan, LISREL, and Mplus) each of which have certain advantages and disadvantages. A couple of citations and a podcast episode are below. I hope this is of some use -- Patrick quantitudepod.org/s2e27-reconnecting-with-discrete-data/ Flora, D. B., & Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9, 466-491. Rhemtulla, M., Brosseau-Liard, P. É., & Savalei, V. (2012). When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions. Psychological Methods, 17, 354-373. Savalei, V., & Rhemtulla, M. (2013). The performance of robust test statistics with categorical data. British Journal of Mathematical and Statistical Psychology, 66, 201-223.

@@centerstat Patrick, Thank you so much for your reply. I've worked through the 2 issues, your suggestions and references. These were very insightful! My likert style scale has 6 options and based on my reading can be considered continuous, and non-normality remains, so at this stage I'm intending to use Mplus with Robust ML. I'll be most interested to see how well the model fits! Thank you for your generous and detailed response. It really helped me to navigate what felt like a a minefield!

@@louiebrown1 Thanks for your very kind note -- I'm glad I could be of even trivial use. Good luck with your work -- Patrick

thanks for the kind words. We have fun, and it's an added bonus if someone actually finds it useful. Good luck with your work -- patrick

ha! I'll remember that one -- the corollary is that a stats lecture is a fail-safe treatment for insomnia. Thanks for the note -- I hope you find this silliness in some way helpful -- Patrick

@@centerstat . . . the silliness!!! an indispensable spontaneity needed for topics like this. Well, your videos...watching them is my guilty pleasure. Especially because I do it during office hours! Hahaha. I do have fun watching you guys. ok, of course I learn growth curve modeling along the way. Seriously, both of you are a student's dream stat professors. I wish I had you in grad school. More power to you.

Miglena -- thanks for your very kind words...we sincerely appreciate it. Good luck with your work -- Patrick

Hi Larissa -- thanks for the note. At its core, SEM assumes independence -- no two residuals are any more or less related than any other two residuals. But as you note, more and more designs involve nested structures -- siblings nested in families, patients nested within physician, etc. There are two ways of handling this in the SEM. The first is to ignore nesting in the analysis itself, but then "adjust" the standard errors and test statistics for violations of independence. The second is to model the dependence directly, and this is often referred to as a multilevel SEM -- these models are challenging to estimate and procedures continue to be developed and perfected. Finally, some prefer to side-step the SEM entirely and bolt together tests of mediation directly within the multilevel model (that is naturally built for nesting). A few exemplar cites are below, but there is much more wonderful work on this topic out on the intertube. Good luck with your work -- Patrick McNeish, D., Stapleton, L. M., & Silverman, R. D. (2017). On the unnecessary ubiquity of hierarchical linear modeling. Psychological methods, 22(1), 114. Preacher, K. J., Zyphur, M. J., & Zhang, Z. (2010). A general multilevel SEM framework for assessing multilevel mediation. Psychological methods, 15(3), 209. Zhang, Z., Zyphur, M. J., & Preacher, K. J. (2009). Testing multilevel mediation using hierarchical linear models: Problems and solutions. Organizational Research Methods, 12(4), 695-719. Zigler, C. K., & Ye, F. (2019). A comparison of multilevel mediation modeling methods: recommendations for applied researchers. Multivariate Behavioral Research, 54(3), 338-359.

You are so welcome -- thanks for the nice note. If you're truly a glutton for punishment, Dan Bauer and I have a completely free 3-day online workshop on SEM -- see centerstat.org for details. Good luck with your work -- Patrick

Hi Holly -- thanks for your note. This one is tricky -- it's not so much that the TVCs are growing and and of themselves, but it's how you person-mean center to isolate within vs between person effects. That is, if your TVC is systematically growing over time (whatever form that might be -- linear, quadratic, etc.) then do the simple person-mean centering will give you a biased estimate of your within-person effects (because the mean assumes no growth over time). I've written a couple of things on this, as have several others -- a few cites are below. Good luck with your work -- patrick Curran, P.J., & Bauer, D.J. (2011). The disaggregation of within-person and between-person effects in longitudinal models of change. Annual Review of Psychology, 62, 583-619. Curran, P.J., Lee, T.H., Howard, A.L., Lane, S.T., & MacCallum, R.C. (2012). Disaggregating within-person and between-person effects in multilevel and structural equation growth models. In J. Harring & G. Hancock (Eds.) Advances in Longitudinal Methods in the Social and Behavioral Sciences (pp. 217-253). Charlotte, NC: Information Age Publishing. Curran, P.J., Howard, A.L., Bainter, S.A., Lane, S.T., & McGinley J.S. (2013). The separation of between-person and within-person components of individual change over time: A latent curve model with structured residuals. Journal of Consulting and Clinical Psychology, 82, 879-894. Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. (2015). A critique of the cross-lagged panel model. Psychological Methods, 20, 102. Wang, L. P., & Maxwell, S. E. (2015). On disaggregating between-person and within-person effects with longitudinal data using multilevel models. Psychological Methods, 20, 63.

Thanks for your question. The reason you're getting perfect fit is that the model is saturated -- that is, you are estimating as many parameters as pieces of information you observed, and thus you have zero degrees-of-freedom. Unfortunately, with 3 time points, a linear trajectory is all you can estimate (there are ways you can trick the model into a quadratic, but we don't recommend this because it will fit your data perfectly). Another option is to abandon the growth model altogether and move to something like a latent change score model or auto-regressive cross-lagged models, but those too have pros and cons. Sorry I can't be more helpful -- patrick

@@centerstat Dear Patrick, thanks for your answer. In fact, i am novice at this analysis method, so i will keep informations you provided in my mind.