So I did a quick calculation. And for the .113 slope works out really perfectly: Model-2, the last model without a slope, is calculating "i" as the average of all the intercepts. Since there are 3 time intervals between time-0 and time-3, the midpoint would be "time-1.5". That is one and a half intervals from time-0. Turns out that if you multiply 1.5 by the slope (.113) you get .169, which would be the growth (change in the intercept) from time-0 to said midpoint. Add that to the "i" in model-2 and you get: 5.352 (the "i' which was initially calculated without a slope). This demonstrates that the intercept calculated in models 1 & 2, really are the average of all intercepts. The graph of model 1&2, before you add the slope, is as if you put a pivot under the slope at its midpoint, and then rotated the slope until it was parallel to the x-axis (so that the gradient would be zero). So in effect, the area under the curve remains the same, whether you account for the slope or not.
Hi Dr. Buchanan. I was wondering whether you might be able to elaborate on the issue of heteroscedasticity. I've followed the steps in your tutorial on some data I've collected, however only get poor fitting models when constraining residual variances like you have. Only in an unconstrained model do I get a very good fitting model--which you've pointed out is concerning as it's probably an issue of heteroscedasticity. I plotted my data and it indeed looks heteroscedastic (in the sense that it's the shape of a reverse trumpet). People start off with great variability in their responses at Time 1, but the variability reduces at time 2, time 3, and time 4. At time 4 everyone has very similar responses. I feel this finding is meaningful in my field of research, but I'm not sure what the implications are of my data being heteroscedastic from a statistical point of view. If the data is naturally heteroscedastic, then constraining residual variances won't really capture these kinds of response patterns in my sample (I think?). In your opinion, would the use of an unconstrained model be academically defensible (and publishable)? And do you have any recommendations of readings I could look at to get a better understanding of why heteroscedasticity is an issue in the context of latent Growth Curve analyses, or what I might be able to do about it? Please forgive my ignorance in this field as I'm relatively new--and thank you for the tutorial, it's extremely informative and one of the best explanations I've come across!
Yeah, that sounds a lot like a sphericity problem, which is pretty common depending on the design type. I'd do exactly what you described here. Explain the models, create some graphics of the variances, show how they decrease, and describe why that's interesting. I don't have any good readings myself, but scholar suggests this article: www.sciencedirect.com/science/article/abs/pii/S000578940480042X
@@StatisticsofDOOM Thank you for your reply! Unfortunately, with a bit more exploring there appears to be many problems with the model (error variances over 1, negative error variances, high correlation between latent intercept and slopes, non-positive definite covariance matrix). I'm trying to investigate what might be causing this, but I suspect that this analysis might just not be suitable for my data. It's been an insightful learning experience nonetheless! Many thanks for your help. Your content is one of a kind :)
Around 8:27 you talk about the improvement of model fit. You say that "anything over 0.1 is significant", but throughout the rest of the example, the CFI improvements are less than 0.1 (0.064, 0.093, 0.005) and you say that Model 3 and Model 4 are significantly better. Could you elaborate on how to statistically determine which model fits better?
If we are talking about change in CFI, the rule > .01, not .10 (this is hu and bentler, 99 I believe) - that is a misspeak on my part, and I can work on updating that. There's a lecture on fit indices and model fit that covers that rule in depth more. I am working on updating this course now in markdown, etc., so thanks for catching that!
@@StatisticsofDOOM Thanks! Looks like Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling: A Multidisciplinary Journal, 6(1), 1-55. doi.org/10.1080/10705519909540118
Hi Dr. Buchanan, my question is related to my own research, and I would really appreciate your opinion. I have measured a variable repeatedly five times among a group of participants. I fit the data in latent growth curve models you talked about in the video. But none of them had good fit indices. The highest CFI is only 0.352 (model 5 with no constraints on the intercept and slope), rmsea is 0.455, srmr is 0.24. What do you think could be the possible reasons for the bad fits? Thanks.
I would check out the error variances - are some of them very high? That might indicate where the model is misfit (or misidentified or just simply the study didn't work).
Would this same approach apply for modeling growth in a latent variable with multiple indicators, just by adding the specification of t1 by x1, x2, xn? What else would be needed?
I'm not sure if I am totally following your question - are you just trying to add more predictors to the slope/intercept or predicting time 1, time 2, etc. ?
@@StatisticsofDOOM Sorry for the confusion, and I think I ended up working it out, but in essence, my dependent variable (loyalty) was a latent variable, represented by 5 indicators (y1, y2, y3..). I wanted to look at change in loyalty over time but wasn't sure how to account for each indicator constraint
So I did a quick calculation. And for the .113 slope works out really perfectly:
Model-2, the last model without a slope, is calculating "i" as the average of all the intercepts. Since there are 3 time intervals between time-0 and time-3, the midpoint would be "time-1.5". That is one and a half intervals from time-0. Turns out that if you multiply 1.5 by the slope (.113) you get .169, which would be the growth (change in the intercept) from time-0 to said midpoint. Add that to the "i" in model-2 and you get: 5.352 (the "i' which was initially calculated without a slope).
This demonstrates that the intercept calculated in models 1 & 2, really are the average of all intercepts. The graph of model 1&2, before you add the slope, is as if you put a pivot under the slope at its midpoint, and then rotated the slope until it was parallel to the x-axis (so that the gradient would be zero). So in effect, the area under the curve remains the same, whether you account for the slope or not.
Hi Dr. Buchanan. I was wondering whether you might be able to elaborate on the issue of heteroscedasticity.
I've followed the steps in your tutorial on some data I've collected, however only get poor fitting models when constraining residual variances like you have. Only in an unconstrained model do I get a very good fitting model--which you've pointed out is concerning as it's probably an issue of heteroscedasticity. I plotted my data and it indeed looks heteroscedastic (in the sense that it's the shape of a reverse trumpet). People start off with great variability in their responses at Time 1, but the variability reduces at time 2, time 3, and time 4. At time 4 everyone has very similar responses. I feel this finding is meaningful in my field of research, but I'm not sure what the implications are of my data being heteroscedastic from a statistical point of view.
If the data is naturally heteroscedastic, then constraining residual variances won't really capture these kinds of response patterns in my sample (I think?).
In your opinion, would the use of an unconstrained model be academically defensible (and publishable)?
And do you have any recommendations of readings I could look at to get a better understanding of why heteroscedasticity is an issue in the context of latent Growth Curve analyses, or what I might be able to do about it?
Please forgive my ignorance in this field as I'm relatively new--and thank you for the tutorial, it's extremely informative and one of the best explanations I've come across!
Yeah, that sounds a lot like a sphericity problem, which is pretty common depending on the design type. I'd do exactly what you described here. Explain the models, create some graphics of the variances, show how they decrease, and describe why that's interesting. I don't have any good readings myself, but scholar suggests this article: www.sciencedirect.com/science/article/abs/pii/S000578940480042X
@@StatisticsofDOOM Thank you for your reply! Unfortunately, with a bit more exploring there appears to be many problems with the model (error variances over 1, negative error variances, high correlation between latent intercept and slopes, non-positive definite covariance matrix). I'm trying to investigate what might be causing this, but I suspect that this analysis might just not be suitable for my data. It's been an insightful learning experience nonetheless! Many thanks for your help. Your content is one of a kind :)
@@abdullaharjmand7318 No problem! Heywood cases are no good, so that's the first thing to take care of. These models seem to do that a lot honestly!
Around 8:27 you talk about the improvement of model fit. You say that "anything over 0.1 is significant", but throughout the rest of the example, the CFI improvements are less than 0.1 (0.064, 0.093, 0.005) and you say that Model 3 and Model 4 are significantly better.
Could you elaborate on how to statistically determine which model fits better?
If we are talking about change in CFI, the rule > .01, not .10 (this is hu and bentler, 99 I believe) - that is a misspeak on my part, and I can work on updating that. There's a lecture on fit indices and model fit that covers that rule in depth more. I am working on updating this course now in markdown, etc., so thanks for catching that!
@@StatisticsofDOOM
Thanks! Looks like Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling: A Multidisciplinary Journal, 6(1), 1-55. doi.org/10.1080/10705519909540118
Hi Dr. Buchanan, my question is related to my own research, and I would really appreciate your opinion. I have measured a variable repeatedly five times among a group of participants. I fit the data in latent growth curve models you talked about in the video. But none of them had good fit indices. The highest CFI is only 0.352 (model 5 with no constraints on the intercept and slope), rmsea is 0.455, srmr is 0.24. What do you think could be the possible reasons for the bad fits? Thanks.
I would check out the error variances - are some of them very high? That might indicate where the model is misfit (or misidentified or just simply the study didn't work).
Would this same approach apply for modeling growth in a latent variable with multiple indicators, just by adding the specification of t1 by x1, x2, xn? What else would be needed?
I'm not sure if I am totally following your question - are you just trying to add more predictors to the slope/intercept or predicting time 1, time 2, etc. ?
@@StatisticsofDOOM Sorry for the confusion, and I think I ended up working it out, but in essence, my dependent variable (loyalty) was a latent variable, represented by 5 indicators (y1, y2, y3..). I wanted to look at change in loyalty over time but wasn't sure how to account for each indicator constraint