What is the difference between "random intercept and trend model' and "random slope"?

[Mulusew Gerbaba]

Dear Statalist families,
I have received a comment from reviewer about random intercept model. He recommended me to consider for random intercept and trend model rather than random intercept model based on RD. Gibbons et. al , 2010) https://www.ncbi.nlm.nih.gov/pmc/art...ihms236714.pdf recommendation . But i was a bit confused about these terminologies? Does he mean that I should take in to account for "random slope"? when he suggest "random intercept and trend model"? Are these two the same? If both random slope and random intercept and trend model are the same, I have already tested the lrtest comparing random slope and random intercept and opted for random intercept model based on the results .

If random intercept and trend model and random slope are different, what is the command in stata taking in to account random intercept and trend model? is it different from random slope?

I appreciate for your help

[Jesse Wursten]

I'm not familiar with this particular literature, but I think this is a confusion in terminology.

Take the equation
y_it = a + b*xit + c*t + e_it

Then
a = constant
b = slope of x
c*t = trend

Hence, there is a difference between a random slope model and a random trend model (though only superficially, as in the end the trend is just special type of slope).

[Mulusew Gerbaba]

Thank you Jesse for your prompt reply
But, what is c? do you mean centering?

[Jesse Wursten]

c is a parameter, just like a and b. E.g. the interpretation of c in this case would be, what is the effect on y_it if we move forward one time period. Just like b is the effect on y_it if x_it increases by one.

I'm guessing in the random trends model this c itself is treated as a random variable, with some assumption on the functional form of its distribution. But as a I said, I'm not familiar with this literature, so don't quote me on that.

[Sebastian Kripfganz]

My understanding of a random intercept and random trend model is the following:
\[y_{it} = \mathbf{x}_{it}' \boldsymbol{\beta} + \alpha_i + \gamma_i t + e_{it}\]
where \(\alpha_i\) is an individual-specific intercept and \(\gamma_i t\) is an individual-specific trend. The interpretation of random versus fixed effects would be similar to a model without individual-specific trend, namely that the regressors \(\mathbf{x}_{it}\) are supposed to be uncorrelated with both \(\alpha_i\) and \(\gamma_i\) in the case of random effects.

[Mulusew Gerbaba]

Thank you Sebastian! Let me be clear, I am trying to see the effect of food insecurity on self rated health. I have three rounds of both dependent variable(health) and predictor(food insecurity). I used the following command to regress random intercept model.

xtmelogit health foodinsec##c.time||personid:,var mle (Model I) AND
estimate store ri

for random slope model
xtmelogit health foodinsec##c.time||personid:time,var mle cov(unstructured) (Model II)
estimate store rc
lrtest ri rc = Prob>chi2 =0.7674

so that I assumed no need to take slope in to account so that I chose Model I. (I will post the result for you)

Now I am asking you how should I type my command on Stata to regress random intercept and trend model (i.e. to include individual-specific trend)?

[Mulusew Gerbaba]

I am sorry that I couldn't get how to paste the Results window of Stata here. I just attached the word file. Below is the outputs from the Stata

[Sebastian Kripfganz]

I do not have any experience with the xtmelogit command (which is superseded by meqrlogit as of Stata 13) but your "random slope model" appears to me like I would probably specify a "random intercept and trend" model (but I might be wrong). The confusion might result from people in different disciplines using different names for the same model, and the same names for different concepts (e.g. "fixed effects" and "random effects").

[Mulusew Gerbaba]

Dear Sebastian and Jesse,
Thank you so much!