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Random Intercept and Slope Models

  • 1.  Random Intercept and Slope Models

    Posted 10-26-2011 05:11
    This message has been cross posted to the following eGroups: Statistics in Epidemiology Section and Statistical Consulting Section .
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    Dear all,

    Good day!

    I appreciate the disccusions.

    First, in a longitudinal data analysis with Random Intercept and Random Slopes Model, what makes the correlation between the random intercept  and one or two of the random coefficients to be near 1 (with sign positive or negative)?

    Second, is it possible to enter a baseline variable in our model as we have done in ANCOVA in the presence of random intercept?

    It seems I made a mistake in my analysis and could not find it? Maybe there is a little bug in what I have done. So, Please let me know your illustrations or hints on the issue.

    I would be also thankful if you introduce me some good references including statistical software codes in STATA.

     Thank you all,

    Amir  


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    Amir Kasaeian
    PhD Student in Biostatistics
    Tehran University of Medical Sciences (TUMS)
    amir_kasaeian@yahoo.com
    akasaeian@razi.tums.ac.ir
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  • 2.  RE:Random Intercept and Slope Models

    Posted 10-26-2011 08:11
    Sounds to me that you should look at the text Generalized Estimating Equations by Hardin and Hilbe.  This book covers longitudinal data analysis with a lot of code written in STATA by Joe Hilbe. 

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    Michael Chernick
    Director of Biostatistical Services
    Lankenau Institute for Medical Research
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    Original Message


  • 3.  RE:Random Intercept and Slope Models

    Posted 10-26-2011 09:15
    Positive correlations between intercept and slope in random coefficient models occur when the subjects with above average intercept also have above average slope and subjects with below average intercept have below average slope.  Negative correlations result from subjects with above average intercept having below average slope and those with below average intercept having above average slope.  In the data with which I have experience, getting a intercept-slope correlation near 1 (or -1) is rare or due to a small sample size.

    If you don't find an example in the citation Michael gave that fits your needs there is an example of using student pretest scores in Littell, Milliken, Stoup, Wolfinger, and Schabenberger (2006) "SAS for Mixed Models". 

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    Jon Baldock
    Baldock Statistical Services
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    Original Message


  • 4.  RE:Random Intercept and Slope Models

    Posted 10-26-2011 10:30
    The request was for a reference with examples in STATA.  That is why I recommended Hardin and Hilbe as they code all examples in STATA and perhaps some in SAS also.  I am not sure whether or not Hardin and Hilbe have the specific example of interest, but the Littell et al book will have examples exclusively in SAS.

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    Michael Chernick
    Director of Biostatistical Services
    Lankenau Institute for Medical Research
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    Original Message


  • 5.  RE:Random Intercept and Slope Models

    Posted 10-26-2011 12:23
    In Analysis of Messy Data: Analysis of Covariance Vol 3, co-authored with Dalles Johnson, there are motivating examples for using random intercepts and slopes as well as examples ran in SAS and JMP.

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    George Milliken
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    Original Message


  • 6.  RE:Random Intercept and Slope Models

    Posted 10-26-2011 12:08

    Suggestion: Doug Bates' stuff on lmer4() for R, including draft chapters of his book.


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    Ralph O'Brien
    Case Western Reserve University
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    Original Message


  • 7.  RE:Random Intercept and Slope Models

    Posted 10-26-2011 13:21

    The book looks interesting.  Ralph or anyone else:  Do you know when Springer will be publishing the book?  Looks like Chapter 3 hasn't been written yet (at least it is not posted on the site).
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    Michael Chernick
    Director of Biostatistical Services
    Lankenau Institute for Medical Research
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    Original Message


  • 8.  RE:Random Intercept and Slope Models

    Posted 10-27-2011 09:19
    I would consult first Rabe-Hesketh and Skrondal's (2008) "Multilevel and Longitudinal Modeling Using Stata" (http://www.stata.com/bookstore/mlmus2.html). It will have all of the example of Stata code (note the proper capitalization and the singular form ;) ) you would need.

    A correlation between the random effects of slopes and intercepts means that you have somewhat of a fan pattern of your individual trajectories. By demeaning (or, more generally, shifting) the variable that enters the model with a random slope, you can achieve any correlation you want or like. If you move the mean of that variable to the origin of the fan, you will achieve approximately zero correlation. Rabe-Hesketh and Skrondal discussed this effect in a continuing education course that the book roughly follows; I have a copy in the office, but not at home at hand to check.

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    Stanislav Kolenikov
    University of Missouri
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  • 9.  RE:Random Intercept and Slope Models

    Posted 11-07-2011 16:53

    Dear all,
    Thank you all your patience,

    Until now, I have a problem regarding the use of random slope without random intercept? I couldn't find a conclusive answer yet.

    I have additional problem, the time variable values are far from zero (for example1985) and regarding the comments I received I centered this variable. I tested three different kind of centering as follows:

    ctime= time-mean(time)

    t_min=time-min(time)

    t_max=time-max(time)

    These different kinds of centering lead to different values for the likelihood or even for p-values. For example the likelihood values are -1275.6719 for ctime, -1132.1318 for t_min and -1259.8309 for t_max.

    I think it is a paradox and because of that I think maybe the use of random slope without random intercept is wrong?!?
    About the mentioned book "    Multilevel and Longitudinal Modeling Using Stata ", seems it's a great book, but I couldn't find it in my area or even its e-book.

    Alternatives books would be helpful.

     Thanks a lot for contributions.

    Amir


    -------------------------------------------
    Amir Kasaeian
    PhD Student in Biostatistics
    Tehran University of Medical Sciences (TUMS)
    amir_kasaeian@yahoo.com
    akasaeian@razi.tums.ac.ir
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    Original Message


  • 10.  RE:Random Intercept and Slope Models

    Posted 11-07-2011 20:36
    When ever you center the covariate or change the origin of the covariate you change the comparisons of intercepts on the analysis of the covariate (a comparison at covariate =0) to a comparison of the regression lines at the covariate = mean(time), or a comparison of the regression lines at covariate = min(time) or or a comparison of the regression lines at covariate = max(time)---all three are different comparisons.  If you change the location of zero for the covariate (like subtracting off the mean) you change the likelihood, so we get different values for the estimates, tests or estimates of the intercepts variance components.  If the intercepts are not random but the slopes are random, you get the similar types of results when you center the covariate or move the origin to some other place.  When I have such a model (which can be called the analysis of covariance model with or without random effects) if the intercepts are not equal, then you need to fit the regression lines at least at 3 values of the covariate.  I generally do the analysis with the covariate-median, the covariate-25%tile of distribution of the covariate and the covariate - 75%tile of the distribution of the covariate.  Sometimes my consultee what to look at the analysis at more values like at 5%tile and 95%tile.  My reference is Analysis of Messy Data: Analysis of Covariance, Milliken and Johnson 2002 Chapman & Hall, chapter 12.

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    George Milliken
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    Original Message


  • 11.  RE:Random Intercept and Slope Models

    Posted 11-07-2011 21:58
    Amir:

      Unless  I misunderstand what you are trying to do, it seems to me that your problem may be with the centering of the data in combination with the assumption of no intercept.  By assuming that the regression model has a zero intercept (which is what you are imposing if your model has no intercept), then you are saying that the response at x=0 (which is time in your case) is y=0.  But if you instead impose the time shift, you are saying that x=0 is at the center in time where I presume your response is not zero.  This would be a big conflict and it could have dramatic consequences for the regression model.

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    Jeffrey Proehl
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    Original Message


  • 12.  RE:Random Intercept and Slope Models

    Posted 11-08-2011 09:42
    Are you trying to use a model with random slopes and and no intercept terms for each batch, ie:

         response = time (fixed)  time*batch (random)

    or a model with random slopes and fixed intercepts, ie:

         response = batch (fixed)  time (fixed)   time*batch (random)

    If so, what is the physical situation that requires such a model?

    If the former, see Jeffrey Proehl's response.  If you are using JMP software for such a model, it will automatically center the time variable to its mean and that probably makes no sense physically.  If you are using such a non-hierarchical model and if it makes sense physically, you can tell JMP to turn off "polynomial centering" by clicking the little red arrow at the top left hand corner of the Fit Model window.  But bear in mind that you really do need to have a good physical reason for omitting the "batch (random)" term and thereby assuming that all of your regression lines absolutely must cross at time=zero.   That assumption can also be dubious if your data doesn't extend all the way to zero because you will also be assuming that your regression lines would stay absolutely straight if you had data going all the way to zero.

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    Emil M Friedman, PhD
    Principal Scientist (Statistician)
    MannKind Corporation
    Danbury, CT 06810
    Disclaimer: The views expressed are mine alone and do not necessarily reflect the views of my employer.

    Original Message


  • 13.  RE:Random Intercept and Slope Models

    Posted 11-08-2011 11:59

    Thank you for this interesting and timely discussion as I am gearing up for a project on yield trends in several agricultural cropping systems.  I am planning to use a random intercept, but fixed slope model because the plots will likely have a range of initial productivities, but the trend over time will likely be due to technological factors unrelated to plots (e.g. genetic improvements, advances in pest control measures, etc).  Thus, the model would be in the formulation of a previous message

    Yield = plot(system)-random   time-fixed  plot(system)*time-fixed

    Does anyone forsee any problems with such an approach?

    Jon Baldock
    Baldock Statistical Services
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    Jon Baldock
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    Original Message


  • 14.  RE:Random Intercept and Slope Models

    Posted 11-09-2011 09:41

    It sounds like you want to learn something about "system".  Therefore, if you have plot within system as a random effect shouldn't you also have system as a fixed effect?  Do you also want system*time as a fixed effect to see whether different slopes are associated with different systems?  Also, if plot within system is a random effect, plot(system)*time should also be random.
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    Emil M Friedman, PhD
    emil.friedman@alum.mit.edu     (work)
    emilfrie@alumni.princeton.edu (home)
    http://www.statisticalconsulting.org

    Original Message


  • 15.  RE:Random Intercept and Slope Models

    Posted 11-09-2011 16:52
    If plot(system) is random then any interaction with plot(system) must also be random, thus  plot(system)*time cannot be fixed.

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    George Milliken
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    Original Message


  • 16.  RE:Random Intercept and Slope Models

    Posted 11-09-2011 21:40

    I think the rather obtuse notation and terminology associated with random intercepts and slopes has obscurred my question.  Certainly plot(system)*time is random if plot is random, but is one required to include that random interaction whenever estimating the fixed portion (system*time).  Until this discussion and recent readings, I would have said yes.  Now it appears that it is acceptable to have an analysis that estimates differing fixed intercepts and slopes (in SAS:  "Model yield = system time system*time;"), but only estimates a random component for intercepts.  That is, the SAS statement would be "Random int / subject= plot(system);" instead of "Random int time / subject=plot(system;"). While such an analysis now seems plausible, I have not seen an example to be sure that it is acceptable.
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    Jon Baldock
    Baldock Statistical Services
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    Original Message


  • 17.  RE:Random Intercept and Slope Models

    Posted 11-14-2011 11:05

    Hi all,

    Thanks for all your considerations and helps.

     Indeed my question is about the second model, that is:

    Response = batch (fixed) time (fixed) time*batch (random)
    My response is child mortality rate (CMR) and it was recorded in 1980, 1985, 1990, 1995, 2000 and 2009 for 52 countries. So my units are countries and the repeated measures are repeated CMR values for these countries. I use the value of CMR in 1980 as a baseline and treat it as a covariate in the model. Other CMR measurements (from 1985 till 2009) are considered as responses. So, as I said, I want to fit a random slope model without random intercept. I think when I have adjusted for the baseline value of CMR (1980) and also used a model that has the ability of considering different slopes for different countries, then why should I consider a model with random intercept? Indeed, I believe when I have adjusted for the baseline of each country then it is redundant to consider a random intercept. So I have fitted a model like this:

    CMR=ß0 [fixed]+ ß1[fixed] (CMR_1980)+ ß2[random] (time) + e

    I've adjusted for CMR value because I believe past CMR is the most important determinant of future CMR and I've also consider a random slope for time because I believe different countries have different trends during the period. Therefore, I conclude that the nature of the data set is more consistent with a random slope model but without random intercept.

    Now, please let me know your comments, thoughts and your helpful advices.

    Bunch of thanks your patience.

     

    Amir





    -------------------------------------------
    Amir Kasaeian
    PhD Student in Biostatistics
    Tehran University of Medical Sciences (TUMS)
    amir_kasaeian@yahoo.com
    akasaeian@razi.tums.ac.ir
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    Original Message


  • 18.  RE:Random Intercept and Slope Models

    Posted 11-14-2011 11:17

    Hi all,

    Thanks for all your considerations and helps.

     Indeed my question is about the second model, that is:

    Response = batch (fixed) time (fixed) time*batch (random)
    My response is child mortality rate (CMR) and it was recorded in 1980, 1985, 1990, 1995, 2000 and 2009 for 52 countries. So my units are countries and the repeated measures are repeated CMR values for these countries. I use the value of CMR in 1980 as a baseline and treat it as a covariate in the model. Other CMR measurements (from 1985 till 2009) are considered as responses. So, as I said, I want to fit a random slope model without random intercept. I think when I have adjusted for the baseline value of CMR (1980) and also used a model that has the ability of considering different slopes for different countries, then why should I consider a model with random intercept? Indeed, I believe when I have adjusted for the baseline of each country then it is redundant to consider a random intercept. So I have fitted a model like this:

    CMR=ß0 [fixed]+ ß1[fixed] (CMR_1980)+ ß2[random] (time) + e

    I've adjusted for CMR value because I believe past CMR is the most important determinant of future CMR and I've also consider a random slope for time because I believe different countries have different trends during the period. Therefore, I conclude that the nature of the data set is more consistent with a random slope model but without random intercept.

    Now, please let me know your comments, thoughts and your helpful advices.

    Bunch of thanks your patience.

     

    Amir



    -------------------------------------------
    Amir Kasaeian
    PhD Student in Biostatistics
    Tehran University of Medical Sciences (TUMS)
    amir_kasaeian@yahoo.com
    akasaeian@razi.tums.ac.ir
    -------------------------------------------






    Original Message