If you have a binary DV the intercept is the negative of the threshold. You have an ordinal DV with 3 categories so two thresholds. You can computed the predicted probability for different random intercept values as shown on slide 66 of our Topic 7 handout, where slides 60-66 deal with understanding two-level logistic regression. See handout and video on our website.
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ljc posted on Monday, September 29, 2014 - 7:07 am
Slide 66 of topic 7 only has the patterns or cluster sizes. Am I looking in the wrong place?
Slide 66 refers to the Larsen-Merlo article - this is a good one to study. The slide looks like:
Understanding The Between-Level Intercept Variance • Intra-class correlation – ICC = 0.807/(π2/3+ 0.807) = 0.20 • Odds ratios – Larsen & Merlo (2005). Appropriate assessment of neighborhood effects on individual health: Integrating random and fixed effects in multilevel logistic regression. American Journal of Epidemiology, 161, 81-88. – Larsen proposes MOR: "Consider two persons with the same covariates, chosen randomly from two different clusters. The MOR is the median odds ratio between the person of higher propensity and the person of lower propensity." 66 MOR = exp( √(2* σ2) * Φ-1 (0.75) ) In the current example, ICC = 0.20, MOR = 2.36 • Probabilities – Compare β0j= -1 SD and β0j= +1 SD from the mean: For males at the aggression mean the probability varies from 0.14 to 0.50
ljc posted on Monday, September 29, 2014 - 11:38 am
Sorry, I hate to be dense, but I don't understand the &# notation.
I think your last sentence has the answer I am looking for which is, the formula for the predicted value for each cluster. I think I am supposed add (or subtract) the standard deviation to something, but I am not sure what.
Just as a note, I only have a random intercept in my particular example.
foe estimator being ML or MLR how to get: 1.) Predicted values (y-hat) and 2.) Residuals (resid)
e.g. I have a latent variable (LV) with three indicators. This LV is a dependent variable Y in the model. So to get the residual of this LV after being predicted by say another independent variable X which also latent with three indicator. i.e. Y by y1 y2 y3; X by x1 x2 x3; Y on X;
Just so, I found an interesting way to get this If X and Y were not latent variables. I can get this from the scatter plot-->save plot data.
However for latent variables this is not there! Please help. (in stata we get this using the predict command for non latent variable regressions)
I am trying to calculate predicted probabilities for a multilevel mediation model with a 4-category ordinal DV. The mediation pathway is not significant, so I am only interested in the level-2 direct effect of x on y. Using MODEL CONSTRAINT and the mean value of x yields implausible results. The probabilities do all add to 1, but the distribution is extremely unlikely. Below is a pared-down version of my model. Am I using the correct equation given the model?
CATEGORICAL = y ; WITHIN = [level-1 IVs] ; BETWEEN = x ; CLUSTER = country ; DEFINE: x = log(x) ;
ANALYSIS: TYPE = TWOLEVEL ; ESTIMATOR = BAYES ;
MODEL: %WITHIN% y ON [individual-level variables] ; m ON [individual-level variables] ; %BETWEEN% m ON x (a) ; y ON m (b) ; y ON x (coef) ; [y$1] (tau1) ; [y$2] (tau2) ; [y$3] (tau3) ;
where VBY is the between variance of Y and Beta*Var(X)*Beta^T is the total variance for the beta*X predictor. I notice that you skipped the within level model completely (but you shouldn't in general). If you skip it that means you condition on all within level X to be zero, which might be inappropriate or irrelevant.
Even the above approach assumes normal distribution for X - it may be best to average P(Y|X) for all X in your data set. We offer that now for single level via the individual predicted values see web note #20 but not for two-level yet.
1) Grand-mean centering mostly worked, and betas were similar in both models--a good sign. I did not grand-mean center the mediator, as doing so switched the sign of the level-2 coefficient. There is no meaningful zero value for the mediator (it's on a 1-5 scale), so it's difficult to tell whether the raw or centered scores are correct. Would grand-mean centering a mediator cause problems with the partitioning of within and between variance, or do you think the raw scores are the more likely problem?
2) Just to verify some figures in the denominator: VBY is the residual variance of y and Beta^T is the squared posterior s.d.?
(All level-1 variables are dummy-coded such that the combined omitted categories represent the benchmark person. I omitted it because I didn't want to clutter the page.)
I now understand the formula--I forgot that SEM must be thought of in terms of matrices. While I understand this theoretically, I'm still at a loss as to which specific numbers to use from the output. Beta*Beta^T suggests Beta is squared to make it conformable with the rest of the equation, but I'm unsure if Var(X) is the residual variance or some other number. Based on page 46 of the Topic 2 handout (https://www.statmodel.com/course_materials.shtml), it does seem to be the residual variance of Beta. Do I understand this correctly?
We don't actually compute Var(X) in the case when X is a covariate. You can compute this separately - it is the variance covariance of all predictors, not residual variance. However, I would still recommend that you figure out how to compute P(Y|X) for one observation. That has no Var(X). It would use the observed values for M, X and the estimated random intercepts factor score.
Like the original poster, I am trying to calculate predicted values in a multilevel ordinal logistic regression. I have watched the video for Topic 7 (which was very helpful), and have been studying the slides. I follow the lesson very clearly, but am then lost on where the values of .14 and .50 come from for values of aggression. They are simply 1 SD below and above the mean, which looks like it would then be 32? I must be missing something, because I do not see where the threshold values then come in when calculating P(uij = 1 | xij) in the formula on the top left of slide 61. Thank you so much for your response and for all of these very helpful resources.
The values you refer to are on slide 66. The formula for the probability is on slide 61. I think I just plugged in
beta_0j = -threshold (threshold= 2.981) beta_1j = 0.060*aggress (to which you have to add 1.071 for Male)
I don't recall what the aggress mean or SD values were.
So the probability was evaluated at the zero mean of the random intercept beta_0j. Not conditioning on the random intercept value, you have to do numerical integration to get the probability because the intercept is normal which doesn't connect nicely with a logistic probability link like it does with a probit probability link.