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Hello,

Using MCMC I have fitted a multivariate normal 3-level cross-classified model with random intercepts at each level. In my data, surgeons have conducted eye-operations, where a patient may have had a single eye operation or two separate eye operations conducted by the same surgeon or different surgeons. The outcome of interest is a 3-component continuous measure of vision. Of interest is if the presence of pre-existing condition diabetes (yes/no) affects the outcome of the surgery.

I want to calculate a 95% interval that tells me the range of values I would see among new eye operations conducted on individuals from the same population of patients and operated on by the same population of surgeons. So, I want to calculate a 95% interval for new eye-operations on people with diabetes and a 95% interval for new eye-operations on people without diabetes.

My thinking on how to do calculate a 95% interval is as follows:
1) After convergence, run the MCMC chain for say 1000 iterations.

2) For each iteration:
(a) Draw a random intercept at level 3 and at level 2 (from a multivariate normal distribution) using the current estimates of the random covariance matrix.
(b) Use the random draws from (a) and current estimates of the fixed effects to generate a new predicted value (e.g., fixed effects + random effect draws)

3) Use the 2.5% and 97.5% of the distribution of predicted values in step 2 as my 95% prediction interval.

Is this a sensible approach? I can see that it accounts for uncertainty in the fixed-effect coefficients and random variance parameters but not the residual variance. Is there a simpler approach?

Thanks in advance
Rach

Hi Rachael,
If you were hoping to get 95% intervals for individuals rather than for the average individual you will also need to have level 1 in your predictions I would expect.
Best wishes,
Bill.

Thanks Bill for your reply. I suspected my idea wasn't quite right.

So to predict for individuals should I amend step 2 (in particular part b) as follows:

2) For each MCMC iteration:
(a) Draw a random intercept at level 3 and at level 2 (from a multivariate normal distribution) using the current estimates of the random covariance matrix.
(b) Draw predicted values from a Normal distribution with
mean=Fixed + random realisation draws from (a)
variance=level 1 residual variance
Best wishes
Rach

Hi Rachael,
Just generate a level 1 residual (using the level 1 variance) and add it to the prediction and you should be sorted.
Bill.