Covariances using orthogonal polynomials

[kaiserdominici]

Hi all,

I am fitting a 3-level repeated measures model with random intercept and slope at levels 2 (ID) and 3 (group), with a fixed quadratic time predictor which can take 3 values corresponding to the 3 measurement occasions.

Something like:

y_ijk = B_0ijk*cons + B_1jk * Time_ijk + B_2 * Time^2_ijk
B_0ijk = B_0 + v_0k + u_0jk + e_0ijk
B_1jk = B_1 + v_1k + u_1jk

I am fitting this model in MLwiN and in R to evaluate whether there are any differences. Now, if I fit the model with the time component "as is" (i.e., the values for Time_ijk = {0,1,2} and Time^2_ijk = {0,1,4}), I have the usual collinearity issues but I get exactly the same output from both programmes (using IGLS), as should be expected.

If I use orthogonal coefficients, MLwiN and R use different contrasts, but the outcomes are comparable in everything (including -2*logLik) except for variances and covariances.

Specifically:

- The total level-2 and level-3 variances are comparable but they are decomposed differently between intercept and Time. For instance, MLwiN may return something like Var(Intercept|Group) = 35 and Var(Time|Group) = 5, whereas R Var(Intercept|Group) = 20 and Var(Time|Group) = 21. Their sum is almost identical but the decomposition differs.

- Even more puzzling, the covariances are not comparable at all. R outputs a level-2 and level-3 covariance which is identical of what both programmes find when using "standard" (i.e., non-orthogonal) coefficients, whereas MLwiN returns something completely different (in this instance, they are both rather large and negative for R and not different from 0 for MLwiN).

To summarise the covariance issue, the situation looks as follows:

Code: Select all

........................|___R___|_MLwiN_|
standard coefficients...|___-2__|___-2__|
orthogonal coefficients.|___-2__|___0.1_|

I appreciate that the two programmes use different contrasts, but I fail to see how this may affect the sign of the covariances or the decomposition of the variances.

Thank you for any thoughts,

k.

[ChrisCharlton]

We are not aware of any reason why this would be the case. Would it be possible to provide some example data and syntax to allow us to replicate and investigate this?

[kaiserdominici]

Hi Chris,

Of course, could I send you the dataset via private message? I will remove all unnecessary and personal information.

Thank you and all the best,

k.

[ChrisCharlton]

Could you please send it via email (my address can be found at http://www.bristol.ac.uk/cmm/team/)?

[kaiserdominici]

Hi Chris,

I've sent it now, the subject of the email is "covariances with orthogonal polynomials".

Thank you and apologies to the rest of the forum community for this semi-private exchange.

k.