Multilevel Discrete time hazard model

[rahulvbb]

Dear all,
I am running the discrete time multilevel model (time to single event (birth of order2) ). I have taken the episodes at level-1 and community at level2 and state at level3. My problem is that, the variances at state and community level are smaller when using null model than when I use time variables in the model. I presume that the variance at both state and community level should decrease as I add time variables in the model. Please suggest. I am Pasting the results from two analysis below

[ChrisCharlton]

Professor Kelvyn Jones has asked me to pass on the following response:

It is quite a common occurrence in a logit model that when you put in predictor variables that the higher level variance goes up. This is because the level 1 variance cannot go down in such a model due to scaling

This is covered in our training materials, see under Discrete models at http://www.bristol.ac.uk/cmm/software/m ... urces.html

This is an extract from something that I have written on this which is not yet published:

“We need, however, to reflect carefully when comparing the higher-level variances from different models as explanatory variables are included. Before we consider discrete outcome binomial models, it is useful to summarize what normally happens in Normal-theory models for the case of a two-level model.

If a predictor variable is measured at level 2, it can only explain the unexplained variation at that higher level; thus the inclusion of important level 2 variables can reduce the level 2 variance, but not the level-1 variance.

If a predictor variable is measured at level 1, it can account for the unexplained variation at level 1 and it can also reduce the higher-level variance if the predictor has an element that systematically varies at the higher-level. For this to happen, two conditions are needed, first if the predictor is treated as a response, there has to be a substantial higher-level variance; second the area mean () of the level 1 predictor has to be related to the original response variable.

It is possible that the level 2 variance can increase with the introduction of a level 1 variable. In research on house prices, contextual neighbourhood effects may being masked so that expensive neighbourhood have small houses, and cheap neighbourhood have large houses, and when we take account of house size, the between district variance increases.

Given this pattern of changes, investigators often fit and report a sequence of models of growing complexity, paying due attention to the nature of the fixed parts estimates, but also inferring what are the effects of predictors by comparing the random parts of the model. These changes are usually given a substantive interpretation; thus in a value-added analysis, a researcher may report that the contextual effects have diminished, once intake ability has been taken into account.

Unfortunately, things are a lot trickier when modelling with a binomial variance. Using the simplifying assumption of the standard logistic distribution; the level 1 variance cannot change even when influential level 1 predictors are included in the model; it remains unchanged at 3.29. The level 2 variance is therefore being estimated relative to a numerically fixed benchmark; it is with model fit being re-scaled to this value. A thought experiment may help here. Imagine a null two level random intercepts model with no predictors, and which the level 2 variance is 0.35 (a VPC of 0.10). Now introduce an important level-1 predictor that has no strong level 2 component; it is ‘pure’ level 1 variable. As it is an important variable it should reduce the level 1 variance and leave the level 2 variance unchanged. But the level 2 variance is really scaled to the level 1 variance, the latter has gone down, but it cannot do so as it is fixed to 3.29. The consequence is that the level-2 variance will appear to go up to keep the relative scaling with the level 1 variance. Of the variance that remains a larger percentage must be at the higher level, as the level 1 value is fixed at 3.29. Sometimes this apparent increase is quite considerable if the level 1 predictor is an important one. In reality the matter is further complicated in that there may be an element of the level 1 variable that varies at the area level, and this might be reducing the level 2 variance but this is not showing as it is being swamped by the rise consequent on the explanatory power of the pure level-1 component of the variable. There is a final and important twist. As the level 2 variance increases the cluster-specific multilevel estimates can be expected to increase in absolute value; so that these constraints affect the fixed part estimates as well as the random part (see Snijders and Bosker,2012, Chapter 17).[1]

In short, in generalised linear models, changes in estimates are in part substantive and in part a technical consequence of scaling to the unchangeable level 1 variance. Adding important level 1 variables will generally increase the estimated level-2 unexplained variance. This will in turn lead to the estimates of already included predictors increasing in absolute size. The advice is to tread very carefully in comparing a sequence of binomial models. It will often be more helpful to include specifically the area means of the individual predictors (see later). It may also be helpful to run a further series of models in which the level 1 predictor is a response so that you ascertain the extent to which this variable varies at level 2. The conservative advice for binomial models is to compare estimates only within a model but not between models.“

[1] Snijders, T. A. B. & Bosker, R. J. (2012) Multilevel Analysis. Second edition,London: Sage.