Hi,

I am running a multilevel negative binomial regression predicting the number of crimes a student commits in a year in school.

In the empty model the variance associated with the random intercept is virtually 0 and tests suggest that this model is not a better fit than a single level model.

When I include an individual level predictor (crime propensity), which is strongly and positively associated with crime, the group level variance increases and tests suggest that the multilevel model is a better fit than the individual model.

Descriptive analysis of my data shows that there are higher levels of criminal behaviour in schools where pupils have higher levels of crime propensity and so I would expect there to be some variance associated with the random intercept in the empty model, which reduces once propensity is added to the model.

As such, I am struggling to interpret these results both statistically and substantively and would appreciate any thoughts on possible interpretations.

Thanks in advanced,

Liam

## Inclusion of level-one variable increases group level variance in a multilevel negative binomial regression

### Re: Inclusion of level-one variable increases group level variance in a multilevel negative binomial regression

Hi Liam,

It is worth noting that for non-Normal multilevel models one has to be careful when interpreting higher level variance terms. The bottom level variance is always fixed by the distributional assumptions made and so when terms are added to the model that fixed variance would normally reduce and as a result there is often an inflation of higher level variances to compensate (effectively a rescaling effect).

It is also possible of course for the level 2 variance to increase in a normal response model with the addition of a predictor. A good example is in repeated measures with say children's height over time (say from ages 1 to 10). If one doesn't control for age then there seems to be a large variation within each child relative to between children but when one includes age in the model that explains loads of the variation and the differences between children become easier to estimate and the between child variance will likely increase.

Hope that this helps as I suspect your scenario is a mixture of the two.

Best wishes,

Bill.

It is worth noting that for non-Normal multilevel models one has to be careful when interpreting higher level variance terms. The bottom level variance is always fixed by the distributional assumptions made and so when terms are added to the model that fixed variance would normally reduce and as a result there is often an inflation of higher level variances to compensate (effectively a rescaling effect).

It is also possible of course for the level 2 variance to increase in a normal response model with the addition of a predictor. A good example is in repeated measures with say children's height over time (say from ages 1 to 10). If one doesn't control for age then there seems to be a large variation within each child relative to between children but when one includes age in the model that explains loads of the variation and the differences between children become easier to estimate and the between child variance will likely increase.

Hope that this helps as I suspect your scenario is a mixture of the two.

Best wishes,

Bill.

### Re: Inclusion of level-one variable increases group level variance in a multilevel negative binomial regression

Many thanks for your response Bill.

### Re: Inclusion of level-one variable increases group level variance in a multilevel negative binomial regression

Hi Bill,

Could you recommend any literature on the rescaling effect?

Many thanks,

Liam

Could you recommend any literature on the rescaling effect?

Many thanks,

Liam

Hi Liam,

If you register on the online LEMMA course there is stuff about this in unit 7 section 7.2 which is worth a read.

Best wishes,

Bill.

If you register on the online LEMMA course there is stuff about this in unit 7 section 7.2 which is worth a read.

Best wishes,

Bill.