ASA Connect

Hi everyone, 

I hope you and your loved ones are staying as safe and healthy as possible in these times of radical uncertainty.  

If anyone can help me wrestle the following interpretation beast into the ground, I would be most grateful. 

The set up is that a set of subjects are monitored yearly for a number of years, with the subjects being nested in some larger groups. (The subjects could be students and the larger groups can be schools.)  For each subject, a count outcome variable is collected each year.  This count outcome variable is modelled via a mixed effects model (e.g., Poisson or Negative Binomial) which includes two variables: Year and Regime.

Depending on the larger group, the Regime can take the value No in each Year or can take the value No in the first year but Yes in all subsequent years.  Here, No stands for no regime change and Yes stands for a regime change.  The model includes a random intercept for subject and a random intercept for the larger group.  

Not all subjects are monitored across all Years.  (Year is coded as numeric, so that the model reports a linear effect of Year.)

After fitting a Poisson mixed effects model, say, there will be a reported estimated effect of Year and a reported estimated effect of Regime (recall that both of these are fixed effect estimates; the model contains no random slopes).  How should one interpret the effect of Regime after exponentiation, given that Regime is time-varying for some subjects but not for others?  Let's say this effect is -0.45 before exponentiation and exp(-0.45) =  0.64 after exponentiation.   Will a subject-specific interpretation be most appropriate - e.g., for a typical subject in a typical larger group, changing the regime from No to Yes is associated with a drop in the (overall level of?) the expected count?    With just two random intercepts in the model, can we also claim a population-level interpretation for the effect of Regime and what would that interpretation look like?  (I know we could claim a population-level interpretation in a Poisson mixed effects model with just one random intercept - not sure that is the case anymore with two random intercepts.)

What if the Poisson mixed effects model were to include an interaction between Year and Regime?  How would we interpret the effect of Regime then? 

How would the above interpretations change for a Negative Binomial mixed effects model? 

Many thanks, 

Isabella 


------------------------------
Isabella R. Ghement, Ph.D.
Ghement Statistical Consulting Company Ltd.
E-mail: Isabella@Ghement.ca
Phone: 604-767-1250
------------------------------

I recognize my answer here addresses only a part of your question but perhaps this thought might help.

Is regimen here really an outcome value? Why not begin the clock after the first year, when it is possible for regimens to change, and then consider regimen a baseline rather than an outcome variable? In this case, only patients still being followed past the first year would be included in the population of inference, so it wouldn't be necessary to address missing regimen values. Doing this would reframe the research question to be something like this: In patients who took the common regimen in the first year and were followed beyond it, what was the effect of the regimen switch?

That way, it wouldn't be necessary to impute missing regimen values. Because decisions to discontinue treatment often have something to do with the treatment effect, assumptions that they don't can be particularly dubious. It might be better to reframe the research question in a way that avoids making such assumptions. 

------------------------------
Jonathan Siegel
Director Clinical Statistics
------------------------------

Original Message:
Sent: 04-23-2020 15:40
From: Isabella Ghement
Subject: Interpretation of effects in a mixed effects model

Hi everyone,

I hope you and your loved ones are staying as safe and healthy as possible in these times of radical uncertainty.

If anyone can help me wrestle the following interpretation beast into the ground, I would be most grateful.

The set up is that a set of subjects are monitored yearly for a number of years, with the subjects being nested in some larger groups. (The subjects could be students and the larger groups can be schools.)  For each subject, a count outcome variable is collected each year.  This count outcome variable is modelled via a mixed effects model (e.g., Poisson or Negative Binomial) which includes two variables: Year and Regime.

Depending on the larger group, the Regime can take the value No in each Year or can take the value No in the first year but Yes in all subsequent years.  Here, No stands for no regime change and Yes stands for a regime change.  The model includes a random intercept for subject and a random intercept for the larger group.

Not all subjects are monitored across all Years.  (Year is coded as numeric, so that the model reports a linear effect of Year.)

After fitting a Poisson mixed effects model, say, there will be a reported estimated effect of Year and a reported estimated effect of Regime (recall that both of these are fixed effect estimates; the model contains no random slopes).  How should one interpret the effect of Regime after exponentiation, given that Regime is time-varying for some subjects but not for others?  Let's say this effect is -0.45 before exponentiation and exp(-0.45) =  0.64 after exponentiation.   Will a subject-specific interpretation be most appropriate - e.g., for a typical subject in a typical larger group, changing the regime from No to Yes is associated with a drop in the (overall level of?) the expected count?    With just two random intercepts in the model, can we also claim a population-level interpretation for the effect of Regime and what would that interpretation look like?  (I know we could claim a population-level interpretation in a Poisson mixed effects model with just one random intercept - not sure that is the case anymore with two random intercepts.)

What if the Poisson mixed effects model were to include an interaction between Year and Regime?  How would we interpret the effect of Regime then?

How would the above interpretations change for a Negative Binomial mixed effects model?

Many thanks,

Isabella


------------------------------
Isabella R. Ghement, Ph.D.
Ghement Statistical Consulting Company Ltd.
E-mail: Isabella@Ghement.ca
Phone: 604-767-1250
------------------------------

Hi Jonathan, 

Thank you so much for your answer.  At this stage, the analyses were conducted by someone else and I just have to double check that the interpretation provided makes sense.  

One of the Poisson mixed effects models fitted looks something like this, where Y_ijk is a count: 

log(E(Y_ijk | Year_ijk, Regime_ijk)) = beta0 + beta1*Year_ijk + beta2*Regime_ijk + v_ij + w_i  

where k indexes the temporal occasion, j indexes the subject and i indexes the larger group within which the subject is located.  The quantities v_ij and w_i are random effects - namely, w_i is a random effect associated with the larger group within which the subject is nested and v_ij is a random effect for the subject.  The pattern of values for Regime_ijk across years can be 0, 0, ..., 0 (all 0's; no regime swicth) or 0, 1, 1, ..., 1 (regime switch after first year only).  

The interpretation provided for beta2 is a population average interpretation (rather than a subject specific one).  So my first question is whether it is correct to provide such an interpretation for beta2 given that the model includes not just one, but two random intercepts.  It seems to me that: 

E(Y_ijk | Year_ijk, Regime_ijk) = exp(beta0 + beta1*Year_ijk + beta2*Regime_ijk + v_ij + w_i ) 
        
                                                  = exp(beta0 + beta1*Year_ijk + beta2*Regime_ijk) * exp( v_ij + w_i )

Even if I don't work out explicitly what exp( v_ij + w_i) looks like, I think this type of decomposition would confirm (?) that beta2 can have a population average interpretation?  If my assertion is correct, can I then invoke a population average interpretation for beta2 to state something like this: 

On any particular year after the first year, the average count among subjects with a regime switch after the first year (regardless of what larger group they come from) differs by a multiplicative factor of exp(beta2) from the average count among subjects without a regime switch after the first year (regardless of what larger group they come from)?

If the above interpretation is valid for the stated Poisson mixed effects model, would it also be valid for a Negative Binomial mixed effects model?  

The Poisson mixed effects model which includes an interaction between Year and Regime trips me up: 

log(E(Y_ijk | Year_ijk, Regime_ijk)) = beta0 + beta1*Year_ijk + beta2*Regime_ijk + beta3* Year_ijk*Regime_ijk + v_ij + w_i  

How to interpret the parameters of this last model?  

Thanks, 

Isabella  



------------------------------
Isabella Ghement
Ghement Statistical Consulting Company Ltd.
------------------------------

Original Message:
Sent: 04-24-2020 08:49
From: Jonathan Siegel
Subject: Interpretation of effects in a mixed effects model

I recognize my answer here addresses only a part of your question but perhaps this thought might help.

Is regimen here really an outcome value? Why not begin the clock after the first year, when it is possible for regimens to change, and then consider regimen a baseline rather than an outcome variable? In this case, only patients still being followed past the first year would be included in the population of inference, so it wouldn't be necessary to address missing regimen values. Doing this would reframe the research question to be something like this: In patients who took the common regimen in the first year and were followed beyond it, what was the effect of the regimen switch?

That way, it wouldn't be necessary to impute missing regimen values. Because decisions to discontinue treatment often have something to do with the treatment effect, assumptions that they don't can be particularly dubious. It might be better to reframe the research question in a way that avoids making such assumptions. 

------------------------------
Jonathan Siegel
Director Clinical Statistics

Original Message:
Sent: 04-23-2020 15:40
From: Isabella Ghement
Subject: Interpretation of effects in a mixed effects model

Hi everyone,

I hope you and your loved ones are staying as safe and healthy as possible in these times of radical uncertainty.

If anyone can help me wrestle the following interpretation beast into the ground, I would be most grateful.

The set up is that a set of subjects are monitored yearly for a number of years, with the subjects being nested in some larger groups. (The subjects could be students and the larger groups can be schools.)  For each subject, a count outcome variable is collected each year.  This count outcome variable is modelled via a mixed effects model (e.g., Poisson or Negative Binomial) which includes two variables: Year and Regime.

Depending on the larger group, the Regime can take the value No in each Year or can take the value No in the first year but Yes in all subsequent years.  Here, No stands for no regime change and Yes stands for a regime change.  The model includes a random intercept for subject and a random intercept for the larger group.

Not all subjects are monitored across all Years.  (Year is coded as numeric, so that the model reports a linear effect of Year.)

After fitting a Poisson mixed effects model, say, there will be a reported estimated effect of Year and a reported estimated effect of Regime (recall that both of these are fixed effect estimates; the model contains no random slopes).  How should one interpret the effect of Regime after exponentiation, given that Regime is time-varying for some subjects but not for others?  Let's say this effect is -0.45 before exponentiation and exp(-0.45) =  0.64 after exponentiation.   Will a subject-specific interpretation be most appropriate - e.g., for a typical subject in a typical larger group, changing the regime from No to Yes is associated with a drop in the (overall level of?) the expected count?    With just two random intercepts in the model, can we also claim a population-level interpretation for the effect of Regime and what would that interpretation look like?  (I know we could claim a population-level interpretation in a Poisson mixed effects model with just one random intercept - not sure that is the case anymore with two random intercepts.)

What if the Poisson mixed effects model were to include an interaction between Year and Regime?  How would we interpret the effect of Regime then?

How would the above interpretations change for a Negative Binomial mixed effects model?

Many thanks,

Isabella


------------------------------
Isabella R. Ghement, Ph.D.
Ghement Statistical Consulting Company Ltd.
E-mail: Isabella@Ghement.ca
Phone: 604-767-1250
------------------------------