Discussion: View Thread

Question on interpretation of a coefficient:

I have an outcome variable which represents the proportion of days spent in hospital for each patient over 30 days. 

After applying the logit transformation to this proportion, I use a linear regression model to relate logit(proportion) to a treatment and another covariate as follows:

logit(proportion) = beta0 + beta1 * Treatment + beta2 * Covariate

What is the meaning of the exp(beta1) in this model presuming that Treatment takes the values 0 (Control) or 1 (Active)?

If the outcome were binary, I'd have no problem understanding the notion of odds ratios. 

But I'm stuck trying to explain the "odds of a proportion" rather than the "odds of an event".

Is there a way to attach meaning to exp(beta1) without having an "event" per say which would enable me to talk about the "odds of an event"?  Or shall I simply stay with the effect of treatment on the logit(proportion)? 

Thanks,

Isabella
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Isabella Ghement
Ghement Statistical Consulting Company Ltd.
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I think it's the phrase "odds of a proportion" that's hanging you up. 

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Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
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...what I said below was perhaps too cryptic, so lemme try again.  Instead of it being the "odds of a proportion", maybe it's merely the odds of being in the hospital, or of being in the hospital on a given day during the 30-day observation period.

Question: does everybody start off being in the hospital & they get out over time?  Or is it more like: they go in, get out, go back in, etc?

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Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
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Original Message:
Sent: 04-01-2014 01:35
From: Eric Siegel
Subject: Logit transformation for proportions

I think it's the phrase "odds of a proportion" that's hanging you up. 

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Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
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Hi Eric,

Thanks so much for your responses.

Yes, every patient has been admitted to hospital subsequent to an ER visit and is being followed for a period of 30 days (except if their medical insurance is cancelled, in which case the follow-up period extends to the date of cancellation of this insurance).  Some patients may died during the follow-up period, but since I only know their month & year of death, I am using the date of cancellation for their medical insurance as a proxy for date of death.

Sounds like your second definition of an "event" makes more sense here (i.e., being in hospital on a given day during the 30-day observation period), unless I am getting hung up again.     

Isabella

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Isabella Ghement
Ghement Statistical Consulting Company Ltd.
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Original Message:
Sent: 04-01-2014 01:56
From: Eric Siegel
Subject: Logit transformation for proportions

...what I said below was perhaps too cryptic, so lemme try again.  Instead of it being the "odds of a proportion", maybe it's merely the odds of being in the hospital, or of being in the hospital on a given day during the 30-day observation period.

Question: does everybody start off being in the hospital & they get out over time?  Or is it more like: they go in, get out, go back in, etc?

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------

Hi, Isabella, I have two additional questions:
(1) if their medical insurance is cancelled on Day 45, do they still get only 30 days of follow-up for discharge?
(2) do some people stay in the hospital the whole 30 days?
Thanks for indulging me :)

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Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
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Original Message:
Sent: 04-01-2014 11:38
From: Isabella Ghement
Subject: Logit transformation for proportions

Hi Eric,

Thanks so much for your responses.

Yes, every patient has been admitted to hospital subsequent to an ER visit and is being followed for a period of 30 days (except if their medical insurance is cancelled, in which case the follow-up period extends to the date of cancellation of this insurance).  Some patients may died during the follow-up period, but since I only know their month & year of death, I am using the date of cancellation for their medical insurance as a proxy for date of death.

Sounds like your second definition of an "event" makes more sense here (i.e., being in hospital on a given day during the 30-day observation period), unless I am getting hung up again.     

Isabella

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







Original Message:
Sent: 04-01-2014 01:56
From: Eric Siegel
Subject: Logit transformation for proportions

...what I said below was perhaps too cryptic, so lemme try again.  Instead of it being the "odds of a proportion", maybe it's merely the odds of being in the hospital, or of being in the hospital on a given day during the 30-day observation period.

Question: does everybody start off being in the hospital & they get out over time?  Or is it more like: they go in, get out, go back in, etc?

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------

Hi Eric,

Thanks for wanting to be indulged.  Here are the answers to your questions:

1.  The follow-up period is defined as [D1, D30], where
       D1 = Date of admission in hospital and
       D30 = Minimum between { D1 + 30, Date of Cancellation of Medical Coverage}

2.   Yes, some people stay in hospital 30 days or even more than 30 days.  

Isabella

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







Original Message:
Sent: 04-01-2014 12:07
From: Eric Siegel
Subject: Logit transformation for proportions

Hi, Isabella, I have two additional questions:
(1) if their medical insurance is cancelled on Day 45, do they still get only 30 days of follow-up for discharge?
(2) do some people stay in the hospital the whole 30 days?
Thanks for indulging me :)

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------




Original Message:
Sent: 04-01-2014 11:38
From: Isabella Ghement
Subject: Logit transformation for proportions

Hi Eric,

Thanks so much for your responses.

Yes, every patient has been admitted to hospital subsequent to an ER visit and is being followed for a period of 30 days (except if their medical insurance is cancelled, in which case the follow-up period extends to the date of cancellation of this insurance).  Some patients may died during the follow-up period, but since I only know their month & year of death, I am using the date of cancellation for their medical insurance as a proxy for date of death.

Sounds like your second definition of an "event" makes more sense here (i.e., being in hospital on a given day during the 30-day observation period), unless I am getting hung up again.     

Isabella

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

Hi Isabella,

Is there a reason you want to stick with a linear regression?  I would be tempted to (ab)use a logistic regression for a problem like this for several reasons.  First, the number of "trials" (days) varies among subjects, so that the proportions have naturally different variances.  Thus, your linear regression should probably have a weighting factor for the number of days observed.  Second, each "trial" has a binary response, which would be appropriate for a logistic regression.  Third, the logit transform does not really stabilize variance, and indeed the variances can be quite different when the proportions are close to 0 or 1. 

The main problem with a logistic regression is the fact that the "trials" (days) are not independent, because if a person is in hospital at day d, they are more likely to be in hospital at d+1 than if they were not in hospital at day d.  The dependence is entirely within-subject, however, and the effects on the model are fairly predictable: the proportions will behave as if they have more variability than the binomial model would expect. That is, they will be overdispersed.  An overdispersion correction can be applied to the analysis to account for the correlation empirically without excess complication.

The usual tools of a linear regression are then at your disposal, including the interpretation of parameters as log-odds-ratios.

-Tom. 

-------------------------------------------
Thomas Loughin
Simon Fraser University
-------------------------------------------







Original Message:
Sent: 04-01-2014 12:37
From: Isabella Ghement
Subject: Logit transformation for proportions

Hi Eric,

Thanks for wanting to be indulged.  Here are the answers to your questions:

1.  The follow-up period is defined as [D1, D30], where
       D1 = Date of admission in hospital and
       D30 = Minimum between { D1 + 30, Date of Cancellation of Medical Coverage}

2.   Yes, some people stay in hospital 30 days or even more than 30 days.  

Isabella

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







Original Message:
Sent: 04-01-2014 12:07
From: Eric Siegel
Subject: Logit transformation for proportions

Hi, Isabella, I have two additional questions:
(1) if their medical insurance is cancelled on Day 45, do they still get only 30 days of follow-up for discharge?
(2) do some people stay in the hospital the whole 30 days?
Thanks for indulging me :)

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------




Original Message:
Sent: 04-01-2014 11:38
From: Isabella Ghement
Subject: Logit transformation for proportions

Hi Eric,

Thanks so much for your responses.

Yes, every patient has been admitted to hospital subsequent to an ER visit and is being followed for a period of 30 days (except if their medical insurance is cancelled, in which case the follow-up period extends to the date of cancellation of this insurance).  Some patients may died during the follow-up period, but since I only know their month & year of death, I am using the date of cancellation for their medical insurance as a proxy for date of death.

Sounds like your second definition of an "event" makes more sense here (i.e., being in hospital on a given day during the 30-day observation period), unless I am getting hung up again.     

Isabella

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

There is one other problem, perhaps, and that is that 0 is an impossible outcome. It sounds like only people admitted to hospital are in the pool, so none would have a 0 day stay.
Perhaps rather than a binomial model, one could consider the number of days in hospital as censored "survival" data, with right censoring occuring on the 30th day. This of course presumes that stays are continuous and not in and out throughout the 30 days....
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Manuela Huso
Research Statistician
US Geological Survey
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Original Message:
Sent: 04-01-2014 15:44
From: Thomas Loughin
Subject: Logit transformation for proportions

Hi Isabella,

Is there a reason you want to stick with a linear regression?  I would be tempted to (ab)use a logistic regression for a problem like this for several reasons.  First, the number of "trials" (days) varies among subjects, so that the proportions have naturally different variances.  Thus, your linear regression should probably have a weighting factor for the number of days observed.  Second, each "trial" has a binary response, which would be appropriate for a logistic regression.  Third, the logit transform does not really stabilize variance, and indeed the variances can be quite different when the proportions are close to 0 or 1. 

The main problem with a logistic regression is the fact that the "trials" (days) are not independent, because if a person is in hospital at day d, they are more likely to be in hospital at d+1 than if they were not in hospital at day d.  The dependence is entirely within-subject, however, and the effects on the model are fairly predictable: the proportions will behave as if they have more variability than the binomial model would expect. That is, they will be overdispersed.  An overdispersion correction can be applied to the analysis to account for the correlation empirically without excess complication.

The usual tools of a linear regression are then at your disposal, including the interpretation of parameters as log-odds-ratios.

-Tom. 

-------------------------------------------
Thomas Loughin
Simon Fraser University
-------------------------------------------







Original Message:
Sent: 04-01-2014 12:37
From: Isabella Ghement
Subject: Logit transformation for proportions

Hi Eric,

Thanks for wanting to be indulged.  Here are the answers to your questions:

1.  The follow-up period is defined as [D1, D30], where
       D1 = Date of admission in hospital and
       D30 = Minimum between { D1 + 30, Date of Cancellation of Medical Coverage}

2.   Yes, some people stay in hospital 30 days or even more than 30 days.  

Isabella

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







Original Message:
Sent: 04-01-2014 12:07
From: Eric Siegel
Subject: Logit transformation for proportions

Hi, Isabella, I have two additional questions:
(1) if their medical insurance is cancelled on Day 45, do they still get only 30 days of follow-up for discharge?
(2) do some people stay in the hospital the whole 30 days?
Thanks for indulging me :)

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------

Beta Regression, that is regression for a dependent variable that is assumed to follow a Beta distribution given the data, does provide log-odd parameter estimates using the logit link. The exponentiation of the the parameter estimate is an odd ratio.

As you state, the inference of a one unit increase in a parameter estimate as being associated with a change in odds of the proportion is a little confusing when compared to the well-known and utilized event / logistic framework.  I have termed that inference not as increase/decrease odds of a proportion, but more simply put as increase/decrease odds of having a larger proportion.  

For example, I used Beta Regression to model volume of stroke lesions as a percentage of total brain volume.  A general inference from this analysis is that increasing age is associated with increasing odds of having a larger percentage of the brain consumed by the stroke - i.e. larger stroke volume. 

I can forward papers offline if you would like.  

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Christopher Swearingen, PhD
Assistant Professor of Pediatrics, Genetics and Biostatistics
Associate Director, Pediatric Biostatistics
University of Arkansas for Medical Sciences, Dept. of Pediatrics
Little Rock, AR
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