How to interpret glm and ols with offset

Question

I ran a few glm and linear models with an offset. Each row in the dataset is a healthcare user. The data contains medical payments and icu days of each user between 2000 to 2007. As the number of years with institute contact differs between user (e.g. some came to the medical institute in 2001-2003 whereas there some who came on all years), I thought I should offset the number of years to account for "observation period". And it is common sense that the longer the years with contact the higher is the payment and icu days.

Gamma:

    Call:
    glm(formula = payment_amt ~ offset(log(years)) + 
        as.factor(gender) + age, 
        family = Gamma(link = "log"), data = pm, control = glm.control(maxit = 50))
    
    Deviance Residuals: 
        Min       1Q   Median       3Q      Max  
    -3.8787  -1.2142  -0.5339   0.1904  15.1442  
    
    Coefficients:
                                       Estimate Std. Error z value Pr(>|z|)    
    (Intercept)                       4.6718536  0.0134132   348.3   <2e-16 ***
    as.factor(gender)M 0.7800695  0.0024625   316.8   <2e-16 ***
    age           0.0642834  0.0001908   337.0   <2e-16 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
    (Dispersion parameter for Gamma family taken to be 1.238685)
    
        Null deviance: 1520252  on 852449  degrees of freedom
    Residual deviance: 1251859  on 852447  degrees of freedom
    AIC: 20497443
    
    Number of Fisher Scoring iterations: 8

OLS:

    Call:
    lm(formula = payment_amt ~ offset(years) + 
        as.factor(gender) + age, 
        data = pm)
    
    Residuals:
        Min      1Q  Median      3Q     Max 
    -170257  -53628  -23808   15835 8808825 
    
    Coefficients:
                                        Estimate Std. Error t value Pr(>|t|)    
    (Intercept)                       -206943.18    1425.83  -145.1   <2e-16 ***
    as.factor(gender)M   48794.00     261.77   186.4   <2e-16 ***
    age              3547.31      20.28   174.9   <2e-16 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
    Residual standard error: 118300 on 852447 degrees of freedom
    Multiple R-squared:  0.07811,   Adjusted R-squared:  0.0781 
    F-statistic: 3.611e+04 on 2 and 852447 DF,  p-value: < 2.2e-16

Poisson:

    Call:
    glm(formula = icu_days ~ offset(log(years)) + as.factor(gender) + 
        age, family = poisson(link = "log"), 
        data = pm)
    
    Deviance Residuals: 
       Min      1Q  Median      3Q     Max  
    -56.95  -15.11   -7.11    3.22  747.64  
    
    Coefficients:
                                        Estimate Std. Error z value Pr(>|z|)    
    (Intercept)                       -5.518e-01  9.058e-04  -609.2   <2e-16 ***
    as.factor(gender)M  6.357e-01  1.341e-04  4738.9   <2e-16 ***
    age            6.395e-02  1.246e-05  5130.9   <2e-16 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
    (Dispersion parameter for poisson family taken to be 1)

Negative binomial:

    Call:
    glm.nb(formula = icu_days ~ offset(log(years)) + 
    as.factor(gender) + 
    age, data = pm, init.theta = 0.9279403178, 
    link = log)
    Deviance Residuals:
    Deviance Residuals: 
            Min       1Q   Median       3Q      Max  
         -2.7720  -1.0788  -0.4652   0.1641  17.2095  

        Coefficients:
                                        Estimate Std. Error z value Pr(>|z|)    
        (Intercept)                       -1.0038131  0.0126237  -79.52   <2e-16 ***
        as.factor(gender)M  0.5977179  0.0023017  259.69   <2e-16 ***
        age            0.0708916  0.0001795  394.97   <2e-16 ***
        ---
        Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

        (Dispersion parameter for Negative Binomial(0.9279) family taken to be 1)

May I know how do i interpret the coefficients?

Answer 1

You show four models, one of them is strange (the one marked OLS) so I will first discuss the three others. What is common is that a response variable (amount of pay for the Gamma, a count of days for the others) is modeled as a function of covariates, but what is modeled is value pr year, that is: $$ \DeclareMathOperator{\E}{\mathbb{E}} \frac{\E Y}{\text{years}} = \exp(\beta^T x) = \exp(\eta), \qquad \text{say} $$ (as you are using a log link function). Taking logarithms (natural) on both sides give: $$ \log \E Y - \log\text{years} = \eta $$ and moving one term above over on the right hand side: $$ \log \E Y = \log\text{years} + \eta $$ and that answers your question: the combination of a log link function, a non-negative response and offset of log of exposure means that you are modeling the ratio of total amount divided into exposure, that is, payment per year or number of days per year. The interpretation of the other variables (linear parameters in the linear predictor $\eta$ should then be clear, it is interpreted as usual in regression models. Please note that the exact choice of distribution family in the explanation above do not play any role!

Your fourth model do not make much sense to me, you can of course use a gaussian model for a non-negative response, that in itself is not a problem. But, the use of identity link function together with offset of log exposure is meaningless, as far as I can see. Let us try to mimic the logic above: $$ \frac{\E Y}{\text{years}} = \beta^T x = \eta, \qquad \text{say} $$ Note we changed to identity link function. But now taking logs doesn't work as above, that worked above only because we had a log link function. So I would not try to interpret the parameters in your OLS model, they are meaningless. But you could try a gaussian family, but with log link! For a more extensive discussion of the arguments above, see Goodness of fit and which model to choose linear regression or Poisson

EDIT  

To the additional question in comments: So does that mean offset only works for log links? No, but log link and rate modeling is the major use case. Most posts on this site mentioning offset is this case. A few posts discussing other uses: Binary Models (Probit and Logit) with a Logarithmic Offset , Using offset in binomial model to account for increased numbers of patients , Offset in Logistic regression: what are the typical use cases?

Answer 2

Your OLS model:

lm(formula = payment_amt ~ offset(years) + 
    as.factor(gender) + age, 
    data = pm)

Is the same as:

lm(formula = payment_amt - years ~ as.factor(gender) + age,
   data = pm)

With a log-link, you can use offsets to model rates because of how math works with logarithms, but for an identity link there's really no point to using an offset.