How to interpret beta coefficients of a generalized linear model using inverse gaussian

Question

I made a generalized linear model with an inverse gaussian link.

glm(lone_total ~ class + age + basic_needs_covered_id,
      data = mod_data_lone,
      family = gaussian(link = "inverse")
      )

I have chosen this family and link because of the diagnostic plots, which were bad for inverse.gaussian(link="1/mu^2") or Gamma(link="log") (See distribution of outcome variable in the image below)

The results with the following coefficients:

    Coefficients:
                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)             0.784873   0.066836  11.743  < 2e-16 ***
class2                 -0.070565   0.067936  -1.039    0.300    
class3                  0.171242   0.203703   0.841    0.401    
age                     0.003912   0.003444   1.136    0.257    
basic_needs_covered_id -0.110093   0.026307  -4.185  4.1e-05 ***

how do I interpret the Estimates? I tried to interpret them like betas from gamma or poisson models, or to run log() and exp() on them, but nothing really made sense to me.

Descriptives of the Data

> str(mod_data_lone)
'data.frame':   229 obs. of  5 variables:
 $ lone_total            : num  0.01 3 3 1 1 1 0.01 3 0.01 0.01 ...
 $ class                 : Factor w/ 3 levels "1","2","3": 3 2 1 3 1 1 1 1 1 2 ...
 $ age                   : num  -14.44 -8.27 7.38 NA 13.99 ...
 $ basic_needs_covered_id: num  0 4 0 2 1 2 0 1 0 1 ...
 $ education_id          : num  6 8 6 6 6 2 6 6 6 6 ...

The following image shows the barplot and density distributions and means by class of the observations:

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another table I got is the following, which shows different types of B (which all do not really make sense to me...)

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Thanks for any help!

Answer 1

I think I found what I need

Y=1/(β2X2+β1X1+β0)

from here: https://cran.r-project.org/web/packages/GlmSimulatoR/vignettes/exploring_links_for_the_gaussian_distribution.html

Example 1: Person from Class 3, mean Age and basic_needs_covered = 4

1/(0.78+0.17+4(-0.11)) = 1.96*

Example 2: Person from Class 2, mean Age and basic_needs_covered = 4

1/(0.78-0.07+4(-0.11)) = 3.70*

Answer 2

In a generalized linear model, a link function $g()$ defines the association between the expected outcome $y$ and the $k$ corresponding predictor values $x_j$:

$$g(y)=\beta_0 + \sum_{j=1}^k \beta_i x_j.$$

If you specify an inverse link in your model, $g(y)= 1/y$, then it will be difficult to have an easy interpretation of associations of individual coefficients with outcome, because you then have:

$$y=\frac{1}{\beta_0 + \sum_{j=1}^k \beta_i x_j} .$$

Even without interactions among predictors, the association of any one predictor with outcome thus depends on the values of the others. You can still use that formula for predictions, however, as your answer illustrates.

With a numeric outcome scale representing increasing values of loneliness, this type of data might instead be modeled via ordinal logistic regression. With a logit link, the individual predictor coefficients then have reasonably straightforward interpretations in terms of changes in the log-odds of changing the outcome by 1 level, given that other predictor values are held constant.