Solved – Computation and Interpretation of Odds Ratio with continuous variables with interaction, in a binary logistic regression model

logisticoddsodds-ratioregression

Question

A binary logistic regression model has been fitted, with the following output;

    Coefficients

    Term                Coef  SE Coef         95% CI          Z-Value  P-Value 
    Constant           -9.80     3.02    (  -17.00,   -4.19)    -3.12    0.001
    A                   0.756    0.510   (  -0.150,   1.552)     1.58    0.103 
    B                   0.0406   0.0201  ( -0.0046,  0.0798)     1.79    0.069
    A*B                -0.00303  0.00298 (-0.00897, 0.00341)    -0.91    0.341 

    Odds Ratios for Continuous Predictors

            Odds    95%
            Ratio    CI
    A          *   (*, *)
    B          *   (*, *)

Where $A,B$ are both continuous values $> 0$.

How would one give and interpret the odds ratios for one of $A$ or $B$ here?

Typically one would state that (for $A$ here) the odds ratio would be
$\exp(0.756 ) = 2.13$, but given there's interaction this is less clear.

As the variables are continuous it also seems more confusing, if $B$ was been
a binary value then could interpret the model as

$$
\text{log} \left(
\frac{p}{1 – p}
\right)
=
\beta_0 + \beta_1 A + \beta_2 B + \beta_3 A \times B
$$

If $B = 0$ then

$$
\text{log} \left(
\frac{p}{1 – p}
\right)
= \beta_0 + \beta_1 A
$$

If $B = 1$ then

$$
\text{log} \left(
\frac{p}{1 – p}
\right)
= \beta_0 + ( \beta_1 + \beta_3 )A
$$

And state that if $B$ is present then an increase of $A$ by one unit increases the odds by a factor of
$\exp(\beta_1 + \beta_3) = \exp(0.756 – 0.00303)$. And
if $B$ is not present then an increase of $A$ by one unit increases the odds by a factor of
$\exp(\beta_1)$

How to go about computation and interpretation of odds ratios in the case of interaction with continuous variables?

Why I don't think this is a duplicate

From exp (coefficients) to Odds Ratio and their interpretation in Logistic Regression with factors
, talks about using categorical variables and has only one factor with three levels (red, orange, blue)

Interaction Test with Odds
Ratio
didn't talk about a similar scenario to my question.

Log odds ratio and unadjusted log odds ratio when we have a continuous
variable
sounds similar, but is really about putting the continuous variables into
buckets rather than the interpretation of them in the context that I have given.

Comparing odds ratios of continuous and discrete
variables
is discussing their design.

Interpretation of continuous by continuous interaction in binary regression
model
doesn't have an answer, in the comments there are links to
this
post
and this
post, neither of which appear to answer the question, or mine.

Odds ratios for continuous independent variables
[duplicate]
has been flagged as a duplicate, although the post that has been linked (
this post
) doesn't appear to answer the problem as it's talking about categorical
variables ( I have mentioned this post already, red, orange, blue).

In the last thread linked above whuber has stated the information is already
there, I feel that it would be a good idea to have an explicit answer in
relation to the example that I have provided as the topic seems to be confusing
to other learners as well as myself.

Best Answer

As others have noted, it is probably easier to interpret this graphically. I will make certain assumptions to demonstrate the thought process for interpreting interactions like this:

$A$ is my predictor of interest, so I will interpret the odds ratio of $A$ at varying levels of $B$, and
$B$ has a range $[0, 100]$ with mean of 50 and most values falling in $[25, 75]$ such that this is the range of interest.

In the scenario I have set up, the actual log odds of $A$, 0.756, is probably not of interest since it is the log odds of $A$ when $B=0$ and $B=0$ applies to so few people in the data that we do not care for it.

I will calculate the log-odds of $A$ when $B=\{25,50,75\}$. This results in:

\begin{align} \beta_1 + \beta_3 \times \{25, 50, 75\}&{}=\\ 0.756 -0.00303 \times \{25, 50, 75\}&{}=\\ \{0.756 -0.07575, 0.756 -0.15150, 0.756 -0.22725\}&{}=\\ \{0.68025, 0.60450, 0.52875\} \end{align}

The odds ratio of $A$ will then be $1.97\ (e^{0.68025})$, $1.83\ (e^{0.60450})$, $1.70\ (e^{0.52875})$ when $B=\{25, 50, 75\}$ respectively.

So you find that the odds ratio of $A$ drops as the value of $B$ increases. We can also graph the set-up at varying values of B:

To create the graph, I used all integer values of $B$ in $[25, 75]$.

Related Solutions

Solved – Relationship between $\beta_1$ and odds in simple logistic regression

For inference in logistic regression, it would be easier to think in terms of log odds instead of odds. A simple logistic regression model is a generalized linear model with the form $$ \newcommand{\logit}{\rm logit} \newcommand{\odds}{\rm odds} \newcommand{\expit}{\rm expit} \logit(\pi_i) = \beta_0 + \beta_1X_{i1}, $$ where $\logit(\pi_i) = \log(\frac{\pi_i}{1-\pi_i}) = \log(\odds(\pi_i))$. Notice that it has the exact same form as linear regression with a Bernoulli-distributed transformed dependent variable. This has the more intuitive interpretation that for every increase in $X_{i1}$, the expected log odds in favor of $X_{i1}$ increase by $\beta_1$.

If you want to interpret the model in terms of odds, you just have to exponentiate the logit, which is what you initially assumed. This gives you $\odds(\pi_i) = \exp(\beta_0 + \beta_1X_{i1})$, and not $\odds(\pi_i) = \frac{\exp(\beta_0 + \beta_1X_{i1})}{1 + \exp(\beta_0 + \beta_1X_{i1})}$. The interpretation in this case is that for every increase in $X_{i1}$, the expected odds in favor of $Y_i = 1$ increase by $\exp(\beta_1)$. I think either you or your professor got odds and probability mixed up in your professor's explanation.

If you want to draw inference on probabilities, you need to invert the $\logit$ transformation by taking the $\expit$ of both sides, where $\expit(x) = \frac{\exp(x)}{1 + \exp(x)}$. In that case, we have $\pi_i = \frac{\exp(\beta_0 + \beta_1X_{i1})}{1 + \exp(\beta_0 + \beta_1X_{i1})}$ as our expected probability that $Y_i=1$. In this case, it is difficult to draw inference in terms of probabilities, which is why we transform back to probabilities by taking the expit of our expected log odds.

Solved – Continuous variable has very large odds ratio in binary logistic regression

After reading Scortchi and Ben's comments, I think I may have found a solution for you.

I think the problem is your scale of predictor. You know a regression coefficient represents the change in Y (outcome variable) relative to a one unit change in the respective independent variable. For your logistic regression your Y is probability of impaired.

For age, it is 1 years change to affect your probability to be impaired.

For your Theta, it is also 1 units change, but attention, your values are 0.1, 0.2,..., so the value "1" might not in a reasonable rage of your measurement. The same as a human can not live for 1000 years, if your unit is 1000 years, then the coefficient will be huge.

I think you may either divide yoru coefficient by 10 directly or may

change your theta power's unit, I don't know what it is.

Such as multiple your theta values by 10 then the results seem more reasonable.

age<-c(77,84,45,47,72,61,78,49,79,77,74,54,65,52,80)
  theta<-10*c(0.117,0.443,0.136,0.285,0.107,0.113,0.263,0.146,0.182,0.299,0.148,0.097,0.091,0.151,0.302)
impaired<-c(0,1,1,1,1,NA,1,0,1,1,1,0,0,0,NA)
mydata<-data.frame(cbind(age,theta,impaired))

logistic <- glm(impaired ~ age+theta, data = mydata, family = "binomial")
summary(logistic)

The results seem much better, but need to be understood according to your new unit.