# Interpretation of multiple logistic regression (continuous variables) with interaction term

I have two independent variables, distance and light intensity (continuous variables) that I want to understand their relationship on a human response (response or no response). Here, light intensity represents the stimulus and distance represents how far people are from it.

Initially, I carried out a t-test of independent means, which indicated a difference in distance and light intensity between response (group 1) and no response (group 2).

I then performed a bivariate logistic regression, to better understand the effect size of distance and light intensity on the response (0 = no response, 1 = response). Note, I performed a linear correlation prior which indicated a weak negative correlation between distance and light intensity (r = -0.18, p = 0.000059).

Bivariate logistic regression results indicate that distance has a larger effect compared to light intensity (see below). So, for every 1-unit increase in distance (meters), the likelihood of a response decreases by 66.9% (exp(-1.1065) - 1 x 100) (or, the closer in distance a person is, the likelihood of a response increases by 33.1%). In contrast, for every 1-unit increase in light intensity (lux), the likelihood of a response increases only by 0.045% (exp(0.0004569)- 1 x 100).

Bivariate logistic regression (light intensity)

Call:
glm(formula = response ~ light.intensity, family = binomial(link = "logit"),
data = df)

Coefficients:
Estimate Std. Error z value             Pr(>|z|)
(Intercept)     -2.7016713  0.2497537 -10.817 < 0.0000000000000002 ***
light.intensity  0.0004569  0.0001209   3.779             0.000158 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 350.35  on 481  degrees of freedom
Residual deviance: 336.97  on 480  degrees of freedom
AIC: 340.97

Number of Fisher Scoring iterations: 5


Bivariate logistic regression (distance)

Call:
glm(formula = response ~ mean.distance, family = binomial(link = "logit"),
data = df)

Coefficients:
Estimate Std. Error z value          Pr(>|z|)
(Intercept)     3.5397     0.5853   6.048 0.000000001469798 ***
mean.distance  -1.1065     0.1491  -7.423 0.000000000000114 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 350.35  on 481  degrees of freedom
Residual deviance: 161.36  on 480  degrees of freedom
AIC: 165.36

Number of Fisher Scoring iterations: 8


In order to explain the relationship between distance and light intensity on the response, I performed a multiple logistic regression with an interaction term to specify a joint effect (see below). After reading, including responses by @Marrtin Buis and @Dave regarding interaction terms, my interpretation of the result is that:

• The likelihood of a response increases by 46.3% for every 1-unit decrease in distance, when light intensity is 0.
• While the effect size of light intensity is small, it does not contribute significantly to this model (p-value = 0.14)
• However, as an interaction term, the interaction effect of light intensity on distance approaches significance (p-value = 0.093162) but the effect is very small. So, for every unit increase in light intensity, the effect of distance decreases only by 0.03% (exp(-0.03) - 1 x 100).
Call:
glm(formula = response ~ distance + light.intensity + distance:light.intensity,
family = binomial(link = "logit"), data = df)

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)               2.4744393  0.9502004   2.604 0.009211 **
distance                 -0.7698838  0.2295094  -3.354 0.000795 ***
light.intensity           0.0008299  0.0005656   1.467 0.142285
distance:light.intensity -0.0002745  0.0001635  -1.679 0.093162 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 350.35  on 481  degrees of freedom
Residual deviance: 157.99  on 478  degrees of freedom
AIC: 165.99

Number of Fisher Scoring iterations: 8


Is the interpretation of the logistic regression results correct?

You write:

Univariate logistic regression results indicate that distance has a larger effect compared to light intensity (see below).

and then give ORs per unit. But this is not correct. If you change the units, you change the ORs, and can reverse the relationship. Change meters to kilometers and the OR for distance will be tiny.

The very small (but highly significant) OR in the bivariate relationship for light intensity is a sign that "lux" may be too large a measure. (Two notes: First, I dislike using "univariate" here, the relationship has two variables. Second, the scale of measurement doesn't change the meaning of the model, but it makes the results easier to interpret).

While the effect size of light intensity is small, it does not contribute significantly to this model (p-value > .5)

First, the p value is 0.14 which is not > 0.5. Maybe this is a typo and you met 0.05? More importantly, p value is not a measure of whether a variable "contributes significantly" to a model.

Your interpretation of the interaction seems right, but the size will be affected if you change the units of light intensity.

• Thank you very much for the feedback. I edited univariate to bivariate and will correct the p-value (it was a typo). To clarify, my understanding of your first comment is that if different scales of measurement is considered, then comparing the bivariate results to each other, as I have, will change. However, if the scale of measurement is context specific (as in, its not relevant to change meters to km), than is my interpretation valid? Regarding the p-value, I will revise my understanding of that thanks!
– kpm
Oct 1 at 12:04
• Scale of measurement, at least for continuous variables, is always subject to change of scale. Even for dummy coded variables, while 0 vs. 1 is most common, some people use other values. For some variables (e.g. IQ) one scale is so common that any change would require a lot of justification. I don't see how that could be the case here. You are talking about two physical measures. Oct 1 at 13:30
• Comparing the importance of different measures in a regression is tricky and has been discussed here and elsewhere. One way to do this is to standardize the variables so that they all have mean 0 and sd 1. I am not a big fan of this, mostly because the SD is specific to your sample. See e.g stats.stackexchange.com/questions/471941/… and stats.stackexchange.com/questions/572293/… Oct 1 at 13:33
• The feedback was valuable. I found an article which discusses the case for standardization for logistic regression: jstor.org/stable/41290135. Thanks again.
– kpm
Oct 1 at 22:37