# How can I interpret natural logarithm transformed value in the binary regression result

I have the binominal regression output result using R programming and x3 below is the natural logarithm of the data which is large. The x3 before imposing the natural logarithm has the following descriptive statistics:

    Min     1065530
Max     7250000000
Mean    842062274.8
Median  500000000


I am trying to provide an economic interpretation of increasing one unit of x3. However, as x3 itself is in natural logarithm and has coefficient of -1.598 as below, I am not sure how to interpret this coefficient in this case. The binary regression output is as below:

    > x_reg=glm(y~x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14, data=DATA, family=binomial(link="logit"))
> summary(x_reg)

Call:
glm(formula = y ~x1 + x2 +x3 +
x4 + x5 + x6 + x7 + x8 +
x9 + x10 + x11 + x12 + x13
+ x14, family = binomial(link = "logit"), data = DATA)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-2.6596  -0.6843  -0.3166   0.7505   2.4914

Coefficients:
Estimate    Std. Error  z value Pr(>|z|)
(Intercept)     27.372022   9.5791     2.857    0.004271    **
x1              0.572836    0.236129    2.426   0.015269    *
x2              0.437914    1.40225     0.312   0.754817
x3             -1.597899    0.491549    -3.251  0.001151    **
x4             -0.591385    0.333672    -1.772  0.076336    .
x5              0.881187    0.264665    3.329   0.00087     ***
x6              0.126759    0.060748    2.087   0.036923    *
x7             -0.270982    0.429146    -0.631  0.52775
x8              0.020369    0.372997    0.055   0.956449
x9             -3.580657    0.932135    -3.841  0.000122    ***
x10             0.22745     0.092035    2.471   0.01346     *
x11            -0.01306     0.020641    -0.633  0.526921
x12             0.009792    0.023543    0.416   0.677482
x13             0.002869    0.01344     0.213   0.830976
x14            -0.010232    0.055211    -0.185  0.852973
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 181.05  on 131  degrees of freedom
Residual deviance: 124.43  on 117  degrees of freedom
(390 observations deleted due to missingness)
AIC: 154.43

Number of Fisher Scoring iterations: 5


May I know how to interpret the x3 which is in natural logarithm?

• Is there any utility in expressing X3 in millions, such that the min becomes 1.06553 and the max becomes 7250? That way no log transform would be needed. Commented Dec 16, 2018 at 23:30

You can linearize the natural logarithm (which has derivative $$1 / x$$) to get to this:

For every one-unit change in the original $$x_3$$ near specific point $$x_3^*$$, there's a change of $$+ \beta_3 / x_3^*$$ in the log odds or $$\times e^{\beta_3 / x_3^*}$$ in the odds.

If you really want one number for the effect of $$x_3$$, then you could plug in the mean for $$x_3^*$$ but note that's an approximation that gets worse as you move away from the mean.

More direct interpretations of $$\beta_3$$ are:

1. The amount the log odds of the event decreases for every $$e$$-fold (approximately 2.7 times) increase in the raw value of $$x_3$$. The multiplier $$e$$ is the base of the logarithm you used.

2. The power $$k$$ of the power-law relationship between the odds of the event and $$x_3$$, while all other covariates are held constant.

The second interpretation comes from: $$\text{log(odds)} = \textit{stuff} + \beta_3 \log(x_3)$$ $$\text{odds} = e^{\textit{stuff}} e^{\log(x_3^{\beta_3})}$$ $$\text{odds} = e^{\textit{stuff}} x_3^{\beta_3}$$ though I don't know it would be especially useful in this case given $$x_3^{\beta_3}$$ is going to be very close to zero for the values in your data set (with a very large intercept to counteract it).

• Let’s say x3 is the firm size measured by total asset of a firm with the descriptive statistics as shown above. If so, how would it be economically interpreted in this case in unit increase wise?
– Eric
Commented Dec 16, 2018 at 22:43
• Could you explain where the "2.7" comes from? There may be some ambiguity concerning whether you are discussing $x_3$ or the original data $\exp(x_3).$
– whuber
Commented Dec 16, 2018 at 23:18
• @whuber, thx for the feedback - attempted to resolve the ambiguities. Eric, I took another shot at it to provide a more direct answer to your question. It's based on a pretty crude approximation so take it with a grain of salt. Commented Dec 17, 2018 at 4:41
• Well I'm afraid we won't have economically practical answer based on the nature of the problem. However, if there are any, it would be awesome.
– Eric
Commented Dec 18, 2018 at 17:39