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

Question

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?

Answer 1

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).