I'd like to understand why my partial dependence plots for a logistic regression model simply show up as straight lines -- even when I'd expect basically a threshold effect from a covariate. I know partial dependence plots are typical of machine learning, but the (excellent) description by the authors of the pdp] package suggest glms are fair game. So why does the relationship between outcome and effort (below) appear to be linear?

Here's a dummy dataset. Note that I forced higher values of effort for outcomes corresponding to 1 (a "win"). Also note that sometimes the algorithm won't converge -- if that's the case, just generate new data.


# Sample game data
outcome <- as.vector(cbind(rep(0,25), rep(1,25)))
effort <- as.vector(cbind(rnorm(25, 25, 5), rnorm(25, 50, 10)))
skill <- rnorm(50, 50, 20)
game <- cbind(outcome, effort, skill) %>% as.data.frame()

# Simple glm
mod <- glm(outcome ~ effort + skill, data = game, family = binomial(link = "logit"))
partial(mod, pred.var = c("effort"), plot = TRUE)

glm(formula = outcome ~ effort + skill, family = binomial(link = "logit"), 
    data = game)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.26979  -0.13985  -0.00751   0.01736   2.34734  

             Estimate Std. Error z value Pr(>|z|)   
(Intercept) -15.12758    5.73393  -2.638  0.00833 **
effort        0.50174    0.19218   2.611  0.00903 **
skill        -0.05414    0.05142  -1.053  0.29231   

Clearly, effort is going to be a strong predictor -- with way more wins (1s) associated with higher effort (given my data assignments). However, the partial dependence plot looks like this:

partial(mod, pred.var = c("effort"), plot = TRUE)


If I use a random forest instead, that threshold effect shows up. (Yes, I know it throws a warning about using <5 unique response values in regression. It also shows up if you force outcome to be a factor.)

rf <- randomForest(outcome ~ effort + skill, data = game)
partial(rf, pred.var = c("effort"), plot = TRUE)

# Random forest

My primary question here is not about which model is a better fit, but why the partial dependence is apparently linear with the logistic regression? Why doesn't that 30-40 range pop out as a threshold in the glm plot? Is that truly representing the relationship between game and effort in the model?

Thanks for any insights!


2 Answers 2


A partial dependence plot for a logistic-type model is constructed by setting all but one feature to fixed, static values, varying the remaining feature throughout a range, and plotting:

$$ t \mapsto \log \left( \frac{p}{1-p} \right) $$

Where $p$ is the (probability) prediction for your model when the varied feature is set to the value $t$. Note that, in particular, the $y$-axis of a partial dependency plot is measured on the log-odds scale, not the probability scale.

For a standard logistic regression, the functional form of your model is:

$$ \log \left( \frac{p}{1-p} \right) = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k $$

So the form of the partial dependence plot is:

$$ t \mapsto \beta_j t + \text{constant} $$

where $j$ is the index of the feature you are constructing the partial dependence plot of. This is why you get a line, the slope of that line is the parameter estimate $\hat \beta_j$ in the regression.

In a random forest the functional form of your model is:

$$ p = \text{average} \left( T_0(x), T_1(x), \ldots, T_{\text{n_trees}}(x) \right) $$

where the $T(x)$'s are the probability predictions from your individual classification trees. So the partial dependence plot is the unwieldy:

$$ t \mapsto \log \left( \frac{p}{1-p} \right) = \frac{\text{average} \left( T_0(t), T_1(t), \ldots, T_{\text{n_trees}}(t) \right)}{1 - \text{average} \left( T_0(t), T_1(t), \ldots, T_{\text{n_trees}}(t) \right)} $$

This can be a very complicated, non-linear function of any individual feature, resulting in a vast multitude of possible shapes for the partial dependence plots. The fact that you are seeing a soft threshold shape is due to the particulars of the problem you are solving, not something structural about partial dependence plots.

  • 1
    $\begingroup$ Thanks @Matthew-Drury. That makes really good sense. Glad to be reminded that the y-axis is on log-odds scale, too. $\endgroup$
    – ltlf653
    Feb 27, 2019 at 20:10
  • $\begingroup$ It should be, though looking at that random forest plot, it looks like R is being inconsistent and the logistic one is on the log-odds scale, but the forest is on the probability scale... : ( I just noticed that... $\endgroup$ Feb 27, 2019 at 20:13
  • $\begingroup$ Perhaps because rf is forcing a regression (vs. classification) but didn't figure out the underlying distribution? $\endgroup$
    – ltlf653
    Feb 27, 2019 at 20:15

enter image description hereThe pdp package allows converting back to probability scale by setting "prob = TRUE". You may also add the rugs to display the distribution of predictors.

partial(mod, pred.var = c("effort"), plot = TRUE, prob = TRUE, rug = TRUE)

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.