Monitoring relationship between score and odds in a propensity scorecard - ePrints Soton
relationships between what an applicant looked like, in terms of predictive variables, at the time of application . Odds at a score of with 40 points to double the odds; this is highly flexible based upon your needs . If employees do not trust .. Using the natural logarithm, you calculate the logarithm of that ratio to create. Outline. • Propensity scoring. • Methodology. – Kalman-filter-based monitoring. – Relationship of score and log odds. • Empirical example. • Conclusions. Under a logistic regression model, for the logit link function, the we obtain a combined Z-score for association of the jth SNP across GWAS by Abbreviation: GRM, genetic relationship matrix. .. Andrew P Morris is a Wellcome Trust Senior Fellow in Basic Biomedical Science (under award WT).
Writing it in an equation, the model describes the following linear relationship. We can manually calculate these odds from the table: Now we can relate the odds for males and females and the output from the logistic regression. The intercept of Using the odds we calculated above for males, we can confirm this: The coefficient for female is the log of odds ratio between the female group and male group: So we can get the odds ratio by exponentiating the coefficient for female.
Most statistical packages display both the raw regression coefficients and the exponentiated coefficients for logistic regression models. The table below is created by Stata. In other words, the odds of being in an honors class when the math score is zero is exp These odds are very low, but if we look at the distribution of the variable math, we will see that no one in the sample has math score lower than In fact, all the test scores in the data set were standardized around mean of 50 and standard deviation of So the intercept in this model corresponds to the log odds of being in an honors class when math is at the hypothetical value of zero.
How do we interpret the coefficient for math? The coefficient and intercept estimates give us the following equation: We will use We can examine the effect of a one-unit increase in math score.
FAQ: How do I interpret odds ratios in logistic regression?
Taking the difference of the two equations, we have the following: We can say now that the coefficient for math is the difference in the log odds. In other words, for a one-unit increase in the math score, the expected change in log odds is. Can we translate this change in log odds to the change in odds? Recall that logarithm converts multiplication and division to addition and subtraction. Its inverse, the exponentiation converts addition and subtraction back to multiplication and division.
If we exponentiate both sides of our last equation, we have the following: Logistic regression with multiple predictor variables and no interaction terms In general, we can have multiple predictor variables in a logistic regression model. Each exponentiated coefficient is the ratio of two odds, or the change in odds in the multiplicative scale for a unit increase in the corresponding predictor variable holding other variables at certain value.
Here is an example. Logistic regression with an interaction term of two predictor variables In all the previous examples, we have said that the regression coefficient of a variable corresponds to the change in log odds and its exponentiated form corresponds to the odds ratio. This is only true when our model does not have any interaction terms. The interpretation of the regression coefficients become more involved. In this simple example where we examine the interaction of a binary variable and a continuous variable, we can think that we actually have two equations: Now we can map the logistic regression output to these two equations.
More explicitly, we can say that for male students, a one-unit increase in math score yields a change in log odds of 0. On the other hand, for the female students, a one-unit increase in math score yields a change in log odds of. A new individualized method we are proposing. An individualized heuristic feature attribution method. A global attribution method based on the average magnitude of the individualized Tree SHAP attributions.
The same method used above in XGBoost, and also equivalent to the Gini importance measure used in scikit-learn tree models. The resulting drop in accuracy of the model when a single feature is randomly permuted in the test data set.
Feature attributions for model A and model B using six different methods. As far we can tell, these methods represent all the tree-specific feature attribution methods in the literature.
All the previous methods other than feature permutation are inconsistent! This is because they assign less importance to cough in model B than in model A. Inconsistent methods cannot be trusted to correctly assign more importance to the most influential features. The astute reader will notice that this inconsistency was already on display earlier when the classic feature attribution methods we examined contradicted each other on the same model.
What about the accuracy property? It is perhaps surprising that such a widely used method as gain gini importance can lead to such clear inconsistency results. This is the error from the constant mean prediction of After splitting on fever in model A the MSE drops toso the gain method attributes this drop of to the fever feature.
Splitting again on the cough feature then leads to an MSE of 0, and the gain method attributes this drop of to the cough feature. In model B the same process leads to an importance of assigned to the fever feature and to the cough feature: Computation of the gain aka.
Typically we expect features near the root of the tree to be more important than features split on near the leaves since trees are constructed greedily. Yet the gain method is biased to attribute more importance to lower splits.
This bias leads to an inconsistency, where when cough becomes more important and it hence is split on at the root its attributed importance actually drops. The individualized Saabas method used by the treeinterpreter package calculates differences in predictions as we descend the tree, and so it also suffers from the same bias towards splits lower in the tree.
As trees get deeper, this bias only grows. In contrast the Tree SHAP method is mathematically equivalent to averaging differences in predictions over all possible orderings of the features, rather than just the ordering specified by their position in the tree. Given that we want a method that is both consistent and accurate, it turns out there is only one way to allocate feature importances.
The details are in our recent NIPS paperbut the summary is that a proof from game theory on the fair allocation of profits leads to a uniqueness result for feature attribution methods in machine learning.
Interpretable Machine Learning with XGBoost – Towards Data Science
The SHAP values we use here result from a unification of several individualized model interpretation methods connected to Shapley values. Armed with this new approach we return to the task of interpreting our bank XGBoost model: Since the impact of hiding a feature changes depending on what other features are also hidden, Shapley values are used to enforce consistency and accuracy.
We can see that the relationship feature is actually the most important, followed by the age feature. However, since we now have individualized explanations for every person, we can do more than just make a bar chart.
FAQ: How do I interpret odds ratios in logistic regression?
We can plot the feature importance for every customer in our data set. The shap Python package makes this easy. We first call shap. Every customer has one dot on each row. By plotting the impact of a feature on every sample we can also see important outlier effects.
For example, while capital gain is not the most important feature globally, it is by far the most important feature for a subset of customers. We can do that for the age feature by plotting the age SHAP values changes in log odds vs. The x-axis is the age of the customer. Here we see the clear impact of age on earning potential as captured by the XGBoost model.
Even though many people in the data set are 20 years old, how much their age impacts their prediction differs as shown by the vertical dispersion of dots at age This means other features are impacting the importance of age.