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Please note that an earlier version of this post had to be retracted because it contained some content which was generated at work. I have since chosen to rewrite the document in a series of posts. Please recognize that this may take some time. Apologies for any inconvenience.

Logistic regression is used to analyze the relationship between a dichotomous dependent variable and one or more categorical or continuous independent variables. It specifies the likelihood of the response variable as a function of various predictors. The model expressed as $log(odds) = \beta_0 + \beta_1*x_1 + ... + \beta_n*x_n$, where $\beta$ refers to the parameters and $x_i$ represents the independent variables. The $log(odds)$, or log of the odds ratio, is defined as $ln[\frac{p}{1-p}]$. It expresses the natural logarithm of the ratio between the probability that an event will occur, $p(Y=1)$, to the probability that an event will not occur, $p(Y=0)$.

The models estimates, $\beta$, express the relationship between the independent and dependent variable on a log-odds scale. A coefficient of $0.020$ would indicate that a one unit increase in $\beta_i$ is associated with a log-odds increase in the occurce of $Y$ by $0.020$. To get a clearer understanding of the constant effect of a predictor on the likelihood that an outcome will occur, odds-ratios can be calculated. This can be expressed as $odds(Y) = \exp(\beta_0 + \beta_1*x_1 + ... + \beta_n*x_n)$, which is the exponentiate of the model. Alongside the odd-ratio, it’s often worth calculating predicted probabilities of $Y$ at specific values of key predictors. This is done through $p = \frac{1}{1 + \exp^{-z}}$ where z refers to the $log(odds)$ regression equation.

Using the GermanCredit dataset in the Caret package, we will construct a logistic regression model to estimate the likelihood of a consumer being a good loan applicant based on a number of predictor variables.

library(caret)
data(GermanCredit)

Train <- createDataPartition(GermanCredit$Class, p=0.6, list=FALSE) training <- GermanCredit[ Train, ] testing <- GermanCredit[ -Train, ] mod_fit_one <- glm(Class ~ Age + ForeignWorker + Property.RealEstate + Housing.Own + CreditHistory.Critical, data=training, family="binomial") summary(mod_fit_one) # estimates exp(coef(mod_fit$finalModel)) # odds ratios
predict(mod_fit_one, newdata=testing, type="response") # predicted probabilities

Great, we’re all done, right? Not just yet. There are some critical questions that still remain. Is the model any good? How well does the model fit the data? Which predictors are most important? Are the predictions accurate? In the next few posts, I’ll provide an overview of how to evaluate logistic regression models in R.  