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Logistic Regression in R Tutorial

Logistic Regression in R Tutorial

Discover all about logistic regression: how it differs from linear regression, how to fit and evaluate these models it in R with the glm() function and more!

Logistic regression is yet another technique borrowed by machine learning from the field of statistics. It's a powerful statistical way of modeling a binomial outcome with one or more explanatory variables. It measures the relationship between the categorical dependent variable and one or more independent variables by estimating probabilities using a logistic function, which is the cumulative logistic distribution.

This R tutorial will guide you through a simple execution of logistic regression:

Tip: if you're interested in taking your skills with linear regression to the next level, consider also DataCamp's Multiple and Logistic Regression course!

Regression Analysis: Introduction

As the name already indicates, logistic regression is a regression analysis technique. Regression analysis is a set of statistical processes that you can use to estimate the relationships among variables. More specifically, you use this set of techniques to model and analyze the relationship between a dependent variable and one or more independent variables. Regression analysis helps you to understand how the typical value of the dependent variable changes when one of the independent variables is adjusted and others are held fixed.

As you already read, there are various regression techniques. You can distinguish them by looking at three aspects: the number of independent variables, the type of dependent variables and the shape of regression line.

Linear Regression

Linear regression is one of the most widely known modeling techniques. It allows you, in short, to use a linear relationship to predict the (average) numerical value of for a given value of with a straight line. This line is called the "regression line".

As a consequence, the linear regression model is . The model assumes that the response variable is quantitative. However, in many situations, the response variable is qualitative or, in other words, categorical. For example, gender is qualitative, taking on values male or female.

Prediciting a qualitative response for an observation can be referred to as classifying that observation, since it involves assigning the observation to a category, or class. On the other hand, the methods that are often used for classification first predict the probability of each of the categories of a qualitative variable, as the basis for making the classification.

Linear regression is not capable of predicting probability. If you use linear regression to model a binary response variable, for example, the resulting model may not restrict the predicted Y values within 0 and 1. Here's where logistic regression comes into play, where you get a probaiblity score that reflects the probability of the occurrence at the event.

Logistic Regression

Logistic regression is an instance of a classification technique that you can use to predict a qualitative response. More specifically, logistic regression models the probability that belongs to a particular category.

That means that, if you are trying to do gender classification, where the response falls into one of the two categories, male or female, you'll use logistic regression models to estimate the probability that belongs to a particular category.

For example, the probability of given can be written as:

The values of (abbreviated as ) will range between 0 and 1. Then, for any given value of , a prediction can be made for .

Given as the explanatory variable and as the response variable, how should you then model the relationship between and ? The linear regression model represents these probabilities as:

The problem with this approach is that, any time a straight line is fit to a binary response that is coded as or , in principle we can always predict for some values of and for others.

To avoid this problem, you can use the logistic function to model that gives outputs between and for all values of :

The logistic function will always produce an S-shaped curve, so regardless of the value of , we will obtain a sensible prediction.

The above equation can also be reframed as:

The quantity

is called the odds ratio, and can take on any value between and . Values of the odds ratio close to and indicate very low and very high probabilities of , respectively.

By taking the logarithm of both sides from the equation above, you get:

The left-hand side is called the logit. In a logistic regression model, increasing X by one unit changes the logit by . The amount that changes due to a one-unit change in X will depend on the current value of X. But regardless of the value of X, if is positive then increasing X will be associated with increasing , and if is negative then increasing X will be associated with decreasing .

The coefficients and are unknown, and must be estimated based on the available training data. For logistic regression, you can use , a powerful statistical technique. Let's refer back to your gender classification example.

You seek estimates for and such that plugging these estimates into the model for yields a number close to 1 for all individuals who are female, and a number close to 0 for all individuals who are not.

This intuition can be formalized using a mathematical equation called a likelihood function:

The estimates and are chosen to maximize this likelihood function. Once the coefficients have been estimated, you can simply compute the probability of being given any instance of having . Overall, maximum likelihood is a very good approach to fit non-linear models.

Multinomial Logistic Regression

So far, this tutorial has only focused on Binomial Logistic Regression, since you were classifying instances as male or female. Multinomial Logistic Regression model is a simple extension of the binomial logistic regression model, which you use when the exploratory variable has more than two nominal (unordered) categories.

In multinomial logistic regression, the exploratory variable is dummy coded into multiple 1/0 variables. There is a variable for all categories but one, so if there are M categories, there will be dummy variables. Each category’s dummy variable has a value of 1 for its category and a 0 for all others. One category, the reference category, doesn’t need its own dummy variable, as it is uniquely identified by all the other variables being 0.

The mulitnomial logistic regression then estimates a separate binary logistic regression model for each of those dummy variables. The result is binary logistic regression models. Each model conveys the effect of predictors on the probability of success in that category, in comparison to the reference category.

Ordinal Logistic Regression

Next to multinomial logistic regression, you also have ordinal logistic regression, which is another extension of binomial logistics regression. Ordinal regression is used to predict the dependent variable with ‘ordered’ multiple categories and independent variables. You already see this coming back in the name of this type of logistic regression, since "ordinal" means "order of the categories".

In other words, it is used to facilitate the interaction of dependent variables (having multiple ordered levels) with one or more independent variables.

For example, you are doing customer interviews to evaluate their satisfaction towards our newly released product. You are tasked to ask a question to respondent where their answer lies between or . To generalize the answers well, you add levels to your responses such as , , , , . This helped you to observe a natural order in the categories.

Logistic Regression in R with glm

In this section, you'll study an example of a binary logistic regression, which you'll tackle with the ISLR package, which will provide you with the data set, and the glm() function, which is generally used to fit generalized linear models, will be used to fit the logistic regression model.

Loading Data

The first thing to do is to install and load the ISLR package, which has all the datasets you're going to use.

source("requirements.R")
knitr::opts_chunk$set(
    dpi = 150, 
    fig.width = 6, 
    fig.height = 4,
    message = FALSE, 
    warning = FALSE)
require(ISLR)

For this tutorial, you're going to work with the Smarket dataset within RStudio. The dataset shows daily percentage returns for the S&P 500 stock index between 2001 and 2005.

Exploring Data

Let's explore it for a bit. names() is useful for seeing what's on the data frame, head() is a glimpse of the first few rows, and summary() is also useful.

names(Smarket)
head(Smarket)
summary(Smarket)

The summary() function gives you a simple summary of each of the variables on the Smarket data frame. You can see that there's a number of lags, volume, today's price, and direction. You will use Direction as a response vairable, as that shows whether the market went up or down since the previous day.

Visualizing Data

Data visualization is perhaps the fastest and most useful way to summarize and learn more about your data. You'll start by exploring the numeric variables individually.

Histograms provide a bar chart of a numeric variable split into bins with the height showing the number of instances that fall into each bin. They are useful to get an indication of the distribution of an attribute.

par(mfrow = c(1, 8))
for (i in 1:8) {
  hist(Smarket[, i], main = names(Smarket)[i])
}

It's extremely hard to see, but most of the variables show a Gaussian or double Gaussian distribution.

You can look at the distribution of the data a different way using box and whisker plots. The box captures the middle 50% of the data, the line shows the median and the whiskers of the plots show the reasonable extent of data. Any dots outside the whiskers are good candidates for outliers.

par(mfrow = c(1, 8))
for (i in 1:8) {
  boxplot(Smarket[, i], main = names(Smarket)[i])
}

You can see that the Lags and Today all has a similar range. Otherwise, there's no sign of any outliers.

Missing data have have a big impact on modeling. Thus, you can use a missing plot to get a quick idea of the amount of missing data in the dataset. The x-axis shows attributes and the y-axis shows instances. Horizontal lines indicate missing data for an instance, vertical blocks represent missing data for an attribute.

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