Problem 6 - An Introduction to Statistical Learning with Applications in R

Question

We continue to consider the use of a logistic regression model to predict the probability of default using income and balance on the Default data set. In particular, we will now compute estimates for the standard errors of the income and balance logistic regression coefficients in two different ways: (1) using the bootstrap, and (2) using the standard formula for computing the standard errors in the glm() function. Do not forget to set a random seed before beginning your analysis. (a) Using the summary() and \(\operatorname{glm}(\) ) functions, determine the estimated standard errors for the coefficients associated with income and balance in a multiple logistic regression model that uses both predictors. (b) Write a function, boot .fn (), that takes as input the Default data set as well as an index of the observations, and that outputs the coefficient estimates for income and balance in the multiple logistic regression model. (c) Use the boot () function together with your boot. fn () function to estimate the standard errors of the logistic regression coefficients for income and balance. (d) Comment on the estimated standard errors obtained using the \(\operatorname{glm}()\) function and using your bootstrap function.

Step-by-Step Solution

Answer

Standard errors from glm() and bootstrap may vary slightly due to different estimation methods.

Step 1: Load required libraries and data

We load the necessary R libraries and data for conducting the logistic regression. Specifically, we need the `boot` library for the bootstrap and ensure the `Default` dataset is available. Also, set a random seed for reproducibility, for example, using `set.seed(123)`. This ensures our results can be consistently regenerated.

Step 2: Fit logistic regression using glm()

Use the `glm()` function to fit a logistic regression model on the `Default` dataset. The dependent variable is `default`, and the predictors are `income` and `balance`. Use the family argument as `binomial` because logistic regression is a type of binary classification model. Example syntax: ```R glm.fit <- glm(default ~ income + balance, data = Default, family = "binomial") ```

Step 3: Calculate standard errors using summary()

The `summary()` function provides detailed information about the fitted model, including the estimates of coefficients and their standard errors. Extract the standard errors from the summary output for the coefficients associated with `income` and `balance`. ```R summary(glm.fit)$coefficients ```

Step 4: Define boot.fn() function

Write a `boot.fn` function that accepts the dataset and an index of observations as inputs. It should recalibrate a logistic regression model on the provided subset of the dataset and return the coefficients for `income` and `balance`. ```R boot.fn <- function(data, index) { fit <- glm(default ~ income + balance, data = data[index,], family = "binomial") return(coef(fit)[2:3]) } ```

Step 5: Perform bootstrap method with boot()

Use the `boot()` function available in the `boot` package to bootstrap the `boot.fn` function. This will estimate the standard errors of the logistic regression coefficients by resampling the data several times. Use a large number of bootstrap samples for accuracy, for instance, 1000. ```R library(boot) boot.results <- boot(data = Default, statistic = boot.fn, R = 1000) boot.results ```

Step 6: Compare standard errors

Compare the standard errors obtained from the `summary()` function and the `boot()` function. Observe any differences or similarities and comment on their implications. Normally, the bootstrapped standard errors may slightly differ due to sampling variance.

Key Concepts

Bootstrap Method

The **Bootstrap Method** is a powerful statistical tool used to estimate the sampling distribution of an estimator by resampling with replacement from the data. This technique is very helpful when the theoretical distribution of a statistic is complex or unknown. Bootstrap allows for the approximation of standard errors and confidence intervals for almost any statistic.
The process involves:

• Randomly selecting samples from the original dataset with replacement.
• Calculating the statistic of interest (like the mean or regression coefficients) for each resample.
• Repeating this process a large number of times to create a distribution of the statistic.

By using the bootstrap method, we can understand the variability of our estimates without relying on strict parametric assumptions. It is particularly useful in logistic regression when evaluating standard errors of coefficients, as demonstrated when using the `boot()` function in R programming.

Standard Errors

**Standard Errors** are a measure of the statistical accuracy of an estimator. They describe how much the estimated value of a coefficient would vary if we were to collect new samples repeatedly. In the context of logistic regression, standard errors help assess the reliability and significance of predictor variables such as income and balance.
In R, standard errors are easily obtained using the `summary()` function after fitting the model with the `glm()` function. These standard errors are based on the assumptions of the logistic regression model being correctly specified and the truth of the underlying data distribution.
However, these assumptions can sometimes be violated in practice. This is why bootstrap standard errors, estimated through repeated resampling as mentioned earlier, offer a robust alternative as they do not depend on the normality of the estimator's sampling distribution.

glm function

The **`glm` function** stands for 'Generalized Linear Model' and is a versatile tool in R for modeling response variables that are not normally distributed, such as binary outcomes. For logistic regression, the `glm()` function is used with a `binomial` family argument, to model binary outcomes like default status.
Using `glm()`, you can perform logistic regression in R with the following syntax: ```R glm.fit <- glm(default ~ income + balance, data = Default, family = "binomial") ``` Here, `default` is the response variable, and `income` and `balance` are the predictors. The fitted model object `glm.fit` stores the model output, which we can inspect to extract estimates and evaluate the fit.
After fitting the model, we can use functions like `summary()` to obtain detailed statistics, including estimated coefficients and standard errors, which are essential for making inferences about the predictors.

R Programming

**R Programming** is a powerful and free software environment for statistical computing and graphics. It is especially popular among statisticians and data miners for developing statistical software and performing data analysis. R provides comprehensive tools not only for statistical analysis but also for data visualization and manipulation.
In the context of logistic regression and bootstrap, R simplifies the process of model fitting and validation through built-in functions and packages, such as `glm()` for logistic regression and `boot` for bootstrap resampling. These tools allow users to perform complex statistical analyses with relatively simple code.

• The `glm()` function enables you to model complex relationships between variables using generalized linear models.
• The `boot` package is used to assess the accuracy of model estimates through resampling techniques.

Overall, R's extensive library of packages and functions makes it an ideal environment for performing and understanding logistic regression analyses, as demonstrated through the exercise's step-by-step methodology.