Simple
Logistic Regression

R Packages

• rcistats
• tidyverse

Heart Disease Data

Variables of Interest

• age: Age of patient
• disease: Indicating if they have heart disease

Modeling Binary Outcomes

• Modeling Binary Outcomes

• Logistic Regression

• Logistic Regression in R

• Prediction with Models

Modeling Binary Outcomes

Modeling Binary Outcomes

Mathematical Model

\[ \left(\begin{array}{c} Yes \\ No \end{array}\right) = \beta_0 + \beta_1X \]

Let …

\[ Y = \left\{\begin{array}{cc} 1 & Yes \\ 0 & No \end{array}\right. \]

Construct a Model

\[ P\left(Y = 1\right) = \beta_0 + \beta_1X \]

Construct a Model

\[ P\left(Y = 1\right) = \frac{e^{\beta_0 + \beta_1X}}{1 + e^{\beta_0 + \beta_1X}} \]

Construct a Model

\[ \frac{P(Y = 1)}{P(Y = 0)} = e^{\beta_0 + \beta_1X} \]

where \(\frac{P(Y = 1)}{P(Y = 0)}\) are considered the odds of observing \(Y = 1\).

The Logistic Model

\[ \log\left\{\frac{P(Y = 1)}{P(Y = 0)}\right\} = \beta_0 + \beta_1X \]

Notation

\[ \log\left\{\frac{P(Y = 1)}{P(Y = 0)}\right\} = \log\left\{odds\ of \ 1\right\} = lo(1) \]

Logistic Regression

• Modeling Binary Outcomes

• Logistic Regression

• Logistic Regression in R

• Prediction with Models

Logistic Regression

Logistic Regression is used to model the association between a predictor and a binary outcome variable.

This is similar Linear Regression which models the association between a predictor and a numerical outcome variable.

Logistic Regression

Logistic Regression uses the logistic model to formulate the relationship between a predictor and the outcome.

More specifically, for an outcome of Y:

\[ Y = \left\{\begin{array}{cc} 1 & \text{Category 1} \\ 0 & \text{Category 2} \end{array}\right. \]

The predictor variable will model the probability of observing category 1 (\(P(Y=1)\))

Logistic Model

\[ \log\left\{\frac{P(Y = 1)}{P(Y = 0)}\right\} = \beta_0 + \beta_1X \]

Regression Coefficients \(\beta\)

The regression coefficients quantify how a specific predictor changes the odds of observing the first category of the outcome (\(Y = 1\))

Estimating \(\beta\)

To obtain the numerical value for \(\beta\), denoted as \(\hat \beta\), we will be finding the values of \(\hat \beta\) that maximizes the likelihood function:

\[ L(\boldsymbol \beta) = \prod_{i=1}^n \left(\frac{e^{\beta_0 + \beta_1X}}{1 + e^{\beta_0 + \beta_1X}}\right)^{Y_i}\left(\frac{1}{1 + e^{\beta_0 + \beta_1X}}\right)^{1-Y_i} \]

The likelihood function can be thought as the probability of observing the entire data set. Therefore, we want to choose the values the \(\beta_0\) and \(\beta_1\) that will result in the highest probability of observing the data.

Estimated Parameters

The values you obtain (\(\hat \beta\)) tell you the relationship between the a predictor variable and the log odds of observing the first category of the outcome \(Y=1\).

Exponentiating the estimate (\(e^{\hat \beta}\)) will give you the relationship between a predictor variable and the odds of observing the first category of the outcome \(Y=1\).

Interpreting \(\hat \beta\)

For a continuous predictor variable:

As X increases by 1 unit, the odds of observing the first category (\(Y = 1\)) increases by a factor of \(e^{\hat\beta}\).

Logistic Regression in R

• Modeling Binary Outcomes

• Logistic Regression

• Logistic Regression in R

• Prediction with Models

Logistic Regression in R

xglm <- glm(Y ~ X,
            data = DATA,
            family = binomial())
xglm

• DATA: Name of the data frame
• Y: Outcome Variable of Interest in the data frame DATA
• X: Predictor Variable in the data frame DATA

Model Information

The model_info function can provide basic information about a linear or logistic regression model.

It can provide what is being modeled (yes, 1) in logistic regression.

It can provide basic info of the predictor variables.

Model Information in R

model_info(MODEL)

• MODEL: fitted model from lm() or glm

Example

Modelling disease by age:

xglm <- glm(disease ~ cp + age + sex + restecg + fbs,
            data = heart_disease,
            family = binomial())
model_info(xglm)
xglm

#> 
#> Call:  glm(formula = disease ~ cp + age + sex + restecg + fbs, family = binomial(), 
#>     data = heart_disease)
#> 
#> Coefficients:
#>                         (Intercept)                   cpNon-anginal Pain  
#>                            -3.97563                             -2.30854  
#>                   cpAtypical Angina                     cpTypical Angina  
#>                            -2.38883                             -2.31736  
#>                                 age                              sexMale  
#>                             0.06316                              1.77141  
#>        restecgST-T wave Abnormality  restecgLeft Ventricular Hypertrophy  
#>                             1.62429                              0.50215  
#>         fbsFasting Blood Sugar >120  
#>                             0.08444  
#> 
#> Degrees of Freedom: 296 Total (i.e. Null);  288 Residual
#> Null Deviance:       409.9 
#> Residual Deviance: 286.6     AIC: 304.6

\[ lo(disease) = -3.0512 + 0.0529 (age) \]

Interpretation

\[ lo(disease) = -3.0512 + 0.0529 (age) \]

exp(0.05291)

#> [1] 1.054335

As age increases by 1 year, the odds of having heart disease increases by a factor of 1.054.

Prediction with Models

• Modeling Binary Outcomes

• Logistic Regression

• Logistic Regression in R

• Prediction with Models

Working with probabilities

As you can see, working with odds may be unintuitive for the average person. It will be better to predict the probability and display those results to individuals.

Predicting Probability

\[ \hat P\left(Y = 1\right) = \frac{e^{\hat\beta_0 + \hat\beta_1X}}{1 + e^{\hat\beta_0 + \hat\beta_1X}} \]

Prediction in R

xglm <- glm(Y ~ X,
            data = DATA,
            family = binomial())
ndf <- data.frame(X = VAL)
predict(xglm,
        ndf,
        type = "response")

Example 1

Predict the probability of having heart disease for a patient who is 40 years old.

xglm <- glm(disease ~ age,
            data = heart_disease,
            family = binomial())
ndf <- data.frame(age = 40)
predict(xglm, ndf, type = "response")

#>         1 
#> 0.2819196

Example 2

Predict the probability of having heart disease for a patient who is 55 years old.

xglm <- glm(disease ~ age,
            data = heart_disease,
            family = binomial())
ndf <- data.frame(age = 55)
predict(xglm, ndf, type = "response")

#>         1 
#> 0.4647183

Example 3

Predict the probability of having heart disease for a patient who is 70 years old.

xglm <- glm(disease ~ age,
            data = heart_disease,
            family = binomial())
ndf <- data.frame(age = 70)
predict(xglm, ndf, type = "response")

#>         1 
#> 0.6575143

Example

Age	40	55	70
Probabiltiy	28.2%	46.5%	65.8%