Logistic Regression

Type of Plot:
Scatterplot

Simple Logistic Regression:
qplot(x = explanatory,
y = response,
data = dataframe,
geom = "point")

Multiple Logistic Regression plots are in dimensions $\ge 3$

$ Y = \beta_0 + \beta_1 X_1 + \ldots + \beta_{k-1} X_{k-1} + \varepsilon $

$ k = 2 $ in Simple Logistic Regression

Simple Logistic Regression
Example Problem

Parameters: $ \beta_i $ for $ i \in { 0, \ldots, k - 1 }$
Point Estimates: $ B_i $ for $ i \in { 0, \ldots, k - 1}$

Hypothesis Test

Hypotheses:
$ H_0: \beta_i = 0 $
1) $ H_a: \beta_i < 0 $
2) $ H_a: \beta_i \ne 0 $
3) $ H_a: \beta_i > 0 $

Test Statistic Random Variable (Assuming $H_0$ is true):
$ Z = \dfrac{B_i - 0}{SE_i} $ where $ SE_i $ is defined here

Observed Test Statistic:
$ z_{obs} = \dfrac{b_{i, obs} - 0}{{SE}_{i, obs}}$ where $ {SE}_{i, obs} $ is defined here

$ \mathbf{\textit{P}}$-value:
1) $ \mathbb{P}(B_i \le b_{i, obs}) = \mathbb{P}(Z \le z_{obs}) $
2) $ \mathbb{P}\left(\big| B_i \big| \ge \big| b_{i, obs} \big| \right) = \mathbb{P}\left(\big| Z \big| \ge \big| z_{obs} \big|\right) $
3) $ \mathbb{P}(B_i \ge b_{i, obs}) = \mathbb{P}(Z \ge z_{obs}) $

Conditions for Distributional Approximation (To $ Z$):

Linear relationship between linked response and predictors
Independent observations, errors, and predictor variables

Confidence Interval

Formula for CI:
$ b_i \pm t_{obs}^* \cdot SE_{obs} $ where $ SE_{obs} $ is defined here

Conditions for Distributional Approximation (To $ Z $):

Linear relationship between linked response and predictors
Independent observations and predictor variables

Multiple Logistic Regression
Example Problem