Key Concepts
The response variable in logistic regression is this type.
What is a binary variable (0/1; Yes/No; ect.)?
Answer to the question: Does logistic regression assume a linear relationship between X and probability?
What is no?
This is the logit form of the logistic regression model.
What is log(pi/(1-pi)) = beta_0 + beta_1X?
If a logistic regression model predicts pi=0.6, where are response variable is equal to 1 if a student passes an exam and 0 if not. This is the interpretation of pi.
What is: there is a 60% chance that the student will pass the exam?
Run the following code in RStudio:
data(mtcars)
mtcars$high_mpg <- ifelse(mtcars$mpg > 20, 1, 0)
head(mtcars)
We would like to make a logistic regression model to predict the probability that a car will have high mpg (over 20) based on its weight (coded as wt). This is the value of the slope (b_0) in the logit form of the model.
What is 19.560?
Answer to the question: can the explanatory variable, X, in single logistic regression be categorical, and if so, how is it used in the model?
What is: Yes. a categorical X can be used by coding categories as indicator variables (0/1) so the model compares groups’ log-odds of the outcome?
Answer to the question: Is normality of errors required for logistic regression?
What is no?
This is the probability form of the logistic regression model.
What is pi = (e^(b_0 + b_1X))/(1+e^(b_0 + b_1X))?
You fit a single logistic regression model and get a p-value of 0.03 for the slope coefficient. This is what the conclusion that we would (at a significance level of 0.05) make based on this evidence.
What is: reject the null hypothesis that beta_1 = 0/there is statistically significant evidence at the 0.05 level that X is associated with the log-odds of the outcome?
Run the following code in RStudio:
data(mtcars)
mtcars$high_mpg <- ifelse(mtcars$mpg > 20, 1, 0)
head(mtcars)
We would like to make a logistic regression model to predict the probability that a car will have high mpg (over 20) based on its weight (coded as wt). Is the slope for wt statistically significant? Why?
What is yes because the p-value is less than 0.05?
This is the transformation of the probability Y = 1 that is modeled as a linear function of X?
What is log(pi/(1-pi))?
In simple linear regression the relationship between X and Y is assumed to be linear.
In logistic regression, this quantity is assumed to be linear in X instead.
What is the log-odds of the response probability?
If log(odds) = 0.99, this is the value of pi?
What is 0.73?
A student predicts the probability of passing an exam is 0.85. These are the corresponding odds and log-odds.
What is: the odds are 5.67 and the log(odds) are 1.73?
Run the following code in RStudio:
data(mtcars)
mtcars$high_mpg <- ifelse(mtcars$mpg > 20, 1, 0)
head(mtcars)
We would like to make a logistic regression model to predict the probability that a car will have high mpg (over 20) based on its weight (coded as wt).
This is the value of the slope coefficient for wt when we convert it to an odds ratio.
What is 0.001676579?
This is what the slope, B_1, represents in logistic regression.
What is the change in log odds of Y=1 for a one unit increase in X?
Answer to the question: Why is it important that observations be independent in logistic regression?
What is: because the model assumes that each outcome provides separate information. Dependence between observations can bias coefficient estimates and standard errors, making inference invalid?
If pi = 0.56, this is the value of the log(odds).
What is 0.24?
This is the way that we can assess whether our logistic regression model fits the data better than a null model with no predictors.
What is by comparing the model deviance (or -2 log-likelihood) of the fitted model to the null model, often using a chi-square test? (G)
Run the following code in RStudio:
data(mtcars)
m
tcars$high_mpg <- ifelse(mtcars$mpg > 20, 1, 0)
head(mtcars)
We would like to make a logistic regression model to predict the probability that a car will have high mpg (over 20) based on its weight (coded as wt).
This is the predicted probability of a car having high mpg if it weighs 3 tons.
What is 0.685?
This is the reason why we usually interpret the slope coefficient, beta_1, by using an exponential instead of using its estimated value?
What is because using an exponential converts the coefficient from change in log-odds to a multiplicative change in odds (odds ratio), which is easier to interpret?
Name at least one reason why we wouldn't want to use linear regression for predicting a binary outcome.
What is we would get probabilities less than 0 or more than 1 (which is not correct), errors are not normally distribution, non-constant variance (different assumptions)?
Answer to the question: if a single logistic regression model gives a slope coefficient, beta_1 = 0.8, how would you explain this to someone who doesn’t understand log-odds?
What is: for each one-unit increase in X, the odds of the event happening are multiplied by about 2.2.
In single logistic regression, these are the null and alternative hypotheses when testing whether the slope beta_1 is associated with the outcome.
What is: H_0: beta_1 = 0 (the predictor X has no effect on the log odds of the response)
H_A: beta_1 does not equal 0 (the predictor X does have an effect on the log odds of the response)?
Run the following code in RStudio:
data(mtcars)
m
tcars$high_mpg <- ifelse(mtcars$mpg > 20, 1, 0)
head(mtcars)
We would like to make a logistic regression model to predict the probability that a car will have high mpg (over 20) based on its weight (coded as wt).
This is the log odds of a car having high mpg if it weighs 5 tons.
What is -12.005?