Assumptions of Logistic Regression

Conditions for simple linear regression

Linearity
Zero Mean
Constant Variance
Independence
Random
Normality

Conditions for simple linear regression

Linearity
Zero Mean
Constant Variance
Independence
Random
Normality

Conditions for linear regression

Linearity
Zero Mean
Constant Variance
Independence
Random
Normality

Conditions for linear regression

How do we check these conditions?

Linearity
Zero Mean
Constant Variance
Independence
Random
Normality

Conditions for linear regression

How do we check these conditions?

Condition	Graph
Linearity	Residuals vs. fits
Zero Mean	by default it’s true
Constant Variance	Residuals vs fits
Independence	No formal check
Random	No formal check
Normality	QQ-plot or histogram of residuals

Conditions for logistic regression

Linearity
Independence
Random

Testing linearity for logistic regression

In logistic regression, the log(odds) are a linear function of \(x\), that is: \(\log(odds) = \beta_0 + \beta_1x\)
You can check this by plotting the empirical logits
Example: ⛳ Examining the relationship between the length of a putt with whether it was made

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134

⛳ Testing for linearity in logistic regression

What is the proportion of success when length is 3?

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134

⛳ Testing for linearity in logistic regression

What is the proportion of sucesses when length is 3?

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134
Probability of success	0.832	0.739	0.565	0.488	0.328

⛳ Testing for linearity in logistic regression

What are the odds of success when length is 3?

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134
Probability of success	0.832	0.739	0.565	0.488	0.328

⛳ Testing for linearity in logistic regression

What are the odds of success when length is 3?

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134
Probability of success	0.832	0.739	0.565	0.488	0.328
Odds of success	4.941	2.839	1.298	0.953	0.489

⛳ Testing for linearity in logistic regression

What are the log(odds) of success when length is 3?

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134
Probability of success	0.832	0.739	0.565	0.488	0.328
Odds of success	4.941	2.839	1.298	0.953	0.489

⛳ Testing for linearity in logistic regression

What are the log(odds) of success when length is 3?

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134
Probability of success	0.832	0.739	0.565	0.488	0.328
Odds of success	4.941	2.839	1.298	0.953	0.489
Empirical logit	1.60	1.04	0.26	−0.05	−0.72

⛳ Testing for linearity in logistic regression

We can plot the empirical logit to examine the linearity assumption

Length	3	4	5	6	7
Number of successes	84	88	61	61	44
Number of failures	17	31	47	64	90
Total	101	119	108	125	134
Probability of success	0.832	0.739	0.565	0.488	0.328
Odds of success	4.941	2.839	1.298	0.953	0.489
Empirical logit	1.60	1.04	0.26	−0.05	−0.72

⛳ Testing for linearity in logistic regression

Code

data <- data.frame(
  length = c(3, 4, 5, 6, 7),
  emp_logit = c(1.6, 1.04, 0.26, -0.05, -0.72)
)
ggplot(data, aes(length, emp_logit)) + 
  geom_point() + 
  labs(y = "log odds of success")

⛳ Testing for linearity in logistic regression

Code

data <- data.frame(
  length = c(3, 4, 5, 6, 7),
  emp_logit = c(1.6, 1.04, 0.26, -0.05, -0.72)
)
ggplot(data, aes(length, emp_logit)) + 
  geom_point() + 
  geom_abline(intercept = 3.26, slope = -0.566, lty = 2) +
  labs(y = "log odds of success")

Testing for linearity in logistic regression

What if the \(x\) variable isn’t discrete?

We can plot the empirical logit to examine the linearity assumption
You can create “bins” and calculate the empirical logit within each bin (for example, count the number of success when x is between 1 and 5: bin 1, count the number of successes when x is between 5 and 10: bin 2, etc)