Steps for modeling

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

• Linearity
• Zero Mean
• Constant Variance
• Independence
• Random
• Normality

Conditions for linear regression

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: $\log \left(odds\right)\approx {\beta }_0+{\beta }_{1}x$
• You can check this by plotting the empirical logits
• Example: ⛳ Examining the relationship between the length of a putt with whether it was made

⛳ Testing for linearity in logistic regression

⛳ Testing for linearity in logistic regression

⛳ Testing for linearity in logistic regression

⛳ Testing for linearity in logistic regression

⛳ Testing for linearity in logistic regression

⛳ Testing for linearity in logistic regression

⛳ Testing for linearity in logistic regression

• We can plot the empirical logit to examine the linearity assumption

⛳ Testing for linearity in logistic regression

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

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

• 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)

Conditions for logistic regression

• Linearity ✅
• Independence
• Random

Randomness

• This cannot be formally "checked" via a graph
• You have to think about the data generating process
• Let's look at some examples

Randomness

During the 1970s, when women were entering the workforce in substantial numbers for the first time since World War II, many men were opposed to the trend. One study chose a random sample of men and asked them to agree or disagree with the statement “Women should stay in the home and let men run the country.” A linear logistic regression relating the proportion of men who agreed to their years of education showed a strong relationship with a negative slope: The more time a man spent in school, the less likely he was to agree.

• Yes! This was a "random sample"

Randomness

One of the authors of our Stat2Data book tried to apply logistic regression to a bluegrass banjo "roll", an eight-note sequence with a fixed pattern. According to the logistic model, the predictor is the time in the sequence when the note is played (1 to 8) and the response is whether the note is picked with the thumb:

• No! There is no randomness because the sequence is fixed. Notice that it is possible to compute p-values and interval estimates. Notice also that a (brainless) computer will do it for you if you ask.

Conditions for logistic regression

• Linearity ✅
• Independence
• Random ✅

Independence

• Even if outcomes are random, they may not be independent.
• For example, if you put tickets numbered 1 to 10 in a box, mix them up, and take them out one at a time, the sequence you get is random, but the individual outcomes are not independent.
• If your first ticket is #9, your second cannot be.
• If you put #9 back and mix again before you grab the next ticket, your outcomes are both random and independent.