Logistic Regression, Probability, and Odds

Question 1

What is regression?

Answer 1

Mathematical equation for straight line that is fitted to data as way of describing relationship between two or more variables.

Question 2

What are the main purposes of regression?

Answer 2

Prediction (to estimate risk).

Control for confounding.

Question 3

What does the regression line do?

Answer 3

Estimates average values for variable on vertical scale (Y) according to values on horizontal scale (X).

Question 4

What happens in simple regression?

Answer 4

For each change in X, there is a change in Y.

Question 5

What is general linear regression?

Answer 5

Refers to conventional linear regression models for a continuous response variable given continuous and/or categorical predictors.

Question 6

What happens with transformation in a GLM?

Answer 6

Without transformation, the coefficient will estimate rate differences.

With transformation, the coefficient will estimate rate ratios.

Question 7

What is the model for simple regression?

Answer 7

Yx=(D | X=x)= α+βx

Y = α + βX + ε (prediction)

Yi = BO + B1Xi + Ei (population)

Question 8

What is a?

Answer 8

Measure when x = 0
Also B0
The y-intercept

Question 9

What does slope coefficient do?

Answer 9

Estimate increase in risk per unit increase in X.

B1

Question 10

How are mean and probability related?

Answer 10

Mean is estimate of probability.

Question 11

Why will linear regression not work?

Answer 11

Probability is scale of 0-1 and regression is not within those constraints.

Linear model assumes risk changed linearly.

Question 12

Why will logistic regression work?

Answer 12

Used for binary outcomes and on 0-1 scale showing odds.

This is because the probability curve becomes linear after taking natural log of a value.

Question 13

What is the model form of logistic regression?

Answer 13

Logit (Y) = ln (Y/(1-Y))

OR

Logit [P (Y)] = ln ((P(Y))/(1-P(Y)))

OR

ln (Px/1-Px) = ln (D|X=x) = a +Bx

Question 14

What is alpha?

Answer 14

Log odds of outcome when X = 0

The y-intercept.

Question 15

What is beta?

Answer 15

Slope

Log odds RATIO comparing two exposure groups differing by one unit on scale of x.

Question 16

In simple logistic regression what do 1 and 0 represent?

Answer 16

1 = exposed, 0 = not exposed.

Question 17

Simple logistic regression, what would 1 (exposure) yield?

Answer 17

Alpha + beta = log odds of exposure

e ^(a+b) is the odds of disease with exposure.

Question 18

Simple logistic regression, what would 0 (no exposure) yield?

Answer 18

Alpha ONLY. Log odds of outcomes when not exposed.

e^(a) is the odds of disease without exposure.

Question 19

How would you get ODDS from log odds/?

Answer 19

Exponentiate both sides:

e ^ (a+b) when exposed
e ^ (a) when unexposed

e is a constant
e = 2.718

Question 20

How would we find odds ratio?

Answer 20

Odds exposed v unexposed:
Odds(a+b)/odds(a)

Therefore:
e^(a+b-a)

Therefore:

e^b

Odds ratio is e^b

Question 21

What happens when you have more than one categorical predictor?

Answer 21

Create dummy variables with a reference group.

Question 22

How do you choose reference group?

Answer 22

Generally the one with the largest sample size so estiamtes with be reasonably precise and stable.

Question 23

What is model for this?

Answer 23

logit(p) = a +b1 + b2

Question 24

What is alpha in this model?

Answer 24

Log odds of disease for reference group.

Question 25

What is b1?

Answer 25

Log odds of disease for group 1 versus reference group.

Question 26

What is b2?

Answer 26

Log odds of disease for group 2 versus reference group.

Question 27

What is model for continuous predictor?

Answer 27

Logit (p ) = alpha + betax

Question 28

What is alpha?

Answer 28

Log odds when X = 0

Question 29

What is beta?

Answer 29

Log OR with 1 unit change in X.

Question 30

How would you calculate odds when X = to a different number?

Answer 30

Multiply that number by BETA and add to ALPHA.

Question 31

What is the Wald hypothesis?

Answer 31

Test of exposed versus unexposed.

Null is that OR = 1 (that e^beta = 1); no difference between the two.

H0: beta = 0
Ha: beta does not equal 0

Question 32

What are advantages to logistic regression?

Answer 32

Obtain estimates and CI for OR (crude and adjusted)

Can handle multiple independent variables.

Question 33

What are disadvantages to logistic regression?

Answer 33

Implicit assumptions difficult to check.

Model selection difficult.