Epidemiology Chapter 3

Question 1

Why is a standard simple linear regression model not appropriate for Bernoulli or binomial responses?

Answer 1

In simple linear regression we assume the errors are normally distributed with mean o and variance sigma square, this implies for fixed x, Y is normally distributed which is not the case with Bernoulli or binomial responses

Mismatch between the two sides of the model

Question 2

Likelihood for the binary logistic regression model

Answer 2

L(α,β│y_1,y_2,….,y_n )=∏(I=1 to n)Pr⁡(Y_i=y_i│α,β)
log⁡(π_i/(1-π_i ))=α+βx_i
π_i=e^( α+βx_i )/(1+e^(α+βx_i ) )
Pr⁡(Y_i=0| α,β)=1-π_i=1/(1+e^(α+βx_i ))
Likelihood function
∏(I=1 to n) ( e^( α+βx_i )/(1+e^(α+βx_i ) ))^y_i x (1/(1+e^(α+βx_i )))^(1-y_i)

Question 3

Log-likelihood for the binary logistic regression

Answer 3

∑_(I=1 to n){y_i log⁡(e^( α+βx_i )/(1+e^(α+βx_i ) ))+(1-y_i )log⁡(1/(1+e^(α+βx_i ) )}

Question 4

Interpretation for one explanatory variable

Model and interpretation

Answer 4

Underlying model
log⁡(π_i/(1-π_i ))=α+βx_i

Define terms
π_i - probability of success for person i
x_i - explanatory variable value for the ith person
α and β are the model parameters

Interpret the parameter estimates
α_hat - log odds of success for someone of age x_i=0, taking exponentials gives the odds of success for someone of x_i= 0, or mean x_i if using centred data
β_hat - a unit increase in x is estimated to change the log odds of success by β_hat, taking exponentials gives the odds ratio of comparing two individuals 1 unit of x apart

Question 5

Interpretation for multiply explanatory variables

Model

Answer 5

Underlying model
log(π_i/(1-π_i)) = α+β_1x_1i+β_2x_2i
Define terms
π_i - probability of success for person i
x_1i - explanatory variable value for the ith person
x_2i - explanatory variable value for the ith person
α, β_1 and β_2 are the model parameters

Question 6

What is Extrapolating?

Answer 6

Making inference outside of our data range

Fix by using centred data

Question 7

Interpretation for one explanatory variable

Hypotheses parameter estimates alpha and beta

Answer 7

Null hypothesis: α=0
Alternative: α≠0
Test statistic:
z=(α_hat-α_0)/se(α_hat)
here α_0 is 0
Under the assumption that H_0 is true, our test statistic will have been approximately sampled form a standard normal distribution
Associated p-value equals, p= 2 x Pr(Z>z)
If p<0.05 reject null at the 5% significance level

Same for beta

Question 8

Interpretation for one explanatory variable

Confidence intervals for parameter estimates alpha and bata

Answer 8

α_hat ± z_(α/2)se(α_hat)
β_hat ± z_(β/2)se(β_hat)

If interval contains 0 cannot reject H0, can’t rule out the possibility that the parameter =0

Question 9

Describe how the AIC is calculated and its uses

Answer 9

AIC = -2 x logloikelihood +2p,
p is the number of parameters
It is used for comparing models, prefer smaller values

Question 10

Describe how the residual deviance have been calculated and their uses

Answer 10

The residual deviance compares the fitted model with the saturated one
2(LL_s-LL_p)

Question 11

Likelihood ratio test

Answer 11

Compares the proposed model with the null model

test statistic = null deviance - residual deviance
To test hypothesis
Null: β=0, suggests replacing the proposed model with the null model
ALternative: β≠0

Under the assumption that H_0 is true, our test statistic follows the chi-squared distribution on one degree of freedom (the difference in number of parameters between the two models being considered)

Question 12

Interpreting a slope coefficient associated with an indicator variable
Multiple binary logistic regression

Answer 12

Interpret the parameter estimates

e^β_2 is the odds ratio for an patient with x_2i = 1 compared to a patient with x_2i =0, all other things being fixed

Question 13

Alternative link functions

Answer 13

The logit function is a link function that links π_i to linear functions of explanatory variables. It is the default in R

The probit link function - based on the inverse cumulative distribution function for the standard normal distribution
ϕ^(-1) (π_i )=α+βx_i
in R: family=binomial(link=”probit”)

The complementary log-log link function maps π_i to the whole real line
log(-log(1-π_i)) = α+βx_i

Question 14

Comparing binary regression models

Answer 14

1. Through the consideration of AIC statistics
2. by considering the interpretability of the models
3. by constructing residual analysis on the fitted models