Generalized Linear Models

Question 1

What is the objective of Generalising Linear Models?

Answer 1

To allow us to do regression in problems where our Yi is not normally distributed

Question 2

What is the stochastic/random part of a model?

Answer 2

The form of the model which characterises the distribution of Yi (eg. Yi ~ N(mu(i), sigma²)

Question 3

What is the structural part of the model?

Answer 3

A function of mu(i) which describes its relationship with the covariates (eg. mu(i) = B0 + B1X1 + B2X2 + … + BPXP)

Question 4

What are the two types of model which we go over in this course?

Answer 4

• Poisson Model (for count outcomes)

- Binomial Model (for binary or binomial outcomes)

Question 5

What is the difference between a binomial outcome and a binary outcome?

Answer 5

Binary (or Bernoulli) outcome is dependent on a single trial where as Binomial outcome is dependent on a number of trials

Question 6

What is a link function?

Answer 6

A function which describes the relationship between the parameter of a distribution and the covariates

Question 7

What is the link function for the Poisson Model?

Answer 7

log(lambda) = linear covariates

*natural logarithm

Question 8

What is the link function for the Binomial Model?

Answer 8

log(odds of success) = linear covariates

*natural logarithm

Question 9

Define the term “odds”?

Answer 9

A quantity which the the ratio of the probably of an event occurring divided by the probability of the event not occurring.

= [p(A)] / [p(not A)]

*in Bernoulli events, “A” is success and “not A” is failure

Question 10

How do you read data from a CSV file into R?

Answer 10

data = read.csv(“filename.csv”)

Question 11

What is the R function for viewing the first few rows of a data object?

Answer 11

head(data)

Question 12

What is the R function for viewing the names of the variables in a data object?

Answer 12

names(data)

Question 13

What is the R code for viewing the values under a specific variable name in a data object?

Answer 13

data$variableName

Question 14

What is the R code for viewing the number of each type of value under a specific variable name in a data object?

Answer 14

table(data$variableName)

Question 15

What is the R code for viewing the proportion of each type of value under a specific variable name in a data object?

Answer 15

prop.table(table(data$variableName))

Question 16

What is the R code for adding a variable name to a data object based on some condition of each row?

Answer 16

data$newVariable = ifelse(data$conditionVariable == “something”, 1 , 0)

Will set newVariable to 1 if condition is true else set newVariable to 0

Question 17

What is the R code for fitting a GLM to a binomial dependent variable and viewing a summary of the model?

Answer 17

model1 = glm(dependent ~ explanatory, family = “binomial”, data = dataObject)

summary(model1)

• the outputs from the summary give the coefficients for the link function (logit in this case)
• standard error gives an idea of the variability in the estimate of that coefficient

Question 18

What is does logit(p) equate to?

Answer 18

log(odds of p)

• natural logarithm

Question 19

How do we know how well the model fits the data?

Answer 19

D = -2(l(c)  - l(f))
D ~ chi-squared with n-k-1
* where n is number of observations
* where k+1 is the parameters estimated
* where l is likelihood
* where c is current
* where f is a full/ideal model which fits all of the data

%Deviance Explained = [Dnull - Dcurrent]/[Dnull]
* where Dnull is the Deviance of the model with just the intercept

Question 20

What is the R code for fitting a GLM to a “count” dependent variable and viewing a summary of the model?

Answer 20

model2 = glm(dependent ~ explanatory, family = “poisson”, data = dataObject)

summary(model2)

• the outputs from the summary give the coefficients for the link function (logit in this case)
• standard error gives an idea of the variability in the estimate of that coefficient

Question 21

What is the criteria for a distribution to be part of the Exponential Family?

Answer 21

The distribution must be able to be written in the form:

exp{a(y).b(theta) + c(theta) + d(y)}

• where theta represents a parameter of the distribution

Question 22

What is the Interaction Component/Term of a distribution written in Exponential form?

Answer 22

a(y).b(theta)

Question 23

What is the Additive Component/Term of a distribution written in Exponential form?

Answer 23

c(theta) + d(y)

Question 24

What is the Natural Parameter of a distribution written in Exponential form?

Answer 24

b(theta)

Question 25

What does e^ln(x) equate to?

Answer 25

x

• we then use this to prove whether a distribution belongs to the Exponential Family. We place the pmf/pdf in the place of x in e^ln(x)

Question 26

Is the Weibul distribution a member of the Exponential Family?

Answer 26

In most cases no, but yes if lambda is told to be some constant

Question 27

What is the expected value of a(y) in the context of Exponential distributions?

Answer 27

[c’(theta)]/b’(theta)]

Question 28

What is the variance of a(y) in the context of Exponential distributions?

Answer 28

[b’’(theta).c’(theta) - b’(theta).c’’(theta)]/[b’(theta)]^3