W4: Poisson Regression

Question 1

What if we are interested in a response variable (which is a count) and its association with more than one binary covariate

Which test do we perform?

Answer 1

Poisson and negative bionominal regression

Question 2

When is Poisson regression suitable?

Answer 2

When we have a response variable in form of count and one or more IV (covariates) which can be discrete or continous

Question 3

Form of Poisson regression equation

Answer 3

Question 4

In Poisson Regression, M in the equation stands for?

Answer 4

Mean count for the individual with covariates values X1, X2…

Question 5

Once we have values of a, b1, b2 we can use Poisson regression equation to make predicts by reversing the log transformation

Prediction:

Answer 5

Question 6

Assumptions of Poisson Regression (3)

Answer 6

1. Observations are independent
2. Disturbition of counts follow a Poisson disturbition
3. The mean and variance of the model as the same

Question 7

Which assumption of Poisson regression is often violated?

Answer 7

Assumption 3: Mean and variance of the model are the same

Question 8

Assumption 3 of the Poisson regression is often violated
This is when

Answer 8

variance of the model is larger than mean –> this is known as overdispersion

Question 9

To assess whether a model is overdispered (Poisson reh) is to look at the

Answer 9

Chi-squared test statistic x^2 divided by model’s DF

Question 10

If Chi-squared (x^2)/DF = 1 then

Poisson/NB

Answer 10

mean and variance of the model are equal

Question 11

If Chi-squared (x^2)/DF > 1 then

Poisson/ NB regression

Answer 11

A value much larger than 1 indicated overdispersion

Question 12

Assessing model fit using (2)

Poisson/NB reg

Answer 12

• AIC value
• Omnibus test

Question 13

What does the Omnibus test the null hypothesis that:

Poisson/NB test

Answer 13

HO = The model is no better than an intercept-only model

Question 14

In other words, what H0 the Omnibus test test?

Poisson/NB

Answer 14

Our Poisson regression with our X variables used to predict this count is no better than guessing.

Question 15

If the Omnibus test is significant then -

Poisson/NB

Answer 15

The model is superior to a null model

Question 16

Testing significance of predictors using __ test in Poisson regression

Answer 16

Wald

Question 17

Testing significance of predictors using Wald test in Poisson regression
We test the null hypothesis that:

Answer 17

Question 18

1. First thing to write up in Poisson regression:

Answer 18

A Poisson regression was performed to assess whether X2(, e.g., Spend on campagin ) and X2 (e.g., Emotiveness of advertisement) affected/predicted the number of Y (e.g., number of clicks)

Question 19

Step 2 of Poisson is writing regression equation (in SPSS) - (2)

Answer 19

The regression equation took the form of:

log (Mean number of clicks) = 3.0414 + 0.372 (Spend) - 0.717 (Emotive)

Question 20

Step 3 of Poisson signifiance of covariates (in SPSS)

Answer 20

Both Spend and Emotive are significant at 0.1% level as predictors of Number of Clicks, and therefore we retain both covariates in the model

Question 21

Step 2 of signifiance of covariates in Poisson regression, what if one covariate is not significant at any level?

Answer 21

Therefore this should be removed and refit the model again

Question 22

Step 2 and 3 of Poisson regression of writing equation and sig of covariates in R

Answer 22

Question 23

Step 3 of Poisson regression: Testing for overdispersion (in SPSS) from goodness of fit table - (2)

Answer 23

• From the goodness of fit table, we see the Chi-squared test statistic divided by its DF is 57.623.
• This is much larger than 1 and therefore we have very strong evidence of overdispersion in the data

Question 24

Step 3 of Poisson regression: Testing for overdispersion (in R) from goodness of fit table - (2)

Answer 24

Divide 15.295/16

Question 25

Step 4 of Poisson is Omnibus test (in SPSS) - (3)

Answer 25

* The Omnibus test is significant at 0.1% level * So model is superior to null model * Including X1 (e.g., Stress) and X2 (e.g., Distance) to predict Y (e.g., Sick days) is better than a null model

Question 26

There is no Omnibus test in Poisson/ Negative NB

Answer 26

R

Question 27

Step 5 of Poisson Regression is writing the AIC (in SPSS) - (2)

Answer 27

The AIC for this model is 1148.239 Cna be used to compare to other models

Question 28

Step 5 of Poisson Regression is writing the AIC (in R)

Answer 28

Question 29

If we have overdispersion in the data, then Poisson regression is not suitable Instead we use:

Answer 29

Negative bionominal regression

Question 30

If we have lots of zero counts then zero-inflated Poisson regression is

Answer 30

suitable

Question 31

The negative bionominal regression equation is same in (5)

Answer 31

* The steps of Poisson regression * Same interpreting AIC value and overdispersion * Same in extracting values for equation * Same regression equation as Poisson regression * Same assumptions

Question 32

In negative bionominal regression we only need to compare the

Answer 32

AIC value to other model (i.e., Poisson regression model)

Question 33

What is AIC? Negative / Poisson Reg

Answer 33

Measure of goodness-of-fit of regression models

Question 34

What is the preferred model based on AIC?

Answer 34

Is one with lowest AIC value

Question 35

Assumptions of negative bionominal regression are: (2)

Answer 35

Observations are independent The counts follow a negative bionominal disturbition

Question 36

Two assumptions of negative bionominal regression is satisfied

Answer 36

when creating scatterplot of predicted values against residuals

Question 37

If negative bionominal regression scatterplot look like this then - (2)

Answer 37

We see a pattern here: variance decreases (range of points lower/less spread of points in residuals) as predicted values increase This provides evidence against our assumptions

Question 38

Check indepndence observation in Poission regression by

Answer 38

plotting a scatterplot of predicted values against residuals

Question 39

Checking indepndence assumption of Poisson in this graph in R, interpret it - (2)

Answer 39

* we don’t have independence in our observations so assumption of independence might not be met as we have got this big gap in the middle of this plot * not obvious why there is this gap