Models for Count Data II

Question 1

When is Poisson regression appropriate?

Answer 1

When the number of events (counts) follows a Poisson distribution, conditional on the predictors

Question 2

What are the ways in which the assumptions for a Poisson regression can be violated?

Answer 2

• Overdispersion (variance > mean)
• Excess zeroes (more zeroes than in a Poisson distribution)
• No zeroes

Question 3

What is equidispersion?

Answer 3

Assumption for Poisson regression - variance = mean

Question 4

In social science, medicine, and health, in what way do count data violate assumptions for Poisson regression?

Answer 4

Overdispersion (variance > mean)

Question 5

Models for count data: equidispersion and zeroes as expected

Answer 5

Poisson

Question 6

Models for count data: equidispersion and excess zeroes

Answer 6

Zero-inflated Poisson

Question 7

Models for count data: Equidispersion and no zeroes

Answer 7

Zero-truncated Poisson

Question 8

Models for count data: Overdispersion and zeros as expected

Answer 8

Negative binominal

Question 9

Models for count data: Overdispersion and excess zeroes

Answer 9

Zero-inflated negative binomial

Question 10

Models for count data: Overdispersion and no zeroes

Answer 10

Zero-truncated negative binomial

Question 11

What about underdispersion?

Answer 11

This can occur in principle, but is rare in practice (variance < mean)

Question 12

What happens if you use the Poisson distribution even though your data are overdispersed? Or use a model that doesn’t consider excess or no zeroes when it should?

Answer 12

Coefficient estimates may be biased and/or misleading (i.e., slope coefficients may not be a good estimate of relationship between predictor(s) and outcome)

Question 13

What are the implications for SEs when not considering overdispersion?

Answer 13

They may be underestimated. This implies that your p-values would be too small and your CIs to narrow, increasing the risk of Type I error

Question 14

What choices do you have when outcome is overdispersed?

Answer 14

• Negative binomial regression (or other models accounting for overdispersion)
• Poisson regression with robust SEs

Question 15

What are robust SEs?

Answer 15

Adjusted so they are robust to violations of Poisson regression. Robust SEs are usually larger than those from a typical Poisson regression. Considered a more cautious way of analysing the data

Question 16

What is the most commonly used overdispersed distribution?

Answer 16

Negative binomial

Question 17

What are the parameters of a negative binomial distribution?

Answer 17

Mean, µ, and a dispersion parameter α
The mean and variance are related (as opposed to in the normal distribution where they are independent): var(Y) = µ + αµ^2
In Poisson, we just have one parameter (mean) as variance is equal to the mean

Question 18

What values can the dispersion parameter α take?

Answer 18

Values of 0 or larger (can never be negative)
- if α = 0, we have a Poisson distribution (with equidispersion)
- if α > 0, we have an overdispersed distribution
The larger the α, the larger the variance relative to the mean

Question 19

Are there other ways of relating the mean to the variance in negative binomial regression?

Answer 19

Yes, different ways of relating the variance to the mean can sometimes slightly change the model or slightly improve your model

Question 20

In the negative binomial distribution, what does larger dispersion imply?

Answer 20

Larger variance

Question 21

What is the shape of the negative binomial distribution?

Answer 21

Tails are much larger compared to when dispersion is equal to 0 (Poisson)

Question 22

Overall comparison of properties of Poisson and negative binomial distributions:

Answer 22

Poisson:
- Equidispersed
- One parameter (µ = mean = variance)
- Var(Y) = µ
Negative binomial:
- Overdispersed
- Two parameters (µ = mean; α = dispersion)
- Var(Y) = µ + αµ^2

Question 23

In the negative binomial distribution, what is this way of specifying the variance called? Var(Y) = µ + αµ^2

Answer 23

NB2-parameterisation
There are other options e.g., the NB1-parameterisation: var(Y) = µ + αµ

Question 24

Negative binomial regression equation:

Answer 24

log(µi) = β0 + β1X1i + β2X2i + … + βkXki
yi ~ NegBin(µi, α), var(yi) = µi + + αµ^2
Where ‘i’ represents each observation
- This looks similar to a Poisson regression. Again, we use a log-transformation of the outcome. The difference is that we now have an additional parameter in the model, the dispersion, α, which we need to estimate. The dispersion parameter governs the extent of overdispersion

Question 25

On what scale are the coefficients from the negative binomial regression?

Answer 25

Log-scale. As with Poisson, they can be exponentiated to get the IRRs and 95% CIS for IRRs

Question 26

Why exponentiate coefficients from the log scale to IRRs?

Answer 26

IRRs are more interpretable than coefficients as IRRs are on the scale of the count variable

Question 27

If coefficient on the log scale is negative, what will the IRR be?

Answer 27

One

Question 28

In an output from negative binomial regression, what does /lnalpha mean?

Answer 28

Log of the alpha and only needs interpreting if using predictors to predict dispersion rather than assuming it's constant

Question 29

How can we get the output on the count scale?

Answer 29

Using Stata to exponentiate the coefficients

Question 30

Do Poisson and negative binomial regression estimated on the same data give the same results?

Answer 30

No - neither estimated coefficients nor SEs will be the same

Question 31

What does zero-truncation often result from?

Answer 31

Zeroes being unobservable or 'impossible' e.g.,: - Number of days in hospital for hospitalised stroke patients - Number of appointments with a psychotherapist

Question 32

How do regression models for zero-truncated data work?

Answer 32

Essentially in the same way as ordinary regression models - There is zero-truncated Poisson and zero-truncated negative binomial regression. In either of these two models, predicted values will have a minimum value of 1. Otherwise, the interpretation of coefficients is the same as in ordinary Poisson or negative binomial regression

Question 33

Does absence of zeroes necessarily imply zero-truncation?

Answer 33

No - may not have observed zeroes 'by chance', even though zeroes are impossible

Question 34

On what basis is the decision to use a zero-truncated model made?

Answer 34

Knowledge about how the data were collected, rather than based on noticing that there are no zeroes

Question 35

With excess zeroes, what are the theoretical bases about the origins of the zeroes?

Answer 35

- Zero-inflation: where zeroes can about in two different ways - Hurdle models: Where the zeroes and the non-zero counts are caused by separate processes

Question 36

In what two ways can zeroes come about?

Answer 36

Structural zeroes vs sampling zeroes

Question 37

Consider a research example of a zero-inflated distribution: "how many joints (cannabis) did you smoke last week?" The answer is zero for:

Answer 37

- Structural zeroes: non-smokers of joints - Sampling zeroes: cannabis users who happened to not smoke last week

Question 38

What are hurdle models?

Answer 38

All zeroes are assumed to be structural, and there are no sampling zeroes The distribution of non-zero counts is zero-truncated, and the zeroes are governed by a totally different process E.g., number of appointments with a psychotherapist after GP referral: - Structural zeroes: Some patients never go to see a therapist - Zero-truncated counts: Those who go to see a therapist have at least one appointments

Question 39

What is a mixture distribution?

Answer 39

One distribution governs zeroes and another governs the counts (zero-truncated Poisson counts)

Question 40

Models for excess zeroes:

Answer 40

Equidispersion: - Zero-inflated Poisson - Poisson hurdle model Overdispersion: - Zero-inflated negative binomial - Negative binomial hurdle model

Question 41

How do you know which model for excess zeroes to use?

Answer 41

Often depends on knowledge/theory of how zeroes come about Sometimes the zero-generating process is unknown; then a pragmatic decision might be made (e.g., based on model fit)

Question 42

Relative to the mean, what indicates overdispersion?

Answer 42

Large counts

Question 43

How can a zero-inflated model be expressed mathematically?

Answer 43

P(Yi = 0) = π + (1 - πi)e^-μi P(Yi = y) = (1 - πi)μi^ye-μi / y! , y ≥ i The first equation describes the probability of observing zero events. This probability is the sum of the probability of a structural zero (πi) and the probability of a sampling zero [(1 - πi)e^-μi The second equation describes the probability of observing 1, 2, 3 or more events

Question 44

How many parameters in a zero-inflated model?

Answer 44

Two - π and μ, each of which appears in both equations. But we model these parameters separately The probability π of having a structural zero is modelled via a logistic regression: logit(πi) = Y0 + Y1X1i + Y2X2i + ... - here, the coefficients are labelled 'Y' to make clear they are not the same coefficients as those in the Poisson part of the model The mean μ of the counts that are not structural zeroes is modelled via a Poisson regression: log(μi) = β0 + β1X1i + β2X2i + ...

Question 45

In a zero-inflated model, do the predictors need to be in both model parts (logistic + Poisson)?

Answer 45

No - you can choose to have different predictors in each part of the model, or to use some predictors in both model parts, and other predictors only in one of them

Question 46

In a ZIP model, how are structural zeroes modelled?

Answer 46

Using a logistic regression

Question 47

In a ZIP model, if a coefficient is positive in the logistic part, what does that indicate?

Answer 47

More zeroes - smaller count of outcome variable (corresponds to a negative coefficient in the Poisson part)

Question 48

In a ZIP model, how are both model parts related?

Answer 48

They are both dependent on one another - changing something in the zero-inflation part will change the estimates in the Poisson part and vice versa

Question 49

How does ZI negative binomial regression work in relation to a ZI poisson model?

Answer 49

In essentially the same way, except that the counts are assumed to follow a negative binomial distribution

Question 50

In a ZIP model, how are coefficients interpreted? Consider an example IRR of 0.89 (95% CI 0.80-0.98) corresponding to a 10 percentage point difference in the proportion of lower class difference in relation to police operations

Answer 50

As in an ordinary Poisson regression, but being mindful that our estimates are conditional on how we adjust for zero-inflation (e.g., if we change predictors in the ZI part, the coefficient estimates in the Poisson part will also change. For example: "Our model estimates that a 10% percentage point difference in the proportion of lower class citizens is associated with fewer police operations by a factor of 0.89, adjusting for zero-inflation, where zeroes are predicted by lower10, vendors, and population."

Question 51

How do you interpret the logistic part in a ZIP? Interpret in relation to police operations

Answer 51

The logistic part predicts zeroes. Thus: - An OR > 1 indicates that a predictor is associated with more zeroes (fewer police operations) - An OR < 1 indicates a predictor is associated with fewer zeroes (more police operations) Interpretation needs to consider how we model the non-zero counts, i.e., the estimates from the logistic part are adjusted for the Poisson part.

Question 52

How are count regression models estimated?

Answer 52

By maximum likelihood

Question 53

What models can be compared using LRTs?

Answer 53

Models of the same type (e.g., negative binomial, ZIP) can be compared using LRTs But: - models without zero-inflation are not nested within zero-inflated models - although the Poisson model is nested within the negative binomial model (as it is a special case of negative binomial regression), the ordinary LRT cannot be applied

Question 54

Is a Poisson model nested within a negative binomial regression?

Answer 54

Yes - it is nested within a negative binomial regression with the same predictors, because if the dispersion α = 0, then the two models are identical But the ordinary LRT would give misleading results (the p-value would be too large and we would reject H0 too rarely)

Question 55

Why can we not use an ordinary LRT with a Poisson and negative binomial regression model?

Answer 55

α cannot be negative - so the test value 0 is "on the boundary of the parameter space." We therefore need to use a special test called the boundary LRT

Question 56

Hypotheses for the boundary LRT to compare Poisson and negative binomial models:

Answer 56

H0: α = 0 Or: "There is no overdispersion" Or: "The NegBin model is no better than the Poisson model." H1: α > 0 Or: "There is overdispersion" Or: "The NegBin model is better than the Poisson model" A small p-value indicates evidence in favour of overdispersion, i.e., evidence against the Poisson model

Question 57

Where is the boundary LRT displayed in Stata?

Answer 57

In the output of a NegBin model alongside for α = 0 It is displayed by default (compares negative binomial regression with Poisson regression model with the same predictors)

Question 58

What statistic does boundary LRT use?

Answer 58

Same as ordinary LRT, but calculates p-value in a different way

Question 59

If one variable is overdispersed, does that necessarily mean that overdispersion also exists in models of the same data?

Answer 59

No - sometimes additional predictors explain away the overdispersion

Question 60

Why can LRTs not be used to assess whether zero-inflation improves a model?

Answer 60

Models without zero-inflation are not nested in models with zero-inflation We can use Akaike's Information Criterion (AIC) instead

Question 61

What is AIC?

Answer 61

Can be used to compare nested and non-nested models Is an information criterion, not a statistical test

Question 62

How is AIC expressed?

Answer 62

AIC = -2 x LL + 2 x k Where: - LL: Log likelihood - k: number of parameters

Question 63

What does AIC do?

Answer 63

Weighs model fit (represented by the LL) against parsimony (represented by k)

Question 64

How to interpret AIC:

Answer 64

A smaller AIC indicates a better model The numeric value of the AIC has no meaning in itself; it is meaningful only when comparing the AICs of different models estimated on the same data (with the same sample size)

Question 65

How many additional parameters does negative binomial regression have in relation to the other models encountered?

Answer 65

One (dispersion)

Question 66

What sentence needs to be included when interpreting zero-inflated models (logistic part)?

Answer 66

"adjusted for all other variables in the model, including both the Poisson and the logit parts"

Question 67

What are differences in the estimates from ZINB and ZIP model?

Answer 67

Estimates from ZIP and ZINB models are similar, but different; different assumptions -> different results SEs are larger in the ZINB model, leading to wider CIs

Question 68

AIC - notes:

Answer 68

- AIC has general applicability beyond count regression - Can be used to compare nested or non-nested models - AIC is one of several information indices: different information indices vary in the way they weight model fit and parsimony

Question 69

Zero-inflated models - summary:

Answer 69

- Excess zeroes can sometimes be accounted for by explanatory variables - Poisson and negative binomial models are not nested in their zero-inflated counterparts - Use AIC (or other information criteria) for comparison