Hi, it's Margriet. So last week we looked

at logistic regression and logistic

regression is an example of a generalized

linear model that you can use if

your outcome variable is categorical.

Now we specifically looked at logistic

regression in the context of a

binomial outcome variable that is,

a two level variable such

as correct versus incorrect,

or looking to the left versus

looking to the right.

Now this week we will look

at Poisson progression.

And that is another type of

generalized linear model that is

particularly useful for count data.

OK, so what is count data?

Now, many researchers in psychology

and behavioral scientists more

generally have theoretical questions

that involve count variables.

Now an account variable is a variable

that takes on discrete values reflecting

the number of occurrences of a particular

event in a fixed period of time.

So some sort of event might

happen none of the time,

once, twice, three times, four times, etc.

Now, accountable can only take on

a positive integer values or zero,

because an event cannot occur,

a negative number of times.

Better some examples on the slide here.

So for instance,

the number of depressive symptoms that

are child exhibits or the number of

alcoholic drinks consumed per day,

or the number of re-admissions to

alcohol detoxification programs.

Or the number of disciplinary incidents

among a group of prison inmates or the

number of fillers such as ur and awe

as a function of politeness context.

If you want kind of a linguistic example.

The the figure on the slide actually

shows and artificial data set where

speech error counts are related

to blood alcohol concentration,

and in the last in last week's lecture,

these speech errors were

treated in a binary fashion.

So present or absent and here

they are treated as a count variable.

OK, so in this figure you can see

the Poisson distribution and for two,

representative parameters.

So the height of the bars indicates

the probability of particular cans.

And the present distribution only

really has one parameter Lambda.

And that specifies the rate

of the counter process.

So if Lambda is high,

then the rate of observing an event

such as speech errors is high.

And so you can see that here

Lambda is 5, then the rate is

higher than for a Lambda that is 2.

The average rate is smaller, right?

Now, the Poisson distribution is

bounded by zero because, as we already

said, counts cannot be negative, right?

So this is the lowest value

that you could ever get 0.

The distribution is discrete, right?

So we only have positive integers.

And. So it was zero one or two

and you can't have like 1 1/2.

Or anything in between.

Now, another peculiar property

of the Poisson distribution

is that the variance, the spread,

of the distribution is

linked to the Lambda.

It's kind of married to the Lambda,

and this is very different

from the normal distribution.

In a normal distribution,

you have a standard deviation,

sigma, as an independent parameter

that needs to be estimated, right?

So you can see in the figure that

the distribution for Lambda that is 5,

so the dark grey lines here,

is wider than the distribution

for Lambda is two,

so this is more scrunched up basically.

So for low low rates the variance

is slow because the distribution

is bounded by zero and with no way

of extending beyond that boundary.

And for high rates the distribution

can extend in both directions towards

the lower and the higher counts.

You can think of this as heteroscedasticity.

So that's so does unequal variance as

being build into this distribution.

OK, so Poisson regression models

the parameter parameter Lambda

as a function of some predictors.

That the problem is that

our familiar equation,

can predict any value, right?

So, our familiar linear predictor equation, here,

can predict any value. Uhm?

But Lambda can only be positive.

Now that means that we need a function

that restricts the output of our

linear predictor to positive values.

And the function that achieves

this is the exponential function.

So in the figure on the slide,

you see a linear relation

between X&Y on the left.

And when you transform the Y

with the exponential function,

the values of Y become restricted to

the positive range, that's on the right.

And the dashed line shows that when

zero is exponentially transformed,

it becomes one.

So here we have our familiar

linear predictor equation.

And around that you have

this exponential function.

So wrapping the exponential function

around the linear predictor of beta

0 plus beta 1 * X will ensure that

no negative values can be predicted.

Now you might remember from last

week that the logarithm is the

inverse of the exponential function.

So if we take the logarithm of both sides

of this equation and it becomes this.

So we've taken the logarithm of

the right side of the equation.

Now that kind of undo's the exponential,

so that's why you can't see it here.

But that means we also have to take the

logarithm of the lambda. So here we have

the logarithm of the lambda.

So, the familiar equation of

beta 0 plus beta 1 * X, that familiar

equation will actually predict log lambda.

Now, similar to logistic regression,

that means that we'll have to be

careful in interpreting the output.

Now let's look at an example.

In this example,

we'll look at the data from

somebody called Nettle,

who looked at linguistic diversity.

So the number of languages that

exist in a particular country.

Nettle's hypothesis was that countries

with low ecological risk, so more

fertile environments, are predicted

to have higher linguistic diversity -

so more languages.

Now on this slide you can see data

for five variables for 74 countries.

So population and area contain

the log 10 of population size

and area of the country.

Then we have MGS or mean growing

season and that is the measure of

ecological risk, that is our predictor.

That variable indicates how many months

per year one can grow crops in a country.

Then we have Langs and that is the

number of languages within a country.

Now, as always, it is a good idea

to get a feel for the data set.

So by for example, checking the range

of values for the MGS variable,

- our predictor.

And if we do that by using

the range function as,

you can see on the slide,

it shows us that there are some countries

in which one cannot grow crops at all.

They have zero months and others where

you can grow crops the entire year,

so 12 months.

So the countries Guyana,

the Solomon Islands,

Suriname and Vanuatu have mean

growing seasons of 12, indicating

kind of minimal ecological risk.

And on the other hand,

countries like Yemen and

Oman have an MGS of zero,

which means maximal ecological risk.

You can already see that

the number of languages in those

countries, is quite a bit lower,

which suggests that Nettle's

hypothesis might actually be correct.

OK, now to model linguistic diversity

as a function of ecological risk

with Poisson regression, we

use that GLM function.

The generalised linear model function,

so the same one as we used last

week for logistic regression.

So, as was the case for logistic regression,

the argument family needs to be specified.

In this case it is Poisson,

not binomial.

So, here we have the estimates of our model.

These represent logarithms,

so we must use exponentiation

to report the predicted mean rates.

If we do that,

Here we have the logarithm for

each of the 12 months.

So or each of the 13 values for MGS

(so 0, 1, 2, 3 months, four months,

five months, etc.)

we've got the intercept and the slope

So for each of the values of MGS,

we have kind of a predicted value.

And now you need to take the

exponential of those to be able

to say something about the rates.

That's what  those logarithms represent.

Based on that, we expect a

country with zero months MGS,

so that's the first location,

zero months, to have about 30 languages

and a country with an MGS of 12,

we would expect a value of 166 languages.

Now in this figure you can see the predicted

Rates on a predicted number of

languages as a function of MGS.

So we have 'number of languages' on

the Y axis, and MGS on the X axis.

And instead of points, we have

told GGplot to use

the names of the countries.

And this thick line

is the Poisson regression fit.

OK, there's something else that

we need to take into account here.

So, in order for the regression of

languages as a function of mean

growing season to be meaningful,

you need to control for country size.

Because larger countries tend to

have a higher number of different languages.

So, in this case a country's area

determines what in Poisson regression

is called 'exposure'.

More area means more opportunities

for observing high counts.

So you can adjust a rate

by an exposure variable,

which in this case is area,

but in another model it could be

something like time. So for example,

if you were to conduct an experiment

where you're counting speech errors

in trials with varying durations,

there are naturally going to be

higher counts for longer trials,

so you would want to control

for trial duration,

so trial duration would be the

exposure variable in that context.

Now, for exposure variables the rate

lambda is split into 2 components.

Those are the mean number of events, mu,

per unit of exposure, tau.

So for example, here we would have...

So here we have a lambda,

here we have mu,

so that is the mean number of events.

And here we have tau, the exposure variable.

So we could have the mean

number of languages

per square mile.

Or to take the speech error example:

the number of speech errors, per second.

Now in R this is easy to do.

You simply wrap the offset function, here,

around the exposure variable of interest.

Let's look at this here.

We have the model that we used previously,

right,

the number of languages as a function

of the mean growing season. I'm using

the Poisson regression.

Now what we're doing now, to take this

exposure variable area into account,

we add it to the model by saying: offset,

brackets, area.

Now, the first thing to notice is that

there is no beta for the exposure

variable in the output or anything.

So the output in terms of the

terms present is the same.

So the exposure variable, the term

doesn't have a coefficient and nothing is

actually being estimated for it, right?

But what you can see is that

compared to the original model...

So this is the original model and

this is the new model that takes

the exposure variable into account.

So compared to the original model,

the slope of the MGS variable, here,

has increased considerably, right?

So after controlling for a country size,

the relationship between ecological

risk and linguistic diversity is

estimated to be even stronger.

So here we have .21 rather than .14.

OK, here's another thing that

we need to talk about in the

context of Poisson regression,

and that is overdispersion.

So as we saw at the start of

this lecture that the variance

of the Poisson distribution

scales with the mean.

So the higher the mean rate,

the more variable the count.

Now it is possible that

in an actual data set,

the variance is larger than it's

theoretically expected for a given Lambda.

Now, if that happens,

you're dealing with what is called

overdispersion or excess variance.

Now you can compensate for

overdispersion by using a variant

of Poisson regression that is

called negative binomial regression.

Now, negative binomial regression

is essentially a variant of

Poisson regression where the variance

is kind of freed from the mean.

So in other words,

the constraints that the mean is

equal to the variance is relaxed.

Other than that,

everything stays the same.

So let's re-fit the

model that we've used previously,

so

with the exposure variable.

So let's refit that,

but this time use a negative

binomial regression.

Now for this you we wanted

to use the glm.nb for

negative binomial function.

So here we have exactly the same model,

so this was a model with the

exposure variable, right?

So here we use the GLM to model language

as a function of MGS plus

ise the offset function to take

our exposure variable into account.

Here we'll be doing exactly the same thing,

except that we use glm.nb

The glm.nb function

is from the mass package. That's why

we have to call that package here.

Now, first thing to notice

is that the standard error,

so here the standard error of our estimate,

so the standard error of the MGS

slope, has increased quite a bit.

compared to the Poisson model.

So here we had a standard error of .0047,

and if we use the negative

binomial regression,

we have a standard error of .034.

So that's substantially

larger than for the Poisson model.

So negative binomial models are generally

the more conservative option if there

actually is overdispersion in the data.

Now you can test whether

there is a significant degree

of overdispersion using what is

called an overdispersion test.

Now in one implementation of

this in R is the odTest.

Here, the odTest function,

from the PSCL package.

Now what this function does is that

it performs a likelihood ratio test

comparing the likelihood of the

negative binomial model against the

likelihood of the corresponding Poisson model.

So if you run that on on our data here,

you would see that in this case

the difference in likelihood here

between the two models significant.

So here we have a P value

that is way smaller than .05.

So that indicates that you

should use a negative binomial

model rather than a Poisson model.

OK, now. This figure summarizes the

different aspects of the generalized

linear model framework that we have

been speaking about last week and

this week well and before actually.

Now there is one bit that you haven't been

exposed to so far, and that is the I

function wrapped around the

predictor of linear regression.

So this bit should look familiar by now.

I'm sure it does: Beta 0 plus beta 1 * X.

That is our linear predictor.

Now here for linear regression,

all of a sudden we have is I function

wrapped around it. What is it?

This is called the identity function.

And that is just a mathematical term

for a function that preserves identity.

Or in other words,

this function does nothing.

It can be translated as:

take this predictor,

'as is', without transformations.

Now you might ask why on Earth do

you add it if it does nothing?

The reason for adding it

is that

in the figure in this way shows

that the parallelism

between the different types of model.

So you can see how all of these different

models that we've looked at are really

kind of versions of the same thing.

So it allows you to see that linear

regression is actually a specific

case of the generalized linear model.

So that is,

a general generalized linear model where

the output of the predictive equation

is not transformed.

Whereas for logistic regression,

it is transformed and for

the Poisson regression it is

transformed in a different way.

Now I'm showing you this to

highlight that each generalized

linear model has three components.

So we have: a distribution for

the data generating process.

We have the normal distribution or the

Bernoulli distribution, or the Poisson distribution.

We have a predictive equation or what

is called the linear predictor, here,

and that is the same in each case.

And then we have a link function.

Which links the linear predictor

to the parameter of interest.

So that function,

(so here and here and here)

makes sure that the linear

predictor predicts sensible

values for each parameter, right?

So the link function makes sure that,

for logistic regression the values fall

between zero and one, the values of

(the probability). Or in case of Poisson

regression, the link function makes

sure that values of the predictor

are only positive so we only

get positive values for Lambda.

Now, maybe somewhat confusingly,

these link functions are named

after their inverses, right?

So logistic regression

uses the logit to log odd link and

Poison regression uses the log link.

And it is because of these link functions,

that you need to do these transformations,

right? So for logistic regression,

it returns a log.

It returns log odds predictions and you

need the logistic function

to transform those to probabilities.

For the Poisson regression,

it returns log predictions and

you needs the exponential

function to transform

those log predictions into rates.

OK, in summary. Today we have spoken

about how to model count data with

Poisson regression and its extension

negative binomial regression.

Coefficients of a Poisson model

are shown as log coefficients,

which means that after calculating

the log predictions, you need to use

exponentiation to interpret your

model in terms of average rates.

Now to control for differential

exposure you can add exposure

variables to the model.

And negative binomial regression:

we use that to account for

something called overdispersion.

Finally, we talked about the

fact that each generalized linear

model has three components,

a distribution for the data generating

process, a linear predictor,

and a link function.

That is it for now.

Thank you very much for your attention.