Data Handling ZO4030 Lecture 5 Count data

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Data Handling ZO4030 Lecture 5 Count data

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Flowering in 5 varieties of perennial plants. 6 dose levels of a growth fertiliser applied ... flowered. What data type is our response variable? ... – PowerPoint PPT presentation

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Title: Data Handling ZO4030 Lecture 5 Count data

1
Data HandlingZO4030Lecture 5 Count data

Andrew Jackson

2
So far

GLMs to study effects on continuous response data
Fixed factors
Linear covariates
Normal errors(residuals)

3
More types of data

Count data
E.g. the number of deaths resulting from IHD
(Ischaemic Heart Disease) in an age group
Positive integers
Proportional data
Number of males born as a proportion of females
Binary data
Species presence (True / False)
Infection status
Behaviour (ate / didnt eat)

4
Proportional counts

Flowering in 5 varieties of perennial plants
6 dose levels of a growth fertiliser applied
Response is number flowered / number of flower
buds

5
What data type is our response variable?

Number of successes (flowers) out of a number of
attempts (flower buds)
Remind you of anything?
Number of heads (successes) out of repeated coin
tosses (trials)

6
The Binomial Distribution

PDF Probability Density Function
Y binom(p,N)
p 0.5
N 6

7
For our flowers

Flowers Binomial(p,Number)
Now, the probability p is surely related to
things like, dose and variety.
So we have something likep b0 b1dose
b2variety

8
But

We have a problem
p is a probability and as such is strictly
bounded by0 p 1

9
Need to transform our relationship

p b0 b1dose b2variety
Aasdf
This looks equationpretty terrible
and it is!

10
Can simplify though..

p is probability of flowering
1-p is probability of not flowering
Odds of flowering p/(1-p)
Ln(p/(1-p)) logit(p)
Logit(p) b0 b1dose b2variety
And we have a straight line again!
Perfect for a GLM (twiddles and all)

11
Running Logistic Regression

Now that its a simple GLM all we have to do is
identify
Response variable
Fixed factors
Covariates

12
Running Logistic Regression in R

In R this becomes
Y lt- cbind(flowers,number-flowers)
Model1 lt- glm(Ydosevariety, familybinomial)
So this is an ANCOVA or a GLM with multiple
slopes and intercepts

13
Interpreting the output
14
But what about the transformation?

When we calculate the effect sizes (coefficients)
they are on the transformed logit(p) scale
So, variety D is 3.18 larger than variety A
BUT.. This means the log(odds) are bigger

15
Back-calculating

To convert back to raw probabilities we use the
equation
For the intercept (i.e. variety A and dose0)p
1 / (1exp(-(-4.6))) 0.001
For variety Dp 1 / (1exp(-(-4.6 3.18)))
0.19

16
Visualising the output
17
Binary Outcome Data
18
Binary Dataset

Fish infected with a parasite
Binary response variable - Infected
Gender as fixed factor
Age and weight as covariates

19
Our hypotheses

That age affects parasite incidence
That gender affects parasite incidence
That weight affects parasite incidence

20
Exactly as before

Number of trials here 1 (just like a single
coin toss)
The probability of having the disease is modelled
as a function of the explanatory variables

21
Visualise the Data
22
Run the model

Coefficients
Estimate Std. Error z
value Pr(gtz)
(Intercept) -0.109124 1.375388
-0.079 0.937
age 0.024128 0.020874
1.156 0.248
weight -0.074156 0.147678
-0.502 0.616
sexmale -5.969109 4.278066
-1.395 0.163
ageweight -0.001977 0.002006
-0.985 0.325
agesexmale 0.038086 0.041325 0.922
0.357
weightsexmale 0.213830 0.343265 0.623
0.533
ageweightsexmale 0.001651 0.003419 -0.483
0.629

23
What do these parameters mean?

Again
We have modelled the log(odds)
And depending on what you want to compare or
describe, you may need to back transform them
using the earlier equation

24
Summary

Non-continuous data can be modelled using under
the same GLM framework by picking the right
family or distribution
Binomial
Proportion data
Binary data
Poisson
Counts
Things that happen at a given rate

25
To do

Revise your notes from last year on the
chi-square test for analysing proportions
R can do this test too
Read up on the following distributions
Binomial
Poisson

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