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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
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Data HandlingZO4030Lecture 5 Count data
- Andrew Jackson
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So far
- GLMs to study effects on continuous response data
- Fixed factors
- Linear covariates
- Normal errors(residuals)
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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)
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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
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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)
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The Binomial Distribution
- PDF Probability Density Function
- Y binom(p,N)
- p 0.5
- N 6
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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
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But
- We have a problem
- p is a probability and as such is strictly
bounded by0 p 1
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Need to transform our relationship
- p b0 b1dose b2variety
- Aasdf
- This looks equationpretty terrible
- and it is!
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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)
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Running Logistic Regression
- Now that its a simple GLM all we have to do is
identify - Response variable
- Fixed factors
- Covariates
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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
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Interpreting the output
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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
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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
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Visualising the output
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Binary Outcome Data
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Binary Dataset
- Fish infected with a parasite
- Binary response variable - Infected
- Gender as fixed factor
- Age and weight as covariates
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Our hypotheses
- That age affects parasite incidence
- That gender affects parasite incidence
- That weight affects parasite incidence
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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
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Visualise the Data
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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
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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
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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
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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
