Logistic regression

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150

Analysis of proportion data

• We know how many times an event occurred, and how many times did not occur.

• We want to know if these proportions are affected by a treatment or a factor

• Examples:Proportion dying

Proportion responding to a treatment

Proportion in a sex

Proportion flowering

The old fashion way:

• People used to model these data using percentage mortality as the response variable

• The problems with this are:• Errors are not normally distributed• The variance is not constant• The response is bounded (1-0)

• We lose information of the size of the sample

However…

• Some data as percentage of plant cover are better analyzed using the conventional models (normal errors and constant variance) following arcsine transformation (the response variable measured in radians)

•

proportion1sin

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

proportiontiontransforma 1sinarcsin_

If the response variable takes the form of percentage change is some

measurement • It is usually better:

• Analysis of covariance, using final weight as the response variable and initial weight as covariate, or

• By specifying the response variable as a relative growth rate, measured as log(final/initial)

Both of which can be analyzed with normal errors without further transformation

Rational for logistic regression

• The traditional transformation of proportion data was arcsine. This transformation took care of the error distribution. There is nothing wrong with this transformation, but a simpler approach is often preferable, and is likely to produce a model easier to interpret

The logistic curve

• The logistic curve is commonly used to describe data on proportions.

• It asymptotes at 0 and 1, so that negative proportions and responses of more than 100 % cannot be predicted.

Binomial errors• If p = proportion of individuals observed to respond in a given

way• The proportion of individuals that respond in alternative ways

is: 1-p and we shall call this proportion q• n is the size of the sample (or number of attempts • An important point is that the variance of the binomial

distribution is not constant. In fact the variance of a binomial distribution with mean np is:

npqs 2

So that the variance changes with the mean like this:

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

S2

The logistic model

X

X

e

ep

10

10

1

0_, pthenx

The logistic model for p as a function of x is given by:

This model is bounded since:

1_, pthenx

The trick of linearizing the logistic model is a simple transformation

X

X

e

ep

10

10

1

Xp

p101

ln

See better description for the logit transformation in the class website

• Small short-lived perennial herb • Narrowly endemic and endangered• Flowers are small and bisexual• Self-compatible, but requires pollinators to set seed

Hypericum cumulicola:

Menges et al. (1999)

Dolan et al. (1999)

Boyle and Menges (2001)

• 15 populations (various patch sizes)

• >80 individuals per population each year

• Data on height and number of reproductive structures

• Survival between August 1994 and August 1995

Demographic data

Histogram of height (cm) Hypericum cumulicola (1994)

Call:glm(formula = survival ~ rep_structures * height, family = binomial)

Deviance Residuals: Min 1Q Median 3Q Max -2.0576 -0.9510 0.5748 0.7394 1.5518

Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 2.043e+00 1.888e-01 10.819 < 2e-16 ***rep_structures -9.112e-03 2.518e-03 -3.619 0.000296 ***height -2.717e-02 7.588e-03 -3.581 0.000343 ***rep_structures:height 1.219e-04 4.096e-05 2.977 0.002912 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 1018.68 on 878 degrees of freedomResidual deviance: 925.22 on 875 degrees of freedomAIC: 933.22

Number of Fisher Scoring iterations: 4

Calculating a given proportion

• You can back-transform from logits (z) to proportions (p) by

)exp(1

1

1

z

p

Survival vs height

Survival vs rep_structures

Height - rep structures interaction0 fruits 100 fruits

200 fruits 1000 fruits

Height (cm)

surv

ival