Logistic regression
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Analysis of proportion data
• We know how many times an event occurred,
and how many times it 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 fashioned way
• People used to model these
data using percentages as the
response variable…
• The problems with this are:
• Errors are not normally
distributed!
• The variance is not constant!
• The response is bounded (0-1)!
• We lose information on the
sample size!
However…
• Some data, such as percentage of plant cover,
is better analyzed using the conventional
models (normal errors and constant variance)
following the arcsine transformation (the
response variable measured in radians)…
sin
1
proportion
arcsin_ transformation  sin
1
proportion
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If the response variable takes the form of
percentage change of some measurement
Usually it is better to:
• Analysis of covariance, using final weight as
the response variable and initial weight as the
covariate
• Specifying the response variable as a relative
growth rate, measured as log(final/initial)
Both can be analyzed with normal errors without
further transformations!
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 that is
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:
0.3
s  npq
2
0.25
So that the variance
changes with the mean like
this:
S2
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The logistic model
The logistic model for p as a function of x is given by:
 0  1 X
e
p
 0  1 X
1 e
This model is bounded since:
x  , then _ p  0
x  , then _ p  1
The trick of linearizing the logistic model is a
simple transformation known as logit…
 0  1 X
e
p
 0  1 X
1 e
 p 
   0  1 X
ln 
 1 p 
See better description for the logit transformation in the class website
Hypericum cumulicola
• Small short-lived perennial herb
• Narrowly endemic and endangered
• Flowers are small and bisexual
• Self-compatible, but requires pollinators to set seed
Menges et al. (1999)
Dolan et al. (1999)
Boyle and Menges (2001)
Demographic data
• 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
Histogram of height (cm)
Hypericum cumulicola (1994)
Call:
glm(formula = survival ~ height, family = binomial)
Deviance Residuals:
Min
1Q
Median
-2.1082 -1.0559
0.5870
3Q
0.7859
Max
1.6166
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.194949
0.170647 12.863
<2e-16 ***
height
-0.043645
0.005198 -8.396
<2e-16 ***
--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
Residual deviance: 941.26
AIC: 945.26
on 878
on 877
degrees of freedom
degrees of freedom
Number of Fisher Scoring iterations: 4
Calculating a given proportion
You can back-transform from logits (z)
to proportions (p) by:
p
1

1 
1  exp( z ) 


Survival vs. height
Survival vs. Rep. Structures