Transcript Logistic regression - UCF Plant Ecology Laboratory

Logistic regression
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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)
•
1
sin
proportion
arcsin_transformation  sin 1 proportion
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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:
0.3
s  npq
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S2
So that the variance
changes with the mean like
this:
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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
 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:
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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 ~ rep_structures * height, family = binomial)
Deviance Residuals:
Min
1Q
Median
-2.0576 -0.9510
0.5748
3Q
0.7394
Max
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 ‘ ’
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1018.68
Residual deviance: 925.22
AIC: 933.22
on 878
on 875
degrees of freedom
degrees of freedom
Number of Fisher Scoring iterations: 4
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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
Height - rep structures interaction
survival
0 fruits
100 fruits
200 fruits
1000 fruits
Height (cm)