Probability, log-odds, and odds
WILD 502- Jay Rotella
To better understand the connections between the log-odds of an outcome, the odds of an outcome, and the
probability of an outcome, it is helpful to work with a range of values on one scale and convert it to the
others. It’s also helpful to visualize the relationships with some plots. Recall that if the probability of an
event is 0.2, that
1. the odds of the event occurring are
odds =0.2
0.8= 0.25
2. the log-odds of the event occurring are
ln0.2
0.8=−1.3863
or
ln(0.25) = −1.3863
3. the probability can be reconstructed as
odds
1 + odds =0.25
1.25 = 0.2
4. the probability can also be reconstructed as
exp(ln(odds))
1 + exp(ln(odds)) =exp(−1.3683)
1 + exp(−1.3683) =0.25
1.25 = 0.2
In R, you can
1.
obtain the odds for a given probability by dividing the probability by 1 minus the probability, e.g.,
odds = 0.2/(1-0.2) = 0.25
2.
obtain the log-odds for a given probability by taking the natural logarithm of the odds, e.g.,
log(0.25)
= -1.3862944 or using the qlogis function on the probability value, e.g., qlogis(0.2) = -1.3862944.
3.
obtain the probability from the log-odds using
exp(x)
1+exp(x)
, where xrepresents the log-odds value either by
writing the expression out, e.g.,
exp(-1.3862944)/(1 + exp(-1.3862944))
, or by using the
plogis
function, e.g., plogis(-1.3862944) = 0.2.
4. obtain the probability from the odds by using odds
1+odds , e.g., 0.25/1.25 = 0.2.
Probability values range from 0 to 1. It turns out that for
exp(x)
1+exp(x)
, values of xranging from -5 to +5 create
probabilities that range from just above 0 to very close to 1. Values of xranging from -1 to +1 create
probabilities that range from about 0.25 to 0.75. The material below will let you explore the relationships for
yourself.
library(ggplot2)
log_odds = seq(from = -5,to=5,by=0.25)
odds = exp(log_odds)
# use 'plogis'function to calculate exp(x)/(1 + exp(x))
p = plogis(log_odds)
# use odds/(1+odds) to calculate p a different way
p2 = odds/(1+ odds)
# store probability of failure (1-p)
q=1- p
# store log_odds and y in data frame for use with ggplot
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