Odds, Odds Ratios, and the Logit Link

Stat 203 Lecture 25

Dr. Janssen

Setup

Odds

Suppose \(A\) is an event. Odds refers to the ratio of the probability that \(A\) occurs to the probability that \(A\) does not occur.

Therefore, the logit link function in a binomial GLM models the logarithm of the odds:

\[ \log(\text{odds}) = \beta_0 + \beta_1 x. \]

Example: `turbines`

Recall:

library(GLMsData); data(turbines)
tr.logit <- glm( Fissures/Turbines ~ Hours, family=binomial, weights=Turbines, data=turbines)
coef(tr.logit)

  (Intercept)         Hours 
-3.9235965551  0.0009992372

Code

LogOdds <- predict( tr.logit ); odds <- exp( LogOdds )
plot( LogOdds ~ turbines$Hours, type="l", las=1,
        xlim=c(0, 5000), ylim=c(-5, 1),
        ylab="Log-odds", xlab="Run-time (in hours)" )
my <- turbines$Fissures; m <- turbines$Turbines
EmpiricalOdds <- (my + 0.5)/(m - my + 0.5) # To avoid log of zeros
points( log(EmpiricalOdds) ~ turbines$Hours)

Code

plot( odds ~ turbines$Hours, las=1, xlim=c(0, 5000), ylim=c(0, 2),
        type="l", ylab="Odds", xlab="Run-time (in hours)")
points( EmpiricalOdds ~ turbines$Hours)

Odds Ratio

Consider the binomial glm with systematic component

\[ \log \frac{\mu}{1-\mu} = \beta_0 + \beta_1 x, \qquad(1)\]

where \(x\) is a dummy variable taking the values 0 or 1.

Then the odds of a success when \(x = 1\) are \(\exp(\beta_1)\) times greater than a success when \(x = 0\). This is the odds ratio.

Explorations

Setup

Does the probability of a save in a soccer match depend upon whether the goalkeeper’s team is behind or not? Roskes et al. (2011) looked at penalty kicks in the men’s World Cup soccer championships from 1982 to 2010, and they assembled data on 204 penalty kicks during shootouts.

	Saves	Scores	Total
Behind	2	22	24
Not Behind	39	141	180
Total	41	163	204

Begin by calculating various odds, including the odds ratio of a successful penalty kick when the shooter’s team is leading.

Odds are one way to quantify a goalkeeper’s performance. Here the odds that a goalkeeper makes a save when his team is behind is 2 to 22 or 0.09 to 1. Or equivalently, the odds that a goal is scored on a penalty kick is 22 to 2 or 11 to 1. An odds of 11 to 1 tells you that a shooter whose team is ahead will score 11 times for every 1 shot that the goalkeeper saves. When the goalkeeper’s team is not behind the odds a goal is scored is 141 to 39 or 3.61 to 1. We see that the odds of a goal scored on a penalty kick are better when the goalkeeper’s team is behind than when it is not behind (i.e., better odds of scoring for the shooter when the shooter’s team is ahead). We can compare these odds by calculating the odds ratio (OR), 11/3.61 or 3.05, which tells us that the odds of a successful penalty kick are 3.05 times higher when the shooter’s team is leading.

In our example, it is also possible to estimate the probability of a goal,
\(p\), for either circumstance. When the goalkeeper’s team is behind, the probability of a successful penalty kick is
\(p = 22/24\) or 0.833. We can see that the ratio of the probability of a goal scored divided by the probability of no goal is \(( 22 / 24 ) / ( 2 / 24 ) = 22 / 2\) or 11, the odds we had calculated above.

THEN Try to set up a logistic regression model in R to calculate the same odds ratio.

Logistic regression model

Code

soccer_model <- glm(formula = cbind(c(2, 22), c(39, 141)) ~ as.factor(c(0,1)), 
family = binomial())
exp(soccer_model$coefficients)

        (Intercept) as.factor(c(0, 1))1 
         0.05128205          3.04255319

Example: `germ`

library(GLMsData)
data(germ); str(germ)

'data.frame':   21 obs. of  4 variables:
 $ Germ   : int  10 23 23 26 17 5 53 55 32 46 ...
 $ Total  : int  39 62 81 51 39 6 74 72 51 79 ...
 $ Extract: Factor w/ 2 levels "Bean","Cucumber": 1 1 1 1 1 2 2 2 2 2 ...
 $ Seeds  : Factor w/ 2 levels "OA73","OA75": 2 2 2 2 2 2 2 2 2 2 ...

plot( Germ/Total ~ Extract, data=germ, las=1, ylim=c(0, 1) )

plot( Germ/Total ~ Seeds,   data=germ, las=1, ylim=c(0, 1) )

A model

library(GLMsData)
data(germ)
gm.m1 <- glm(Germ/Total ~ Seeds + Extract, family=binomial,
               data=germ, weights=Total)
printCoefmat(coef(summary(gm.m1)))

                Estimate Std. Error z value  Pr(>|z|)    
(Intercept)     -0.70048    0.15072 -4.6475 3.359e-06 ***
SeedsOA75        0.27045    0.15471  1.7482   0.08044 .  
ExtractCucumber  1.06475    0.14421  7.3831 1.546e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

exp( coef(gm.m1) )

    (Intercept)       SeedsOA75 ExtractCucumber 
      0.4963454       1.3105554       2.9001133

Exploration: Health Expenditures

Head to

https://prof.mkjanssen.org/glm/S203_Lecture26.html

Odds, Odds Ratios, and the Logit Link

Setup

Odds

Example: turbines

Odds Ratio

Explorations

Setup

Logistic regression model

Example: germ

A model

Exploration: Health Expenditures

Example: `turbines`

Example: `germ`