outline

1) Motivation

2) Model

3) Model fitting

4) Model interpretation

logistic regression

Logistic regression is a specific type of GLM in which the response variable follows a binomial distribution and the link function is (usually) the logit

Logistic regression is used to model a binary response variable as a function of explanatory variables

Examples:

• Is presence of a focal species related to habitat type?

• Is survival a function of body condition?

• Is disease prevalence related to population density?

logistic regression

$$\large y_i \sim Bernoulli(p_i)$$

$$\large log\bigg(\frac{p_i}{1 - p_i}\bigg) = logit(p_i) = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + ...$$

where:

• $\large y_i$ is the response (coded as 0/1)

• $\large p_i$ is the probability of success for sample unit i

• $\large \frac{p_i}{1 - p_i}$ is the odds of success

• $\large log\big(\frac{p_i}{1 - p_i}\big)$ is the log odds

logit link

The logit link allows us to transform the probability $p_i$ (constrained to $0$ to $1$) to the real scale ( $-\infty$ to $\infty$)

$$\large \log\bigg(\frac{0.5}{1-0.5}\bigg) = log(1) = 0$$ The inverse logit $\frac{e^\mu}{1+e^\mu}$ allows us to transform the linear predictor to the probability scale

$$\large \frac{e^0}{1+e^0} = \frac{1}{2}$$

The odds ratio $\frac{p}{1-p}$ varies from $0$ to $\infty$ and indicate the probability of success relative to the probability of failure

• $1:2$ odds = $0.33/(1 - 0.33)$ = failure is twice as likely as success

• $4:1$ odds = $0.8/(1 - 0.8)$ = success if four times a likely as failure

logit link

example

Imagine we are interested in the effects of elevation and habitat on the probability of occurrence for a rare orchid

data("orchidata")
head(orchiddata, n=12)

##    presence abundance elevation habitat
## 1         0         0        58     Oak
## 2         1         7       191     Oak
## 3         0         0        43     Oak
## 4         1        11       374     Oak
## 5         1        11       337     Oak
## 6         1         1        64     Oak
## 7         1         4       195     Oak
## 8         1         6       263     Oak
## 9         0         0       181     Oak
## 10        1         1        59     Oak
## 11        1        50       489   Maple
## 12        1         5       317   Maple

raw data

ggplot(orchiddata, aes(x = elevation, y = presence)) +
  geom_point() +
  scale_y_continuous("Orchid Occurrence") + scale_x_continuous("Elevation")

raw data

library(dplyr)
orchiddata %>% group_by(habitat) %>% summarise(group.prob = mean(presence)) %>%
  ggplot(., aes(x = habitat, y = group.prob)) +
  geom_col(fill = "grey70", color = "black") +
  scale_y_continuous("Proportion of sites with orchids") + scale_x_discrete("Habitat")

the glm function

fm1 <- glm(presence ~ habitat + elevation, 
           family=binomial(link="logit"), 
           data = orchiddata)
broom::tidy(fm1)

intepreting parameters

• The probability of occurrence in maple-dominated forest at sea level = 0.27 (plogis(-0.996))

• The probability of occurrence in oak-dominated forest at sea level = 0.25 (plogis(-0.996 - 0.09))

  • The odds of occurrence in oak (relative to maple) are exp(-0.09) = 0.91

• The probability of occurrence in pine-dominated forest at sea level = 0.21 (plogis(-0.996 - 0.34))

  • The odds of occurrence in pine (relative to maple) are exp(-0.34) = 0.71

intepreting parameters

What about elevation?

First, let's visualize the fitted relationship

example

example

What's the change in probability of occurrence at 1m vs 0m elevation (in maple habitat)?

• plogis(-0.996 + 0.014 * 1) - plogis(-0.996 + 0.014 * 0) = 0.002

What's the change in probability of occurrence at 101m vs 100m elevation ?

• plogis(-0.996 + 0.014 * 101) - plogis(-0.996 + 0.014 * 100) = 0.004

Change in probability is not linear

example

Change in probability is not linear.

But the change is odds is

• Odds ratio: $\frac{p_2}{1 - p_2}/\frac{p_1}{1 - p_1}$

p1 <- plogis(-0.996 + 0.014 * 0)
p2 <- plogis(-0.996 + 0.014 * 1)
(OR1 <- (p2/(1-p2))/(p1/(1-p1)))

## [1] 1.014

p3 <- plogis(-0.996 + 0.014 * 400)
p4 <- plogis(-0.996 + 0.014 * 401)
(OR2 <- (p4/(1-p4))/(p3/(1-p3)))

## [1] 1.014

What is the change in odds for one unit change in elevation?

• $e^{\beta_{Elev}}$ = exp(0.014) = 1.0141

summary

Logistic regression is used when we want to model a binary response as a function of explanatory variables

The response $y_i$ is modeled as arising from a Bernoulli distribution with probability $p_i$ and the logit link is used to map the linear predictor to the probability scale

All of the linear modeling concepts we have learned this semester (continuous/categorical explanatory variables, multiple regression, interactions, random effects) can be used within the logistic regression framework

Parameter estimates from the logistic regression measure the change in log odds for one unit change in the explanatory variables

• Log odds are constant across all values of $x$, but the probabilities are not

looking ahead

Next time: Poisson regression

Reading: Fieberg chp. 15