You can use the log() function in R to calculate the log and the exp() function to calculate \(e^x\). For example, if I wanted to calculate log(10), I would run:

log(10)
## [1] 2.302585

If I wanted to calculate \(e^10\), I would run:

exp(10)
## [1] 22026.47

If I wanted to calculate \(e^{log(10)}\), I would run:

exp(log(10))
## [1] 10

The probability of an event occuring is 0.5, calculate the odds.

0.5 / (1 - 0.5)
## [1] 1

The probability of an event occuring is 0.25, calculate the odds.

0.25 / (1 - 0.25)
## [1] 0.3333333

The probability of an event occuring is 0.9, calculate the log(odds)

log(.9 / (1 - .9))
## [1] 2.197225

The odds of an event are 2:1, calculate the probability.

2 / (1 + 2)
## [1] 0.6666667

The odds of an event are 1:5, calculate the probability.

(1 / 5) / (1 + (1 / 5))
## [1] 0.1666667

Using the following model: log(odds) = 0.2 + 2 x, what is the log(odds) given x is 1

0.2 + 2 * 1
## [1] 2.2

Using the same model, what is the probability that the Outcome is Yes (1), given x is 1?

exp(0.2 + 2 * 1) / (1 + exp(0.2 + 2 * 1))
## [1] 0.9002495
library(Stat2Data)
library(tidyverse)
library(broom)
data("MedGPA")
glimpse(MedGPA)
## Observations: 55
## Variables: 11
## $ Accept     <fct> D, A, A, A, A, A, A, D, A, A, A, A, A, D, D, A, D, A,…
## $ Acceptance <int> 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1,…
## $ Sex        <fct> F, M, F, F, F, M, M, M, F, F, F, F, M, M, M, F, M, M,…
## $ BCPM       <dbl> 3.59, 3.75, 3.24, 3.74, 3.53, 3.59, 3.85, 3.26, 3.74,…
## $ GPA        <dbl> 3.62, 3.84, 3.23, 3.69, 3.38, 3.72, 3.89, 3.34, 3.71,…
## $ VR         <int> 11, 12, 9, 12, 9, 10, 11, 11, 8, 9, 11, 11, 8, 9, 11,…
## $ PS         <int> 9, 13, 10, 11, 11, 9, 12, 11, 10, 9, 9, 8, 10, 9, 8, …
## $ WS         <int> 9, 8, 5, 7, 4, 7, 6, 8, 6, 6, 8, 4, 7, 4, 6, 8, 8, 9,…
## $ BS         <int> 9, 12, 9, 10, 11, 10, 11, 9, 11, 10, 11, 8, 10, 10, 7…
## $ MCAT       <int> 38, 45, 33, 40, 35, 36, 40, 39, 35, 34, 39, 31, 35, 3…
## $ Apps       <int> 5, 3, 19, 5, 11, 5, 5, 7, 5, 11, 6, 9, 5, 8, 15, 6, 6…

Fit the appropriate model predicting MCAT from GPA

lm(MCAT ~ GPA, data = MedGPA) %>%
  tidy()
## # A tibble: 2 x 5
##   term        estimate std.error statistic   p.value
##   <chr>          <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept)     3.92      6.92     0.567 0.573    
## 2 GPA             9.10      1.94     4.69  0.0000197
glm(MCAT ~ GPA, data = MedGPA) %>%
  tidy()
## # A tibble: 2 x 5
##   term        estimate std.error statistic   p.value
##   <chr>          <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept)     3.92      6.92     0.567 0.573    
## 2 GPA             9.10      1.94     4.69  0.0000197
glm(MCAT ~ GPA, data = MedGPA, family = gaussian) %>%
  tidy()
## # A tibble: 2 x 5
##   term        estimate std.error statistic   p.value
##   <chr>          <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept)     3.92      6.92     0.567 0.573    
## 2 GPA             9.10      1.94     4.69  0.0000197

Fit the appropriate model predicting Accept from GPA

levels(MedGPA$Accept)
## [1] "A" "D"
MedGPA <- MedGPA %>%
  mutate(Accept = fct_relevel(Accept, c("D", "A")))

glm(Accept ~ GPA, data = MedGPA, family = binomial) %>%
  tidy()
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)   -19.2       5.63     -3.41 0.000644
## 2 GPA             5.45      1.58      3.45 0.000553

How do you think you interpret the coefficient for GPA in the second model?

One unit change in GPA yields a 5.45 change in the log odds of acceptance to medical school.

How can I change this from log odds to odds

exp(5.45)
## [1] 232.7582

One unite change in GPA yields a 232 change in the odds of acceptance to medical school