what are the odds

• Go to RStudio Cloud and open what are the odds

Outcome variable

• So far, we've only had continuous (numeric, quantitative) outcome variables ( $y$ )
• We've just learned about categorical and binary explanatory variables ( $x$ )
• What if we have a binary outcome variable?

Outcome variable

• So far, we've only had continuous (numeric, quantitative) outcome variables ( $y$ )
• We've just learned about categorical and binary explanatory variables ( $x$ )
• What if we have a binary outcome variable?

Let's look at an example

• 446 teens were asked "On an average school night, do you get at least 7 hours of sleep"
• Outcome is [1 = "Yes", 0 = "No"]
• Is Age related to this outcome?
• What if I try to fit this as a linear regression model?

Let's look at an example

• 446 teens were asked "On an average school night, do you get at least 7 hours of sleep"
• Outcome is [1 = "Yes", 0 = "No"]
• Is Age related to this outcome?
• What if I try to fit this as a linear regression model?

Let's look at an example

• 446 teens were asked "On an average school night, do you get at least 7 hours of sleep"
• Outcome is [1 = "Yes", 0 = "No"]
• Is Age related to this outcome?
• What if I try to fit this as a linear regression model?

Let's look at an example

• Perhaps it would be sensible to find a function that would not extend beyond 0 and 1?

Let's look at an example

• Perhaps it would be sensible to find a function that would not extend beyond 0 and 1?

Let's look at an example

• Perhaps it would be sensible to find a function that would not extend beyond 0 and 1?

• this line is fit using logistic regression model

How does this compare to linear regression?

• $\pi$ is the probability that $y=1$ ( $P(y=1)$ )

Notation

• $\log \left(\frac{\pi }{1-\pi }\right)$: the "log odds"
• $\pi$ is the probability that $y=1$ - the probability that your outcome is 1.
• $\frac{\pi }{1-\pi }$ is a ratio representing the odds that $y=1$
• $\log \left(\frac{\pi }{1-\pi }\right)$ is the log odds
• The transformation from $\pi$ to $\log \left(\frac{\pi }{1-\pi }\right)$ is called the logistic or logit transformation

A bit about "odds"

• The "odds" tell you how likely an event is
• 👛 if I flip a fair coin, what is the probability that I'd get heads?
  • $p=0.5$
• What is the probability that I'd get tails?
  • $1-p=0.5$
  • The odds are 1:1, 0.5:0.5
• the odds can be written as $\frac{p}{1-p}=\frac{0.5}{0.5}=1$

A bit about "odds"

• The "odds" tell you how likely an event is
• ☔ Let's say there is a 60% chance of rain today
• What is the probability that it will rain?
  • $p=0.6$
• What is the probability that it won't rain?
  • $1-p=0.4$
• What are the odds that it will rain?
  • 3 to 2, 3:2, $\frac{0.6}{0.4}=1.5$

Transforming logs

• How do you "undo" a $\log$ base $e$?
• Use $e$! For example:
  • $e^{\log \left(10\right)}=10$
  • $e^{\log \left(1283\right)}=1283$
  • $e^{\log \left(x\right)}=x$

Transforming logs

• How do you "undo" a $\log$ base $e$?
• Use $e$! For example:
  • $e^{\log \left(10\right)}=10$
  • $e^{\log \left(1283\right)}=1283$
  • $e^{\log \left(x\right)}=x$
  • $e^{\log \left(odds\right)}$ = odds

Transforming odds

• odds = $\frac{\pi }{1-\pi }$
• Solving for $\pi$
  • $\pi =\frac{\text{odds}}{1+\text{odds}}$
• Plugging in $e^{\log \left(odds\right)}$ = odds
  • $\pi =\frac{e^{\log \left(odds\right)}}{1+e^{\log \left(odds\right)}}$
• Plugging in $\log \left(odds\right)={\beta }_0+{\beta }_{1}x$
  • $\pi =\frac{e^{{\beta }_0+{\beta }_{1}x}}{1+e^{{\beta }_0+{\beta }_{1}x}}$

The logistic model

• ✌️ forms

The logistic model

The logistic model

• ✌️ forms
• log(odds): $l\approx {\beta }_0+{\beta }_{1}x$
• P(Outcome = Yes): $\pi \approx \frac{e^{{\beta }_0+{\beta }_{1}x}}{1+e^{{\beta }_0+{\beta }_{1}x}}$

what are the odds

• Go to RStudio Cloud and open what are the odds

Example

• We are interested in the probability of getting accepted to medical school given a college student's GPA

data("MedGPA")
ggplot(MedGPA, aes(Accept, GPA)) + 
  geom_boxplot() + 
  geom_jitter()

Example

• We are interested in the probability of getting accepted to medical school given a college student's GPA

Example

• $\log \left(odds\right)={\beta }_0+{\beta }_{1}GPA$
• P(Accept) $\approx \frac{e^{{\beta }_0+{\beta }_{1}GPA}}{1+e^{{\beta }_0+{\beta }_{1}GPA}}$
• We are interested in the probability of getting accepted to medical school given a college student's GPA

Example

• We are interested in the probability of getting accepted to medical school given a college student's GPA

glm(Accept ~ GPA, data = MedGPA,
    family = "binomial")

## 
## Call:  glm(formula = Accept ~ GPA, family = "binomial", data = MedGPA)
## 
## Coefficients:
## (Intercept)          GPA  
##       19.21        -5.45  
## 
## Degrees of Freedom: 54 Total (i.e. Null);  53 Residual
## Null Deviance:        75.8 
## Residual Deviance: 56.8     AIC: 60.8

Example

• We are interested in the probability of getting accepted to medical school given a college student's GPA

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

Example

• We are interested in the probability of getting accepted to medical school given a college student's GPA

glm(Accept ~ GPA, data = MedGPA,
    family = "binomial") %>%
  predict()

##      1      2      3      4      5      6      7      8      9     10 
## -0.538 -1.737  1.590 -0.919  0.771 -1.083 -2.010  0.990 -1.028 -2.010 
##     11     12     13     14     15     16     17     18     19     20 
## -2.447  0.171 -1.356 -0.483  1.208 -0.101 -0.701 -0.101  1.480 -2.010 
##     21     22     23     24     25     26     27     28     29     30 
## -1.028 -1.356 -2.119 -1.956 -0.865 -0.210  0.444 -0.319  0.662 -1.628 
##     31     32     33     34     35     36     37     38     39     40 
## -0.538  2.353 -2.010 -0.974  1.535 -1.847 -0.101  0.662 -1.901  2.080 
##     41     42     43     44     45     46     47     48     49     50 
##  0.826  0.771 -0.538 -2.283  0.826  0.881 -2.447  2.626  1.262 -0.810 
##     51     52     53     54     55 
##  4.371 -0.210  0.226  3.935  0.444

Example

• We are interested in the probability of getting accepted to medical school given a college student's GPA

glm(Accept ~ GPA, data = MedGPA,
    family = "binomial") %>%
  predict(type = "response")

##      1      2      3      4      5      6      7      8      9     10 
## 0.3688 0.1496 0.8306 0.2851 0.6838 0.2529 0.1181 0.7290 0.2634 0.1181 
##     11     12     13     14     15     16     17     18     19     20 
## 0.0797 0.5428 0.2049 0.3815 0.7699 0.4747 0.3315 0.4747 0.8146 0.1181 
##     21     22     23     24     25     26     27     28     29     30 
## 0.2634 0.2049 0.1072 0.1239 0.2963 0.4476 0.6093 0.4208 0.6598 0.1640 
##     31     32     33     34     35     36     37     38     39     40 
## 0.3688 0.9132 0.1181 0.2741 0.8227 0.1363 0.4747 0.6598 0.1300 0.8890 
##     41     42     43     44     45     46     47     48     49     50 
## 0.6955 0.6838 0.3688 0.0925 0.6955 0.7069 0.0797 0.9325 0.7794 0.3078 
##     51     52     53     54     55 
## 0.9875 0.4476 0.5563 0.9808 0.6093

what are the odds

• Go to RStudio Cloud and open what are the odds
• load the Stat2Data, tidyverse, and broom libraries
• load data("MedGPA")
• fit the appropriate model predicting MCAT from GPA
• fit the appropriate model predicting Accept from GPA
• How do you think you interpret the coefficient for GPA in the second model?