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Lecture 4: Logistic regression & NFL kickers
Skidmore College, MA 276
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Goals
I Kicker statistics, NFLI Extensions: Expected pointsI Tools: Logistic regression
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Review: multivariate linear regression
Model:yi = β0 + β1 ∗ xi1 + β2 ∗ xi2 + . . .+ βp−1 ∗ xi,p−1 + εiAssumptions:
I εi ∼ N(0, σ2)I εi ,εi′ independent for all i , i ′I Linear relationship between y and x
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Example: NFL kickers
library(RCurl); library(mosaic)url <- getURL("https://raw.githubusercontent.com/statsbylopez/StatsSports/master/Kickers.csv")nfl.kick <- read.csv(text = url)head(nfl.kick)
## Team Year GameMinute Kicker Distance ScoreDiff Grass Success## 1 PHI 2005 3 Akers 49 0 FALSE 0## 2 PHI 2005 29 Akers 49 -7 FALSE 0## 3 PHI 2005 51 Akers 44 -7 FALSE 1## 4 PHI 2005 14 Akers 43 14 TRUE 0## 5 PHI 2005 60 Akers 23 0 TRUE 1## 6 PHI 2005 39 Akers 34 -3 TRUE 1
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Example: NFL kickers
fit.0 <- lm(Success ~ Distance, data = nfl.kick)xyplot(jitter(Success) ~ Distance, data = nfl.kick)
Distance
jitte
r(S
ucce
ss)
0.0
0.5
1.0
20 30 40 50 60 70
What are the problems?
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Example: NFL kickers
fit.0 <- lm(Success ~ Distance, data = nfl.kick)qqnorm(fit.0$resid)
−4 −2 0 2 4
−1.
0−
0.5
0.0
0.5
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
What are the problems?
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Logistic regression model
Model: log( P(y=1)1−P(y=1) ) = β0 + β1 ∗ x1 + β2 ∗ x2 + . . .+ βp−1 ∗ xp−1
Comments:I Dependent variable: log-odds
I What are odds?I Model checks more complexI Uses z test statistics for parameters
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Logistic regression model
Model: log( P(y=1)1−P(y=1) ) = β0 + β1 ∗ x1 + β2 ∗ x2
Extract probabilities:I P(y = 1):
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Estimated logistic regression model
Estimated model:log( P(y=1)
1−P(y=1) ) = β̂0 + β̂1 ∗ x1 + β̂2 ∗ x2 + . . .+ ˆβp−1 ∗ xp−1Slope interpretation:
I β̂1:I eβ̂1 :
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Ex: Field goal kicking by distance
Model: log( P(Success=1)1−P(Success=1) ) = β0 + β1 ∗ Distance
fit.1 <- glm(Success ~ Distance, data = nfl.kick, family = "binomial")msummary(fit.1)
## Coefficients:## Estimate Std. Error z value Pr(>|z|)## (Intercept) 5.724620 0.137223 41.72 <2e-16 ***## Distance -0.102615 0.003135 -32.73 <2e-16 ***#### (Dispersion parameter for binomial family taken to be 1)#### Null deviance: 10105.0 on 11186 degrees of freedom## Residual deviance: 8748.4 on 11185 degrees of freedom## AIC: 8752.4#### Number of Fisher Scoring iterations: 5
Slope interpretation: eβ̂1
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Ex: Field goal kicking by distance
msummary(fit.1)
## Coefficients:## Estimate Std. Error z value Pr(>|z|)## (Intercept) 5.724620 0.137223 41.72 <2e-16 ***## Distance -0.102615 0.003135 -32.73 <2e-16 ***#### (Dispersion parameter for binomial family taken to be 1)#### Null deviance: 10105.0 on 11186 degrees of freedom## Residual deviance: 8748.4 on 11185 degrees of freedom## AIC: 8752.4#### Number of Fisher Scoring iterations: 5
Estimate the probability of a successful 50-yard field goal:
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Ex: Field goal kicking by distance
msummary(fit.1)
## Coefficients:## Estimate Std. Error z value Pr(>|z|)## (Intercept) 5.724620 0.137223 41.72 <2e-16 ***## Distance -0.102615 0.003135 -32.73 <2e-16 ***#### (Dispersion parameter for binomial family taken to be 1)#### Null deviance: 10105.0 on 11186 degrees of freedom## Residual deviance: 8748.4 on 11185 degrees of freedom## AIC: 8752.4#### Number of Fisher Scoring iterations: 5
Estimate the probability of a successful 51-yard field goal:
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Ex: Field goal kicking by distance
Use your answers on the previous slides to estimate the odds of a 51-yardfield goal relative to the odds of a 50-yard field goal. Where else do yousee this number?
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Model checking
I Model checking for logistic regression relies on assessment of fitI Are the predicted probabilities accurate?I Ex: 48 to 52 yard field goals
long.FG <- filter(nfl.kick, Distance >= 48, Distance <= 52)tally(Success ~ Distance, format = "proportion", data = long.FG)
## Distance## Success 48 49 50 51 52## 0 0.3307494 0.3061889 0.3538462 0.3590734 0.4156379## 1 0.6692506 0.6938111 0.6461538 0.6409266 0.5843621
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Expected points
Probability: E (X ) =∑
i=1 xi ∗ P(X = xi)1. What is expected points from a 50 yard field goal?2. What are expected points for 20, 30, 40, 50, 60 yard field goals?
Distance <- data.frame(Distance = c(20, 30, 40, 50, 60))pred <- predict(fit.1, Distance, type = "response")round(pred, 2)*3
## 1 2 3 4 5## 2.94 2.79 2.49 1.92 1.17
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Why it matters?
Offensive decision making:1. Maximize number of points scored on a drive
I What does this entail?I Should we kick the field goal or punt?I Should we kick the field goal or go for it?I When to go for it on fourth down?
2. Maximize chances of winning the gameI When does this differ from (1)?