World Series

Read in data

baseball = read.csv(“baseball.csv”) str(baseball)

‘data.frame’: 1232 obs. of 15 variables:

$ Team : chr “ARI” “ATL” “BAL” “BOS” …

$ League : chr “NL” “NL” “AL” “AL” …

$ Year : int 2012 2012 2012 2012 2012 2012 2012 2012 2012 2012 …

$ RS : int 734 700 712 734 613 748 669 667 758 726 …

$ RA : int 688 600 705 806 759 676 588 845 890 670 …

$ W : int 81 94 93 69 61 85 97 68 64 88 …

$ OBP : num 0.328 0.32 0.311 0.315 0.302 0.318 0.315 0.324 0.33 0.335 …

$ SLG : num 0.418 0.389 0.417 0.415 0.378 0.422 0.411 0.381 0.436 0.422 …

$ BA : num 0.259 0.247 0.247 0.26 0.24 0.255 0.251 0.251 0.274 0.268 …

$ Playoffs : int 0 1 1 0 0 0 1 0 0 1 …

$ RankSeason : int NA 4 5 NA NA NA 2 NA NA 6 …

$ RankPlayoffs: int NA 5 4 NA NA NA 4 NA NA 2 …

$ G : int 162 162 162 162 162 162 162 162 162 162 …

$ OOBP : num 0.317 0.306 0.315 0.331 0.335 0.319 0.305 0.336 0.357 0.314 …

$ OSLG : num 0.415 0.378 0.403 0.428 0.424 0.405 0.39 0.43 0.47 0.402 …

MLB Glossary

    Team: A code for the name of the team
    League: The Major League Baseball league the team belongs to, either AL (American League) or NL (National League)
    Year: The year of the corresponding record
    RS: The number of runs scored by the team in that year
    RA: The number of runs allowed by the team in that year
    W: The number of regular season wins by the team in that year
    OBP: The on-base percentage of the team in that year
    SLG: The slugging percentage of the team in that year
    BA: The batting average of the team in that year
    Playoffs: Whether the team made the playoffs in that year (1 for yes, 0 for no)
    RankSeason: Among the playoff teams in that year, the ranking of their regular season records (1 is best)
    RankPlayoffs: Among the playoff teams in that year, how well they fared in the playoffs. The team winning the World Series gets a RankPlayoffs of 1.
    G: The number of games a team played in that year
    OOBP: The team’s opponents’ on-base percentage in that year
    OSLG: The team’s opponents’ slugging percentage in that year

#Each row in the baseball dataset represents a team in a particular year.How many team/year pairs are there in the whole dataset? nrow(baseball)

[1] 1232

#Though the dataset contains data from 1962 until 2012, we removed several years with shorter-than-usual seasons. Using the table() function, identify the total number of years included in this dataset. table(baseball$Year)

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1973 1974 1975 1976 1977 1978

20 20 20 20 20 20 20 24 24 24 24 24 24 24 26 26

1979 1980 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1996 1997

26 26 26 26 26 26 26 26 26 26 26 26 26 28 28 28

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

#The baseball dataset contains 47 years (1972, 1981, 1994, and 1995 are missing). We can count the number of years in the table, or use the command length(table(baseball\(Year)) directly to get the answer. length(table(baseball\)Year))

[1] 47

Limiting to Teams Making the Playoffs Because we’re only analyzing teams that made the playoffs, we can use the subset() function to replace baseball with a data frame limited to teams that made the playoffs (so our subsetted data frame should still be called “baseball”). How many team/year pairs are included in the new dataset?

baseball = subset(baseball, Playoffs == 1) nrow(baseball)

[1] 244

#Through the years, different numbers of teams have been invited to the playoffs. table(baseball$Year)

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1973 1974 1975 1976 1977 1978

2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4

1979 1980 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1996 1997

4 4 4 4 4 4 4 4 4 4 4 4 4 4 8 8

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

8 8 8 8 8 8 8 8 8 8 8 8 8 8 10

The number of teams in the postseason has changed.

table(table(baseball$Year))

2 4 8 10

7 23 16 1

Adding an Important Predictor It’s much harder to win the World Series if there are 10 teams competing for the championship versus just two. Therefore, we will add the predictor variable NumCompetitors to the baseball data frame. NumCompetitors will contain the number of total teams making the playoffs in the year of a particular team/year pair. For instance, NumCompetitors should be 2 for the 1962 New York Yankees, but it should be 8 for the 1998 Boston Red Sox.

#We start by storing the output of the table() function that counts the number of playoff teams from each year:

PlayoffTable = table(baseball$Year)

#You can output the table with the following command:

PlayoffTable

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1973 1974 1975 1976 1977 1978

2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4

1979 1980 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1996 1997

4 4 4 4 4 4 4 4 4 4 4 4 4 4 8 8

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

8 8 8 8 8 8 8 8 8 8 8 8 8 8 10

str(names(PlayoffTable))

chr [1:47] “1962” “1963” “1964” “1965” “1966” “1967” “1968” “1969” “1970” …

#Which function call returns the number of playoff teams in 1990 and 2001? PlayoffTable[c(“1990”, “2001”)]

1990 2001

4 8

#Putting it all together, we want to look up the number of teams in the playoffs for each team/year pair in the dataset, and store it as a new variable named NumCompetitors in the baseball data frame. baseball\(NumCompetitors = PlayoffTable[as.character(baseball\)Year)] baseball$NumCompetitors

[1] 10 10 10 10 10 10 10 10 10 10 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

[26] 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

[51] 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

[76] 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

[101] 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

[126] 8 8 8 8 8 8 8 8 8 8 8 8 8 4 4 4 4 4 4 4 4 4 4 4 4

[151] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

[176] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

[201] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

[226] 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2

#How many playoff team/year pairs are there in our dataset from years where 8 teams were invited to the playoffs? table(baseball$NumCompetitors)

2 4 8 10

14 92 128 10

Bivariate Models for Predicting World Series Winner In this problem, we seek to predict whether a team won the World Series; in our dataset this is denoted with a RankPlayoffs value of 1. Add a variable named WorldSeries to the baseball data frame, by typing the following command in your R console:

baseballWorldSeries=as.numeric(baseballRankPlayoffs == 1)

WorldSeries takes value 1 if a team won the World Series in the indicated year and a 0 otherwise.

#How many observations do we have in our dataset where a team did NOT win the World Series? baseball\(WorldSeries = as.numeric(baseball\)RankPlayoffs == 1) table(baseball$WorldSeries)

0 1

197 47

Bivariate Models for Predicting World Series Winner When we’re not sure which of our variables are useful in predicting a particular outcome, it’s often helpful to build bivariate models, which are models that predict the outcome using a single independent variable. Which of the following variables is a significant predictor of the WorldSeries variable in a bivariate logistic regression model? To determine significance, remember to look at the stars in the summary output of the model. We’ll define an independent variable as significant if there is at least one star at the end of the coefficients row for that variable (this is equivalent to the probability column having a value smaller than 0.05). Note that you have to build 12 models to answer this question! Use the entire dataset baseball to build the models

#Which of the following variables is a significant predictor of the WorldSeries variable in a bivariate logistic regression model? #Varibales to use as predictors for each bivariate model(Year, RS, RA, W, OBP, SLG, BA, RankSeason, OOBP,OSLG, NumCompetitors, League) model1<-glm(WorldSeries~Year, data=baseball, family=“binomial”) summary(model1)

Call:

glm(formula = WorldSeries ~ Year, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0297 -0.6797 -0.5435 -0.4648 2.1504

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 72.23602 22.64409 3.19 0.00142 **

Year -0.03700 0.01138 -3.25 0.00115 **

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 228.35 on 242 degrees of freedom

AIC: 232.35

Number of Fisher Scoring iterations: 4

Year is a significant predictor!

model2<-glm(WorldSeries~RS, data=baseball, family=“binomial”) summary(model2)

Call:

glm(formula = WorldSeries ~ RS, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.8254 -0.6819 -0.6363 -0.5561 2.0308

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 0.661226 1.636494 0.404 0.686

RS -0.002681 0.002098 -1.278 0.201

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 237.45 on 242 degrees of freedom

AIC: 241.45

Number of Fisher Scoring iterations: 4

RS is not significant

model3<-glm(WorldSeries~RA, data=baseball, family=“binomial”) summary(model3)

Call:

glm(formula = WorldSeries ~ RA, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.9749 -0.6883 -0.6118 -0.4746 2.1577

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 1.888174 1.483831 1.272 0.2032

RA -0.005053 0.002273 -2.223 0.0262 *

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 233.88 on 242 degrees of freedom

AIC: 237.88

Number of Fisher Scoring iterations: 4

RA is a significant predictor!

model4<-glm(WorldSeries~W, data=baseball, family=“binomial”) summary(model4)

Call:

glm(formula = WorldSeries ~ W, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0623 -0.6777 -0.6117 -0.5367 2.1254

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -6.85568 2.87620 -2.384 0.0171 *

W 0.05671 0.02988 1.898 0.0577 .

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 235.51 on 242 degrees of freedom

AIC: 239.51

Number of Fisher Scoring iterations: 4

W is not significant

model5<-glm(WorldSeries~OBP, data=baseball, family=“binomial”) summary(model5)

Call:

glm(formula = WorldSeries ~ OBP, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.8071 -0.6749 -0.6365 -0.5797 1.9753

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 2.741 3.989 0.687 0.492

OBP -12.402 11.865 -1.045 0.296

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 238.02 on 242 degrees of freedom

AIC: 242.02

Number of Fisher Scoring iterations: 4

OBP is not significant

model6<-glm(WorldSeries~SLG, data=baseball, family=“binomial”) summary(model6)

Call:

glm(formula = WorldSeries ~ SLG, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.9498 -0.6953 -0.6088 -0.5197 2.1136

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 3.200 2.358 1.357 0.1748

SLG -11.130 5.689 -1.956 0.0504 .

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 235.23 on 242 degrees of freedom

AIC: 239.23

Number of Fisher Scoring iterations: 4

SLG is not significant

model7<-glm(WorldSeries~BA, data=baseball, family=“binomial”) summary(model7)

Call:

glm(formula = WorldSeries ~ BA, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.6797 -0.6592 -0.6513 -0.6389 1.8431

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -0.6392 3.8988 -0.164 0.870

BA -2.9765 14.6123 -0.204 0.839

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 239.08 on 242 degrees of freedom

AIC: 243.08

Number of Fisher Scoring iterations: 4

BA is not significant

model8<-glm(WorldSeries~RankSeason, data=baseball, family=“binomial”) summary(model8)

Call:

glm(formula = WorldSeries ~ RankSeason, family = “binomial”,

data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.7805 -0.7131 -0.5918 -0.4882 2.1781

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -0.8256 0.3268 -2.527 0.0115 *

RankSeason -0.2069 0.1027 -2.016 0.0438 *

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 234.75 on 242 degrees of freedom

AIC: 238.75

Number of Fisher Scoring iterations: 4

RankSeason is significant!

model9<-glm(WorldSeries~OOBP, data=baseball, family=“binomial”) summary(model9)

Call:

glm(formula = WorldSeries ~ OOBP, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.5318 -0.5176 -0.5106 -0.5023 2.0697

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -0.9306 8.3728 -0.111 0.912

OOBP -3.2233 26.0587 -0.124 0.902

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 84.926 on 113 degrees of freedom

Residual deviance: 84.910 on 112 degrees of freedom

(130 observations deleted due to missingness)

AIC: 88.91

Number of Fisher Scoring iterations: 4

OOBP is not significant

model10<-glm(WorldSeries~OSLG, data=baseball, family=“binomial”) summary(model10)

Call:

glm(formula = WorldSeries ~ OSLG, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.5610 -0.5209 -0.5088 -0.4902 2.1268

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -0.08725 6.07285 -0.014 0.989

OSLG -4.65992 15.06881 -0.309 0.757

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 84.926 on 113 degrees of freedom

Residual deviance: 84.830 on 112 degrees of freedom

(130 observations deleted due to missingness)

AIC: 88.83

Number of Fisher Scoring iterations: 4

OSLG is not significant

model11<-glm(WorldSeries~NumCompetitors, data=baseball, family=“binomial”) summary(model11)

Call:

glm(formula = WorldSeries ~ NumCompetitors, family = “binomial”,

data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.9871 -0.8017 -0.5089 -0.5089 2.2643

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 0.03868 0.43750 0.088 0.929559

NumCompetitors -0.25220 0.07422 -3.398 0.000678 ***

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 226.96 on 242 degrees of freedom

AIC: 230.96

Number of Fisher Scoring iterations: 4

NumCompetitors is significant!

model12<-glm(WorldSeries~League, data=baseball, family=“binomial”) summary(model12)

Call:

glm(formula = WorldSeries ~ League, family = “binomial”, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.6772 -0.6772 -0.6306 -0.6306 1.8509

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -1.3558 0.2243 -6.045 1.5e-09 ***

LeagueNL -0.1583 0.3252 -0.487 0.626

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 238.88 on 242 degrees of freedom

AIC: 242.88

Number of Fisher Scoring iterations: 4

League is not significant

In short, the significant predictors are Year, RA, and NumCompetitors.

Multivariate Models for Predicting World Series Winner

In this section, we’ll consider multivariate models that combine the variables we found to be significant in bivariate models. Build a model using all of the variables that you found to be significant in the bivariate models. How many variables are significant in the combined model?

#How many variables are significant in the combined model? LogModel = glm(WorldSeries ~ Year + RA + RankSeason + NumCompetitors, data=baseball, family=binomial) summary(LogModel)

Call:

glm(formula = WorldSeries ~ Year + RA + RankSeason + NumCompetitors,

family = binomial, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0336 -0.7689 -0.5139 -0.4583 2.2195

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 12.5874376 53.6474210 0.235 0.814

Year -0.0061425 0.0274665 -0.224 0.823

RA -0.0008238 0.0027391 -0.301 0.764

RankSeason -0.0685046 0.1203459 -0.569 0.569

NumCompetitors -0.1794264 0.1815933 -0.988 0.323

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 226.37 on 239 degrees of freedom

AIC: 236.37

Number of Fisher Scoring iterations: 4

Looking at summary(LogModel), we can see that none of the variables are significant in the multivariate model. Often, variables that were significant in bivariate models are no longer significant in multivariate analysis due to correlation between the variables. Which of the following variable pairs have a high degree of correlation (a correlation greater than 0.8 or less than -0.8)?

#We can compute all pair-wise correlations between these variables with: cor(baseball[c(“Year”, “RA”, “RankSeason”, “NumCompetitors”)])

Year RA RankSeason NumCompetitors

Year 1.0000000 0.4762422 0.3852191 0.9139548

RA 0.4762422 1.0000000 0.3991413 0.5136769

RankSeason 0.3852191 0.3991413 1.0000000 0.4247393

NumCompetitors 0.9139548 0.5136769 0.4247393 1.0000000

While every pair was at least moderately correlated, the only strongly correlated pair was Year/NumCompetitors, with correlation coefficient 0.914.

Let us build all six of the two variable models listed in the previous problem. Together with the four bivariate models that were significant, we should have 10 different logistic regression models to analyze. Which model has the best AIC value (the minimum AIC value)?

#The two-variable models can be built with the following commands:

model13 = glm(WorldSeries ~ Year + RA, data=baseball, family=binomial) summary(model13)

Call:

glm(formula = WorldSeries ~ Year + RA, family = binomial, data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0402 -0.6878 -0.5298 -0.4785 2.1370

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 63.610741 25.654830 2.479 0.0132 *

Year -0.032084 0.013323 -2.408 0.0160 *

RA -0.001766 0.002585 -0.683 0.4945

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 227.88 on 241 degrees of freedom

AIC: 233.88

Number of Fisher Scoring iterations: 4

model14 = glm(WorldSeries ~ Year + RankSeason, data=baseball, family=binomial) summary(model14)

Call:

glm(formula = WorldSeries ~ Year + RankSeason, family = binomial,

data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0560 -0.6957 -0.5379 -0.4528 2.2673

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 63.64855 24.37063 2.612 0.00901 **

Year -0.03254 0.01231 -2.643 0.00822 **

RankSeason -0.10064 0.11352 -0.887 0.37534

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 227.55 on 241 degrees of freedom

AIC: 233.55

Number of Fisher Scoring iterations: 4

model15 = glm(WorldSeries ~ Year + NumCompetitors, data=baseball, family=binomial) summary(model15)

Call:

glm(formula = WorldSeries ~ Year + NumCompetitors, family = binomial,

data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0050 -0.7823 -0.5115 -0.4970 2.2552

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 13.350467 53.481896 0.250 0.803

Year -0.006802 0.027328 -0.249 0.803

NumCompetitors -0.212610 0.175520 -1.211 0.226

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 226.90 on 241 degrees of freedom

AIC: 232.9

Number of Fisher Scoring iterations: 4

model16 = glm(WorldSeries ~ RA + RankSeason, data=baseball, family=binomial) summary(model16)

Call:

glm(formula = WorldSeries ~ RA + RankSeason, family = binomial,

data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.9374 -0.6933 -0.5936 -0.4564 2.1979

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 1.487461 1.506143 0.988 0.323

RA -0.003815 0.002441 -1.563 0.118

RankSeason -0.140824 0.110908 -1.270 0.204

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 232.22 on 241 degrees of freedom

AIC: 238.22

Number of Fisher Scoring iterations: 4

model17 = glm(WorldSeries ~ RA + NumCompetitors, data=baseball, family=binomial) summary(model17)

Call:

glm(formula = WorldSeries ~ RA + NumCompetitors, family = binomial,

data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0433 -0.7826 -0.5133 -0.4701 2.2208

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 0.716895 1.528736 0.469 0.63911

RA -0.001233 0.002661 -0.463 0.64313

NumCompetitors -0.229385 0.088399 -2.595 0.00946 **

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 226.74 on 241 degrees of freedom

AIC: 232.74

Number of Fisher Scoring iterations: 4

model18 = glm(WorldSeries ~ RankSeason + NumCompetitors, data=baseball, family=binomial) summary(model18)

Call:

glm(formula = WorldSeries ~ RankSeason + NumCompetitors, family = binomial,

data = baseball)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0090 -0.7592 -0.5204 -0.4501 2.2562

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 0.12277 0.45737 0.268 0.78837

RankSeason -0.07697 0.11711 -0.657 0.51102

NumCompetitors -0.22784 0.08201 -2.778 0.00546 **

—

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 239.12 on 243 degrees of freedom

Residual deviance: 226.52 on 241 degrees of freedom

AIC: 232.52

Number of Fisher Scoring iterations: 4

None of the models with two independent variables had both variables significant, so none seem promising as compared to a simple bivariate model. Indeed the model with the lowest AIC value is the model with just NumCompetitors as the independent variable. This seems to confirm the claim made by Billy Beane in Moneyball that all that matters in the Playoffs is luck, since NumCompetitors has nothing to do with the quality of the teams!