Logit Analysis
Building Logistic Regression Model
# building the model
Model1 <- glm(Won ~ TossWon
+ BatFrist
+ HomeMatch
+ PPRuns
+ PPWickets
+ FourCount
+ SixCount
+ WicketsLost
+ TotelRuns
+ Year
+ Team,
data = IPLData, family = binomial())
# summary of the model
summary(Model1)##
## Call:
## glm(formula = Won ~ TossWon + BatFrist + HomeMatch + PPRuns +
## PPWickets + FourCount + SixCount + WicketsLost + TotelRuns +
## Year + Team, family = binomial(), data = IPLData)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.3241 -0.8824 -0.2143 0.8511 2.2801
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.765059 0.872891 0.876 0.3808
## TossWon1 0.175730 0.311800 0.564 0.5730
## BatFrist1 -0.352356 0.315292 -1.118 0.2638
## HomeMatch1 0.802882 0.205678 3.904 9.48e-05 ***
## PPRuns 0.022711 0.010665 2.129 0.0332 *
## PPWickets -0.025834 0.113906 -0.227 0.8206
## FourCount 0.072760 0.037190 1.956 0.0504 .
## SixCount 0.042510 0.047148 0.902 0.3673
## WicketsLost -0.423578 0.052777 -8.026 1.01e-15 ***
## TotelRuns -0.003345 0.006855 -0.488 0.6256
## YearB2018 0.218858 0.316903 0.691 0.4898
## YearC2017 0.262339 0.350292 0.749 0.4539
## YearD2016 0.193095 0.351213 0.550 0.5825
## YearE2015 0.267262 0.372516 0.717 0.4731
## YearF2014 0.449776 0.324002 1.388 0.1651
## TeamDC -0.999901 0.434966 -2.299 0.0215 *
## TeamGL -0.859330 0.618370 -1.390 0.1646
## TeamKKR -0.544720 0.442733 -1.230 0.2186
## TeamKXIP -0.700140 0.436910 -1.602 0.1090
## TeamMI -0.329963 0.436539 -0.756 0.4497
## TeamRCB -0.882062 0.439100 -2.009 0.0446 *
## TeamRPS -0.701445 0.608046 -1.154 0.2487
## TeamRR -0.553484 0.473415 -1.169 0.2424
## TeamSH 12.541240 882.743514 0.014 0.9887
## TeamSPS 11.577445 882.743529 0.013 0.9895
## TeamSRH -0.329923 0.442357 -0.746 0.4558
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 848.41 on 611 degrees of freedom
## Residual deviance: 654.29 on 586 degrees of freedom
## AIC: 706.29
##
## Number of Fisher Scoring iterations: 13Mixed Effect Logistic Regression Model
## Loading required package: MatrixModel2 <- glmer(Won ~ TossWon
+ BatFrist
+ HomeMatch
+ PPRuns
+ PPWickets
+ FourCount
+ SixCount
+ WicketsLost
+ TotelRuns
+ Year
+ (1 | Team),
data = IPLData, family = binomial,
control = glmerControl(optimizer = "bobyqa"), nAGQ = 1)## boundary (singular) fit: see ?isSingular# print the model results without correlations among fixed effects
print(summary(Model2), corr = TRUE)## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula:
## Won ~ TossWon + BatFrist + HomeMatch + PPRuns + PPWickets + FourCount +
## SixCount + WicketsLost + TotelRuns + Year + (1 | Team)
## Data: IPLData
## Control: glmerControl(optimizer = "bobyqa")
##
## AIC BIC logLik deviance df.resid
## 695.6 766.2 -331.8 663.6 596
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.9607 -0.7157 -0.1722 0.6838 3.2551
##
## Random effects:
## Groups Name Variance Std.Dev.
## Team (Intercept) 0 0
## Number of obs: 612, groups: Team, 12
##
## Fixed effects:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.267304 0.819656 0.326 0.744336
## TossWon1 0.153016 0.306873 0.499 0.618041
## BatFrist1 -0.315705 0.310701 -1.016 0.309579
## HomeMatch1 0.747774 0.200822 3.724 0.000196 ***
## PPRuns 0.022951 0.010443 2.198 0.027966 *
## PPWickets -0.029972 0.113054 -0.265 0.790924
## FourCount 0.060284 0.036544 1.650 0.099019 .
## SixCount 0.025291 0.045808 0.552 0.580878
## WicketsLost -0.434106 0.052018 -8.345 < 2e-16 ***
## TotelRuns -0.001389 0.006775 -0.205 0.837603
## YearB2018 0.219835 0.314966 0.698 0.485199
## YearC2017 0.139841 0.324555 0.431 0.666562
## YearD2016 0.036444 0.325688 0.112 0.910904
## YearE2015 0.255970 0.369062 0.694 0.487952
## YearF2014 0.393555 0.321133 1.226 0.220380
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1##
## Correlation of fixed effects could have been required in summary()##
## Correlation of Fixed Effects:
## (Intr) TssWn1 BtFrs1 HmMtc1 PPRuns PPWckt ForCnt SixCnt WcktsL
## TossWon1 -0.353
## BatFrist1 -0.275 0.763
## HomeMatch1 -0.019 -0.033 -0.140
## PPRuns -0.291 -0.031 0.090 0.003
## PPWickets -0.276 -0.052 0.025 -0.062 0.359
## FourCount 0.206 -0.068 -0.038 0.018 -0.164 -0.051
## SixCount 0.286 -0.100 -0.060 -0.022 -0.003 -0.063 0.532
## WicketsLost -0.219 -0.036 -0.089 -0.053 -0.204 -0.362 0.096 0.155
## TotelRuns -0.549 0.109 -0.018 -0.031 -0.161 0.063 -0.699 -0.743 -0.024
## YearB2018 -0.106 0.011 0.010 0.052 -0.041 -0.061 0.066 -0.019 -0.048
## YearC2017 -0.071 0.001 -0.004 0.036 -0.085 -0.097 0.080 0.101 -0.025
## YearD2016 -0.177 -0.012 -0.008 0.016 0.031 -0.007 0.090 0.158 0.052
## YearE2015 -0.172 -0.008 -0.005 0.020 0.048 0.000 0.028 0.098 -0.017
## YearF2014 -0.209 -0.020 -0.031 0.150 0.081 -0.009 0.156 0.169 0.019
## TtlRns YB2018 YC2017 YD2016 YE2015
## TossWon1
## BatFrist1
## HomeMatch1
## PPRuns
## PPWickets
## FourCount
## SixCount
## WicketsLost
## TotelRuns
## YearB2018 -0.056
## YearC2017 -0.094 0.492
## YearD2016 -0.115 0.469 0.472
## YearE2015 -0.050 0.414 0.415 0.423
## YearF2014 -0.149 0.487 0.484 0.502 0.437
## convergence code: 0
## boundary (singular) fit: see ?isSingularAIC, BIC and Log-likelihood test for the Model1 & Model2
## df AIC
## Model1 26 706.2949
## Model2 16 695.5801## df BIC
## Model1 26 821.1299
## Model2 16 766.2479## 'log Lik.' 654.2949 (df=26)## 'log Lik.' 663.5801 (df=16)Logistic regression Assumptions
The logistic regression method assumes that:
The outcome is a binary or dichotomous variable like yes vs no, positive vs negative, 1 vs 0.
There is a linear relationship between the logit of the outcome and each predictor variables. Recall that the logit function is logit(p) = log(p/(1-p)), where p is the probabilities of the outcome.
There is no influential values (extreme values or outliers) in the continuous predictors
There is no high intercorrelations (i.e. multicollinearity) among the predictors.
Linearity Assumption
library(tidyverse)
library(broom)
# Predict the probability
probabilities <- predict(Model2, type = "response")
logit <- log(probabilities/(1-probabilities))
mydata <- IPLData %>%
dplyr::select_if(is.numeric)
predictors <- colnames(mydata)
mydata <- mydata %>%
mutate(logit = log(probabilities/(1-probabilities))) %>%
gather(key = "predictors", value = "predictor.value", -logit)
# Create the scatter plots:
ggplot(mydata, aes(logit, predictor.value))+
geom_point(size = 0.5, alpha = 0.5) +
geom_smooth(method = "loess") +
theme_bw() +
facet_wrap(~predictors, scales = "free_y")
Multicollinearity
## Registered S3 methods overwritten by 'car':
## method from
## influence.merMod lme4
## cooks.distance.influence.merMod lme4
## dfbeta.influence.merMod lme4
## dfbetas.influence.merMod lme4## GVIF Df GVIF^(1/(2*Df))
## TossWon 2.658700 1 1.630552
## BatFrist 2.719590 1 1.649118
## HomeMatch 1.121307 1 1.058918
## PPRuns 1.617735 1 1.271902
## PPWickets 1.397512 1 1.182164
## FourCount 2.534929 1 1.592146
## SixCount 2.746477 1 1.657250
## WicketsLost 1.277906 1 1.130445
## TotelRuns 4.349602 1 2.085570
## Year 1.678021 5 1.053125
## Team 1.750530 11 1.025777## GVIF Df GVIF^(1/(2*Df))
## TossWon 2.615618 1 1.617287
## BatFrist 2.683231 1 1.638057
## HomeMatch 1.087224 1 1.042700
## PPRuns 1.582959 1 1.258157
## PPWickets 1.388087 1 1.178171
## FourCount 2.468067 1 1.571008
## SixCount 2.630357 1 1.621837
## WicketsLost 1.251017 1 1.118488
## TotelRuns 4.353196 1 2.086431
## Year 1.220991 5 1.020167INFERENCE
Probability of winning for HomeMatch
attach(IPLData)
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "1",
PPRuns = mean(PPRuns),
PPWickets = mean(PPWickets),
FourCount = mean(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbF1 <- predict(Model2, Fdata, type = "response")
ProbF1## 1
## 0.5420654Probability of winning for non-HomeMatch
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "0",
PPRuns = mean(PPRuns),
PPWickets = mean(PPWickets),
FourCount = mean(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbF2 <- predict(Model2, Fdata, type = "response")
ProbF2## 1
## 0.3591366Summary
homeMatch <- c("0", "1")
Probability <- c(round(ProbF2,2),round(ProbF1,2))
cbind(homeMatch, Probability)## homeMatch Probability
## 1 "0" "0.36"
## 1 "1" "0.54"Probability of Winning with (mean - sd) PPRuns
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "1",
PPRuns = mean(PPRuns)- sd(PPRuns),
PPWickets = mean(PPWickets),
FourCount = mean(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbG1 <- predict(Model2, Fdata, type = "response")
ProbG1## 1
## 0.4732919Probability of Winning with (mean + sd) PPRuns
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "1",
PPRuns = mean(PPRuns) + sd(PPRuns),
PPWickets = mean(PPWickets),
FourCount = mean(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbG2 <- predict(Model2, Fdata, type = "response")
ProbG2## 1
## 0.6092723Summary
PowerPlayRuns <- c("mean-sd", "mean", "mean + sd")
probability <- c(round(ProbG1,2),round(ProbF1,2),round(ProbG2,2))
cbind(PowerPlayRuns, probability)## PowerPlayRuns probability
## 1 "mean-sd" "0.47"
## 1 "mean" "0.54"
## 1 "mean + sd" "0.61"Probability of Winning with (mean - sd) FourCount
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "1",
PPRuns = mean(PPRuns),
PPWickets = mean(PPWickets),
FourCount = mean(FourCount)- sd(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbH1 <- predict(Model2, Fdata, type = "response")
ProbH1## 1
## 0.4783956Probability of Winning with (mean + sd) FourCount
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "1",
PPRuns = mean(PPRuns) ,
PPWickets = mean(PPWickets),
FourCount = mean(FourCount) + sd(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbH2 <- predict(Model2, Fdata, type = "response")
ProbH2## 1
## 0.6043903Summary
FourCounts <- c("mean-sd", "mean", "mean + sd")
probability <- c(round(ProbH1,2),round(ProbF1,2),round(ProbH2,2))
cbind(FourCounts, probability)## FourCounts probability
## 1 "mean-sd" "0.48"
## 1 "mean" "0.54"
## 1 "mean + sd" "0.6"Probability of Winning with (mean - sd) WicketsLost
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "1",
PPRuns = mean(PPRuns),
PPWickets = mean(PPWickets),
FourCount = mean(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost) - sd(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbI1 <- predict(Model2, Fdata, type = "response")
ProbI1## 1
## 0.7710066Probability of Winning with (mean + sd) WicketsLost
# creating single value dataframe
Fdata <- data.frame(
TossWon = "1",
BatFrist = "1",
HomeMatch = "1",
PPRuns = mean(PPRuns) ,
PPWickets = mean(PPWickets),
FourCount = mean(FourCount),
SixCount = mean(SixCount),
WicketsLost = mean(WicketsLost)+ sd(WicketsLost),
TotelRuns = mean(TotelRuns),
Year = "A2019",
Team = "KKR")
# predicting probability for female attrition
ProbI2 <- predict(Model2, Fdata, type = "response")
ProbI2## 1
## 0.2938655Summary
WicketLost <- c("mean-sd", "mean", "mean + sd")
probability <- c(round(ProbI1,2),round(ProbF1,2),round(ProbI2,2))
cbind(WicketLost, probability)## WicketLost probability
## 1 "mean-sd" "0.77"
## 1 "mean" "0.54"
## 1 "mean + sd" "0.29"