Titanic using machine learning with logistic regression
MAHMOUD ELAGAMY
Logistic regression is employed in this analysis to foretell which passengers will survive the sinking of the Titanic.
Introduction
According to Wikipedia, RMS Titanic was a British passenger liner, operated by the White Star Line, which sank in the North Atlantic Ocean on April 15, 1912, after striking an iceberg during her maiden voyage from Southampton, England, to New York City, United States. Of the estimated 2,224 passengers and crew aboard, more than 1,500 died, making it the deadliest sinking of a single ship up to that time. It remains the deadliest peacetime sinking of an ocean liner or cruise ship.
By taking into account each passenger’s sex, age, fare, ticket class, and the number of siblings or parents who were also on board, we hope to create a binary logistic regression model that will represent the pattern of survivors on the Titanic. and we’ll use it to forecast whether some people will survive or not, whose data we won’t include in our model.
Loading libraries
Firstly I will start by loading some packages that I will use during the analysis
library(tidyverse)
library(Hmisc)
library(rms)
library(mice)Getting the data ~~~ ##Getting the data from Hmisc and rms libraries getHdata(titanic3) ##assign it to a variable called “titanic” titanic<-titanic3 ~~~
Exploration of the data ~~~ ##the structure of the data glimpse(titanic) ~~~
Rows: 1,309
Columns: 14
$ pclass <fct> 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st, 1st,…
$ survived <labelled> 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, …
$ name <labelled> "Allen, Miss. Elisabeth Walton", "Allison, Master. Hudson Trevor", "A…
$ sex <fct> female, male, female, male, female, male, female, male, female, male, male…
$ age <labelled> 29.0000, 0.9167, 2.0000, 30.0000, 25.0000, 48.0000, 63.0000, 39.0000,…
$ sibsp <labelled> 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, …
$ parch <labelled> 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, …
$ ticket <labelled> "24160", "113781", "113781", "113781", "113781", "19952", "13502", "1…
$ fare <labelled> 211.3375, 151.5500, 151.5500, 151.5500, 151.5500, 26.5500, 77.9583, 0…
$ cabin <fct> B5, C22 C26, C22 C26, C22 C26, C22 C26, E12, D7, A36, C101, , C62 C64, C62…
$ embarked <fct> Southampton, Southampton, Southampton, Southampton, Southampton, Southampt…
$ boat <fct> 2, 11, , , , 3, 10, , D, , , 4, 9, 6, B, , , 6, 8, A, 5, 5, 5, 4, 8, , 7, …
$ body <labelled> NA, NA, NA, 135, NA, NA, NA, NA, NA, 22, 124, NA, NA, NA, NA, NA, NA,…
$ home.dest <labelled> "St Louis, MO", "Montreal, PQ / Chesterville, ON", "Montreal, PQ / Ch…Below is the description of our data variables from “kaggle.com”
survival Survival 0 = No, 1 = Yes
pclass Ticket class 1 = 1st, 2 = 2nd, 3 = 3rd
sex Sex
Age Age in years
sibsp number of siblings / spouses aboard the Titanic
parch number of parents / children aboard the Titanic
ticket Ticket number
fare Passenger fare
cabin Cabin number
embarked Port of Embarkation C = Cherbourg, Q = Queenstown, S = Southampton
##Eliminate the variables that we wouldn't be using in our analysis.
titanic<-titanic%>%
select(-name,-ticket,-cabin,-embarked,-boat,-body,-home.dest)##summarize the data
summary(titanic) pclass survived sex age sibsp parch
1st:323 Min. :0.000 female:466 Min. : 0.1667 Min. :0.0000 Min. :0.000
2nd:277 1st Qu.:0.000 male :843 1st Qu.:21.0000 1st Qu.:0.0000 1st Qu.:0.000
3rd:709 Median :0.000 Median :28.0000 Median :0.0000 Median :0.000
Mean :0.382 Mean :29.8811 Mean :0.4989 Mean :0.385
3rd Qu.:1.000 3rd Qu.:39.0000 3rd Qu.:1.0000 3rd Qu.:0.000
Max. :1.000 Max. :80.0000 Max. :8.0000 Max. :9.000
NA's :263
fare
Min. : 0.000
1st Qu.: 7.896
Median : 14.454
Mean : 33.295
3rd Qu.: 31.275
Max. :512.329
NA's :1 ##basic description of the data
describe(titanic)titanic
7 Variables 1309 Observations
----------------------------------------------------------------------------------------------
pclass
n missing distinct
1309 0 3
Value 1st 2nd 3rd
Frequency 323 277 709
Proportion 0.247 0.212 0.542
----------------------------------------------------------------------------------------------
survived : Survived
n missing distinct Info Sum Mean Gmd
1309 0 2 0.708 500 0.382 0.4725
----------------------------------------------------------------------------------------------
sex
n missing distinct
1309 0 2
Value female male
Frequency 466 843
Proportion 0.356 0.644
----------------------------------------------------------------------------------------------
age : Age [Year]
n missing distinct Info Mean Gmd .05 .10 .25 .50
1046 263 98 0.999 29.88 16.06 5 14 21 28
.75 .90 .95
39 50 57
lowest : 0.1667 0.3333 0.4167 0.6667 0.7500, highest: 70.5000 71.0000 74.0000 76.0000 80.0000
----------------------------------------------------------------------------------------------
sibsp : Number of Siblings/Spouses Aboard
n missing distinct Info Mean Gmd
1309 0 7 0.67 0.4989 0.777
lowest : 0 1 2 3 4, highest: 2 3 4 5 8
Value 0 1 2 3 4 5 8
Frequency 891 319 42 20 22 6 9
Proportion 0.681 0.244 0.032 0.015 0.017 0.005 0.007
----------------------------------------------------------------------------------------------
parch : Number of Parents/Children Aboard
n missing distinct Info Mean Gmd
1309 0 8 0.549 0.385 0.6375
lowest : 0 1 2 3 4, highest: 3 4 5 6 9
Value 0 1 2 3 4 5 6 9
Frequency 1002 170 113 8 6 6 2 2
Proportion 0.765 0.130 0.086 0.006 0.005 0.005 0.002 0.002
----------------------------------------------------------------------------------------------
fare : Passenger Fare [British Pound (\243)]
n missing distinct Info Mean Gmd .05 .10 .25 .50
1308 1 281 1 33.3 38.61 7.225 7.567 7.896 14.454
.75 .90 .95
31.275 78.051 133.650
lowest : 0.0000 3.1708 4.0125 5.0000 6.2375, highest: 227.5250 247.5208 262.3750 263.0000 512.3292
----------------------------------------------------------------------------------------------A First Look at the Data
let’s plot the survival marginal distributions.
##The survival marginal distributions
ggplot(titanic,aes(x=as.factor(survived),fill=as.factor(survived)))+
geom_bar(width = 0.6)+
theme(legend.position = "none")+
scale_x_discrete(labels=c("No","Yes"))+
xlab("survived")+
ggtitle("The survival marginal distributions")
As we see, just about 38% of the passengers made it out alive.
##The survival marginal distributions according to sex
ggplot(titanic,aes(x=sex,fill=as.factor(survived)))+
geom_bar(width = 0.6)+
guides(fill=guide_legend(title="survived"))+
ggtitle("The survival marginal distributions according to sex")+
l marginal distributions according to sex
The survival marginal distributions according to class
##The survival marginal distributions according to class
ggplot(titanic,aes(x=pclass,fill=as.factor(survived)))+
geom_bar(width = 0.6)+
guides(fill=guide_legend(title="survived"))+
ggtitle("The survival marginal distributions according to class")
The survival marginal distributions according to class ~~~ ##The survival marginal distributions according to class ggplot(titanic,aes(x=pclass,fill=as.factor(survived)))+ geom_bar(width = 0.6)+ guides(fill=guide_legend(title=“survived”))+ ggtitle(“The survival marginal distributions according to class”) ~~~
Clearly, sex and passenger class have a significant effect on the possibility of survival.
using variables to investigate trends
age and the probability of survival
##age and the probability of survival
ggplot(titanic,aes(x=age,y=survived))+
geom_smooth(method="loess")+
ylim (0,1)
We will employ restricted cubic splines on age with 3, 4, or 5 knots since it is unclear how age affects survival probability and it is obvious that there is not a linear relationship.
Does sex make the effect of age more obvious?
##Does sex make the effect of age more obvious?
ggplot(titanic,aes(x=age,y=survived,col=sex))+
geom_smooth(method="loess")+
ylim (0,1)
The impact of age is more pronounced when each sex is taken into account separately. where the older female is more likely to survive than the younger, and the opposite is true for a male.
Let’s see the impact of passenger class along with age. ~~~ ##Let’s see the impact of passenger class along with age. ggplot(titanic,aes(x=age,y=survived,col=pclass))+ geom_smooth(method=“loess”)+ ylim (0,1) ~~~
Considering that the lines aren’t parallel, we’ll include a passenger class/age interaction term.
Split the passenger class and age groups based on gender. ~~~ ##Split the passenger class and age groups based on gender. ggplot(titanic,aes(x=age,y=survived,col=pclass))+ facet_grid(~sex)+ geom_smooth(method=“loess”)+ ylim (0,1) ~~~
the impact of fare
##the impact of fare
ggplot(titanic,aes(x=fare,y=survived))+
geom_smooth(method="loess")+
ylim (0,1)
very positive impact on the probability of survival; investing for outliers
##investing for outliers
ggplot(titanic,aes(fare,fill=fare))+
geom_histogram()
ggplot(titanic,aes(fare))+
geom_histogram()+
facet_grid(~pclass)
There are some outliers among passengers in first class.
Data cleaning and imputation
first we need to convert the survived variable to factor ~~~ ##convert the survived variable to factor titanic\(survived<-as.factor(titanic\)survived) ~~~
Checking for NAs
colSums(is.na(titanic)) pclass survived sex age sibsp parch fare
0 0 0 263 0 0 1 There is one NA in fare variable and 263 in age.
pattern of NAs ~~~ md.pattern(titanic) ~~~
Let’s look at the observation of the one missing value on fare.
titanic[which(is.na(titanic$fare)),] pclass survived sex age sibsp parch fare
1226 3rd 0 male 60.5 0 0 NAAnalysis of Variance between the fare and class of Passengers ~~~ summary(aov(fare~pclass,data=titanic)) ~~~
Df Sum Sq Mean Sq F value Pr(>F)
pclass 2 1272986 636493 372.7 <2e-16 ***
Residuals 1305 2228415 1708
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
1 observation deleted due to missingnessIt is statistically significant that the means of the fares differ amongst the classes.
Tukey test to see where the difference is ~~~ TukeyHSD(aov(fare~pclass,data=titanic)) ~~~
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = fare ~ pclass, data = titanic)
$pclass
diff lwr upr p adj
2nd-1st -66.329795 -74.26990 -58.389693 0.0000000
3rd-1st -74.206103 -80.71643 -67.695772 0.0000000
3rd-2nd -7.876308 -14.74783 -1.004781 0.0198199The difference between each class is statistically significant, so the third class fare’s median will be used to fill in the fare’s missing value.
titanic$fare[which(is.na(titanic$fare))]<-median(titanic$fare[titanic$pclass=="3rd"],na.rm = TRUE)Age imputation
Let’s start by attempting to determine why there are so many missing values for the age field.
test the theory of the missing age owing to not surviving.
summary(is.na(age)~survived,data=titanic)is.na(age) N= 1309
+--------+-+----+----------+
| | | N|is.na(age)|
+--------+-+----+----------+
|survived|0| 809| 0.2348578|
| |1| 500| 0.1460000|
+--------+-+----+----------+
| Overall| |1309| 0.2009167|
+--------+-+----+----------+There is no evidence that the age value was lost as a result of not surviving.
Check it against all the factors
age.na<-glm(is.na(age)~sex+pclass+survived+parch+sibsp,data=titanic)
summary(age.na)Call:
glm(formula = is.na(age) ~ sex + pclass + survived + parch +
sibsp, data = titanic)
Deviance Residuals:
Min 1Q Median 3Q Max
-0.3222 -0.2845 -0.1228 -0.0410 1.0389
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.157060 0.036134 4.347 1.49e-05 ***
sexmale -0.002480 0.026931 -0.092 0.92665
pclass2nd -0.069131 0.031976 -2.162 0.03080 *
pclass3rd 0.161677 0.027340 5.914 4.27e-09 ***
survived1 -0.034264 0.027185 -1.260 0.20775
parch -0.039656 0.013551 -2.927 0.00349 **
sibsp 0.001743 0.011134 0.157 0.87561
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for gaussian family taken to be 0.1493451)
Null deviance: 210.16 on 1308 degrees of freedom
Residual deviance: 194.45 on 1302 degrees of freedom
AIC: 1234.7
Number of Fisher Scoring iterations: 2The only predictor that has a statistically significant impact on missing values in age is “parch”, so we will use the random forest method in the mice function to impute the missing values.
We’ll now divide the data into training and testing data and impute the missing age values to each of them separately.
##make it reproducible
set.seed(113)
#use 80% of dataset as training set and 20% as test set
sample <- sample(c(TRUE, FALSE), nrow(titanic), replace=TRUE, prob=c(0.8,0.2))
train.titanic <- titanic[sample, ]
test.titanic <- titanic[!sample, ]##imputing age into the training set
set.seed(113)
imputed_data<-mice(test.titanic,m=5,maxit = 50,method = 'rf')The one whose mean and standard deviation are closest to the original data will be selected from the five we generate.
mean(titanic$age,na.rm = TRUE) ##the original data's mean value
[1] 29.88113
lapply(imputed_data$imp$age,mean) ##each imputed set's mean
$`1`
[1] 30.40678
$`2`
[1] 27.5
$`3`
[1] 28.52542
$`4`
[1] 28.72881
$`5`
[1] 26.01412
sd(titanic$age,na.rm = TRUE) ##the original data's standard deviation value
[1] 14.4135
lapply(imputed_data$imp$age,sd) ##each imputed set's standard deviation
$`1`
[1] 14.64184
$`2`
[1] 12.71267
$`3`
[1] 13.8727
$`4`
[1] 11.49975
$`5`
[1] 11.72532The first set will be the one we choose.
titanic.train.complete<-complete(imputed_data,1)##assign the survived variable from the test.titanic to to "actual"
actual<-test.titanic$survived
##drop it from the dataset
test.titanic<-select(test.titanic,-survived)##imputing age into the testing data
set.seed(113)
imputed_data.test<-mice(test.titanic,m=5,maxit = 50,method = 'rf')
mean(titanic$age,na.rm = TRUE) ##the original data's mean value
[1] 29.88113
lapply(imputed_data.test$imp$age,mean) ##each imputed set's mean
$`1`
[1] 25.61017
$`2`
[1] 28.33475
$`3`
[1] 28.42938
$`4`
[1] 28.79237
$`5`
[1] 28.34322
sd(titanic$age,na.rm = TRUE) ##the original data's standard deviation value
[1] 14.4135
lapply(imputed_data.test$imp$age,sd) ####each imputed set's standard deviation
$`1`
[1] 11.51126
$`2`
[1] 13.07831
$`3`
[1] 13.13758
$`4`
[1] 13.33263
$`5`
[1] 13.33414The fourth set will be the one we choose.
titanic.test.complete<-complete(imputed_data.test,4)##Change the sibsp and sibsp variables to factor
titanic.train.complete$sibsp<-as.factor(titanic.train.complete$sibsp)
titanic.train.complete$parch<-as.factor(titanic.train.complete$parch)
titanic.test.complete$sibsp<-as.factor(titanic.test.complete$sibsp)
titanic.test.complete$parch<-as.factor(titanic.test.complete$parch)Fitting and testing the model
Let’s look at how each variable impacts the probability of surviving once more before we create our model.
plot(summary(survived~sex+pclass+age+parch+sibsp,data=titanic3))
sur
dd <- datadist (titanic.train.complete)
options(datadist = 'dd')
##our first model
fit1<-lrm(survived~age+sex+pclass+sibsp+parch+fare,data = titanic.train.complete)
fit1Logistic Regression Model
lrm(formula = survived ~ age + sex + pclass + sibsp + parch +
fare, data = titanic.train.complete)
Model Likelihood Discrimination Rank Discrim.
Ratio Test Indexes Indexes
Obs 1023 LR chi2 429.68 R2 0.465 C 0.843
0 624 d.f. 18 R2(18,1023)0.331 Dxy 0.686
1 399 Pr(> chi2) <0.0001 R2(18,730.1)0.431 gamma 0.687
max |deriv| 0.07 Brier 0.147 tau-a 0.327
Coef S.E. Wald Z Pr(>|Z|)
Intercept 3.0416 0.3994 7.61 <0.0001
age -0.0254 0.0069 -3.69 0.0002
sex=male -2.5289 0.1777 -14.23 <0.0001
pclass=2nd -1.1240 0.2714 -4.14 <0.0001
pclass=3rd -1.9147 0.2678 -7.15 <0.0001
sibsp=1 -0.1618 0.2042 -0.79 0.4282
sibsp=2 -0.3075 0.4664 -0.66 0.5096
sibsp=3 -1.9282 0.6324 -3.05 0.0023
sibsp=4 -1.8224 0.7512 -2.43 0.0153
sibsp=5 -9.4166 53.5468 -0.18 0.8604
sibsp=8 -8.9065 55.8148 -0.16 0.8732
parch=1 0.5163 0.2696 1.91 0.0555
parch=2 0.3341 0.3343 1.00 0.3175
parch=3 -0.0044 0.9354 0.00 0.9963
parch=4 -8.3219 65.3777 -0.13 0.8987
parch=5 -1.2721 1.1556 -1.10 0.2710
parch=6 -8.3988 65.1696 -0.13 0.8975
parch=9 -8.6391 67.3570 -0.13 0.8979
fare 0.0019 0.0022 0.89 0.3732 The following models will start to include restricted cubic splines on age as well as interaction terms with age and sex.
fit2<-lrm(survived~pclass+sex*rcs(age,3)+sibsp+parch+fare,data=titanic.train.complete)
fit2Logistic Regression Model
lrm(formula = survived ~ pclass + sibsp + sex * rcs(age, 3) +
parch + fare, data = titanic.train.complete)
Model Likelihood Discrimination Rank Discrim.
Ratio Test Indexes Indexes
Obs 1023 LR chi2 454.44 R2 0.486 C 0.850
0 624 d.f. 21 R2(21,1023)0.345 Dxy 0.700
1 399 Pr(> chi2) <0.0001 R2(21,730.1)0.448 gamma 0.701
max |deriv| 0.07 Brier 0.141 tau-a 0.334
Coef S.E. Wald Z Pr(>|Z|)
Intercept 2.3887 0.5330 4.48 <0.0001
pclass=2nd -1.2219 0.2868 -4.26 <0.0001
pclass=3rd -1.9886 0.2798 -7.11 <0.0001
sibsp=1 -0.1968 0.2082 -0.94 0.3447
sibsp=2 -0.2332 0.4597 -0.51 0.6119
sibsp=3 -1.7161 0.6139 -2.80 0.0052
sibsp=4 -1.8432 0.7021 -2.63 0.0087
sibsp=5 -9.4349 57.2107 -0.16 0.8690
sibsp=8 -9.4003 56.7384 -0.17 0.8684
sex=male -0.3041 0.5451 -0.56 0.5769
age 0.0041 0.0201 0.20 0.8398
age' 0.0016 0.0340 0.05 0.9623
parch=1 0.3442 0.2835 1.21 0.2247
parch=2 0.2199 0.3404 0.65 0.5184
parch=3 -0.3863 0.9536 -0.41 0.6854
parch=4 -8.7418 61.9833 -0.14 0.8878
parch=5 -1.6564 1.1700 -1.42 0.1568
parch=6 -8.7588 61.9897 -0.14 0.8876
parch=9 -9.0449 67.5271 -0.13 0.8934
fare 0.0015 0.0022 0.68 0.4934
sex=male * age -0.0954 0.0254 -3.75 0.0002
sex=male * age' 0.0722 0.0409 1.77 0.0775
Trying to increase the number of knots on restricted cubic age, eliminating the parch variable due to its insignificance, and defining interaction terms between pclass and sex.
fit3<-lrm(survived~sex*rcs(age,5)+sibsp+pclass*sex+fare,data=titanic.train.complete)
fit3
Logistic Regression Model
lrm(formula = survived ~ sex * rcs(age, 5) + sibsp + pclass *
sex + fare, data = titanic.train.complete)
Model Likelihood Discrimination Rank Discrim.
Ratio Test Indexes Indexes
Obs 1023 LR chi2 504.92 R2 0.528 C 0.868
0 624 d.f. 20 R2(20,1023)0.378 Dxy 0.736
1 399 Pr(> chi2) <0.0001 R2(20,730.1)0.485 gamma 0.737
max |deriv| 0.1 Brier 0.135 tau-a 0.351
Coef S.E. Wald Z Pr(>|Z|)
Intercept 4.8729 0.9310 5.23 <0.0001
sex=male -1.4096 1.0473 -1.35 0.1783
age -0.0409 0.0455 -0.90 0.3691
age' 0.1177 0.2909 0.40 0.6859
age'' -0.4607 1.9946 -0.23 0.8173
age''' 0.2249 3.0893 0.07 0.9420
sibsp=1 -0.2850 0.2131 -1.34 0.1810
sibsp=2 -0.1314 0.4992 -0.26 0.7924
sibsp=3 -1.9996 0.6822 -2.93 0.0034
sibsp=4 -2.1627 0.7274 -2.97 0.0029
sibsp=5 -9.0033 32.1878 -0.28 0.7797
sibsp=8 -8.1088 34.2258 -0.24 0.8127
pclass=2nd -1.7139 0.7652 -2.24 0.0251
pclass=3rd -4.0339 0.7232 -5.58 <0.0001
fare 0.0011 0.0022 0.49 0.6252
sex=male * age -0.2272 0.0611 -3.72 0.0002
sex=male * age' 1.0723 0.3768 2.85 0.0044
sex=male * age'' -6.2594 2.5076 -2.50 0.0126
sex=male * age''' 8.6736 3.8019 2.28 0.0225
sex=male * pclass=2nd -0.0391 0.8237 -0.05 0.9621
sex=male * pclass=3rd 2.5798 0.7466 3.46 0.0005 trying to eliminate the fare variable
fit4<-lrm(survived~sex*rcs(age,5)+sibsp+pclass*sex,data=titanic.train.complete)
fit4Logistic Regression Model
lrm(formula = survived ~ sex * rcs(age, 5) + sibsp + pclass *
sex, data = titanic.train.complete)
Model Likelihood Discrimination Rank Discrim.
Ratio Test Indexes Indexes
Obs 1023 LR chi2 504.68 R2 0.528 C 0.868
0 624 d.f. 19 R2(19,1023)0.378 Dxy 0.737
1 399 Pr(> chi2) <0.0001 R2(19,730.1)0.486 gamma 0.738
max |deriv| 0.02 Brier 0.135 tau-a 0.351
Coef S.E. Wald Z Pr(>|Z|)
Intercept 4.9757 0.9082 5.48 <0.0001
sex=male -1.4395 1.0452 -1.38 0.1684
age -0.0405 0.0455 -0.89 0.3734
age' 0.1156 0.2908 0.40 0.6910
age'' -0.4411 1.9941 -0.22 0.8249
age''' 0.1859 3.0886 0.06 0.9520
sibsp=1 -0.2676 0.2100 -1.27 0.2025
sibsp=2 -0.1012 0.4948 -0.20 0.8379
sibsp=3 -1.9606 0.6737 -2.91 0.0036
sibsp=4 -2.1372 0.7250 -2.95 0.0032
sibsp=5 -8.9599 32.1937 -0.28 0.7808
sibsp=8 -8.0443 34.2281 -0.24 0.8142
pclass=2nd -1.8066 0.7424 -2.43 0.0150
pclass=3rd -4.1360 0.6938 -5.96 <0.0001
sex=male * age -0.2272 0.0610 -3.72 0.0002
sex=male * age' 1.0718 0.3767 2.85 0.0044
sex=male * age'' -6.2724 2.5071 -2.50 0.0124
sex=male * age''' 8.7128 3.8011 2.29 0.0219
sex=male * pclass=2nd -0.0029 0.8207 0.00 0.9972
sex=male * pclass=3rd 2.6159 0.7434 3.52 0.0004 In his high ROC area, where c=0.868, we will use model 4 (fit4).
anova(fit4)
Wald Statistics Response: survived
Factor Chi-Square d.f. P
sex (Factor+Higher Order Factors) 192.56 7 <.0001
All Interactions 53.24 6 <.0001
age (Factor+Higher Order Factors) 54.13 8 <.0001
All Interactions 17.29 4 0.0017
Nonlinear (Factor+Higher Order Factors) 33.21 6 <.0001
sibsp 15.68 6 0.0156
pclass (Factor+Higher Order Factors) 94.46 4 <.0001
All Interactions 31.19 2 <.0001
sex * age (Factor+Higher Order Factors) 17.29 4 0.0017
Nonlinear 13.73 3 0.0033
Nonlinear Interaction : f(A,B) vs. AB 13.73 3 0.0033
sex * pclass (Factor+Higher Order Factors) 31.19 2 <.0001
TOTAL NONLINEAR 33.21 6 <.0001
TOTAL INTERACTION 53.24 6 <.0001
TOTAL NONLINEAR + INTERACTION 69.83 9 <.0001
TOTAL 235.51 19 <.0001let’s take a look how the model work at sibsp is zero
ggplot(Predict(fit4, age , sex , pclass, sibsp=0 , fun=plogis))
sibsp are two
ggplot(Predict(fit4, age , sex , pclass, sibsp=0 , fun=plogis))
It’s time to predict the survived of the remaining 20%.
test<-predict(fit4,titanic.test.complete) ##apply the model to the test data
test<-ifelse(test4>0.5,1,0) ##convert the probability value to survive or not.
confusionMatrix(table(test4,actual)) ##compare the model's output to the actual dataConfusion Matrix and Statistics
actual
test4 0 1
0 175 49
1 10 52
Accuracy : 0.7937
95% CI : (0.7421, 0.8391)
No Information Rate : 0.6469
P-Value [Acc > NIR] : 4.384e-08
Kappa : 0.5051
Mcnemar's Test P-Value : 7.530e-07
Sensitivity : 0.9459
Specificity : 0.5149
Pos Pred Value : 0.7813
Neg Pred Value : 0.8387
Prevalence : 0.6469
Detection Rate : 0.6119
Detection Prevalence : 0.7832
Balanced Accuracy : 0.7304
'Positive' Class : 0 Sensitivity appears to be good (95%) but specificity isn’t (only 51%), and overall accuracy appears to be decent (79%).