credit card data statistical analysis

cc_projgroup 2December 3, 2016

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##Data Preprocessing library(faraway)

## Warning: package 'faraway' was built under R version 3.2.5

library(AER)

## Warning: package 'AER' was built under R version 3.2.5

## Loading required package: car

## ## Attaching package: 'car'

## The following objects are masked from 'package:faraway': ## ## logit, vif

## Loading required package: lmtest

## Loading required package: zoo

## ## Attaching package: 'zoo'

## The following objects are masked from 'package:base': ## ## as.Date, as.Date.numeric

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## Loading required package: sandwich

## Loading required package: survival

## ## Attaching package: 'survival'

## The following object is masked from 'package:faraway': ## ## rats

data(CreditCard) head(CreditCard)

## card reports age income share expenditure owner selfemp ## 1 yes 0 37.66667 4.5200 0.033269910 124.983300 yes no ## 2 yes 0 33.25000 2.4200 0.005216942 9.854167 no no ## 3 yes 0 33.66667 4.5000 0.004155556 15.000000 yes no ## 4 yes 0 30.50000 2.5400 0.065213780 137.869200 no no ## 5 yes 0 32.16667 9.7867 0.067050590 546.503300 yes no ## 6 yes 0 23.25000 2.5000 0.044438400 91.996670 no no ## dependents months majorcards active ## 1 3 54 1 12 ## 2 3 34 1 13 ## 3 4 58 1 5 ## 4 0 25 1 7 ## 5 2 64 1 5 ## 6 0 54 1 1

summary(CreditCard)

## card reports age income ## no : 296 Min. : 0.0000 Min. : 0.1667 Min. : 0.210 ## yes:1023 1st Qu.: 0.0000 1st Qu.:25.4167 1st Qu.: 2.244 ## Median : 0.0000 Median :31.2500 Median : 2.900 ## Mean : 0.4564 Mean :33.2131 Mean : 3.365 ## 3rd Qu.: 0.0000 3rd Qu.:39.4167 3rd Qu.: 4.000 ## Max. :14.0000 Max. :83.5000 Max. :13.500 ## share expenditure owner selfemp ## Min. :0.0001091 Min. : 0.000 no :738 no :1228 ## 1st Qu.:0.0023159 1st Qu.: 4.583 yes:581 yes: 91 ## Median :0.0388272 Median : 101.298 ## Mean :0.0687322 Mean : 185.057 ## 3rd Qu.:0.0936168 3rd Qu.: 249.036 ## Max. :0.9063205 Max. :3099.505 ## dependents months majorcards active ## Min. :0.0000 Min. : 0.00 Min. :0.0000 Min. : 0.000 ## 1st Qu.:0.0000 1st Qu.: 12.00 1st Qu.:1.0000 1st Qu.: 2.000 ## Median :1.0000 Median : 30.00 Median :1.0000 Median : 6.000 ## Mean :0.9939 Mean : 55.27 Mean :0.8173 Mean : 6.997 ## 3rd Qu.:2.0000 3rd Qu.: 72.00 3rd Qu.:1.0000 3rd Qu.:11.000 ## Max. :6.0000 Max. :540.00 Max. :1.0000 Max. :46.000

CreditCard$income <‐ round(CreditCard$income, digits = 2) CreditCard$expenditure <‐ round(CreditCard$expenditure, digits = 2) CreditCard$age <‐ round(CreditCard$age + .01) ##Knowing Our Data library("ggplot2")

## Warning: package 'ggplot2' was built under R version 3.2.5

oldpar <‐ par(mfrow=c(1,1)) ## Density Plot for Expenditure plot(density(CreditCard$expenditure), main = "expenditure")

## Density Plot for Income plot(density(CreditCard$income), main = "income")

##QQ Normal Plot for expenditure qqnorm(CreditCard$expenditure, xlab = "Theoretical", ylab = "Actual values") qqline(CreditCard$expenditure, col = "blue")

##QQ Normal Plot for income qqnorm(CreditCard$income, xlab = "Theoretical", ylab = "Actual values") qqline(CreditCard$income, col = "blue")

##Boxplot boxplot(expenditure~owner, data=CreditCard, xlab = "House Owner", ylab = "Expenditure")

##Linear Model Selection ##Adjusted R square=0.8108 g1<‐ lm(expenditure~., data = CreditCard) summary(g1)

## ## Call: ## lm(formula = expenditure ~ ., data = CreditCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## ‐671.01 ‐31.86 4.00 31.42 1870.37 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) ‐173.21651 15.10314 ‐11.469 <2e‐16 *** ## cardyes 0.12999 9.75430 0.013 0.9894 ## reports ‐1.55943 2.84062 ‐0.549 0.5831 ## age 0.57416 0.38897 1.476 0.1402 ## income 51.27075 2.18281 23.488 <2e‐16 *** ## share 2465.69482 37.87678 65.098 <2e‐16 *** ## owneryes ‐4.60817 7.72632 ‐0.596 0.5510 ## selfempyes ‐5.01405 13.02388 ‐0.385 0.7003 ## dependents 4.39018 2.86144 1.534 0.1252 ## months ‐0.13467 0.05526 ‐2.437 0.0149 * ## majorcards ‐3.08646 8.59266 ‐0.359 0.7195 ## active 0.82229 0.56700 1.450 0.1472 ## ‐‐‐ ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 118.4 on 1307 degrees of freedom ## Multiple R‐squared: 0.8124, Adjusted R‐squared: 0.8108 ## F‐statistic: 514.4 on 11 and 1307 DF, p‐value: < 2.2e‐16

##Adjusted R square=0.8108 g2<‐ lm(expenditure ~ income + share + months, data = CreditCard) summary(g2)

## ## Call: ## lm(formula = expenditure ~ income + share + months, data = CreditCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## ‐666.87 ‐28.91 3.47 30.12 1872.37 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) ‐157.3660 8.0093 ‐19.648 <2e‐16 *** ## income 53.1772 1.9442 27.352 <2e‐16 *** ## share 2460.0934 34.5478 71.208 <2e‐16 *** ## months ‐0.1018 0.0497 ‐2.049 0.0407 * ## ‐‐‐ ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 118.4 on 1315 degrees of freedom ## Multiple R‐squared: 0.8112, Adjusted R‐squared: 0.8108 ## F‐statistic: 1884 on 3 and 1315 DF, p‐value: < 2.2e‐16

##Adjusted R square=0.08431 g3<‐ lm(expenditure ~ age + owner + income, data = CreditCard) summary(g3)

## ## Call: ## lm(formula = expenditure ~ age + owner + income, data = CreditCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## ‐538.70 ‐137.16 ‐67.59 60.48 2509.26 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 99.011 25.888 3.825 0.000137 *** ## age ‐2.547 0.782 ‐3.257 0.001155 ** ## owneryes 16.422 15.979 1.028 0.304277 ## income 48.581 4.607 10.545 < 2e‐16 *** ## ‐‐‐ ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 260.5 on 1315 degrees of freedom ## Multiple R‐squared: 0.08639, Adjusted R‐squared: 0.08431 ## F‐statistic: 41.45 on 3 and 1315 DF, p‐value: < 2.2e‐16

##We reject the subset models and choose the bigger model g1 for our further analysis. ##Checking non‐constant variance, non‐normality ##We can see the variance of the residuals is increasing as the fitted values get larger par(mfrow=c(1,1)) plot(fitted(g1), residuals(g1), xlab="Fitted", ylab="Residuals") abline(h=0)

##An approximate test of nonconstant variance ##There is linear relationship between residuals and the fitted value. summary(lm(abs(residuals(g1)) ~ fitted(g1)))

## ## Call: ## lm(formula = abs(residuals(g1)) ~ fitted(g1)) ## ## Residuals: ## Min 1Q Median 3Q Max ## ‐288.72 ‐41.06 ‐9.76 25.68 1569.37 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 15.250276 2.938698 5.189 2.44e‐07 *** ## fitted(g1) 0.232487 0.009565 24.307 < 2e‐16 *** ## ‐‐‐ ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 85.2 on 1317 degrees of freedom ## Multiple R‐squared: 0.3097, Adjusted R‐squared: 0.3092 ## F‐statistic: 590.8 on 1 and 1317 DF, p‐value: < 2.2e‐16

##Checking for non‐normal errors ##QQ‐plots for detecting nonnormality par(mfrow=c(1,1)) qqnorm(residuals(g1), ylab="Residuals") qqline(residuals(g1))

##The histogram for detecting nonnormality hist(residuals(g1))

##A test of normal versus nonnormal errors shapiro.test(residuals(g1))

## ## Shapiro‐Wilk normality test ## ## data: residuals(g1) ## W = 0.63898, p‐value < 2.2e‐16

##Checking for unusual observations ##Checking for influential outliers ##The leverage measure for detecting influential outliers library("faraway") sales<‐ row.names(CreditCard) halfnorm(lm.influence(g1)$hat, labs=sales, ylab="Leverages")

##Cook's Distance for detecting influential outliers cook <‐ cooks.distance(g1) ##Half normal plot of Cook's Distance with labels of three largest value halfnorm(cook,3,labs=sales,ylab="Cook's distance")

##Model fit excluding observation with largest Cook's Distance CreditCard2 <‐ CreditCard[‐447,] g4<‐lm(expenditure~. ,data = CreditCard2) ##Comparison of model fits with and without influential observation coef(g1); coef(g4); summary(g1); summary(g4)

## (Intercept) cardyes reports age income ## ‐173.2165091 0.1299925 ‐1.5594275 0.5741566 51.2707456 ## share owneryes selfempyes dependents months ## 2465.6948184 ‐4.6081681 ‐5.0140531 4.3901807 ‐0.1346725 ## majorcards active ## ‐3.0864627 0.8222948


## ## Call: ## lm(formula = expenditure ~ ., data = CreditCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## ‐671.01 ‐31.86 4.00 31.42 1870.37 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) ‐173.21651 15.10314 ‐11.469 <2e‐16 *** ## cardyes 0.12999 9.75430 0.013 0.9894 ## reports ‐1.55943 2.84062 ‐0.549 0.5831 ## age 0.57416 0.38897 1.476 0.1402 ## income 51.27075 2.18281 23.488 <2e‐16 *** ## share 2465.69482 37.87678 65.098 <2e‐16 *** ## owneryes ‐4.60817 7.72632 ‐0.596 0.5510 ## selfempyes ‐5.01405 13.02388 ‐0.385 0.7003 ## dependents 4.39018 2.86144 1.534 0.1252 ## months ‐0.13467 0.05526 ‐2.437 0.0149 * ## majorcards ‐3.08646 8.59266 ‐0.359 0.7195 ## active 0.82229 0.56700 1.450 0.1472 ## ‐‐‐ ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 118.4 on 1307 degrees of freedom ## Multiple R‐squared: 0.8124, Adjusted R‐squared: 0.8108 ## F‐statistic: 514.4 on 11 and 1307 DF, p‐value: < 2.2e‐16

## ## Call: ## lm(formula = expenditure ~ ., data = CreditCard2) ## ## Residuals: ## Min 1Q Median 3Q Max ## ‐634.84 ‐31.07 2.77 29.50 1083.73 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) ‐157.44703 13.56086 ‐11.610 < 2e‐16 *** ## cardyes 7.04707 8.74830 0.806 0.42066 ## reports ‐0.23040 2.54626 ‐0.090 0.92792 ## age 0.49416 0.34855 1.418 0.15650 ## income 45.93365 1.97828 23.219 < 2e‐16 *** ## share 2415.38198 34.05304 70.930 < 2e‐16 *** ## owneryes ‐2.25150 6.92400 ‐0.325 0.74510 ## selfempyes ‐0.29942 11.67230 ‐0.026 0.97954 ## dependents 6.49011 2.56651 2.529 0.01156 * ## months ‐0.14118 0.04951 ‐2.851 0.00442 ** ## majorcards ‐1.93620 7.69926 ‐0.251 0.80148 ## active 0.37043 0.50865 0.728 0.46659 ## ‐‐‐ ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 106.1 on 1306 degrees of freedom ## Multiple R‐squared: 0.8351, Adjusted R‐squared: 0.8337 ## F‐statistic: 601.4 on 11 and 1306 DF, p‐value: < 2.2e‐16

##Indicator of Model fit, adjusted R square increases by eliminating the 447th point. ##Added Variable Plot d <‐ residuals(lm(expenditure~card+reports+age+share+owner+selfemp+dependents+majorcards+active, CreditCard)) m <‐ residuals(lm(income~card+reports+age+share+owner+selfemp+dependents+majorcards+active, CreditCard)) plot(m,d,xlab="income residuals", ylab="expenditure residuals") coef(lm(d~m))

## (Intercept) m ## 7.617251e‐15 5.136060e+01

coef(g1)


abline(0,coef(g1)['income'])

#The plot indicates that income should be entered linearly into the model. #ROBUST REGRESSION #OLS g.ols <‐ lm(expenditure ~ ., CreditCard) library(MASS) g.huber <‐ rlm(expenditure ~ ., psi = psi.huber, data = CreditCard)#huber g.hampel <‐ rlm(expenditure ~ ., psi = psi.hampel, init = "lts", data = CreditCard) #hampel g.bisquare <‐ rlm(expenditure ~ ., psi = psi.bisquare, init = "lts", data = CreditCard) #bisquare

## Warning in rlm.default(x, y, weights, method = method, wt.method = ## wt.method, : 'rlm' failed to converge in 20 steps

library(robustbase)

## ## Attaching package: 'robustbase'

## The following object is masked from 'package:survival': ## ## heart

## The following object is masked from 'package:faraway': ## ## epilepsy

g.lts <‐ ltsReg(expenditure ~ ., data = CreditCard) # LTS

## Warning in covMcd(X, alpha = alpha, use.correction = use.correction): The 665‐th order statistic of the absolute deviation of variable 1 ## is zero. ## There are 1023 observations (in the entire dataset of 1319 obs.) ## lying on the hyperplane with equation a_1*(x_i1 ‐ m_1) + ... + ## a_p*(x_ip ‐ m_p) = 0 with (m_1, ..., m_p) the mean of these ## observations and coefficients a_i from the vector a <‐ c(1, 0, 0, ## 0, 0, 0, 0, 0, 0, 0, 0)

library(quantreg)

## Warning: package 'quantreg' was built under R version 3.2.5

## Loading required package: SparseM

## ## Attaching package: 'SparseM'

## The following object is masked from 'package:base': ## ## backsolve

## ## Attaching package: 'quantreg'

## The following object is masked from 'package:survival': ## ## untangle.specials

g.lad <‐ rq(expenditure ~ ., data = CreditCard) # LAD library(car) coefs <‐ compareCoefs(g.ols, g.huber, g.hampel, g.bisquare, g.lts, g.lad, se = FALSE)

## Warning in compareCoefs(g.ols, g.huber, g.hampel, g.bisquare, g.lts, ## g.lad, : models to be compared are of different classes

## ## Call: ## 1: lm(formula = expenditure ~ ., data = CreditCard) ## 2: rlm(formula = expenditure ~ ., data = CreditCard, psi = psi.huber) ## 3: rlm(formula = expenditure ~ ., data = CreditCard, psi = psi.hampel, ## init = "lts") ## 4: rlm(formula = expenditure ~ ., data = CreditCard, psi = ## psi.bisquare, init = "lts") ## 5: ltsReg.formula(formula = expenditure ~ ., data = CreditCard) ## 6: rq(formula = expenditure ~ ., data = CreditCard) ## Est. 1 Est. 2 Est. 3 Est. 4 Est. 5 Est. 6 ## (Intercept) ‐1.73e+02 ‐9.84e+01 ‐7.68e+01 ‐5.69e+01 ‐8.11e+01 ## cardyes 1.30e‐01 1.66e+00 9.70e+00 4.75e+00 6.00e+00 ‐4.53e+00 ## reports ‐1.56e+00 ‐7.13e‐01 ‐4.49e‐01 ‐6.43e‐01 ‐1.40e‐01 ‐7.52e‐01 ## age 5.74e‐01 1.69e‐01 2.24e‐01 1.21e‐01 7.18e‐02 9.04e‐02 ## income 5.13e+01 3.33e+01 2.29e+01 1.80e+01 4.01e+00 3.07e+01 ## share 2.47e+03 2.26e+03 2.19e+03 2.17e+03 2.07e+03 2.25e+03 ## owneryes ‐4.61e+00 1.37e‐01 4.48e‐01 ‐3.70e‐01 ‐7.00e‐01 ‐2.15e‐01 ## selfempyes ‐5.01e+00 ‐4.06e+00 ‐6.27e+00 ‐3.08e+00 9.24e‐01 ‐1.08e+00 ## dependents 4.39e+00 2.50e+00 4.04e+00 1.67e+00 5.42e‐01 6.63e‐01 ## months ‐1.35e‐01 ‐2.22e‐02 ‐2.73e‐03 1.36e‐02 ‐6.80e‐03 ‐1.10e‐02 ## majorcards ‐3.09e+00 2.92e‐01 1.33e+00 1.95e+00 ‐1.15e‐01 3.60e‐01 ## active 8.22e‐01 ‐1.85e‐01 ‐3.52e‐01 ‐1.58e‐01 1.80e‐02 ‐1.04e‐01 ## Intercept ‐1.56e+01

colnames(coefs) <‐ c("OLS", "Huber", "Hampel", "Bisquare", "LTS", "LAD") coefs

## OLS Huber Hampel Bisquare ## (Intercept) ‐173.2165091 ‐98.42281790 ‐7.683350e+01 ‐56.93246290 ## cardyes 0.1299925 1.66296788 9.699189e+00 4.75250636 ## reports ‐1.5594275 ‐0.71321874 ‐4.492695e‐01 ‐0.64294070 ## age 0.5741566 0.16876448 2.235484e‐01 0.12083572 ## income 51.2707456 33.32804912 2.287674e+01 17.99766048 ## share 2465.6948184 2263.93528379 2.191596e+03 2174.47842947 ## owneryes ‐4.6081681 0.13661070 4.484377e‐01 ‐0.36960925 ## selfempyes ‐5.0140531 ‐4.06380494 ‐6.270421e+00 ‐3.07652355 ## dependents 4.3901807 2.49797375 4.035395e+00 1.66759545 ## months ‐0.1346725 ‐0.02224799 ‐2.733177e‐03 0.01360463 ## majorcards ‐3.0864627 0.29179580 1.331047e+00 1.94670914 ## active 0.8222948 ‐0.18469793 ‐3.517037e‐01 ‐0.15819744 ## Intercept NA NA NA NA ## LTS LAD ## (Intercept) NA ‐81.07011726 ## cardyes 6.003973e+00 ‐4.53115627 ## reports ‐1.396422e‐01 ‐0.75240228 ## age 7.180315e‐02 0.09038923 ## income 4.007000e+00 30.69346829 ## share 2.074940e+03 2250.32089452 ## owneryes ‐6.996942e‐01 ‐0.21510403 ## selfempyes 9.237160e‐01 ‐1.07628232 ## dependents 5.419621e‐01 0.66323380 ## months ‐6.796691e‐03 ‐0.01099476 ## majorcards ‐1.151345e‐01 0.36028882 ## active 1.801497e‐02 ‐0.10395183 ## Intercept ‐1.555717e+01 NA

##Comments:: All three M‐estimation methods, Huber, Bisquare, and Hample are quite similar to each other, and different from OLS and both LTS's.LAD is similar to OLS. LTS is recommended since it has the best breakdown. #CHECK FOR LEAST MODEL g.step <‐ step(g1)

## Start: AIC=12606.27 ## expenditure ~ card + reports + age + income + share + owner + ## selfemp + dependents + months + majorcards + active ## ## Df Sum of Sq RSS AIC ## ‐ card 1 2 18326814 12604 ## ‐ majorcards 1 1809 18328620 12604 ## ‐ selfemp 1 2078 18328890 12604 ## ‐ reports 1 4226 18331037 12605 ## ‐ owner 1 4988 18331799 12605 ## <none> 18326811 12606 ## ‐ active 1 29492 18356303 12606 ## ‐ age 1 30551 18357363 12606 ## ‐ dependents 1 33007 18359818 12607 ## ‐ months 1 83288 18410100 12610 ## ‐ income 1 7736017 26062828 13069 ## ‐ share 1 59421549 77748361 14510 ## ## Step: AIC=12604.27 ## expenditure ~ reports + age + income + share + owner + selfemp + ## dependents + months + majorcards + active ## ## Df Sum of Sq RSS AIC ## ‐ majorcards 1 1809 18328623 12602 ## ‐ selfemp 1 2088 18328902 12602 ## ‐ owner 1 5009 18331823 12603 ## ‐ reports 1 5432 18332246 12603 ## <none> 18326814 12604 ## ‐ active 1 30355 18357168 12604 ## ‐ age 1 30551 18357364 12604 ## ‐ dependents 1 33155 18359969 12605 ## ‐ months 1 83286 18410100 12608 ## ‐ income 1 7807034 26133848 13070 ## ‐ share 1 68280506 86607320 14651 ## ## Step: AIC=12602.4 ## expenditure ~ reports + age + income + share + owner + selfemp + ## dependents + months + active ## ## Df Sum of Sq RSS AIC ## ‐ selfemp 1 2088 18330710 12600 ## ‐ owner 1 5170 18333793 12601 ## ‐ reports 1 5328 18333951 12601 ## <none> 18328623 12602 ## ‐ active 1 29157 18357780 12602 ## ‐ age 1 30715 18359337 12603 ## ‐ dependents 1 33672 18362295 12603 ## ‐ months 1 82170 18410792 12606 ## ‐ income 1 7853017 26181640 13071 ## ‐ share 1 68411348 86739970 14651 ## ## Step: AIC=12600.55 ## expenditure ~ reports + age + income + share + owner + dependents +

## months + active ## ## Df Sum of Sq RSS AIC ## ‐ owner 1 5074 18335784 12599 ## ‐ reports 1 5341 18336052 12599 ## <none> 18330710 12600 ## ‐ active 1 29117 18359828 12601 ## ‐ age 1 30039 18360749 12601 ## ‐ dependents 1 33683 18364393 12601 ## ‐ months 1 82936 18413647 12604 ## ‐ income 1 7885891 26216602 13070 ## ‐ share 1 68749553 87080263 14654 ## ## Step: AIC=12598.91 ## expenditure ~ reports + age + income + share + dependents + months + ## active ## ## Df Sum of Sq RSS AIC ## ‐ reports 1 4186 18339970 12597 ## ‐ active 1 25223 18361007 12599 ## ‐ age 1 26418 18362202 12599 ## <none> 18335784 12599 ## ‐ dependents 1 29711 18365495 12599 ## ‐ months 1 88821 18424605 12603 ## ‐ income 1 8016819 26352603 13075 ## ‐ share 1 68772952 87108736 14652 ## ## Step: AIC=12597.21 ## expenditure ~ age + income + share + dependents + months + active ## ## Df Sum of Sq RSS AIC ## ‐ active 1 22090 18362060 12597 ## ‐ age 1 26693 18366663 12597 ## <none> 18339970 12597 ## ‐ dependents 1 29784 18369754 12597 ## ‐ months 1 89980 18429950 12602 ## ‐ income 1 8036240 26376210 13074 ## ‐ share 1 70719037 89059006 14680 ## ## Step: AIC=12596.8 ## expenditure ~ age + income + share + dependents + months ## ## Df Sum of Sq RSS AIC ## <none> 18362060 12597 ## ‐ dependents 1 31895 18393955 12597 ## ‐ age 1 32293 18394353 12597 ## ‐ months 1 87594 18449654 12601 ## ‐ income 1 8245145 26607205 13084 ## ‐ share 1 70725375 89087435 14678

summary(g.step)

## ## Call: ## lm(formula = expenditure ~ age + income + share + dependents + ## months, data = CreditCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## ‐674.42 ‐29.70 4.84 31.05 1879.22 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) ‐172.91959 12.35476 ‐13.996 <2e‐16 *** ## age 0.57505 0.37842 1.520 0.1289 ## income 51.27696 2.11179 24.281 <2e‐16 *** ## share 2468.67451 34.71399 71.115 <2e‐16 *** ## dependents 4.20062 2.78151 1.510 0.1312 ## months ‐0.13683 0.05467 ‐2.503 0.0124 * ## ‐‐‐ ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 118.3 on 1313 degrees of freedom ## Multiple R‐squared: 0.812, Adjusted R‐squared: 0.8113 ## F‐statistic: 1134 on 5 and 1313 DF, p‐value: < 2.2e‐16

g.step$coef

## (Intercept) age income share dependents ## ‐172.9195852 0.5750485 51.2769573 2468.6745073 4.2006201 ## months ## ‐0.1368346

#ANOVA test anova(g.step, g1)

## Analysis of Variance Table ## ## Model 1: expenditure ~ age + income + share + dependents + months ## Model 2: expenditure ~ card + reports + age + income + share + owner + ## selfemp + dependents + months + majorcards + active ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 1313 18362060 ## 2 1307 18326811 6 35249 0.419 0.8668

#p‐value is 0.8668, greater than 0.05, which means the 5 extra variables in big model do not contribute in determining credicard exenditures. #CROSS VALIDATIONlibrary(DAAG)

## Warning: package 'DAAG' was built under R version 3.2.5

## Loading required package: lattice

## ## Attaching package: 'lattice'

## The following object is masked from 'package:faraway': ## ## melanoma

## ## Attaching package: 'DAAG'

## The following object is masked from 'package:robustbase': ## ## milk

## The following object is masked from 'package:MASS': ## ## hills

## The following object is masked from 'package:survival': ## ## lung

## The following object is masked from 'package:car': ## ## vif

## The following objects are masked from 'package:faraway': ## ## orings, ozone, vif

oldpar <‐ par(mfrow=c(1,2)) p1 <‐ CVlm(data = CreditCard, form.lm=g1, printit=F)

## Warning in CVlm(data = CreditCard, form.lm = g1, printit = F): ## ## As there is >1 explanatory variable, cross‐validation ## predicted values for a fold are not a linear function ## of corresponding overall predicted values. Lines that ## are shown for the different folds are approximate

p2 <‐ CVlm(data = CreditCard, form.lm=g.step, printit=F)

## Warning in CVlm(data = CreditCard, form.lm = g.step, printit = F): ## ## As there is >1 explanatory variable, cross‐validation ## predicted values for a fold are not a linear function ## of corresponding overall predicted values. Lines that ## are shown for the different folds are approximate

par(oldpar) #Compare the two models using the mse's from the cross‐validations #with number of folds equal to 3. #Which model gives the better mse? seed <‐ round(runif(1, min=0, max=100)) oldpar1 <‐ par(mfrow=c(1,2)) mse.g1 <‐ CVlm(data = CreditCard, form.lm=g1, m=3, seed=seed, printit=F, main = "g1")

## Warning in CVlm(data = CreditCard, form.lm = g1, m = 3, seed = seed, printit = F, : ## ## As there is >1 explanatory variable, cross‐validation ## predicted values for a fold are not a linear function ## of corresponding overall predicted values. Lines that ## are shown for the different folds are approximate

mse.g.step <‐ CVlm(data = CreditCard, form.lm=g.step, m=3, seed=seed, printit=F, main = "g.step")

## Warning in CVlm(data = CreditCard, form.lm = g.step, m = 3, seed = seed, : ## ## As there is >1 explanatory variable, cross‐validation ## predicted values for a fold are not a linear function ## of corresponding overall predicted values. Lines that ## are shown for the different folds are approximate

par(oldpar1) data.frame(mse.g1=attr(mse.g1, "ms"), mse.g.step=attr(mse.g.step, "ms"))

## mse.g1 mse.g.step ## 1 14325.28 14186.83

library("scatterplot3d") attach(CreditCard) s3d <‐ scatterplot3d(income,dependents,expenditure, pch=16, highlight.3d=TRUE, type="h", main="3D Scatterplot") fit <‐ lm(expenditure ~ income + dependents) s3d$plane3d(fit)

##This helps us detecting outliers and influential points and it helps us #understanding a confidence region for average and predicted response values. ##Individual CIs confint(g1, level = 0.95)

## 2.5 % 97.5 % ## (Intercept) ‐202.8455645 ‐143.587454 ## cardyes ‐19.0057968 19.265782 ## reports ‐7.1320925 4.013238 ## age ‐0.1889249 1.337238 ## income 46.9885446 55.552947 ## share 2391.3888920 2540.000745 ## owneryes ‐19.7655171 10.549181 ## selfempyes ‐30.5640541 20.535948 ## dependents ‐1.2233452 10.003707 ## months ‐0.2430761 ‐0.026269 ## majorcards ‐19.9433837 13.770458 ## active ‐0.2900267 1.934616

##Joint Confidence Interval for share and active credit cards library(ellipse)

## ## Attaching package: 'ellipse'

## The following object is masked from 'package:car': ## ## ellipse

plot(ellipse(g1, c("share", "active")), type = "l", main = "Joint Confidence Region") points(0,0) points(coef(g1)["share"], coef(g1)["active"], pch=18) abline(v=confint(g1)["share",], lty=2) abline(h=confint(g1)["active",], lty=2)

##Since zero does not fall in any limit, we reject the null hypothesis and accept the alternate hypothesis.

Note that the echo = FALSE parameter was added to the code chunk to prevent printing of the R code thatgenerated the plot.

credit card data statistical analysis

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