DATA 605 Multiple Regression
Warner Alexis
2024-04-14
Data Analysis
The attached who.csv dataset contains real-world data from 2008. The variables included follow:
Country: name of the country
LifeExp: average life expectancy for the country in years
InfantSurvival: proportion of those surviving to one year or more Under5Survival: proportion of those surviving to five years or more TBFree: proportion of the population without TB. PropMD: proportion of the population who are MDs PropRN: proportion of the population who are RNs PersExp: mean personal expenditures on healthcare in US dollars at average exchange rate GovtExp: mean government expenditures per capita on healthcare, US dollars at average exchange rate TotExp: sum of personal and government expenditures.
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(ggplot2)
library(cowplot)
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recodelibrary(corrplot)## corrplot 0.92 loadeddata <- read.csv('who.csv', stringsAsFactors = F)
head(data)## Country LifeExp InfantSurvival Under5Survival TBFree PropMD
## 1 Afghanistan 42 0.835 0.743 0.99769 0.000228841
## 2 Albania 71 0.985 0.983 0.99974 0.001143127
## 3 Algeria 71 0.967 0.962 0.99944 0.001060478
## 4 Andorra 82 0.997 0.996 0.99983 0.003297297
## 5 Angola 41 0.846 0.740 0.99656 0.000070400
## 6 Antigua and Barbuda 73 0.990 0.989 0.99991 0.000142857
## PropRN PersExp GovtExp TotExp
## 1 0.000572294 20 92 112
## 2 0.004614439 169 3128 3297
## 3 0.002091362 108 5184 5292
## 4 0.003500000 2589 169725 172314
## 5 0.001146162 36 1620 1656
## 6 0.002773810 503 12543 13046str(data)## 'data.frame': 190 obs. of 10 variables:
## $ Country : chr "Afghanistan" "Albania" "Algeria" "Andorra" ...
## $ LifeExp : int 42 71 71 82 41 73 75 69 82 80 ...
## $ InfantSurvival: num 0.835 0.985 0.967 0.997 0.846 0.99 0.986 0.979 0.995 0.996 ...
## $ Under5Survival: num 0.743 0.983 0.962 0.996 0.74 0.989 0.983 0.976 0.994 0.996 ...
## $ TBFree : num 0.998 1 0.999 1 0.997 ...
## $ PropMD : num 2.29e-04 1.14e-03 1.06e-03 3.30e-03 7.04e-05 ...
## $ PropRN : num 0.000572 0.004614 0.002091 0.0035 0.001146 ...
## $ PersExp : int 20 169 108 2589 36 503 484 88 3181 3788 ...
## $ GovtExp : int 92 3128 5184 169725 1620 12543 19170 1856 187616 189354 ...
## $ TotExp : int 112 3297 5292 172314 1656 13046 19654 1944 190797 193142 ...summary(data)## Country LifeExp InfantSurvival Under5Survival
## Length:190 Min. :40.00 Min. :0.8350 Min. :0.7310
## Class :character 1st Qu.:61.25 1st Qu.:0.9433 1st Qu.:0.9253
## Mode :character Median :70.00 Median :0.9785 Median :0.9745
## Mean :67.38 Mean :0.9624 Mean :0.9459
## 3rd Qu.:75.00 3rd Qu.:0.9910 3rd Qu.:0.9900
## Max. :83.00 Max. :0.9980 Max. :0.9970
## TBFree PropMD PropRN PersExp
## Min. :0.9870 Min. :0.0000196 Min. :0.0000883 Min. : 3.00
## 1st Qu.:0.9969 1st Qu.:0.0002444 1st Qu.:0.0008455 1st Qu.: 36.25
## Median :0.9992 Median :0.0010474 Median :0.0027584 Median : 199.50
## Mean :0.9980 Mean :0.0017954 Mean :0.0041336 Mean : 742.00
## 3rd Qu.:0.9998 3rd Qu.:0.0024584 3rd Qu.:0.0057164 3rd Qu.: 515.25
## Max. :1.0000 Max. :0.0351290 Max. :0.0708387 Max. :6350.00
## GovtExp TotExp
## Min. : 10.0 Min. : 13
## 1st Qu.: 559.5 1st Qu.: 584
## Median : 5385.0 Median : 5541
## Mean : 40953.5 Mean : 41696
## 3rd Qu.: 25680.2 3rd Qu.: 26331
## Max. :476420.0 Max. :482750ggplot(data, aes(x = LifeExp ,y = TotExp)) +
geom_point() +
labs(x = "Average Life Expectancy", y = "Total Average Healthcare Expenditure") +
geom_smooth(method=lm)## `geom_smooth()` using formula = 'y ~ x'
set.seed(123)
linear_lm <- lm(LifeExp ~ TotExp, data= data)
summary(linear_lm)##
## Call:
## lm(formula = LifeExp ~ TotExp, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.764 -4.778 3.154 7.116 13.292
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.475e+01 7.535e-01 85.933 < 2e-16 ***
## TotExp 6.297e-05 7.795e-06 8.079 7.71e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.371 on 188 degrees of freedom
## Multiple R-squared: 0.2577, Adjusted R-squared: 0.2537
## F-statistic: 65.26 on 1 and 188 DF, p-value: 7.714e-14pvalue for TotExp is very small which is lower than 0.05 indicates that it is significant for the prefiction of the LifeExp. the adjusted R-squared 0.2537 is too low which show us that the model need a lot of more works. We can assume there is a linear relationship betwen the feature and the target variables, but not a strong one since thye model has low pvalue.
x_e = 4.6
y_e = 0.06
df <- data %>%
mutate(LifeExpT = LifeExp^x_e,
TotExpT = TotExp^y_e)
ggplot(df, aes(x = LifeExpT ,y = TotExpT)) +
geom_point() +
labs(x = "Average Life Expectancy", y = "Total Average Healthcare Expenditure") +
geom_smooth(method=lm)## `geom_smooth()` using formula = 'y ~ x'
set.seed(123)
linear_lm2 <- lm(LifeExpT ~ TotExpT, data= df)
summary(linear_lm2)##
## Call:
## lm(formula = LifeExpT ~ TotExpT, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -308616089 -53978977 13697187 59139231 211951764
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -736527910 46817945 -15.73 <2e-16 ***
## TotExpT 620060216 27518940 22.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 90490000 on 188 degrees of freedom
## Multiple R-squared: 0.7298, Adjusted R-squared: 0.7283
## F-statistic: 507.7 on 1 and 188 DF, p-value: < 2.2e-16pvalue is significantly low which means TotExpT is a statistically significant predictor of LifeExpT. The Rsquare value is close to 1 which means the model did very good and we can use it predict LifeExpT. The feature variable and target variables (LifeExpT ~ TotExpT) has a correlation of 0.85 which means they are related. The residual from y axis are scatted randomly. We can see most fitted value has good correlated with a residual.
ctrd <- cor(df[, sapply(df, is.numeric)])
corrplot(ctrd
, method = 'color' # I also like pie and ellipse
, order = 'hclust' # Orders the variables so that ones that behave similarly are placed next to each other
, addCoef.col = 'black'
, number.cex = .6 # Lower values decrease the size of the numbers in the cells
)
par(mfrow = c(2, 2), mar = c(2,2,2,2))
plot(linear_lm2)
prediction1 <- predict(linear_lm2, newdata = data.frame(TotExpT = 1.5))^(1/4.6)
prediction2 <- predict(linear_lm2, newdata = data.frame(TotExpT = 2.5))^(1/4.6)
cat(
'Prediction with 1.5: ',
scales::comma(prediction1),
'\nPrediction with 2.5: ',
scales::comma(prediction2),
sep = ''
)## Prediction with 1.5: 63
## Prediction with 2.5: 87set.seed(123)
multiple_lm <- lm(LifeExp ~ PropMD + TotExp + PropMD:TotExp, data= data)
summary(multiple_lm)##
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD:TotExp, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.320 -4.132 2.098 6.540 13.074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.277e+01 7.956e-01 78.899 < 2e-16 ***
## PropMD 1.497e+03 2.788e+02 5.371 2.32e-07 ***
## TotExp 7.233e-05 8.982e-06 8.053 9.39e-14 ***
## PropMD:TotExp -6.026e-03 1.472e-03 -4.093 6.35e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.765 on 186 degrees of freedom
## Multiple R-squared: 0.3574, Adjusted R-squared: 0.3471
## F-statistic: 34.49 on 3 and 186 DF, p-value: < 2.2e-16par(mfrow = c(2, 2), mar = c(2,2,2,2))
plot(multiple_lm)## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
the Rsquare is 0.3471 which means the model didn’t perfom better that the previous model. we need more work to evaluate the model and do hyperprameters tiunning on the features. The residuals are clustered on the y-axis which means the model didn’t predict the value corerctly.
newdata = data.frame(PropMD = 0.03, TotExp = 14)
predict(multiple_lm, newdata = newdata)## 1
## 107.696Based on the life expextancy recorded in the databased, the presiction seems to be irrelevant because the model didnt do a good job. The predicted life expancy is way higher than the one in the dataset