Introduction
The who.csv dataset contains real-world data from 2008. The data set include multiple measured data points related to healthcare and life expectancy.
Libraries
library(tidyverse)Data
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.
who_df <- read.csv("https://raw.githubusercontent.com/engine2031/Data-Sets/main/who.csv")
head(who_df)## 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 13046summary(who_df)## 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. :482750Excercise 1
Provide a scatter plot of LifeExp~TotExp, and run simple linear regression. Do not transform the variables. Provide and interpret the F statistics, R^2, standard error,and p-values only. Discuss whether the assumptions of simple linear regression met.
Looking at the scatter plot of these two variables, it can be seen that there is no linear relationship between the variables.
ggplot(data = who_df, aes(x = TotExp, y = LifeExp)) +
geom_point(color="steelblue")+
theme_minimal()+
geom_smooth(method = lm, se=FALSE)+
labs(x = "Total Healthcare Expenditures", y = "Life Expectancy")
We create a linear regression model for these two variables and see what
the summary statistics tell us. The P-value is close to zero and
indicates that the Total Healthcare Expenditure is statistically
significant. The Multiple R squared value is on the lower end of the
scale from 0 to 1 indicating that the linear model does not describe the
measure data well. We see that the slope estimate and the Total
Healthcare Expenditure coefficients have little variability since the
ratio of the coefficient to the standard error is high(above 5.)
who_lm <- lm(LifeExp ~ TotExp, data = who_df)
summary(who_lm)##
## Call:
## lm(formula = LifeExp ~ TotExp, data = who_df)
##
## 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-14Excercise 2
Raise life expectancy to the 4.6 power (i.e., LifeExp^4.6). Raise total expenditures to the 0.06 power (nearly a log transform, TotExp^.06). Plot LifeExp^4.6 as a function of TotExp^.06, and r re-run the simple regression model using the transformed variables. Provide and interpret the F statistics, R^2, standard error,and p-values. Which model is “better?”
When performing the power transformation on our two variables we can see a improvement between the two variables on our scatterplot.
who_df$LifeExp <- who_df$LifeExp^4.6
who_df$TotExp <- who_df$TotExp^.06
ggplot(data = who_df, aes(x = TotExp, y = LifeExp)) +
geom_point(color="steelblue")+
theme_minimal()+
geom_smooth(method = lm, se=FALSE)+
labs(x = "Total Healthcare Expenditures", y = "Life Expectancy")## `geom_smooth()` using formula 'y ~ x'
In our new linear model we see that our results improve. The p-value, and standard error remain favorable. The major improvement in this new model is the Multiple R-squared value. In this model we are above .5, indicating that the model describes the measured data well. The only remaining statistic is the F-statistic which is not relevant for our model since we have a one factor regression model.
who_lm <- lm(LifeExp ~ TotExp, data = who_df)
summary(who_lm)##
## Call:
## lm(formula = LifeExp ~ TotExp, data = who_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 ***
## TotExp 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-16Excercise 3
Using the results from 2, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.
# Linear Model
LifeExp <- -736527910 + 620060216*1.5
LifeExp## [1] 193562414LifeExp <- -736527910 + 620060216*.06
LifeExp## [1] -699324297Excercise 4
Build the following multiple regression model and interpret the F Statistics, R^2, standard error, and p-values. How good is the model? LifeExp = b0+b1 x PropMd + b2 x TotExp +b3 x PropMD x TotExp
In our new model we see that our ratio of coefficient to std. error are acceptable, the P-value is significant but our R-squared value is below .05. The r-squared value indicates that the model does not describe our measured data well. We see the our F-statistic is greater than our critical F-value. Indicating that at least one of our coefficients related to the predictor variables is non-zero for the alternate hypothesis.
who_df <- read.csv("https://raw.githubusercontent.com/engine2031/Data-Sets/main/who.csv")
who_lm <- lm(LifeExp ~ PropMD + TotExp + PropMD*TotExp, data = who_df)
summary(who_lm)##
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = who_df)
##
## 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-16Critical F Value
qf(p=.05, df1=3, df2=186, lower.tail=FALSE)## [1] 2.653165Excercise 5
Forecast LifeExp when PropMD =.03 and TotExp = 14. Does this forecast seem realistic? Why or why not? No, this forecast does not seem realistic. Our maximum life expectancy is 84 years in the base data set. The total expenditure of 14 is near the minimum for the data set. Given that total expenditure is a significant value predictor we would expect a low life expectancy. Instead this model generates a value higher than the max of the measured data.
PropMD <- .03
TotExp <- 14
LifeExp <- 6.277e+01 + 1.497e+03*PropMD + 7.233e-05*TotExp - 6.026e-03*PropMD*14
LifeExp## [1] 107.6785