data605_hw12
Vijaya Cherukuri
11/13/2019
Data set :
library(ggplot2)
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, unionReading data :
who <- read.csv("who.csv")
head(who)## Country LifeExp InfantSurvival Under5Survival TBFree
## 1 Afghanistan 42 0.835 0.743 0.99769
## 2 Albania 71 0.985 0.983 0.99974
## 3 Algeria 71 0.967 0.962 0.99944
## 4 Andorra 82 0.997 0.996 0.99983
## 5 Angola 41 0.846 0.740 0.99656
## 6 Antigua and Barbuda 73 0.990 0.989 0.99991
## PropMD PropRN PersExp GovtExp TotExp
## 1 0.000228841 0.000572294 20 92 112
## 2 0.001143127 0.004614439 169 3128 3297
## 3 0.001060478 0.002091362 108 5184 5292
## 4 0.003297297 0.003500000 2589 169725 172314
## 5 0.000070400 0.001146162 36 1620 1656
## 6 0.000142857 0.002773810 503 12543 130461. Provide a scatterplot 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.
REGRESSION MODEL :
m1 = lm(LifeExp ~ TotExp, data=who)
# linear regression
ggplot(m1, aes(TotExp, LifeExp)) + geom_point(colour="blue", size=2) + geom_abline(aes(slope=round(m1$coefficients[2], 4), intercept=round(m1$coefficients[1], 4))) + labs(title = "Total Expenditures vs. Life Expetancy") + xlab("Total Expenditures") + ylab("Life Expectancy")
RESIDUAL PLOT :
# residual plot
ggplot(m1, aes(.fitted, .resid)) + geom_point(color = "darkgreen", size=2) +labs(title = "Fitted Values vs Residuals") +labs(x = "Fitted Values") +labs(y = "Residuals")
# normal plot
qqnorm(resid(m1))
qqline(resid(m1))
Conclusion :
summary(m1)##
## Call:
## lm(formula = LifeExp ~ TotExp, data = who)
##
## 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-14F−Statistic is 65.26 and the Standard Error is 9.371. The p−value is almost 0.R2 is 0.2577. The data points are not around the abline.Q_Q plot is not normal. This shows a week relationship between variables TotExp and LifeExp.
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?”
REGRESSION MODEL :
lifeexp2 <- who$LifeExp^4.6
Totexp2 <- who$TotExp^0.06
m2 <- lm(lifeexp2 ~ Totexp2)
# linear regression
ggplot(m2, aes(Totexp2,lifeexp2)) + geom_point(colour="green", size=2) + geom_abline(aes(slope=round(m2$coefficients[2], 4), intercept=round(m2$coefficients[1], 4))) + labs(title = "Transformed") + xlab("Total Expenditures") + ylab("Life Expectancy")
RESIDUAL PLOT :
# residual plot
ggplot(m2, aes(.fitted, .resid)) + geom_point(color = "red", size=2) +labs(title = "Fitted Values vs Residuals") +labs(x = "Fitted Values") +labs(y = "Residuals")
# normal plot
qqnorm(resid(m2))
qqline(resid(m2))
Conclusion :
summary(m2)##
## Call:
## lm(formula = lifeexp2 ~ Totexp2)
##
## 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 ***
## Totexp2 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-16F−Statistic is 507.7 and the Standard Error is 90490000. The p−value is again nearly 0. The correlation is 0.8543 which is much better than the previous case and R2 is 0.7298.
In this new model, we notice that the data points are around the abline.Q-Q plot is almost normal. This shows a strong relationship between transformed variables TotExp and LifeExp. Therefore, we can say this transformed model is better than the previous model.
3. Using the results from 3, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5
life_exp <- function(x)
{ y <- -736527910 + 620060216 * (x)
y <- y^(1/4.6)
print(y)
}
#Life expectancy when TotExp^.06 =1.5
life_exp(1.5)## [1] 63.31153#Life expectancy when TotExp^.06 =2.5
life_exp(2.5)## [1] 86.50645Conclusion:
When TotExp=1.5, the forecast life expectancy is 63.31 years and when the TotExp=2.5, the life expectancy is 86.51 years.
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
REGRESSION MODEL :
m3 = lm(LifeExp ~ PropMD + TotExp + PropMD*TotExp, data = who)
# linear regression
ggplot(m3, aes(TotExp, LifeExp)) + geom_point(colour="orange", size=2) + geom_abline(aes(slope=round(m1$coefficients[2], 4), intercept=round(m1$coefficients[1], 4))) + labs(title = "Total Expenditures vs. Life Expetancy") + xlab("Total Expenditures") + ylab("Life Expectancy")
RESIDUAL PLOT :
# residual plot
ggplot(m3, aes(.fitted, .resid)) + geom_point(color = "red", size=2) +labs(title = "Fitted Values vs Residuals") +labs(x = "Fitted Values") +labs(y = "Residuals")
# normal plot
qqnorm(resid(m3))
qqline(resid(m3))
CONCLUSION :
summary(m3)##
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = who)
##
## 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-16F−Statistic is 34.49 and the Standard Error is 8.765. The p−value is again nearly 0. The R2 is 0.3574. The model explains only 35.74% of variability.
In this new model, we notice that the residuals and Q-Q plot are not normally distributed. This model is not a good model to describe the relationships between variables TotExp, PropMd and LifeExp.
5.Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
summary(m3)$coefficients[1] + .03* summary(m3)$coefficients[2] + 14*summary(m3)$coefficients[3] + (.03*14)*summary(m3)$coefficients[4]## [1] 107.696