Data 605 HW 12
Matthew Lucich
4/13/2022
df_who <- read.csv("who.csv")
summary(df_who)## 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. :482750glimpse(df_who)## Rows: 190
## Columns: 10
## $ Country <chr> "Afghanistan", "Albania", "Algeria", "Andorra", "Angola…
## $ LifeExp <int> 42, 71, 71, 82, 41, 73, 75, 69, 82, 80, 64, 74, 75, 63,…
## $ InfantSurvival <dbl> 0.835, 0.985, 0.967, 0.997, 0.846, 0.990, 0.986, 0.979,…
## $ Under5Survival <dbl> 0.743, 0.983, 0.962, 0.996, 0.740, 0.989, 0.983, 0.976,…
## $ TBFree <dbl> 0.99769, 0.99974, 0.99944, 0.99983, 0.99656, 0.99991, 0…
## $ PropMD <dbl> 0.000228841, 0.001143127, 0.001060478, 0.003297297, 0.0…
## $ PropRN <dbl> 0.000572294, 0.004614439, 0.002091362, 0.003500000, 0.0…
## $ PersExp <int> 20, 169, 108, 2589, 36, 503, 484, 88, 3181, 3788, 62, 1…
## $ GovtExp <int> 92, 3128, 5184, 169725, 1620, 12543, 19170, 1856, 18761…
## $ TotExp <int> 112, 3297, 5292, 172314, 1656, 13046, 19654, 1944, 1907…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.
plot(df_who$TotExp, df_who$LifeExp)
m1 <- lm(LifeExp ~ TotExp, data=df_who)
summary(m1)##
## Call:
## lm(formula = LifeExp ~ TotExp, data = df_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-14plot(m1)
Interpret the F statistics, R^2, standard error,and p-values only.
F statistic is indicating that the model overall is significant (better than guessing the mean) at a 99% confidence level. Adjusted R-squared is rather low, but I would usually take this as an indicator that the model is worth attempting to improve (ignoring the obviously failed assumptions for the moment), rather than abandoning. The p-values for the intercept, TotExp, and the model overall are all significant at a 99% confidence level.
Discuss whether the assumptions of simple linear regression met.
Based on the scatterplot we can see the relationship is not linear. Based on the residual plots we can further conclude the relationship is non-linear and the residuals are non-normal. There may be other assumptions not being met, but we can reject this model before proceeding further.
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?”
plot(df_who$TotExp^.06, df_who$LifeExp^4.6)
m2 <- lm(I(LifeExp^4.6) ~ I(TotExp^.06), data=df_who)
summary(m2)##
## Call:
## lm(formula = I(LifeExp^4.6) ~ I(TotExp^0.06), data = df_who)
##
## 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 ***
## I(TotExp^0.06) 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-16Provide and interpret the F statistics, R^2, standard error, and p-values. Which model is “better?”
The F statistic is indicating the model is extremely significant (better than guessing the mean). Adjusted R-squared is substantially better than the previous model and overall .728 is interpreted as large (Cohen 1988). This model is significantly better.
Using the results from 3, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.
(-736527910 + 620060216 * 1.5)^(1/4.6)## [1] 63.31153(-736527910 + 620060216 * 2.5)^(1/4.6)## [1] 86.50645Build 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
m3 <- lm(LifeExp ~ PropMD + TotExp + PropMD*TotExp, data=df_who)
summary(m3)##
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = df_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-16Interpret the F Statistics, R^2, standard error, and p-values. How good is the model?
The F statistic is indicating the model is very significant (better than guessing the mean). Adjusted R-squared is improved over the first model, but a step back from the second model and overall 0.347 is interpreted as moderate (Cohen 1988). This model is better than the first and worse than the second.
Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
new <- data.frame(PropMD = c(0.3), TotExp = c(14))
predict(m3, new)## 1
## 511.9966This does not seem realistic as human life expectancy is far below 512 and is not expected to approach that in the near future despite the number of MDs and government expenditures. This is another reminder that models should avoid extrapolation when possible and if it is necessary do not exceed significantly beyond the data values you do have.