Regression on HDI 2018

hdi = read.csv("https://ericwfox.github.io/data/hdi2018.csv")

Variable descriptions: • hdi 2018: HDI for the year 2018 • median age: Median age (years) in 2015 • pctpop65: Percent of population 65 and older in 2018 • pct internet: Percent of population that uses the internet in 2017-2018 • pct labour: Percent of country’s working-age population that engages actively in the labour market, either by working or looking for work in 2018

head(hdi)

Exercise 1 (a) Fit a multiple linear regression model with hdi 2018 as the response, and the other four variables as predictors. Also, make a scatterplot matrix to visualize the relationships between the variables.

lm1<-lm(hdi_2018~ median_age  + pctpop65 + pct_internet + pct_labour, data=hdi)
summary(lm1)

Call:
lm(formula = hdi_2018 ~ median_age + pctpop65 + pct_internet + 
    pct_labour, data = hdi)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.194838 -0.034699  0.003272  0.031096  0.122529 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)   0.3374494  0.0319098  10.575  < 2e-16 ***
median_age    0.0080796  0.0011337   7.127  2.7e-11 ***
pctpop65     -0.0697020  0.1022759  -0.682    0.496    
pct_internet  0.0028967  0.0002451  11.817  < 2e-16 ***
pct_labour   -0.0001738  0.0003809  -0.456    0.649    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05193 on 172 degrees of freedom
Multiple R-squared:  0.8882,    Adjusted R-squared:  0.8856 
F-statistic: 341.5 on 4 and 172 DF,  p-value: < 2.2e-16

pairs(hdi_2018 ~ median_age  + pctpop65 + pct_internet + pct_labour, data=hdi)

2. Using the model fit in (a), is there evidence of a relationship between hdi 2018 and at least one of the predictor variables? Write the null and alternative hypotheses, report the F-test statistic and p-value, and state your conclusion.

Yes, there is evidence of a relationship between hdi 2018 with median age and pct internet based on the p-value being less than 0.05.

H0 :β1 =β2 =β3 =β4 =0 HA :atleastone βj ̸=0

F-statistic: 341.5 on 4 and 172 DF p-value: < 2.2e-16

Based off the data, when all predictor variables are taken into consideration we can see a correlation between hdi 2018 with median age and pct internet meaning we reject our null.

3. Using the model fit in (a), which predictor variables are statistically significant according to the individual t-tests?

The predictor variables that are statistically significant according to the individual t-test are median age and pct internet based on the p-values being extremely small.

4. Fit a reduced model with median age and pct internet as predictors. Use the anova() function to conduct a partial F-test that compares this reduced model with the full model specified in (a). Make sure to write the null and alternative hypotheses, report the p-value, and state your conclusion.

H0 :β1 =β2 =β3 =β4 =0 HA :atleastoneβj ̸=0

lm_full<-lm(hdi_2018~ median_age  + pctpop65 + pct_internet + pct_labour, data=hdi)
lm_2<-lm(hdi_2018~ median_age + pct_internet, data=hdi)

anova(lm_2, lm_full)

Analysis of Variance Table

Model 1: hdi_2018 ~ median_age + pct_internet
Model 2: hdi_2018 ~ median_age + pctpop65 + pct_internet + pct_labour
  Res.Df     RSS Df Sum of Sq      F Pr(>F)
1    174 0.46552                           
2    172 0.46380  2 0.0017236 0.3196 0.7269

The p-value = 0.73 is large, so we do not reject the null hypothesis that H0 : β1 = β4 = 0. So we can remove both predictors, median_age and pct_internet, from the model.

5. According to the adjusted-R2, how does the full model in (a) compare with the reduced model in (d)? Is this consistent with your conclusion for the partial F-test?

s1 <- summary(lm_full)
s2 <- summary(lm_3)

s1$adj.r.squared
s2$adj.r.squared

As we can see looking at the adjusted r squared for the full model we got a much higher value (.8855) compared to the reduced model (0.6859). This agrees with the conclusion of the F-test. So the adjusted-R2 also indicates that we can remove median_age and pct_internet.