Regression - Discussion 11

Swiss Education vs Agriculture Regression

I am using the built-in R dataset of Swiss socioeconomic and fertility factors from 1888. I will focus on the relationship between education and agriculture.

data(swiss)
head(swiss, 6)

##              Fertility Agriculture Examination Education Catholic
## Courtelary        80.2        17.0          15        12     9.96
## Delemont          83.1        45.1           6         9    84.84
## Franches-Mnt      92.5        39.7           5         5    93.40
## Moutier           85.8        36.5          12         7    33.77
## Neuveville        76.9        43.5          17        15     5.16
## Porrentruy        76.1        35.3           9         7    90.57
##              Infant.Mortality
## Courtelary               22.2
## Delemont                 22.2
## Franches-Mnt             20.2
## Moutier                  20.3
## Neuveville               20.6
## Porrentruy               26.6

Visualization

A scatter plot of agriculture as a function of education. We can take the log of education to fit the a linear regression.

ggplot(swiss, aes(Education,Agriculture)) + geom_point() + 
  ggtitle("Education vs Agriculture")

ggplot(swiss, aes(log(Education),Agriculture)) + geom_point() + 
  ggtitle("Log Education vs Agriculture")

Linear Model

Fit agriculture ~ log(education).

attach(swiss)
lm <- lm(Agriculture ~ log(Education))
lm

## 
## Call:
## lm(formula = Agriculture ~ log(Education))
## 
## Coefficients:
##    (Intercept)  log(Education)  
##          91.28          -19.35

eq = paste0("y = ", round(lm[1]$coefficients[2],3), "*x + ", round(lm[1]$coefficients[1],3))
ggplot(swiss, aes(log(Education),Agriculture)) + geom_point() + 
  geom_abline(intercept = lm[1]$coefficients[1], slope = lm[1]$coefficients[2]) +
  ggtitle(paste("Log Education vs Agriculture:",eq))

Conditions

The Swiss towns which we used to fit the line are independent.
The relationship between the log education and agriculture is linear.
The residuals from the regression line are nearly normal.
The variability of education/agriculture observations around the regression line is constant.

Model Quality

This model has an intercent of -19.348 and a slope of 91.277.
To determine the model quality, we look at the Multiple R-squared value. R-squared values closer to one indicate better model quality, so R^2 = 0.4569 indicates that this model is not ideal.

summary(lm)

## 
## Call:
## lm(formula = Agriculture ~ log(Education))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -37.181 -14.089  -0.326  13.974  28.291 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      91.277      7.048  12.950  < 2e-16 ***
## log(Education)  -19.348      3.145  -6.152 1.85e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.92 on 45 degrees of freedom
## Multiple R-squared:  0.4569, Adjusted R-squared:  0.4448 
## F-statistic: 37.85 on 1 and 45 DF,  p-value: 1.854e-07

Residual Analysis

The residuals do not have a trend, which indicates that a linear model fits the data well.

swiss$resid <- resid(lm)
ggplot(swiss, aes(log(Education),resid)) + geom_point() + 
  geom_hline(yintercept=0, linetype="dashed", color = "blue") +
  ggtitle("Residual Plot")

ggplot(data=swiss, aes(swiss$resid)) + 
  geom_histogram(binwidth=5) +
  ggtitle("Residual Histogram")

Computational Mathematics - Assignment 11

Mary Anna Kivenson

November 9, 2019