SixPointReg

Data

The six point example uses six data points. Yes, it is small and boring.

  x <- c(3, 4, 4, 5, 6, 6)
  y <- c(3, 3, 4, 6, 5, 7)
  sixpts <- data.frame(x,y)
  sixpts

##   x y
## 1 3 3
## 2 4 3
## 3 4 4
## 4 5 6
## 5 6 5
## 6 6 7

Plots and Regression

The points can be plotted.

  p_load(ggplot2)
  ggplot(sixpts, aes(x = x, y = y)) + 
    geom_point() +
    theme_bw() +  # Add theme for cleaner look 
    coord_cartesian(xlim = c(0,7), ylim=c(-1,8))+
    annotate(geom="segment", 
             y=seq(-1,8,0.25), 
             yend = seq(-1,8,0.25), 
             x=-0.25, 
             xend=7.25, 
             col="lightgrey") +
    annotate(geom="segment", 
             x=seq(-0.25,7.25,0.25), 
             xend = seq(-0.25,7.25,0.25),
             y=-1, 
             yend=8, 
             col="lightgrey") +
    coord_fixed()

## Coordinate system already present.
## ℹ Adding new coordinate system, which will replace the existing one.

The “best fit” linear regression line can be added.

  ggplot(sixpts, aes(x = x, y = y)) + 
    geom_smooth(method = "lm", se = FALSE, 
                color = "darkgrey") +  # Plot regression slope
    geom_point() +
    theme_bw() +
    coord_cartesian(xlim = c(0,7), ylim=c(-1,8))+
    annotate(geom="segment", y=seq(-1,8,0.25), yend = seq(-1,8,0.25), 
             x=-0.25, xend=7.25, col="lightgrey") +
    annotate(geom="segment", x=seq(-0.25,7.25,0.25), 
             xend = seq(-0.25,7.25,0.25), y=-1, yend=8, col="lightgrey") +
    coord_fixed()

## Coordinate system already present.
## ℹ Adding new coordinate system, which will replace the existing one.
## `geom_smooth()` using formula = 'y ~ x'

The vertical errors (residuals) can be observed.

  sixpts.lm = lm(y ~ x, data=sixpts)
  sixpts$predicted <- predict(sixpts.lm)
  ggplot(sixpts, aes(x = x, y = y)) + 
    geom_smooth(method = "lm", se = FALSE, 
                color = "darkgrey") +  # Plot regression slope
    geom_segment(aes(xend = x, yend = predicted), 
                 alpha = .2) + # alpha to fade lines
    geom_point() +
    geom_point(aes(y = predicted), shape = 1) +
    theme_bw() +  # Add theme for cleaner look 
    coord_cartesian(xlim = c(0,7), ylim=c(-1,8))+
    annotate(geom="segment", y=seq(-1,8,0.25), yend = seq(-1,8,0.25), 
             x=-0.25, xend=7.25, col="lightgrey") +
    annotate(geom="segment", x=seq(-0.25,7.25,0.25), xend = seq(-0.25,7.25,0.25), 
             y=-1, yend=8, col="lightgrey") +
    coord_fixed()

## Coordinate system already present.
## ℹ Adding new coordinate system, which will replace the existing one.
## `geom_smooth()` using formula = 'y ~ x'

We now show that the regression line goes through \((\bar{x},\ \bar{y})\) = (4.6667, 4.6667).

  ggplot(sixpts, aes(x = x, y = y)) +
    geom_smooth(method = "lm", se = FALSE, 
                color = "darkgrey") +  # Plot regression slope
    geom_segment(aes(xend = x, yend = predicted), 
                 alpha = .2) + # alpha to fade lines
    geom_point() +
    geom_point(aes(y = predicted), shape = 1) +
    theme_bw() +  # Add theme for cleaner look 
    geom_hline(yintercept=mean(y), lty = 3, col = 1) +
    geom_vline(xintercept=mean(x), lty = 3, col = 1) +
    coord_cartesian(xlim = c(0,7), ylim=c(-1,8))+
    annotate(geom="segment", y=seq(-1,8,0.25), yend = seq(-1,8,0.25), 
             x=-0.25, xend=7.25, col="lightgrey") +
    annotate(geom="segment", x=seq(-0.25,7.25,0.25), xend = seq(-0.25,7.25,0.25), 
             y=-1, yend=8, col="lightgrey") +
    coord_fixed()

## Coordinate system already present.
## ℹ Adding new coordinate system, which will replace the existing one.
## `geom_smooth()` using formula = 'y ~ x'

The regression equation and associated measures of its quality are easy to access.

  sixpts.lm = lm(y ~ x, data=sixpts)
  summary(sixpts.lm)

## 
## Call:
## lm(formula = y ~ x, data = sixpts)
## 
## Residuals:
##        1        2        3        4        5        6 
##  0.22727 -0.90909  0.09091  0.95455 -1.18182  0.81818 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  -0.6364     1.7405  -0.366   0.7332  
## x             1.1364     0.3629   3.131   0.0351 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9828 on 4 degrees of freedom
## Multiple R-squared:  0.7102, Adjusted R-squared:  0.6378 
## F-statistic: 9.804 on 1 and 4 DF,  p-value: 0.03515

  anova(sixpts.lm)

## Analysis of Variance Table
## 
## Response: y
##           Df Sum Sq Mean Sq F value  Pr(>F)  
## x          1 9.4697  9.4697  9.8039 0.03515 *
## Residuals  4 3.8636  0.9659                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1