A number of graphs

Y = β_{0}^{} + β_{1}^{} X_{}^{} + ε

library(ggplot2)

residuals <- rnorm(numbervalues, 0, standres)

XLine <- runif(numbervalues, 0, 20);          # Simulates X-data 0 - 20,
    
YLine <- Line_B0 + Line_B1*XLine + residuals
    
myData <- data.frame(YLine, XLine)

zero <- rep(0, 2*numbervalues)                         # To plot Y at x = 0.
 
Y_at_Zero <- data.frame(zero, YLine)                   # To plot the grey dots.
         
model <- lm( YLine ~ XLine )

summary(model)                                         # Shows result from analysis.
residuals <- resid(model)                              # Shows the residuals.
myData['Residuals'] <- residuals                       # Includes residuals into data frame.
   
mat_coeff <- summary(model)$coefficients               # All estimated coefficients.

b0 <- mat_coeff[1, 1]   # (Intercept)                  # Extracts coefficients.
b1 <- mat_coeff[2, 1]   # X
 
fun1 <- function(XLine) b0 + b1*XLine

myDiagram <- ggplot(myData, aes(x = XLine, y = YLine)) + labs(x = "X-values", y = "Y-values") +
     annotate("text", x = 10.0, y = Inf, label = "Estimated model", vjust = 2, fontface = "bold", color = "blue", size = 5) + 
     geom_point(colour = "blue") + stat_function(fun = fun1, color = "blue")                              # Adds line
myDiagram <- myDiagram + geom_point(data = Y_at_Zero, aes(x = zero, y= YLine), colour = "lightgrey")
myDiagram

# ---------------------------------------------------------------------------- Histograms of Y and residuals.

justY <- YLine - mean(YLine)                           # Gives mean 0, same as 'residuals'.

Group <- c(rep('Y', numbervalues), rep('Res', numbervalues))

YoRes  <- c(justY, residuals)

HistData <- data.frame(YoRes, Group)

xmin <- min(YoRes)

hist <- ggplot(HistData, aes(x = YoRes, fill = Group)) + geom_histogram(bins=15, color = 1, alpha = 0.5, position = "identity", show.legend = FALSE)
hist <- hist + labs(x = "Y-values and residuals", y = "") + scale_fill_manual(values = c("blue", "lightgrey"))
hist <- hist +  annotate("text", x=xmin, y=Inf, label="Y-values (grey) and\nresiduals (blue)", hjust = 0, vjust=2, fontface="bold", color="blue", size=4)
hist

Start values Exercise 1 Exercise 2

Info, referenser, m.m...

Show/hide program code for R

Show/hide comments

Show/hide diagrams

48. A simple regression model

Correlation coefficient – a simple but sometimes misused statistic

r = -0.815

r² = 0.665

Var (grey) = 114.5

Var (res) = 38.5

An 'ordinary-least-squares (OLS)' model

β₀

β₁

(-10 – 10): -5.5

(-2 – 2): -1.6

(Number of values: 30 – 100): 40

(Residual sigma: 0 – 10): 6.5

More regression (Sw)

Start values.

Line_B0      <- -5.5
Line_B1      <- -1.6
numbervalues <- 40
standres     <- 6.5