psychology 302

1

Bivariate Regression

Straight Lines

¾ Simple way to describe a relationship ¾ Remember the equation for a straight line?

z y = mx + b ¾ What is m? What is b?

¾ How do you compute the equation?

(x1,y1)

(x2,y2)

What if every point is not on the line?

¾ Straight line may be good description even if not all points are on the line

Computing the line when points are scattered

¾ = a + bX ¾ Y-hat means predicted value of Y ¾ Computing the slope:

¾ b = 𝑋−𝑋 𝑌−𝑌 𝑋−𝑋

¾ I ill ri e/r n, b no e al o consider variability in X and Y

Computing the intercept

¾ a = - bX ¾ Need o pl g in al e of (X, ) ¾ Can e j an Y or X!

z Line would be very different depending on which ones you chose

¾ Must have X and Y that we know are on the line z mean of X and mean of Y

2

Computing the intercept

¾ Regression line will always go through the mean of X and mean of Y

¾ A = 𝑌 - b𝑋

¾ Le r it with our example from before

X (# of kids)

Y (hours of

housework) 𝑋 𝑋 𝑌 𝑌 𝑋 𝑋 𝑌 𝑌 𝑋 𝑋 1 1 -1.75 -2.5 4.375 3.063

1 2 -1.75 -1.5 2.625 3.063

1 3 -1.75 -0.5 0.875 3.063

2 6 -0.75 2.5 -1.875 0.563

2 4 -0.75 0.5 -0.375 0.563

2 1 -0.75 -2.5 1.875 0.563

3 5 0.25 1.5 0.375 0.063

3 0 0.25 -3.5 -0.875 0.063

4 6 1.25 2.5 3.125 1.563

4 3 1.25 -0.5 -0.625 1.563

5 7 2.25 3.5 7.875 5.063

5 4 2.25 0.5 1.125 5.063

MX=2.75 MY=3.5 = 0 = 0 = 18.5 = 24.25

Computing the equation

¾ b = . .

.76

¾ a = 3.5 - .76(2.75) ¾ = 1.41

¾ = 1.41 + .76X

Interpreting the coefficients ¾ Slope

z For a one unit increase in X, we predict a b unit increase in Y

What does that mean for this study?

¾ Intercept z The predicted value of Y when X = 0

What does that mean for this study?

Interpreting the coefficients ¾ Slope

z For each additional child, we predict parents will do an additional .76 hours of housework per day

¾ Intercept z For a family with zero kids, we predict they

will do 1.41 hours of housework per day

Drawing the regression line

¾ Need to plot two points z 𝑋, 𝑌 z Y-intercept