Slides 7: Correlation and Regression

Regression

tests for linear relationship between x and y, two numeric variables, each point in graph = one observational unit

Null hypothesis Regression

β = 0, no linear relationship in pop.

Regression equation and variable meanings

ȳ = a + bx

a = y intercept

b = slope

x = independent variable value

How do you calculate the amount of variation explained by the regression line?

SSregression = SStotal - SSresidual

Regression lines always have what degree of freedom?

1

SS/df =

MS

What does p represent in regression?

the probability of fitting the line as well as sample did if Ho is true (no relationship)

What is r²?

coefficient of determination, % variation in y explained by x

How do you calculate r²?

SSregression / SStotal

What value do you use to draw conclusions from?

p value

What does p > alpha mean for regression?

fail to reject Ho, no linear relationship, x does not predict y

What does p < alpha mean for regression?

reject Ho, linear relationship, find value from best fit equation, r² to say how well data fits

Regression assumptions

data from random representative sample, independent observations, x and y relationship linear, residuals normally distributed, residual variation similar over all values of x

if r² = 0.486 that means

48.6% variation explained by x, other 51.4% explained by other factors

Differences between correlation and regression

both: test for relationship between 2 numeric,

different: correlation only association, neither causes other, can determine direction and strength; regression x is causative, can find equation describing

Correlation questions

Shape of relationship (pos or neg slope)? strength of relationship (close points are to line)?

p (rho)

correlation coefficient for pop.(between -1 and 1)

r statistic

correlation coefficient in sample (between -1 and 1)

When correlation Ho true, r follows a _____ with df = _____

t distribution, n - 2

where t = r x root((n-2)/1-r²))

If correlation p > alpha

no association, rho = 0

If correlation p < alpha then

is association, use r to decide of pos or neg, weak or strong association

(usually) r closer to one means

p value lower