Bivariate Regression  Assumptions  Each variable is interval/ratio  There is linear (straight line) relationship between the variables.  Normal distribution for each variable.

Regression Equation (for the population) Y (a value of the DV) = a + b(X) + e  a = Y intercept (a constant value: The point (value) where the line crosses the Y axes when X is 0)  b = slope of X Tells us the amount of change in Y for a one unit change in X. Tells us the amount of change in Y for a one unit change in X.  e = error term is the amount of Y that cannot be accounted for by a and b(x).

Predicted Ŷ for a Sample Example: Sentence Length (in months) & Prior Convictions (in # of convictions) Ŷ= a + b(X)  a = a base line amount given all defendants The length of sentence when there are 0 priors. The length of sentence when there are 0 priors.  b = the unit change in sentence length for each prior conviction.  X = prior convictions for a given respondent

Calculating a & b For Prison Sentence and Prior Convictions  a = mean of Y – b (mean of X)  b = SP/SSx  SP = Sum (X – mean of X)(Y- mean of Y)  SSx = Sum (X – mean of x) ²  Turn to overhead for calculations

Plug into formula  If prior convictions are two  a = 14  b = 3  Ŷ = (X)  Ŷ = (2) =  Ŷ = 20 months

 Regression line is the line that minimizes the distance from the actual values of y and Ŷ.  Errors in prediction: e = Y – Ŷ  EXAMPLE (on overhead)

SPSS Example Civil Liberties and Education Civlib:A scale ranging from (high tolerance) 8 (low tolerance) recoded to read 4 (low tolerance) 8 (high tolerance)  Let’s read the output.  a is the value where X crosses the Y axis.  What is the value on the Civ Lib scale when someone has 0 years of educ?

Civlib and Educ  What is the value on the Civ Lib scale when someone has 0 years of education?  How much of an increase on the Civ Lib scale occurs with a one year increase in education?  What would the predicted score be for someone who has 12 years of educ? How about 16 yrs of educ?

Education & Occupational Prestige  What is the occupational prestige for someone with 0 years of education?  What would the predicted occ prestige score be for someone with 12yrs of educ? With 18 yrs. of educ?

Final Example  How many kids does someone with 0 years of education have?  How many kids would we predict that someone with 10 years of education has? With 15 years of education?