• purposes:

assess cumulative association of set of independent variables with dependent variable and improve prediction

evaluate association between particular independent variable and dependent variable, controlling for other independent variables

• i.e., determine how magnitude of independent association changes after holding other variables constant

• e.g., 50 states in 1996
• dv = crime rate
• ivs = metropolitan, poverty, and college education rates
• all show linear bivariate relationships

• mean crime rate = 504.8
• crime = 57.7 + 6.6 metropolitan
• r = .56  r² = .31
• crime = 379.0 + 5.4 metropolitan + -9.2 college + 24.8 poverty
• multiple R = .72  R² = .52
• multiple R = multiple correlation
• multiple R^{2 =} PRE given multiple independent variables
• South Dakota (SD) crime = 177

 
• error in pred. w/o ivs = 177 - 505 = 328
• pred. w/ metropolitan only = 57.7 + 6.6 (33.3) = 277.5
• error in pred. = 177 - 277.5 = 100.5
• pred. w/ all 3 ivs = 379 + 5.4(33.3) + -9.2(72.8) + 24.8(11.8) = 181.7
• error in pred. = 177 - 181.7 = -4.7
• errors in pred. not always reduced for each case or w/ each additional i.v., but overall predictions can only improve with each additional i.v.

 
• partial correlation - correlation between two variables holding other variable(s) constant -- degree of independent association

• e.g., 1991-6 GSS, women > age 40
• dv = number of children
• ivs = education, ideal number of children, number of siblings
• R² = .08

• related techniques available for categorical variables

• visually display patterns of similarity/difference among variables

• multidimensional scaling - the closer two points are in the picture the more similar or closely related they are

• hierarchical clustering - the farther down in the tree diagram two points are joined, the more similar they are (imagine how far up branches one must go to find a common branch for two leaves)