Multiple Regression Spring 2006

Gore Likeability Example Suppose: –Gore’s* likeability is a function of Clinton’s likeability and not directly a function of party –Clinton’s likeability is a function of one’s partisan identification plus other factors –What would the regression of Gore likeability on Clinton likeability look like? Party ID Clinton Likeability Gore Likeability e1 e2 *This example probably works better if we’re predicting the likeability of Socks the cat.

Democratic picture

Independent picture

Republican picture

Combined data picture

Combined data picture with regression Clinton thermometer

Combined data picture with “true” regression lines overlaid Clinton thermometer

Tempting yet wrong normalizations Subtract the Gore therm. from the avg. Gore therm. score Subtract the Clinton therm. from the avg. Clinton therm. score

Summary: Why we control Remove confounding effects Improve efficiency

Look at actual data graph7 clinton gore party3, matrix

Look at actual data (jitter) graph7 gore clinton party3, matrix jitter(5)

Gore vs. Clinton Overall Within party Rep. Ind. Dem.

Gore vs. party Overall Within party Clinton low Clinton med. Clinton high

Back to the basic data

3D Relationship

3D Linear Relationship

3D Relationship: Clinton

3D Relationship: party Rep Ind Dem

The Linear Relationship between Three Variables

The Slope Coefficients

The Slope Coefficients More Simply

The Intercept Note: Add “hats” (^) over all the Greek letters

The Matrix form y1y1 y2y2 … ynyn 1x 1,1 x 2,1 …x k,1 1x 1,2 x 2,2 …x k,2 1………… 1x 1,n x 2,n …x k,n

What Difference Does This Make? One Regression vs. a separate regression for each independent variable

Consider two regression coefficients When does ? Obviously, when

Separate regressions (1)(2)(3) Intercept Clinton Party

Why did the Clinton Coefficient change from 0.62 to corr gore clinton party,cov (obs=1745) | gore clinton party gore | clinton | party3 |

The Calculations. corr gore clinton party,cov (obs=1745) | gore clinton party gore | clinton | party3 |

Accounting for total effects (i.e., regression coefficient when we regress X 2 (as dep. var.) on X 1 (as ind. var.)

Accounting for the total effect Total effect = Direct effect + indirect effect Y X1X1 X2X2

Accounting for the total effects in the Gore thermometer example EffectTotalDirectIndirect Clinton Party

The Output. reg gore clinton party3 Source | SS df MS Number of obs = F( 2, 1742) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = gore | Coef. Std. Err. t P>|t| [95% Conf. Interval] clinton | party3 | _cons |

Drinking and Greek Life Example Why is there a correlation between living in a fraternity/sorority house and drinking? –Greek organizations often emphasize social gatherings that have alcohol. The effect is being in the Greek organization itself, not the house. –There’s something about the House environment itself.

Dependent variable: Times Drinking in Past 30 Days

. infix age residence 16 greek 24 screen 102 timespast howmuchpast gpa studying 281 timeshs 325 howmuchhs 326 socializing 283 stwgt_ weight using da3818.dat,clear (14138 observations read). recode timespast30 timeshs (1=0) (2=1.5) (3=4) (4=7.5) (5=14.5) (6=29.5) (7=45) (timespast30: 6571 changes made) (timeshs: changes made). replace timespast30=0 if screen<=3 (4631 real changes made)

. tab timespast30 timespast30 | Freq. Percent Cum | 4, | 2, | 2, | 1, | 1, | | Total | 13,

Three Regressions Dependent variable: number of times drinking in past 30 days Live in frat/sor house4.44 (0.35) (0.38) Member of frat/sor (0.16) 2.44 (0.18) Intercept4.54 (0.56) 4.27 (0.059) 4.27 (0.059) R N13,876 Note: Corr. Between living in frat/sor house and being a member of a Greek organization is.42

The Picture Drinks per 30 day period Living in frat house Member of fraternity

Accounting for the effects of frat house living and Greek membership on drinking EffectTotalDirectIndirect Member of Greek org (85%) 0.44 (15%) Live in frat/ sor. house (51%) 2.18 (49%)