Presentation on theme: "Partial and Semipartial Correlation"— Presentation transcript:
1Partial and Semipartial Correlation Working With Residuals
2QuestionsGive a concrete example (names of vbls, context) where it makes sense to compute a partial correlation. Why a partial rather than semipartial?Why is the squared semipartial always less than or equal to the squared partial?Give a concrete example where it makes sense to compute a semipartial correlation. Why semi rather than partial?Why is regression more closely related to semipartials than partials?How could you use ordinary regression to compute 3rd order partials?
3Partial CorrelationPeople differ in many ways. When one difference is correlated with an outcome, cannot be sure the correlation is not spurious.Would like to hold third variables constant, but cannot manipulate.Can use statistical control.Statistical control is based on residuals. If we regress X2 on X1 and take residuals of X2, this part of X2 will be uncorrelated with X1, so anything X2 resids correlate with will not be explained by X1.
4Example of Partials Use SAT to predict grades (HS & College Fresh) HS= *SAT; F= *SAT.(HS) (F)PersonSAT-VHSGPAFGPAPFGPAE1E215003.02.82.86-0.01-0.0625503.23.05-0.02-0.0534502.670.010.1344002.52.22.48-0.08-0.2856003.33.24-0.240.0666503.83.430.15-0.1377003.93.53.610.03-0.1283.70.580.6593.4-0.15-0.03103.12.9R2 for HS = .76; R2 for F = .62 (fictional data).
5Example Partials (2)There are 2 sets of predicted values; one for each GPA, however, they correlate 1.0 with each other, so only 1 is presented.High correlationsSAT-VHSGPAFPE1(HS)E2 (F)1HSGPA.87FGPA.81.921.00.00.50.45E188.8.131.52Note that P and SAT are perfectly correlated. P & SAT do not correlate with E1 or E2 (residuals).A partial correlation; the correlation between the residuals of the two GPAs. The correlation between HS GPA and FGPA holding SAT constant.
6The Meaning of Partials The partial is the result of holding constant a third variable via residuals.It estimates what we would get if everyone had same value of 3rd variable, e.g., corr b/t 2 GPAs if all in sample have SAT of 500.Some examples of partials? Control for SES, prior experience, what else?
7Computing Partials from Correlations Although you compute partials via residuals, sometimes it is handy to compute them with correlations. Also looking at the formulas is (could be?) informative.Notation. The partial correlation is r12.3 where variable 3 is being partialed from the correlation between 1 and 2. In our example,The partial correlation can be a little or a lot bigger or smaller than the original.
8The Order of a PartialIf you partial 1 vbl out of a correlation, the resulting partial is called a first order partial correlation.If you partial 2 vbls out of a correlation, the resulting partial is called a second order partial correlation. Can have 3rd, 4th, etc., order partials.Unpartialed (raw) correlations are called zero order correlations because nothing is partialed out.Can use regression to find residuals and compute partial correlations from the residuals, e.g. for r12.34, regress 1 and 2 on both 3 and 4, then compute correlation between 2 sets of residuals.
9Partials from Multiple Correlation We can compute squared partial correlations from various R2 values.is the R2 from the regression in which 1 is the DV and 2 and 3 are the Ivs.Alternative (possibly friendlier) notation.
10Squared Partials from R2 - Venn Diagrams Here we want the partial correlationBetween Y and X1 holding X2 constant.184.108.40.206.
11Exercise – Find a Partial 1231 ANX2 Fam History.203 DOC Visit.35.15What is the correlation between trait anxiety and the number of doctor visits controlling for family medical history?
12Find a partial1231 ANX2 Fam History.203 DOC Visit.35.15
13Semipartial Correlation With partial correlation, we find the correlation between X and Y holding Z constant for both X and Y. Sometimes, we want to hold Z constant for just X or just Y. Instead of holding constant for both, hold for only one, therefore it’s a semipartial correlation instead of a partial. With a semipartial, we find the residuals of X on Z or Y on Z but the other is the original, raw variable. Correlate one raw with one residual.In our example, we found the correlation between E1 (HSGPA) and FGPA to be This is the semipartial correlation between HSGPA and FGPA holding SAT constant for HSGPA only.
14Semipartials from Correlations Note that r1(2.3) means the semipartial correlation between variables 1 and 2 where 3 is partialled only from 2. In our example:Agrees with earlier results within rounding error.
15Squared Semipartials from Multiple Correlations Squared semipartial is an increment in R2.
16Partial vs. Semipartial Why is the squared partial larger than the squared semipartial? Look at the respective areas for Y.
17Regression and Semipartial Correlation Regression is essentially about semipartialsEach X is residualized on the other X variables.For each X we add to the equation, we ask, “What is the unique contribution of this X above and beyond the others?” Increment in R2 when added last.We do NOT residualize Y, just X.Semipartial because X is residualized but Y is not.b is the slope of Y on X, holding the other X variables constant.
18Semipartial and Regression 2 Standardized regression coefficientSemipartial correlationThe difference is the square root in the denominator. The regression coefficient can exceed 1.0 in absolute value; the correlation cannot.
19Uses of Partial and Semipartial The partial correlation is most often used when some third variable z is a plausible explanation of the correlation between X and Y.Job characteristics and job sat by NACog ability and grades by SESThe semipartial is most often used when we want to show that some variable adds incremental variance in Y above and beyond other X variablePilot performance and Cog ability, motor skillsPatient well being and surgery, social support
20ReviewGive a concrete example (names of vbls, context) where it makes sense to compute a partial correlation. Why a partial rather than semipartial?Give a concrete example where it makes sense to compute a semipartial correlation. Why semi rather than partial?
21Suppressor Effects Hard to understand, but Inspection of r not enough to tell valueNeed to know to avoid looking dumbShow problems with Venn diagramsThink of observed variable as composite of different stuff, e.g., satisfaction with car (price, prestige, etc.)
22Suppressor Effects (2)Note that X2 is correlated with X1 but NOT with Y. Will X2 be useful in a regression equation?YX1X21.50.00If we solve for beta weights, we find, beta1=.667 and beta2 = Notice that the beta weight for the first is actually larger than r (.50), and the second has become negative. Can also happen that r is (usually slightly) positive and beta is negative. This is a suppressor effect. Always inspect your correlations along with your regression weights to see if this is happening.What does it mean that beta2 is negative? Sometimes people forget that there are other X variables in the equation. “The results mean that we should feed people more to get them to lose weight.”
23Suppressor Effects (3) Can also happen in path analysis, CSM. Explanation – X2 is a measure of prediction error in X1. If we subtract X2, will have a more useful measure of X1. X2 ‘suppresses’ the correlation of Y and X1.Inspection of correlation matrix not sufficient to see value of variables.Looking dumb.Venn diagram.
24ReviewWhy is the squared semipartial always less than or equal to the squared partial?Why is regression more closely related to semipartials than partials?How could you use ordinary regression to compute 3rd order partials?
25Exercise – Find a Semipartial YX1X220.127.116.11What is the correlation between Y and X1 holding X2 constant only for X1?
26Find a SemipartialThe correlation of X1 with Y after controlling for X2 (from X1 only) is rather small.YX1X18.104.22.168
27Computer Exercise Go to labs and download 2IV Example. Find the partial correlation between hassles and well being holding gender and anger constant (2nd order partial).Find the squared semipartial for anger when well being is the DV and gender and hassles are the other IVs, that is, find the increment in R-square when anger is added to the equation after gender and hassles.