Download - Partial Correlation

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Partial & Semi-PartialCorrelation

andMultiple Regression

Relationships among > 2 variables

Correlation & Regression♦Both test simple linear relationships

between 2 variables∗ Correlation: non-directional∗ Regression: directional

♦Both can be extended to more than 2variables∗ Partial correlation: non-directional∗ Semi-partial correlation: “directional”∗ Multiple regression: directional

Dealing with Data♦ Imagine the ETS calls you up and says

they think there is a relationship betweenthe hours a student spends preparing forthe SAT and the score on the SAT. Theyhave asked recent SAT-takers to providean estimate of the hours spent preparing(including classes). They provide youwith these data as well as each student’sGPA and the final score on the SAT.

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ETS Example♦What data do you have?

∗ Hours of prep∗ GPA∗ SAT score

♦What kinds of predictions might you makeabout the relationship between hours ofpreparation and SAT score?

♦How can you examine the relationship(s)?

Simple Correlation

♦Goal: determine therelationship between2 variables(e.g. y and x1)

♦r2yx1 is the shared

variance between yand x1

Y X1r2yx1

ETS Example

♦Can look at simplecorrelation betweeneach pair of variables∗ prep hours & SAT∗ prep hours & GPA∗ GPA & SAT

Y X1r2yx1

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ETS Example♦Prep hours x SAT

Prep hours SAT score GPA

15 1040 2.8

6 1450 3.75

12 1000 2.6

2 1510 3.8

18 1230 3.2

30 1160 2.75

26 1580 3.15

15 1240 2.4

10 1329 3.3

20 1470 3.5

5 1460 3.4

30 1020 2.4

12 1390 3.6

16 1200 2.87

25 1060 2.9

7 1040 2.65

24 1340 2.67

10 1280 3.5

14 1290 3.23

22 1450 3.0

ETS Example♦GPA x SAT

Prep hours SAT score GPA 15 1040 2.8

6 1450 3.75

12 1000 2.6

2 1510 3.8

18 1230 3.2

30 1160 2.75

26 1580 3.15

15 1240 2.4

10 1329 3.3

20 1470 3.5

5 1460 3.4

30 1020 2.4

12 1390 3.6

16 1200 2.87

25 1060 2.9

7 1040 2.65

24 1340 2.67

10 1280 3.5

14 1290 3.23

22 1450 3.0

ETS Example♦GPA x prep hours

Prep hours SAT score GPA

15 1040 2.8

6 1450 3.75

12 1000 2.6

2 1510 3.8

18 1230 3.2

30 1160 2.75

26 1580 3.15

15 1240 2.4

10 1329 3.3

20 1470 3.5

5 1460 3.4

30 1020 2.4

12 1390 3.6

16 1200 2.87

25 1060 2.9

7 1040 2.65

24 1340 2.67

10 1280 3.5

14 1290 3.23

22 1450 3.0

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Prep Hrs

SAT

GPAPrepHrs

ETS Example

♦GPA & SAT: notsurprising

♦GPA & Prep hours:huh?

♦GPA & Prep hours:∗ People with lower

GPAs prep more(why?)

∗ Could explain theGPA & Prep hrs

Three (or more) Variables♦3 variables = 3 relationships

∗ Each can effectthe other two

♦Partial & semi-partialcorrelation--removecontributions of3rd variable

Y X1r2yx1

X2

r2yx1r2

yx1.x2

Partial Correlation

♦ Find the correlationbetween two variableswith the third heldconstant in BOTH

♦ That is, we remove theeffect of x2 from both yand x1

Y X1

X2r2yx1.x2 is the shared

variance of y & x1 with x2removed

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Partial Correlation

♦ y without x2 & x1 without x2(residuals)

♦We can put this in terms ofsimple corr. coefficients:

r2yx1.x2Y X1

X2

ryx1.x2 =ryx1 - ryx2rx1x2

√ (1 - r2yx2)(1 - r2

x1x2)

Simple correlationbetween y and x1

Product of the corr.between y & x2 andthe corr. of x1 & x2

These represent all the variancewithout the partialled out relationships

Partial Correlationr2

yx1.x2Y X1

X2

♦The significance of ryx1.x2can be calculated using t∗ H0 : ρxy = 0 (no relationship)∗ H1 : ρxy ≠ 0 (either positive or negative corr.)∗ t(N-3) = ryx1.x2 - ρyx1.x2

√(1 - r2yx1.x2)/N-3

• 1- r2yx1.x2 is the unexplained variance

• N-3 = degrees of freedom (three variables)• √(1 - ryx1.x22)/N-3 = standard error of ryx1.x2

ETS Example♦Correlation between prep hours and SAT

score with GPA partialled out:ryx1.x2 =

ryx1 - ryx2rx1x2√ (1 - r2

yx2)(1 - r2x1x2)

=-0.21 -(-0.54*0.71)√ (1 - (-0.542))(1 - 0.712)

= 0.28

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ETS Example♦The partial correlation between prep

hours and SAT score with effect of GPAremoved: ryx1.x2 = 0.28, r2

yx1.x2 = 0.08

Significant? t0.05(17) = 2.11∴ t(17) = 1.23 is not significant

t (N-3) =ryx1.x2

√ (1 - r2yx1.x2)/N-3

t (17) =0.28

√ (1 - 0.08)/17= 1.23

r2yx1r2

y(x1.x2)

Semi-Partial Correlation

♦ Find the correlationbetween two variableswith the third heldconstant in one of thevariables

♦ That is, we remove theeffect of x2 from x1

Y X1

X2r2y(x1.x2) is the shared

variance of y & x1 with x2removed from x1

Semi-Partial Correlation♦Why semi-partial?♦Generally used with

multiple regression toremove the effect of onepredictor from anotherpredictor without removingthat variability in thepredicted variable

♦NOT typically reported asthe only analysis

r2yx1r2

y(x1.x2)Y X1

X2

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r2y(x1.x2)Y X1

X2


♦ y & x1 without x2(residuals)

♦ Put in terms of simple correlationcoefficients:

ry(x1.x2) =ryx1 - ryx2rx1x2

√ (1 - r2x1x2)

Simple correlationbetween y and x1

Product of the corr.between y & x2 andthe corr. of x1 & x2

Same as partial except the sharedvariance of y & x2 is left in


♦Which will be larger, the partial or the semi-partial correlation?


√ (1 - r2x1x2)

ryx1.x2 =ryx1 - ryx2rx1x2

√ (1 - r2yx2)(1 - r2

x1x2)partial semi-partial

ETS Example♦Going back to the SAT example, suppose

we partial out GPA from hours of preponly


√ (1 - r2x1x2)

=-0.21 -(-0.54*0.71)√ (1 - 0.542)

= 0.20

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Significance of Semi-partial♦Same as for partial correlation, just

substitute the ry(x1.x2)

♦df = N-3

t (N-3) =ry(x1.x2)

√ (1 - r2y(x1.x2))/N-3

ETS Example♦The semi-partial correlation between prep

hours and SAT score with effect of GPAremoved: ry(x1.x2) = 0.20, r2

y(x1.x2) = 0.04

Significant? t0.05(17) = 2.11∴ t(17) = 0.84 is not significant

t (N-3) =ry(x1.x2)

√ (1 - r2y(x1.x2))/N-3

t (17) =0.20

√ (1 - 0.04)/17

= 0.84

Multiple Regression

♦Simple regression: y’ = a + bx♦Multiple regression: General Linear Model

∗ y’ = a + b1x1 + b2x2 (2 predictors)∗ Therefore, the general formula:

y’ = a + b1x1 + … + bkxk (k predictors)• The problem is to solve for k+1 coefficients

» k predictors (regressors) + the intercept» We are most concerned with the predictors

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ETS Example♦Prep hours (x1), GPA

(x2), & SAT (y)∗ Use Prep hours and

GPA to predict SATscore

♦Simple regressionsy’ = -4.79x1 + 1353y’ = 300x2 + 355

ETS Example

♦Use bothprep hoursand GPAto predictSAT score

♦Now findequationfor 3-Drelationship

Finding Regression Weights

♦What do we minimize?∗ ∑(y-y’)2 (least square principle)

♦For multiple regression, it is easier to thinkin terms of standardized regressioncoefficients*

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♦What do we minimize?∗ ∑(y-y’)2 (least square principle)

♦For multiple regression, it is easier to thinkin terms of standardized regressioncoefficients*∗ zy’ = β1zx1 + β2zx2∗ The goal is to find β’s that minimizes:

1N

∑(zy - zy’)2 1N

∑(zy - β1zx1 - β2zx2)2=

Finding Regression Weights♦Using differential calculus, we find 2

“normal equations” for 2 regressors:β1 + rx1x2β2 - rx1y = 0rx1x2 β1 + β2 - rx2y = 0

♦These can be converted to:

β1 =rx1y - rx2yrx1x2

1 - r2x1x2


1 - r2x1x2

Notice thatthese are like

the semi-partial

correlation

Finding Regression Weights♦ In practice, the raw scores are used:

zy’ = β1zx1 + β2zx2

y’ - y

est σy=

x1 - x1

est σx1

x2 - x2

est σx2β2β1 +

… which is equivalent to:

y’ =est σy

est σx1

est σy

est σx2β2β1 +x1 x2

est σy

est σx1

est σy

est σx2β2β1 -x1 x2+ y -

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Finding Regression Weights♦Look at each segment...

∴ we have the regression equation with...

y’ =est σy

est σx1

est σy

est σx2β2β1 +x1 x2

est σy

est σx1

est σy

est σx2β2β1 -x1 x2+ y -

y’ = b1x1 + b2x2 + a

est σy

est σx2b2 = β2

est σy

est σx1b1 = β1

a = y - b1x1 - b2x2

Note: RAWregression

weights

ETS Example♦Use the r’s to get the β’s

rx1x2 = -0.54 rx1y = -0.22 rx2y = 0.72

est σy

est σx2b2 = β2

est σy

est σx1b1 = β1

a = y - b1x1 - b2x2


1 - r2x1x2


1 - r2x1x2

β1 = 0.24 β2 = 0.84Use the β’s to get the coefficients

= 5.16 = 353

= 110

Finding Regression Weights♦ For >2 predictors, the same principle apply

∗ Use normal equations will minimize (y - y’)2

(deviation of actual from predicted)∗ The equations can be expressed in matrix form as:

RijBj - Rjy = 0∗ Rij = k x k matrix of the correlation among the

different independent variables (x’s)∗ Bj = a column vector of the k unknown β values (1 β

for each x)∗ Rjy = a column vector of the correlation coefficient for

each k predictor and the dependent variable (y)

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Finding Regression WeightsRijBj - Rjy = 0

♦Rij and Rjy are known∗ each rxixj and each ryxi

♦Therefore, we can solve for BjBj = Rij

-1 Rjy(in matrix form, this is really easy!)

♦Don’t worry about actually calculatingthese, but be sure you understand theequation!


♦For each independent variable, we canuse the relationship of b to β:

est σy

est σxjbj = βj

♦The same principle for obtaining theintercept in simple regression applies aswell:

a = y - ∑ bjxj

Explained Variance (Fit)♦ For 2 predictors, equation defines a plane

y’ = 5.16x1 + 353x2 + 110 (ETS example)♦ How far are the points in 3-D space from the

plane defined by the equation?

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Explained Variance

♦ In addition to simple (rxy), partial (ryx1.x2), &semi-partial (ry(x1.x2)) correlation coefficients,we can have a multiple correlationcoefficient (Ry.x1x2)

♦Ry.x1x2 = correlation between observedvalue of y and predicted value of y∗ can be expressed in terms of beta weights and

simple correlation coefficients

Ry.x1x2 = β1ryx1 + β2ryx2

√ R2y.x1x2 = β1ryx1

+ β2ryx2 OR

Explained VarianceR2

y.x1x2 = β1ryx1 + β2ryx2

♦Any βi represents the contribution ofvariable xi to predicting y

♦The more general version of this equationis simply:R2 = ∑ βjryxj or in matrix form... R2 = BjRjy

(Just add up the products of the β’s and the r’s)♦How are βi’s and R2 related to the simple

correlation coefficients?

Explained Variance

R2y.x1x2 = β1ryx1

+ β2ryx2

♦ If x1 and x2 are uncorrelated:

Y

X2X1

β1 = ryx1β2 = ryx2

R2y.x1x2 = ryx1ryx1

+ ryx2ryx2

= r2yx1

+ r2yx2

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Explained Variance

R2y.x1x2 = β1ryx1

+ β2ryx2

♦ If x1 and x2 are correlated:

Y

X2X1

βi’s are correctedso that overlap isnot counted twice

Adjusted R2

♦R2 is a biased estimate of the populationR2 value

♦ If you want to estimate the population,use Adjusted R2

∗ Most stats packages calculate both R2 andAdjusted R2

∗ If not, the value can be obtained from the R2:

(k)(1 - R2)

N-k-1Adj R2 = R2 -

Significance Tests♦ In multiple regression, there are 3

different statistical tests that are ofinterest∗ Significance of R2

• Is the fit of the regression model significant?∗ Significance for increments to R2

• How much does adding a variable improve the fitof the regression model?

∗ Significance of the regression coefficients• βj is the contribution of xj. Is this different from 0?

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Partitioning Variance

♦ Same as in simple regression!♦ The only difference is that y’ is generated by a

linear function of several independent variables(k predictors)

♦ Note: SStotal = SSregression + SSresidual

∑(y-y)2 = ∑(y’-y)2 + ∑(y-y’)2

Total variancein y (aka SStotal)

Explained Variance(aka SSreg)

UnexplainedVariance (aka SSres)

MSReg

MSResF =

SSReg/dfReg

SSRes/dfRes=

SSReg = ∑(y’-y)2; dfReg = k (# of regressors - 1)

SSRes = ∑(y-y’)2; dfRes = N-k-1 (# obs - # reg)

Significance of R2

♦Need a ratio of variances (F value)

♦Where do these values come from?

♦F for the overall model reflects this ratio

Significant Increments to R2

♦As variables (predictors) are added to theregression R2 can…∗ …stay the same; additional variable has NO

contribution∗ … increase; additional variable has some

contribution♦ If R2 increases, we want to know if that

increase is significant

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Significant Increments to R2

♦ Use an F∆R2

RL2 - RS

2/kL - kS

(1-RL2)/(N-kL-1)

F∆R2 =

♦Making sense of the equation∗ L = larger model; S = smaller model∗ ALL variables in smaller model (S) must also

be in larger model (L)∗ Therefore, L is model S + one or more

additional variables

SSRes/N-k-1

SSj(1-Rj2)

est σbj =

Significance of Coefficients♦Think about bj in t terms: bj/est σbj

♦bj/est σbj is distributed as a t with N-k-1degrees of freedom, where...

♦SSj = sum of squares for variable xj

♦Rj2 = squared multiple correlation for

predicting j from remaining k-1 predictors(treating xj as the predicted variable)

Significance of Coefficients

SSRes/N-k-1

SSj(1-Rj2)

est σbj =

♦As Rj2 increases, the denominator of the t

equation approaches 0; that is, est σbjbecomes larger

♦As the remaining x’s account for xj, bj isless likely to reach significance

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Importance of IVs (x’s)♦Uncorrelated IVs: simple ryxj’s work♦Correlated IVs:

∗ simple correlation coefficients includevariance shared among IVs (over-estimated)

∗ regression weights can involve predictorintercorrelations or suppressors (more later)

∗ Best measure: squared semi-partialcorrelation srj

2

∗ BUT srj2 comes in different forms for different

types of regression

Multiple Regression Types

♦Several types of regression available♦How do they differ?

∗ Method for entering variables• What variables are in the model• What variables are held constant

∗ Use different types of R2 values to assessimportance

∗ Use of different measures to assessimportance of IVs


♦Simultaneous Regression (most common)∗ Single regression model with all variables

• All predictors are entered “simultaneously”• All variables treated equally

∗ Each predictor is assessed as if it wasentered last

• Each predictor is evaluated in terms of what itadds to the prediction of the dependent variable,over and above the other variables

• Key test: srj2 for each xj with all other x’s held

constant

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Multiple Regression Types♦Hierarchical Regression

∗ Multiple models calculated• Start with one predictor• Add predictors• Order specified by researcher

∗ Each predictor is assessed in terms of what itadds at the time it is entered

• Each predictor is evaluated in terms of what itadds to the prediction of the dependent variable,over and above the other variables that havealready been entered

• Key test: ∆R2 at each step

Multiple Regression Types♦Hierarchical Regression

∗ Used when the researcher has a priorireasons for entering variables in a certainorder

• Specific hypothesis about the components oftheoretical models

• Practical concerns about what it is important toknow

Multiple Regression Types♦Stepwise & Setwise Regressions

∗ Multiple models calculated (like hierarchical)• Use statistical criteria to determine order• Limit final model to meaningful regressors

∗ Recommended for exploratory analyses ofvery large data sets (> 30 predictors)

• With lots of predictors, keeping all but oneconstant may make it difficult to find anysignificant

• These procedures capitalize on chance to find themeaningful variables

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♦Stepwise Regression: Forward∗ Step 1: enter xj with largest simple ryxj

∗ Step 2: partial out first variable and choose xjwith highest partial ryxj.x1

∗ Step 3: partial out x1 and x2…∗ Stop when resulting model reaches some

criteria (e.g., min R2)

Multiple Regression Types♦Stepwise Regression: Backward

∗ Step 1: Start with complete model (all xj’s)∗ Step 2: remove xj based on some criterion

• Smallest R2

• Smallest F∗ Stop removing variables when some criteria

is reached• All regressors significant• Min R2

Multiple Regression Types♦Setwise Regression

∗ Test several simultaneous models∗ Finds the best possible subset of variables

• Setwise(#): for a given set size• Setwise Full: for all possible set sizes

∗ For example, with 8 variables:• Look at all possible combinations of say 5

variables• Figure out which combo has the largest R2

• Can be done for sets of 2, 3, 4, 5, 6, or 7 variables• In each case, find the set with the largest R2

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Importance of Regressors

♦βi’s primarily serve to help define theequation for predicting y

♦Squared semi-partial correlation (sr2) moreappropriate for practical importance∗ Put in terms of variance explained by each

regressor∗ Compare how variance much each regressor

explains

Importance of IVs (x’s)♦For simultaneous or setwise regression

∗ srj2 is the amount R2 would be reduced if

variable xj were not included in the regressionequation

∗ In terms of the regression statistics:Fj

dfRessrj

2 = (1-R2)

∗ When the IVs are correlated, the srj2’s for all

of the xj’s will not sum to the R2 for the fullmodel

Importance of IVs (x’s)♦For hierarchical or stepwise regression

∗ srj2 is the increment to R2 added when xj is

entered into the equation.∗ Because each variable is added separately,

the srj2 will reflect that variables contribution

AT A PARTICULAR POINT in the model∗ The sum of the srj

2 values WILL sum to R2

∗ The importance of the different variables mayvary depending on the order in which thevariables are entered

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Potential Problems♦Several assumptions

(see Berry & Feldman pp. 10-11 in book)

∗ Random variables, interval scale∗ No perfect collinear relationships

♦Also practical concerns♦Focus on most relevant/prevalent…

Multicollinearity♦Perfect collinearity: when one

independent variable is perfectly linearlyrelated to one or more of the otherregressors∗ x1 = 2.3x2 + 4 : x1 is perfectly predicted by x2

∗ x1 = 4.1x3 + .45x4 + 11.32 ; x1 is perfectlypredicted by linear combination of x3 and x4

∗ Any case where there is an R2 value of 1.00among the regressors (NOT including y)

∗ Why might this be a problem?

Multicollinearity♦Perfect collinearity

(simplest case)∗ One variable is a

linear function ofanother

∗ We’d be thrilled(and skeptical) tosee this in a simpleregression

∗ However...

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(simplest case)∗ Problem in multiple

regression∗ y values will line up

in single planerather thanvarying abouta plane


(simplest case)∗ No way to

determine theplane that fits the yvalues best

∗ Many possibleplanes

Multicollinearity♦ In practice

∗ perfect collinearity violates assumptions ofregression

∗ less-than-perfect collinearity is more common• not an all or nothing situation• can have varying degrees of multicollinearity• dealing with multicollinearity depends on what you

want to know

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Multicollinearity♦Consequences

∗ If the only goal is prediction, not a problem• plugging in known numbers will give you the

unknown value• although specific regression weights may vary, the

final outcome will not∗ We usually want to explain the data

• can identify the contributions of the regressors thatare NOT collinear

• cannot identify the contributions of the regressorsthat are collinear because regression weights willchange from sample to sample

Multicollinearity♦Detecting collinearities

∗ Some clues• full model is significant but none of the individual

regressors reach significance• instability of weights across multiple samples• look at simple regression coefficients for all pairs• cumbersome way: regress each independent

variable on all other independent variables to see ifany R2 values are close to 1

Multicollinearity♦What can you do about it?

∗ Increase the sample size• reduce the error• offset the effects of multicollinearity

∗ If you know the relationship, you can use thatinformation to offset the effect (yea right!)

∗ Delete one of the variables causing the problem• which one? If one is predicted by group of others• logical rationale?• presumably, the variables were there for theoretical

reasons

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Multicollinearity♦Detecting collinearities

∗ SPSS: Collinearity diagnostics & follow-up• Tolerance: 1-R2 for the regression of each IV against

the remaining regressors.» Collinearity: tolerance close to 0» Use this to locate the collinearity

• VIF: variance inflation factor = instability of β(reciprocal of Tolerance)

• To locate collinearity» removing the variable with the lowest tolerance

• To resolve original regression» Run a Forward regression on the variable with the lowest

tolerance

Suppression♦Special case of multicollinearity♦Suppressor variables are variables that

increase the values of R2 by virtue of theircorrelations with other predictors andNOT the dependent variable

♦The best way to explain this is by way ofan example...

Suppression Example♦Predicting course grade in a multivariate

statistics course with GRE verbal andquantitative∗ The multiple correlation R was 0.62

(reasonable, right?)∗ However, the β’s were 0.58 for GRE-Q and

-0.24 for GRE-V∗ Does this mean that higher GRE-V scores

were associated with lower courseperformance? Not exactly

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Suppression Example♦Why was the β for GRE-V negative?

∗ The GRE-V alone actually had a smallpositive correlation with course grade

∗ The GRE-V and GRE-Q are highly correlatedwith each other

∗ The regression weights indicate that for agiven score on the GRE-Q, the lower aperson scores on the GRE-V, the higher thepredicted course grade

Suppression Example♦Another way to put it...

∗ The GRE-Q is a good predictor of coursegrade, but part of the performance on GRE-Qis determined by GRE-V, so it favors peopleof high verbal ability.

∗ Suppose we have 2 people who scoreequally on GRE-Q but differently on GRE-V

• Bob scores 500 on GRE-Q and 600 on GRE-V• Jane scores 500 on GRE-Q and 500 on GRE-V• What happens to the predictions about course

grade?

Suppression Example♦Another way to put it...

∗ Bob: 500 on GRE-Q and 600 on GRE-V∗ Jane: 500 on GRE-Q and 500 on GRE-V∗ Based on the verbal scores, we would predict

that Bob should have better quantitative skillsthan Jane, but he does not score better

∗ Thus, Bob must actually have LESSquantitative knowledge than Jane, so wewould predict his course grade to be lower.

∗ This is equivalent to giving GRE-V a negativeregression weight, despite positive correlation

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Suppression♦More generally...

∗ If x2 is a better measure of the source oferrors in x1 than in y, then giving x2 anegative regression weight will improve ourpredictions of y

∗ x2 subtracts out/corrects for (i.e., suppresses)sources of error in x1

∗ Suppression seems counterintuitive, butactually improves the model

Suppression♦More generally...

∗ Suppressor variables usually considered“bad”--can cause misinterpretation(GRE example)

∗ However, careful exploration• enlighten understanding of interplay of variables• improve our prediction of y

∗ Easy to identify• Significant regression weights• b/β (reg) & r (simple corr) have opposite signs

Practical Issues♦Number of cases

∗ Must exceed number of predictors (N > k)∗ Acceptable N/k ratio depends on

• reliability of data• researcher’s goals

∗ Larger samples required for:• more specific conclusions vs vague conclusions• post-hoc vs. a priori tests• designs with interactions• collinear predictors

∗ Generally, more is better

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Practical Issues♦Outliers

∗ Correlation is extremely sensitive to outliers∗ Easiest to show with simple correlation

rxy = -0.03

rxy = +0.59• Outliers should be

assessed for DV andall IVs

• Ideally, we wouldidentify multivariateoutliers, but this isnot practical

Practical Issues♦Linearity

∗ Multiple regression assumes linearrelationship between DV and each IV

∗ Relationships can be non-linear, multipleregression may not be appropriate

∗ Transformations may rectify non-linearity• logs• reciprocals

Practical Issues♦Normality

∗ Normally distributed relationship between y’and residuals (y-y’)

∗ violation affects statistical conclusions, butnot validity of model

♦Homoscedasticity∗ multivariate version of homogeneity of

variance∗ violation affects statistical conclusions, but

not validity of model

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Assumptions Met Normality Violated

Linearity ViolatedHomoscedasticity

Violated

What do you report?♦Correlation analyses

∗ Always state what’s happening in the data∗ Report the r value and its corresponding p

value (either actual p or p < α)∗ Qualify simple correlation with partial

correlation coefficient, if multiple variables∗ Authors may include r2 values, stating “xx%

of the variance was accounted for by thisrelationship.”

What do you report?♦Regression analyses

∗ Report the correlations first∗ For simple regression: state the equation, the

r2 value, and significance of the regressionweight (sometimes a table will work)

∗ For multiple regression• state equation (not always in manuscripts)• state practical importance of each regressor (sr2)• state the relative relationship among regressors• state the significance of each regressor

Download - Partial Correlation

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