partial correlation

1

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.

2

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

3

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

4

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

5

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

6

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

7

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

8

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

9

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*

10


♦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 -

11

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)

12

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?

13

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

14

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?

15

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

16

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

17

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

18

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

19


♦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

20

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

21

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...

22


(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

23

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

24

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

25

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

26

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

27

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

28

.

0

y' valuesRe

sidu

als

(y-y

')Re

sidu

als

(y-y

’)

.

0

y' values

Resi

dual

s (y

-y')

Resi

dual

s (y

-y’)

.

0

y' values

Resi

dual

s (y

-y')

Resi

dual

s (y

-y’)

.

0

y' valuesRe

sidu

als

(y-y

')Re

sidu

als

(y-y

’)

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

partial correlation

Documents