psyc3010 lecture 8uqwloui1/stats/3010 for post... · 2018-08-02 · 1 1 psyc3010 lecture 8 standard...

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psyc3010 lecture 8psyc3010 lecture 8

standard multiple regressionstandard multiple regressionhierarchical multiple regressionhierarchical multiple regression

last lecture: analysis of covariance (ANCOVA)last lecture: analysis of covariance (ANCOVA)next lecture: moderated multiple regressionnext lecture: moderated multiple regression

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assignment 2assignment 2all materials posted on Blackboard this week

due on 23 May by 12 noon (Week 12)tutorials this week and next week will teach concepts and SPSS skills required for the assignment tutorials in Week 10 will provide feedback on Assignment 1 and tips for Assignment 2tutorials in Week 11 will provide consultation for the assignment

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last lectures last lectures this lecturethis lecture

previously:previously:–– review of correlation and regression, focusing review of correlation and regression, focusing

on underlying principle of partitioning variance on underlying principle of partitioning variance into into SSSSregressionregression ((ŶŶ –– Y) and Y) and SSSSerrorerror (Y (Y –– ŶŶ))

–– introduced multiple regression and touched on introduced multiple regression and touched on indicators of predictor importance (b, r, etc.)indicators of predictor importance (b, r, etc.)

this lecture:this lecture:–– more about multiple regressionmore about multiple regression–– introduce hierarchical regressionintroduce hierarchical regression

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topics for this lecturetopics for this lecture

indices of predictor importancestandard multiple regression (SMR)– linear model– preliminary statistics– test of overall model– tests of individual predictors

hierarchical multiple regression (HMR)– order of entering predictors– test of overall model– tests of individual predictors

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indices of predictor importance (1)r – Pearson or zero-order correlation

• standardised (scale-free) covariance between two variables

r2 – the coefficient of determination• proportion of variability in one variable accounted for by

another variable

b – unstandardised slope / regression coefficient• (scale-dependent) slope of regression line: change in units

of Y expected with a 1 unit increase in X

β – standardised slope / regression coefficient• (scale-free) slope of the regression line if all variables were

standardised: change in standard deviations in Y expected with a 1 standard deviation increase in X, controlling for all other predictors

• β = r in bivariate regression (when only one IV)

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indices of predictor importance (2)spr2 or sr2 – semi-partial correlation squared

• scale-free measure of association between two variables, independent of other IVs

• proportion of total variance in DV uniquely accounted for by IV

pr2 – partial correlation squared• scale-free measure of association between two variables,

independent of other IVs• proportion of residual variance in DV (after other IVs are

controlled for) uniquely accounted for by IV

unique IV : DV variance total DV variance

unique IV : DV varianceunique IV-DV variance + unexplained DV variance

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comparing the different comparing the different rrsszerozero--order (Pearson’s) correlationorder (Pearson’s) correlation between IV and DV between IV and DV ignores the extent to which IV is correlated with other IVsignores the extent to which IV is correlated with other IVs

semisemi--partial correlationpartial correlation deals with unique effect of IV on deals with unique effect of IV on total variance in DV total variance in DV –– usually what we are interested inusually what we are interested in–– conceptually similar to ‘eta squared’ (effect size measure)conceptually similar to ‘eta squared’ (effect size measure)

SPSS calls the semiSPSS calls the semi--partial r the “part correlation”partial r the “part correlation”

SSeffectSStotal

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comparing the different comparing the different rrsspartial correlationpartial correlation deals with unique effect of the IV on deals with unique effect of the IV on residual variance in DVresidual variance in DV–– conceptually similar to ‘partial eta squared’ (effect size)conceptually similar to ‘partial eta squared’ (effect size)–– more difficult to interpret: pr for each IV is based on a more difficult to interpret: pr for each IV is based on a

different denominatordifferent denominator–– most useful when other IVs = control variablesmost useful when other IVs = control variables

generally r > generally r > srsr and pr > and pr > srsr

SSeffectSSeffect + SSerror

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multiple regressionmultiple regression

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1111

criterion

predictor1

bivariate vs multiple regression:model coefficient of determination

predictor2

criterion

predictor

r2

R2

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the linear model the linear model –– one predictor one predictor (2D space)(2D space)

predictor (X)

criter

ion

(Y)

Ŷ = bX + a

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the linear model the linear model –– two predictors two predictors (3D space)(3D space)

criter

ion

(Y) Ŷ = b1X1 + b2X2 + a

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the linear model the linear model –– two predictorstwo predictorscriterion scores are predicted using the best criterion scores are predicted using the best linear combinationlinear combination of the predictorsof the predictors–– similar to similar to lineline--ofof--bestbest--fitfit, but now , but now planeplane--ofof--bestbest--fitfit–– equation derived according to the leastequation derived according to the least--squares criterion:squares criterion:

such that such that ΣΣ (Y(Y--Ŷ)Ŷ)2 2 (deviations of dots from plane) is (deviations of dots from plane) is minimizedminimized

•• bb11 is the slope of the plane relative to the is the slope of the plane relative to the XX11 axis, axis, •• bb22 is the slope relative to theis the slope relative to the XX22 axis, axis, •• aa is the point where the plane intersects the is the point where the plane intersects the YY axis axis

(when(when XX11 and and XX22 are equal to zero)are equal to zero)idea extends to 3+ predictors but becomes tricky to idea extends to 3+ predictors but becomes tricky to represent graphically represent graphically (i.e., hyperspace)(i.e., hyperspace)

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example example –– new studynew study

how can we improve academic achievement?how can we improve academic achievement?possible factors:possible factors:–– minutes spent studying per week minutes spent studying per week –– motivationmotivation–– anxiety anxiety

how much variance how much variance (R2 ) can the predictors can the predictors explain as a set? explain as a set? what is the relative importance what is the relative importance (r, b, β, pr2, sr2) of each predictor?of each predictor?

continuousmeasures

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data table data table participant study motivation anxiety GPAparticipant study motivation anxiety GPA

(X(X11)) (X(X22)) (X(X33)) (Y)(Y)1 104 12 1 5.52 109 13 9 5.73 123 9 2 5.54 94 15 11 5.35 114 15 2 6.16 91 7 9 4.97 100 5 1 4.5

29 107 10 6 5.930 119 8 2 6.0

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overview of preliminary statisticsoverview of preliminary statistics

MeanMean SDSD NN alphaalphastudy timestudy time 97.96797.967 8.9158.915 3030 .88.88motivationmotivation 14.53314.533 4.3924.392 3030 .75.75anxietyanxiety 4.2334.233 1.4551.455 3030 .85.85GPAGPA 5.5515.551 2.1632.163 3030 .82.82

STST MOTMOT ANXANX GPAGPAstudy timestudy time 1.001.00motivationmotivation .313.313 1.001.00anxietyanxiety .256.256 .536.536 1.001.00

GPAGPA .637.637 .653.653 .505.505 1.001.00

Descriptive Statistics

Correlations

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more about descriptive statisticsmore about descriptive statistics

MeanMean SDSD NN alphaalphastudy timestudy time 97.96797.967 8.9158.915 3030 .88.88motivationmotivation 14.53314.533 4.3924.392 3030 .75.75anxietyanxiety 4.2334.233 1.4551.455 3030 .85.85GPAGPA 5.5515.551 2.1632.163 3030 .82.82

IQIQ MOTMOT ANXANX GPAGPAIQIQ 1.001.00motivationmotivation .313.313 1.001.00anxietyanxiety .256.256 .536.536 1.001.00

GPAGPA .637.637 .653.653 .505.505 1.001.00


Correlations• means and standard deviations are used to obtain regression estimates

• reported as preliminary statistics for MR

• needed to interpret coefficients, although descriptively they are not as critical for MR as they are for t-tests and ANOVA

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more about more about Cronbach’sCronbach’s alphaalpha

MeanMean SDSD NN alphaalphaIQIQ 97.96797.967 8.9158.915 3030 .88.88motivationmotivation 14.53314.533 4.3924.392 3030 .75.75anxietyanxiety 4.2334.233 1.4551.455 3030 .85.85GPAGPA 5.5515.551 2.1632.163 3030 .82.82

STST MOTMOT ANXANX GPAGPAstudy timestudy time 1.001.00motivationmotivation .313.313 1.001.00anxietyanxiety .256.256 .536.536 1.001.00

GPAGPA .637.637 .653.653 .505.505 1.001.00


Correlations

• Cronbach’s α is an index of internal consistency (reliability) for a continuous scale

• index of how well items “hang together”

• best to use scales with high reliability (α > .70) if available – less error variance

2020

more about correlations

MeanMean SDSD NN alphaalphaIQIQ 97.96797.967 8.9158.915 3030 .88.88motivationmotivation 14.53314.533 4.3924.392 3030 .75.75anxietyanxiety 4.2334.233 1.4551.455 3030 .85.85GPAGPA 75.53375.533 15.16315.163 3030 .82.82

STST MOTMOT ANXANX GPAGPASTST 1.001.00motivationmotivation .313.313 1.001.00anxietyanxiety .256.256 .536.536 1.001.00

GPAGPA .637.637 .653.653 .505.505 1.001.00


Correlations

correlation matrix tells you:

•• relationship between each predictor and the criterion relationship between each predictor and the criterion (validities)(validities)

•• intercorrelations among predictors (collinearities) intercorrelations among predictors (collinearities)

to maximise R2 we want predictors that have highvalidities and low collinearity

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principle of parsimony

predictor1

predictor3

predictor2

criterionR2

predictors are:

- highly correlated with criterion

- have low(er) correlations with one another

high parsimony is good: IVs explain unique variance

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principle of parsimony

predictor1

predictor3

predictor2

criterionR2

predictors are:

- highly correlated with criterion

- highly correlated with one another

low parsimony is bad: IVs don’t explain much unique variance and so appear to be redundant. Delete redundant ones (ideally based on theory)

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•• 1 predictor:1 predictor: model Ŷ with a line described by 2 parameters model Ŷ with a line described by 2 parameters ((bXbX + a) + a)

•• 2 predictors:2 predictors: model Ŷ as a plane described by 3 parameters model Ŷ as a plane described by 3 parameters (b(b11XX11 + b+ b22XX22 + a)+ a)

•• p predictors:p predictors: model Ŷ as a pmodel Ŷ as a p--dimensional hyperspace blob dimensional hyperspace blob with p + 1 parameters (1 slope per IV + constant)with p + 1 parameters (1 slope per IV + constant)

Ŷ is modeled with a linear composite formed by multiplyingŶ is modeled with a linear composite formed by multiplyingeach predictor by its regression weight / slope / coefficient each predictor by its regression weight / slope / coefficient (just like a linear contrast) and adding the constant :(just like a linear contrast) and adding the constant :

Ŷ = .79Ŷ = .79STST + 1.45+ 1.45MOTMOT + 1.68+ 1.68ANXANX –– 95.0295.02

the criterion (GPA) is regressed on this linear compositethe criterion (GPA) is regressed on this linear composite

regression solutionregression solution

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the linear model the linear model –– two predictors two predictors (3D space)(3D space)

criter

ion

(Y) Ŷ = b1X1 + b2X2 + a

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ŶŶ

the linear compositethe linear composite

Study

Anxiety

Motivationcriterion

ŶŶ = = .79.79STST + + 1.451.45MOTMOT + + 1.681.68ANXANX –– 95.0295.02

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the linear compositethe linear composite

criterionŶŶ

so we end up with two overlapping variables just like so we end up with two overlapping variables just like in bivariate regression in bivariate regression (only one is blue and weird and wibbly,

graphically symbolising that underlying the linear relationship between the DV and Y hat, the linear composite, is a 4-dimensional

space defined by the 3 IVs and the DV)

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the model:the model: RR and and RR22

despite the underlying complexity, the despite the underlying complexity, the multiple multiple correlation coefficientcorrelation coefficient ((RR) is just a ) is just a bivariatebivariate correlation correlation between the criterion (GPA) and the best linear between the criterion (GPA) and the best linear combination of the predictors (combination of the predictors (ŶŶ))

i.e., R2 = r2YŶ

where Ŷ = .79ST + 1.45MOT + 1.68ANX – 95.02

accordingly, we can treat the model R exactly like r:accordingly, we can treat the model R exactly like r:

i.i. calculate R adjusted:calculate R adjusted:

ii.ii. square R to obtain amount of variance accounted for square R to obtain amount of variance accounted for in Y by our linear composite (in Y by our linear composite (Ŷ)Ŷ)

iii.iii. test for statistical significancetest for statistical significance

2N)1N)(R1(1

2

−−−

−

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((ii) calculating R adjusted) calculating R adjustedin this example, R = .81in this example, R = .81–– so R so R adjadj ==

= .798= .798

(ii) calculating R(ii) calculating R22

RR22 = .65 (.638 adjusted)= .65 (.638 adjusted)“ therefore, 65% of the variance in participants’ GPA “ therefore, 65% of the variance in participants’ GPA

was explained by the combination of their study was explained by the combination of their study time, motivation, and anxiety.” time, motivation, and anxiety.”

230)130)(65.1(1

−−−

−

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(iii) testing overall model ((iii) testing overall model (RR22) for significance) for significanceHH00 : relationship between predictors (as a group) : relationship between predictors (as a group)

and criterion is zeroand criterion is zero


F = df = p, N – p – 1

=

= variance accounted for / dfvariance not accounted for (error) / df

= MS REGRESSIONMS RESIDUAL

)1()1(2

2

RpRp

−−−

)1/()1(/

2

2

−−− pRpR

What we know (can account for)

What we don’t know (can’t account for)

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analysis of regression analysis of regression –– test of test of R2

23.16)6518.1(36518.)1330()R1(pR)1pN(

2

2

=−×

×−−=

−−−

=F 1p,p −−=df

* calculating F from R2

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SSSSYY = = SSSSRegressionRegression + + SSSSResidualResidual

SSSSYY == ∑∑ (Y (Y -- Y)Y)2 2 = (5.5 = (5.5 -- 5.551)5.551)22 + (5.7 + (5.7 -- 5.551)5.551)2 2

= 6667.46= 6667.46

SSSSRegressionRegression == ∑∑ ((Ŷ Ŷ -- YY))2 2 = (6.22 = (6.22 –– 5.551)5.551)22 + + = 4346.03= 4346.03

SSSSResidualResidual == ∑∑ (Y (Y -- ŶŶ))2 2

= = SSSSYY -- SSSSRegressionRegression = 6667.46 = 6667.46 -- 4346.03 4346.03 = 2321.43= 2321.43

analysis of regression analysis of regression –– test of test of R2

* calculating F from sums of squares

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Summary Table for Analysis of Regression:

1p,p −−=df

ModelModelSums of Sums of SquaresSquares dfdf

Mean Mean SquareSquare FF sigsig

RegressionRegression 4346.034346.03 33 1448.681448.68 16.2316.23 .000.000

ResidualResidual 2321.432321.43 2626 89.2989.29

TotalTotal 6667.466667.46 2929

“The model including study time, motivation, and anxiety “The model including study time, motivation, and anxiety accounted for significant variation in participants’ GPAaccounted for significant variation in participants’ GPA, F(3, 26) = 16.23, p < .001, R2 = .65.”

analysis of regression – test of R2

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individual predictorsindividual predictors

we already have our we already have our bivariatebivariate correlations (r) correlations (r) between each predictor and the criterionbetween each predictor and the criterion

in addition, SPSS gives us:in addition, SPSS gives us:–– b : (b : (unstandardisedunstandardised) partial regression coefficient) partial regression coefficient–– ββ : standardised partial regression coefficient: standardised partial regression coefficient–– pr : partial correlation coefficientpr : partial correlation coefficient–– srsr : semi: semi--partial correlation coefficient (which it calls partial correlation coefficient (which it calls

part correlation)part correlation)

calculations for these all involve matrix algebracalculations for these all involve matrix algebra

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criterion

predictor 1

predictor2

pr2

pr – partial correlation coefficientcorrelation between predictor p and the criterion, with variance shared with other predictors partialled out

r01.2 = partial r between 0 and 1 excluding shared variance with 2

pr2 = proportion of residual variancein the criterion (DV variance left unexplained by the other predictors) that is explained by predictor p

prST = .581; prST2 = .337 = 33.7%

prMOT = .562 ; prMOT2 = .316 = 31.6%

prANX = .293; prANX2 = .085 = 8.5%

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pred2

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predictor2

sr – semi-partial correlation coefficient

sr2

predictorp

criterion

sr = correlation between predictor p and criterion, with the variance shared with the other predictors partialled out of predictor p

r0(1.2) = partial r b/w 0 and (1 excluding 2)

sr2 = unique contribution to total variancein DV explained by predictor p

srST = .469; srST2 = = .219 = 21.9%

srMOT = .411; srMOT2 = .169 = 16.9%

srANX = .224; srANX2 = .050 = 5%

shared variance ≈ 21% (R2 - ∑sr2 = 65 - 44)

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pred2

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Ŷ Ŷ = = bb11XX11+ + bb22XX22 + + aa

ŶŶ = = bbYY1.21.2XX11 + + bbYY2.12.1XX22 + + aabbYY1.21.2 is a is a firstfirst--orderorder coefficient coefficient bbYY1.21.2 ≠ ≠ bbYY11 unless unless rr1212 = 0 = 0

ŶŶ = = bb11XX11 + + bb22XX22 + + bb33XX33 + + aaŶŶ = = bbYY1.231.23XX11 + + bbYY2.132.13XX22 + + bbYY3.123.12XX33 + + aa

bbYY1.231.23 is a is a secondsecond--orderorder coefficientcoefficient

zero-order coefficient : doesn’t take other IVs into account

takes 1 other IV into account

takes 2 other IVs into account

all reported coefficients (e.g. in SPSS) are highest-order coefficients possible

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ttbb11 = .789247 = 3.785* ST= .789247 = 3.785* ST.208497 .208497

ttbb22 = 1.453540 = 3.000* MOT= 1.453540 = 3.000* MOT.484508.484508

ttbb33 = 1.678871 = 1.168 ANX= 1.678871 = 1.168 ANX1.4372211.437221

tests of tests of bbss•• test importance of test importance of

each predictor each predictor in the in the contextcontext of all other of all other predictorspredictors

•• divide divide bb by its by its standard errorstandard error

•• dfdf = = NN –– p p –– 11

•• ST contributes significantly to prediction of DV, ST contributes significantly to prediction of DV, independently of the other predictorsindependently of the other predictors

•• MOT also contributes significantly to prediction of DV, MOT also contributes significantly to prediction of DV, independently of the other predictors independently of the other predictors

•• ANX is a valid zeroANX is a valid zero--order predictor of DV, but it is order predictor of DV, but it is notnota significant predictor independent of ST and MOTa significant predictor independent of ST and MOT

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assessing the importance of predictorsassessing the importance of predictors

• can't rely on rs (zero-order), because the predictors are interrelated

a predictor with a significant r may contribute nothing, once others are included (e.g., ANX)

• partial regression coefficient (bs) are adjusted for correlation of the predictor with the other predictors

BUT• can't use relative magnitude of bs, because they

are scale-bound importance of a given b depends on unit and variability of measure

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standardized regression standardized regression coefficients (coefficients (β β ss))

•• rough estimate of relative contribution of rough estimate of relative contribution of predictors, because use same metricpredictors, because use same metric

•• can compare can compare ββ ss within a regression equation within a regression equation

•• cannot necessarily compare across groups & cannot necessarily compare across groups & settings: in this case, the standard deviation of settings: in this case, the standard deviation of variables may changevariables may change

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when IVs are not correlated:when IVs are not correlated: ββ == rrwhen IVs are correlated:when IVs are correlated:β β s (magnitudes, signs) are affected by s (magnitudes, signs) are affected by pattern of correlations among predictorspattern of correlations among predictors

ZZYY = = ββ11ZZ11 + + ββ22ZZ22 + + ββ33ZZ33 +... + +... + ββppZZpp

ZZYY = .46 = .46 ZZSTST + .42 + .42 ZZMOTMOT + .16 + .16 ZZANXANX

a onea one--SD increase in ST SD increase in ST (with all other variables held constant) is associated with an increase of .46 SDs in DV

standardized regression coefficientsstandardized regression coefficients

Yss1β 1 = b1 .

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hierarchical regressionhierarchical regression

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standard regression– all predictors are entered simultaneously– each predictor is evaluated in terms of what it adds to

prediction beyond that afforded by all others

hierarchical regression– predictors are entered sequentially in a pre-specified

order based on logic and/or theory– each predictor is evaluated in terms of what it adds to

prediction at its point of entry (i.e., independently of all other predictors in the model)

– order of prediction based upon logic and/or theory

standard vs hierarchical regressionstandard vs hierarchical regression

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standard multiple regression

criterion

predictor1

predictor2model

predictor1

predictor2

criterion

b for each IV based on unique contribution

IV1IV1IV2

model R2 assessed in 1 step

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hierarchical multiple regression

predictor1

predictor2

criterion

step 1

step 2

each step adds more IVs

model R2 assessed in more than 1 step

b at each step based on unique contribution, controlling for other IVs in current and earlier steps but not later ones

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• rationales for order of entry:1. partial out effect of control variable

exactly the same idea as ANCOVA:predictor at step 1 is like the covariate

2. build a sequential model according to theorye.g., broad measure of personality entered at step 1, more

specific/narrow attitudinal measure entered at step 2

• order is crucial to outcome and interpretation• predictors entered singly or in blocks of > 1• for each step possible to report R2 (or R), b or β,

and sometimes pr2 or sr2

• also test increment in prediction at each block:R2 change and F change

order of entering predictors

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4747


predictor1

predictor2

criterionR

R2

Fstep 1

model 1

4848


predictor1

predictor2

criterion

step 1

R ch

R2 ch

F chstep 1

step 2

model 2

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predictor1

predictor2

criterion

step 1

step 2

step 1

step 2

model 2R

R2

F

final model

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testing hierarchical models

ƒ = full(er) model (with more variables added)r = reduced model

)1/)1()/()R-(R

2

22

−−−

−=

f

rf

pfRpprf

Fchange

1N,df −−−= frf ppp

rRfRchangeR 222 −=

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5252

• suppose we want to repeat our GPA study using hierarchical regression

• further suppose our real interest is motivation and study timestudy time, we just want to control for anxiety:

– enter anxiety at step 1– enter motivation and study time (ST)study time (ST) at step 2

• preliminary statistics would be same as before

• model would be assessed sequentially– step 1: prediction by anxiety– step 2: prediction by motivation and STST above

and beyond that explained by anxiety

back to our exampleback to our example

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model summary

ModelModel RR RR22 RR22adjadj

RR22 chch F chF ch df1df1 df2df2 sig F sig F chch

11 .505.505 .255.255 .228.228 .255.255 9.5849.584 11 2828 .004.004

22 .813.813 .652.652 .612.612 .397.397 14.83614.836 22 2626 .000.000

change statistics

for model 1:R and R2 are the same as for the bivariate rbetween GPA and Anxiety (as anxiety is the only predictor in this model)

5454



11 .505.505 .255.255 .228.228 .255.255 9.5849.584 11 2828 .004.004

22 .813.813 .652.652 .612.612 .397.397 14.83614.836 22 2626 .000.000

change statistics

for model 1:R2 ch = R2 because it simply reflects the change from zero

model summary

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5555



11 .505.505 .255.255 .228.228 .255.255 9.5849.584 11 2828 .004.004

22 .813.813 .652.652 .612.612 .397.397 14.83614.836 22 2626 .000.000

change statistics

for model 2:R and R2 are the same as our full standard multiple regression conducted earlier

model summary

5656



11 .505.505 .255.255 .228.228 .255.255 9.5849.584 11 2828 .004.004

22 .813.813 .652.652 .612.612 .397.397 14.83614.836 22 2626 .000.000

change statistics

for model 2:R2 ch tells us that including ST and Motivation increases the amount of variance accounted for in GPA by 40%

standard MR can’t do that

model summary

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5757



11 .505.505 .255.255 .228.228 .255.255 9.5849.584 11 2828 .004.004

22 .813.813 .652.652 .612.612 .397.397 14.83614.836 22 2626 .000.000

change statistics

for model 2:alternatively, R2 ch tells us that after controlling for anxiety, ST and Motivation explain 40% of the variance in GPA

model summary

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11 .505.505 .255.255 .228.228 .255.255 9.5849.584 11 2828 .004.004

22 .813.813 .652.652 .612.612 .397.397 14.83614.836 22 2626 .000.000

change statistics

for model 2:and F ch tells us that this increment in the

variance accounted is significantH0 is that R2 ch = 0

model summary

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test of R2

1p,p −−=df


Mean SquareMean SquareFF sigsig

1 Regression1 Regression 1702.9011702.901 11 1702.9011702.901 9.5849.584 .004.004


TotalTotal 6667.466667.46 2929



TotalTotal 6667.466667.46 2929

for model 1: details are the same as reported in the change statistics section (as the change was relative to zero)

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1p,p −−=df


Mean SquareMean SquareFF sigsig



TotalTotal 6667.466667.46 2929



TotalTotal 6667.466667.46 2929

model 2: F tests the overall significance of the model (thus, exactly the same as if we had done a standard multiple regression)

test of R2

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tests of coefficients

1p,p −−=df

ModelModel BB SESE ββ tt sigsig

1 constant1 constant --80.23380.233 7.5957.595 7.0097.009 .000.000

ANXANX 5.2685.268 1.7001.700 .505.505 3.0093.009 .004.004

2 constant2 constant --95.0295.02 33 1448.681448.68 16.2316.23 .000.000

ANXANX 1.6781.678 1.4371.437 .16.16 1.1681.168 .253.253

STST .789.789 .208.208 .42.42 3.7853.785 .000.000

MOTMOT 1.4531.453 .484.484 .46.46 3.0003.000 .005.005

for model 1:coefficient for anxiety as sole predictor of GPA (i.e., the variable included at step 1)

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ModelModel BB SESE ββ tt sigsig

1 constant1 constant --80.23380.233 7.5957.595 7.0097.009 .000.000

ANXANX 5.2685.268 1.7001.700 .505.505 3.0093.009 .004.004

2 constant2 constant --95.0295.02 33 1448.681448.68 16.2316.23 .000.000

ANXANX 1.6781.678 1.4371.437 .16.16 1.1681.168 .253.253

STST .789.789 .208.208 .42.42 3.7853.785 .000.000

MOTMOT 1.4531.453 .484.484 .46.46 3.0003.000 .005.005

for model 2:identical to the coefficients table we would get in standard multiple regression if all predictors were entered simultaneously

tests of coefficients

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some uses for hierarchical multiple regression (HMR)

to account for control (nuisance) variablesto account for control (nuisance) variables–– logic is same as for ANCOVAlogic is same as for ANCOVA

to test mediation to test mediation to test moderated relationships to test moderated relationships (interactions)(interactions)

Ŷ = b1X1 + b2X2 + b3X1X2 + c

next lecture

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structure of SMR and HMR testsstructure of SMR and HMR tests

hierarchical MR:1.1. Tests overall Tests overall model R2 automatically2.2. Tests each Block separately (2+ sets of Fs)Tests each Block separately (2+ sets of Fs)3.3. Tests unique effect of each IV in its block (only control for Tests unique effect of each IV in its block (only control for

variables entered in its block and earlier variables entered in its block and earlier –– shared variance with shared variance with IVs IVs that haven’t been entered yet is given to early variables))

4.4. Does not test for interactions automatically Does not test for interactions automatically ––but can use HMR to test manually but can use HMR to test manually

standard MRstandard MR::1.1. Tests overall model RTests overall model R22 automaticallyautomatically2.2. Does not test groups of variables (blocks)Does not test groups of variables (blocks)3.3. Tests unique effect of each IV Tests unique effect of each IV 4.4. Does not test for interactions automaticallyDoes not test for interactions automatically

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reporting SMR and HMR testsreporting SMR and HMR testsreport for hierarchical MR:

•• each block each block RR22 change with change with FF testtest

•• IVs’ IVs’ ββs with s with pp--values from each values from each block as enteredblock as entered

•• final model final model RR22 with with FF testtest•• any relevant followany relevant follow--upsups

report for standard MRreport for standard MR::

•• Model Model RR22 with with FF testtest•• each IV’s each IV’s ββ with with ppvaluesvalues•• any relevant followany relevant follow--upsups

more about reporting hierarchical MR:more about reporting hierarchical MR:

•• depending on theory, you may or may not report betas for depending on theory, you may or may not report betas for IVs from earlier blocks again if they change in later blocksIVs from earlier blocks again if they change in later blocks

•• usually not if early block = controlusually not if early block = control•• definitely yes if mediation testdefinitely yes if mediation test

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• occurs when predictors are highly correlated (>.80 - 90)• diagnosed with high intercorrelations of IVs (collinearities)

and a statistic called tolerance• tolerance = (1 - R2

x)• RR22

xx is the overlap between a particular predictor and all the other predictors

• low tolerance = multicollinearity singularity • high tolerance = relatively independent predictors

multicollinearity leads to unstable calculation of regression coefficients (b), even though R2 may be significant

multicollinearity & singularitymulticollinearity & singularity

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assumptions of multiple regressionassumptions of multiple regressiondistribution of residualsdistribution of residuals–– conditional Y values are conditional Y values are normallynormally distributed around the distributed around the

regression lineregression line–– homoscedasticityhomoscedasticity:: variance of Y values are constant variance of Y values are constant

across different values of across different values of Ŷ (Ŷ (homogeneity of variance)homogeneity of variance)–– No No linearlinear relationship between relationship between Ŷ and errors of predictionŶ and errors of prediction–– independenceindependence of errorsof errors

scales (predictor and criterion scores)scales (predictor and criterion scores)–– variables are normally distributedvariables are normally distributed–– linear linear relationship between predictors and criterionrelationship between predictors and criterion–– predictors are predictors are not singularnot singular (extremely highly correlated)(extremely highly correlated)–– measured using a measured using a continuouscontinuous scale (interval or ratio)scale (interval or ratio)

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readingsreadingsStandard & Hierarchical Multiple Regression (this lecture)

Field (3rd ed): Chapter 7Field (2nd ed): Chapter 5Howell (all eds): Chapter 15

Moderated Multiple Regression (next lecture)Field (2nd or 3rd ed): no new readingsHowell (all eds): Chapter 15

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