multiple linear regression ii & analysis of variance i

8/4/2019 Multiple Linear Regression II & Analysis of Variance I

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Lecture 9Survey Research & Design in Psychology

James Neill, 2011

Multiple Linear Regression II

& Analysis of Variance I


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Overview

1. Multiple Linear Regression II2. Analysis of Variance I


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Multiple LinearRegression II

Summary of MLR I

Partial correlations Residual analysis Interactions Analysis of change


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1. Howell (2009).Correlation & regression[Ch 9]

2. Howell (2009).Multiple regression[Ch 15; not 15.14 Logistic Regression]

3. Tabachnick & Fidell (2001). Standard & hierarchical regression inSPSS (includes example write-ups)

[Alternative chapter from eReserve]

Readings - MLR As perpreviouslecture


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Summary of MLR I

Check assumptions LOM, N,Normality, Linearity, Homoscedasticity,Collinearity, MVOs, Residuals

Choose type Standard, Hierarchical,Stepwise, Forward, Backward

Interpret Overall ( R 2, Changes in R 2 (ifhierarchical)), Coefficients (Standardised &unstandardised), Partial correlations

Equation If useful (e.g., is the studypredictive?)


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Partial correlations ( r p)

r p between X and Y after controlling for(partialling out) the influence of a 3rd variablefrom both X and Y.


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Partial correlations ( r p): Examples

Does years of marriage (IV 1)predict marital satisfaction (DV)

after number of children iscontrolled for (IV2)?

Does time management (IV 1)predict university studentsatisfaction (DV) after general lifesatisfaction is controlled for (IV

2

)?


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Partial correlations ( r p) in MLR

When interpreting MLR coefficients,compare the 0-order and partialcorrelations for each IV in each

model draw a Venn diagram Partial correlations will be equal to or

smaller than the 0-order correlations If a partial correlation is the same as

the 0-order correlation, then the IV

operates independently on the DV.


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Partial correlations ( r p) in MLR

To the extent that a partial correlationis smaller than the 0-ordercorrelation, then the IV's explanation

of the DV is shared with other IVs. An IV may have a sig. 0-order

correlation with the DV, but a non-sig. partial correlation. This wouldindicate that there is non-sig. uniquevariance explained by the IV.


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Semi-partial correlations ( sr 2)in MLR

The sr 2 indicate the %s of variance inthe DV which are uniquely explained

by each IV . In PASW, the sr s are labelled part.

You need to square these to get sr 2. For more info, see Allen and Bennett

(2008) p. 182


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Part & partial correlations in SPSSIn Linear Regression - Statistics dialog box,check Part and partial correlations


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Coefficientsa

-.521 -.460

-.325 -.178

Worry

Ignore the Problem

Model1

Zero-order Partial

Correlations

Dependent Variable: Psychological Distressa.

Multiple linear regression -Example


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.18 .32.46

.52

.34

Y

X1X2


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Residual analysis


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Residual analysis

Three key assumptions can be testedusing plots of residuals:1. Linearity : IVs are linearly related to DV2. Normality of residuals

3. Equal variances (Homoscedasticity)


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Residual analysis

Assumptions about residuals:

Random noise Sometimes positive, sometimesnegative but, on average, 0

Normally distributed about 0


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Residual analysis


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Residual analysis


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Residual analysisThe Normal P-P (Probability) Plot ofRegression Standardized

Residuals can be used toassess the assumption ofnormally distributedresiduals.If the points cluster

reasonably tightly along thediagonal line (as they dohere), the residuals arenormally distributed.Substantial deviations from

the diagonal may be causefor concern.Allen & Bennett, 2008, p. 183


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Residual analysis


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The Scatterplot of standardised residuals against standardisedpredicted values can be used to assess the assumptions ofnormality, linearity and homoscedasticity of residuals. Theabsence of any clear patterns in the spread of points indicates

that these assumptions are met.Allen & Bennett, 2008, p. 183


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Why the big fussabout residuals?

assumption violation

Type I error rate(i.e., more false positives)


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Why the big fussabout residuals?

Standard error formulae (which areused for confidence intervals and sig. tests)

work when residuals are well-behaved.

If the residuals dont meet

assumptions these formulae tend tounderestimate coefficient standarderrors giving overly optimistic p -

values and too narrow CIs .


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Interactions


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Interactions

Additivity refers to the assumptionthat the IVs act independently, i.e.,they do not interact.

However, there may also beinteraction effects - when themagnitude of the effect of one IV ona DV varies as a function of asecond IV.

Also known as a moderation effect .


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InteractionsPhysical exercise in naturalenvironments may providemultiplicative benefits in reducingstress e.g.,

Natural environment StressPhysical exercise Stress

Natural env. X Phys. ex. Stress


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Interactions

Y = b 1 x 1 + b 2 x 2 + b 12 x 12 + a + e b 12 is the product of the first two slopes

( b 1 x b 2 ) b 12 can be interpreted as the amount of

change in the slope of the regression ofY on b 1 when b 2 changes by one unit.


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Interactions Conduct Hierarchical MLR Step 1 :

Pseudoephedrine

Caffeine Step 2 : Pseudo x Caffeine (cross-product)

Examine R 2, to see whether theinteraction term explains additionalvariance above and beyond the direct

effects of Pseudo and Caffeine.


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Interactions

Possible effects of Pseudo and Caffeine onArousal: None

Pseudo only (incr./decr.) Caffeine only (incr./decr.) Pseudo + Caffeine (additive inc./dec.) Pseudo x Caffeine (synergistic inc./dec.) Pseudo x Caffeine (antagonistic inc./dec.)


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Interactions

Cross-product interaction terms maybe highly correlated (multicollinear) with the corresponding simple IVs,

creating problems with assessing therelative importance of main effectsand interaction effects.

An alternative approach is to runseparate regressions for each level ofthe interacting variable .


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Interactions SPSS example


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Analysis of Change


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Analysis of change

Example research question:In group-based mental healthinterventions, does the quality of

social support from groupmembers (IV1) and group leaders(IV2) explain changes inparticipants mental healthbetween the beginning and end ofthe intervention (DV)?


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Analysis of change

Hierarchical MLR DV = Mental health after the intervention

Step 1 IV1 = Mental health before the intervention

Step 2 IV2 = Support from group members IV3 = Support from group leader


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Analysis of change

Results of interest Change in R 2 how much

variance in change scores isexplained by the predictors

Regression coefficients for

predictors in step 2


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Summary (MLR II)

Partial correlationUnique variance explained by IVs;calculate and report sr 2.

Residual analysisA way to test key assumptions.


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Summary (MLR II)

InteractionsA way to model (rather than ignore)interactions between IVs.

Analysis of changeUse hierarchical MLR to partial outbaseline scores in Step 1 in order touse IVs in Step 2 to predict changesover time.


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Student questions

?


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ANOVA I

Analysing differences t -tests

One sample Independent Paired


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Howell (2010): Ch3 The Normal Distribution Ch4 Sampling Distributions and

Hypothesis Testing Ch7 Hypothesis Tests Applied to

Means

Readings Analysing differences


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Analysing differences

Correlations vs. differences Which difference test? Parametric vs. non-parametric


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Correlational vsdifference statistics

Correlation and regressiontechniques reflect the

strength of association Tests of differences reflect

differences in central tendency of variables between groupsand measures.


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In MLR we see the world asmade of covariation.

Everywhere we look, we seerelationships . In ANOVA we see the world as

made of differences.Everywhere we look we seedifferences .


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LR/MLR e.g.,What is the relationshipbetween gender and height inhumans?

t -test/ANOVA e.g.,What is the difference between the heights of human males and

females? Are the differences in a sample


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Are the differences in a samplegeneralisable to a population?

Whi h diff ? (2 )


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How many groups?(i.e. categories of IV)

More than 2 groups =ANOVA models

2 groups:Are the groups

independent ordependent?

Independent groups Dependent groups

1 group =one-sample t -test

Para DV =Independent samples t-test Para DV =

Paired samplest -test

Non-para DV =Mann-Whitney U Non-para DV =

Wilcoxon

Which difference test? (2 groups)

Parametric s


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Parametric vs.non-parametric statistics

Parametric statistics inferential test that assumes certain characteristicsare true of an underlying population,especially the shape of its distribution.

Non-parametric statistics inferential test that makes few or noassumptions about the populationfrom which observations were drawn(distribution-free tests).


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There is generally at least onenon-parametric equivalent test for

each type of parametric test. Non-parametric tests aregenerally used when

assumptions about the underlyingpopulation are questionable (e.g.,non-normality).

P t i


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Parametric statistics commonlyused for normally distributed intervalor ratio dependent variables.

Non-parametric statistics can beused to analyse DVs that are non-

normal or are nominal or ordinal. Non-parametric statistics are less powerful that parametric tests.


S when d I


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So, when do I use a non-parametric test?

Consider non-parametric tests when(any of the following):

Assumptions, like normality, havebeen violated.

Small number of observations ( N). DVs have nominal or ordinal levels

of measurement.

Some Commonly Used

Some commonly used parametric & non-parametric tests


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Some Commonly UsedParametric & Nonparametric

Tests

Compares groupsclassified by two

different factors

Friedman; 2 test of

independence

2-way ANOVA

Compares three ormore groupsKruskal-Wallis1-way ANOVA

Compares two relatedsamples

Wilcoxon matchedpairs signed-rankt test (paired)

Compares twoindependent samples

Mann-Whitney U;Wilcoxon rank-sum

t test(independent)

PurposeNon-parametricParametric

Some commonly used parametric & non parametric tests


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Why a t -test or ANOVA?

A t -test or ANOVA is used todetermine whether a sample ofscores are from the same population

as another sample of scores. These are inferential tools for

examining differences between

group means. Is the difference between two sample

means real or due to chance?

t


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t -tests One-sample

One group of participants, compared with fixed, pre-existing value (e.g.,population norms)

IndependentCompares mean scores on the samevariable across different populations

(groups) Paired

Same participants, with repeated

measures


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Use of t in t tests


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Use of t in t -tests t reflects the ratio of between group

variance to within group variance Is the t large enough that it is unlikely

that the two samples have come from

the same population? Decision: Is t larger than the critical

value for t ? (see t tables depends on critical and N)


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68%95%

99.7%

Ye good ol normal distribution

l l


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One-tail vs. two-tail tests Two-tailed test rejects null hypothesis if

obtained t -value is extreme is eitherdirection

One-tailed test rejects null hypothesis ifobtained t -value is extreme is onedirection (you choose too high or too low)

One-tailed tests are twice as powerful as two-tailed, but they are only focusedon identifying differences in onedirection.


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One sample t -test

Compare one group (a sample) with a fixed, pre-existing value(e.g., population norms)

Do uni students sleep less than therecommended amount?e.g., Given a sample of N = 190 uni studentswho sleep M = 7.5 hrs/day ( SD = 1.5), doesthis differ significantly from 8 hours hrs/day ( = .05)?

O l t t t


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One-sample t -test

I d d


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Independent groups t -test

Compares mean scores on thesame variable across differentpopulations (groups)

Do Americans vs. Non-Americansdiffer in their approval of Barack

Obama? Do males & females differ in theamount of sleep they get?


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Do males and females differ in


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Do males and females differ inin amount of sleep per night?

D l d f l diff i


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Group Statistics

1189 7.34 2.109 .061

1330 8.24 2.252 .062

gender_R Genderof respondent1 Male

2 Female

immrec immediaterecall-numbercorrect_wave 1

N Mean Std. DeviationStd. Error

Mean

Independent Samples Test

4.784 .029 -10.268 2517 .000 -.896 .087 -1.067 -.725

-10.306 2511.570 .000 -.896 .087 -1.066 -.725

F Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% Confidence

Interval of theDifference

t-test for Equality of Means

Do males and females differ inmemory recall?

Adolescents'


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Group Statistics

323 4.9995 .7565 4.209E-02168 4.9455 .7158 5.523E-02

Type of SchoolSingle SexCo-Educational

SSRN Mean Std. Deviation

Std. ErrorMean


.017 .897 .764 489 .445 5.401E-02 7.067E-02 -8.48E-02 .1929

.778 355.220 .437 5.401E-02 6.944E-02 -8.26E-02 .1906

Equal variancesassumedEqual variances

not assumed

SSRF Sig.

Levene's Test for

Equality of Variances



95% ConfidenceInterval of the

Difference


AdolescentsSame Sex Relations in

Single Sex vs. Co-Ed Schools

Adolescents'


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.017 .897 .764 489 .445 5.401E-02 7.067E-02 -8.48E-02 .1929

.778 355.220 .437 5.401E-02 6.944E-02 -8.26E-02 .1906

Equal variancesassumedEqual variancesnot assumed

SSRF Sig.

Levene's Test for

Equality of Variances




Difference


Group Statistics

327 4.5327 1.0627 5.877E-02172 3.9827 1.1543 8.801E-02

Type of SchoolSingle SexCo-Educational

OSRN Mean Std. Deviation

Std. ErrorMean

AdolescentsOpposite Sex Relations in

Single Sex vs. Co-Ed Schools

Independent samples t test


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Independent samples t -test 1-way ANOVA

Comparison b/w means of 2 independent sample variables = t -test(e.g., what is the difference in EducationalSatisfaction between male and female students?)

Comparison b/w means of 3+ independent sample variables = 1-way

ANOVA(e.g., what is the difference in EducationalSatisfaction between students enrolled in fourdifferent faculties?)


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Paired samples t -test

Same participants, with repeatedmeasures

Data is sampled within subjects Pre- vs. post- treatment ratings Different factors e.g., Voters

approval ratings of candidates X andY

Assumptions


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Assumptions(Paired samples t -test)

LOM: IV: Two measures from same participants(w/in subjects)

a variable measured on two occasions or two different variables measured on the same

occasion

DV: Continuous (Interval or ratio)

Normal distribution of differencescores (robust to violation with larger samples)

Independence of observations (oneparticipants score is not dependent on anothers score)

D i i h ff ?


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Does an intervention have an effect?

There was no significant difference between pretestand posttest scores ( t (19) = 1.78, p = .09).

Ad l t ' O it S


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Paired Samples Test

.7289 .9645 3.128E-02 .6675 .7903 23.305 950 .000SSR - OSRPair 1Mean Std. Deviation

Std. ErrorMean Lower Upper


Difference

Paired Differences

t df Sig. (2-tailed)

Paired Samples Statistics

4.9787 951 .7560 2.451E-024.2498 951 1.1086 3.595E-02

SSROSR

Pair1

Mean N Std. DeviationStd. Error

Mean

Adolescents' Opposite Sexvs. Same Sex Relations


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Paired samples t -test

1-way repeated measures ANOVA Comparison b/w means of 2 within

subject variables = t -test Comparison b/w means of 3+ within

subject variables = 1-way ANOVA(e.g., what is the difference in Campus, Social,and Education Satisfaction?)

Summary


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Su a y(Analysing Differences)

Non-parametric and parametrictests can be used for examiningdifferences between the centraltendency of two of more variables

Develop a conceptualisation of

when to each of the parametrictests from one-sample t -testthrough to MANOVA (e.g. decisionchart).

Summary


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Summary(Analysing Differences)

t -tests One-sample

Independent-samples Paired samples

What will be covered in


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What will be covered inANOVA II?

1. 1-way ANOVA2. 1-way repeated measures ANOVA

3. Factorial ANOVA4. Mixed design ANOVA5. ANCOVA6. MANOVA7. Repeated measures MANOVA


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References

Allen, P. & Bennett, K. (2008). SPSS for thehealth and behavioural sciences . SouthMelbourne, Victoria, Australia: Thomson.

Francis, G. (2007). Introduction to SPSS for Windows: v. 15.0 and 14.0 with Notes for Studentware (5th ed.). Sydney: PearsonEducation.

Howell, D. C. (2010). Statistical methods for psychology (7th ed.). Belmont, CA:Wadsworth.

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multiple linear regression ii & analysis of variance i

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