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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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Correlational vsdifference statistics
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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Correlational vsdifference statistics
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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Parametric vs.non-parametric statistics
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.
Parametric vs.non-parametric statistics
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
Independent Samples Test
.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
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
AdolescentsSame Sex Relations in
Single Sex vs. Co-Ed Schools
Adolescents'
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Independent Samples Test
.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
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
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
95% ConfidenceInterval of the
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.
Open Office Impress
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Open Office Impress This presentation was made using
Open Office Impress. Free and open source software. http://www.openoffice.org/product/impress.html
http://www.openoffice.org/product/impress.htmlhttp://www.openoffice.org/product/impress.html