Multiple Regression Extension of Simple Linear Regression –
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Transcript of Multiple Regression Extension of Simple Linear Regression –
Multiple Regression
Extension of Simple Linear Regression –
using multiple predictors
each predictor could help predict or explain additional variability in the response/criterion variable
However:What should be the effect of using any additional
predictors?
Multiple Regression
What should be the effect of using additional
predictors?
Logically, unless correlation with DV is 0,
each predictor will improve prediction (explain additional variance in DV)
So just adding variables as predictors at
random will usually “improve” model
Creates potential for misuse of the strategy
IDEALLY
each predictor should be
- correlated with the DV
- uncorrelated with other predictors(r over .8 undesirable)
each predictor should explain some
unique variability in DV
each predictor should make sense!
Best situation CLEAR THEORY or LOGIC determines the predictors selected
Examples
Relationship Commitmentsatisfaction with outcomes (+)investments in relationship (+)attractiveness of available alternatives (-)
Job Satisfactionsalaryphysical conditionssocial conditions
Simple linear regression
Yp = a + bX (+ residuals)
Multiple regression
Yp = a + b1X1 + b2X2 (+ residuals)
a is value of Y when all X = 0 (regression constant)
b’s are ‘partial regression coefficients’
slope for each predictor when
other predictors held constant
Graph of relationship when two predictors are used
Now try to fit a plane rather than a line – to minimize the errors of prediction
Multiple regression
Yp = a + b1X1 + b2X2 + b3X3 (+ residuals)
Commitment = a +
b1 (satisfaction) +
b2 (investments) +
- b3 (alternatives)
(+ residuals)
A weighted linear combination of predictorscomparison to ANOVA – main effects only model
• Let’s return to the question of predicting Exam 2 grades – using multiple predictors
• Undergraduate GPA (0-4 scale)• GRE Verbal (200-800 scale)• GRE Quantitative (200-800 scale)• Exam 1 grade (0-100 scale)• Mean Homework grade (0-10 scale)
Note variety of scales for predictors, weights (partial regression coefficients) will be variable to take those into account
Ideally, all predictors are related to the criterion, and are unrelated to each other
Variables Entered/Removedb
homework, grev, gpatot, greq, exam1a . EnterModel1
Variables EnteredVariablesRemoved Method
All requested variables entered.a.
Dependent Variable: exam2b.
Model Summary
.735a .540 .518 4.61879Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), homework, grev, gpatot, greq,exam1
a.
ANOVAb
2609.715 5 521.943 24.466 .000a
2218.658 104 21.333
4828.373 109
Regression
Residual
Total
Model1
Sum ofSquares df Mean Square F Sig.
Predictors: (Constant), homework, grev, gpatot, greq, exam1a.
Dependent Variable: exam2b.
Coefficientsa
16.059 8.406 1.911 .059
.192 1.226 .011 .157 .876 .127 .015 .010
-7.4E-005 .006 -.001 -.011 .991 .137 -.001 -.001
.001 .007 .006 .086 .932 .078 .008 .006
.440 .067 .500 6.538 .000 .637 .540 .435
3.673 .681 .390 5.394 .000 .564 .468 .359
(Constant)
gpatot
grev
greq
exam1
homework
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig. Zero-order Partial Part
Correlations
Dependent Variable: exam2a.
Exam 2 (Pred) = 16.06 + .19 (gpa) - .00(grev) + .00(greq) +.44(exam1) +3.67(homework)
Using just Exam 1 score, the correlation between Exam 1 and Exam 2 was r = .637, r2 = .406
Now the R, between the set of predictors and Exam 2 is .735,
and R2 = .540
Since gpatot, grev, greq were all not significant, should they be excluded from the equation?
Assumptions - never likely to satisfy all
Essentially same as for r, but at a multivariate level
Independent ObservationsInterval/Ratio Data – or at least pretend
Normality – all Predictors (X’s) and Response (Y) - errors of prediction are normally
distributedLinearity
all X’s have linear relationship with Y- errors of prediction/predicted scores are
linearEquality of Variances (Homoscedasticity)
- variability of errors of Y are same at all values of X
Assumptions can be evaluated within SPSS at the multivariate level.
In the Regression window, choose Plots and request zresid (Y) and zpred (X).
The tables at the right demonstrate the patterns that would indicate each violation. Although deciding when there is ‘enough’ discrepancy is still subjective.
From Tabachnick & Fidell (2007). Using multivariate statistics (5th). Boston: Allyn & Bacon. Example to follow
Predicting Rated Distress : (1) none to Extreme (9) - when partner is emotionally unfaithful using Age and Rated Distress over Sexual Infidelity as predictors. All 3 variables are skewed.
Other Considerations in Multiple Regression -- Truncated Range – same as with r, can lead
to poor assessment of ‘real’ R
-- Outliers due to multivariate deviation
Discrepancy (distance) –outlier on criterion
Leverage – outlier on predictors
Influence – combines D & L to assess influence on solution (change in regression coefficients if case
deleted)
How these would appear in a simple linear regression situation
From Tabachnick & Fidell (2007). Using multivariate statistics (5th). Boston: Allyn & Bacon.
Other Considerations in Multiple Regression
-- Outliers due to multivariate deviation
Simple diagnostic for Influence is to request Cook’s Distance statistic in Regression window, Save option. Values over 1 would suggest potentially strong influence.
Linear Regression
0.00 10.00 20.00 30.00
years
2.00
4.00
6.00
pu
bs
Cook’s distance = 92.6Note that residual for the outlier is not great, but it has strong influence on solution
Linear Regression
0.00 10.00 20.00 30.00
years
2.00
4.00
6.00
8.00
pu
bs
Line with outlier
Line without outlier
Other Considerations in Multiple Regression
– Sample Size – if too small may get good, but meaningless prediction – too little variability
Minimum sample sizes recommended (to detect moderate effect sizes, 13%
with power of approximately .80)
(Green, S. B. (1991) How many subjects does it take to do a regression analysis?
Multivariate Behavioral Research, 26, 499-510.)
• For test of a model n= 50 +8p• For test of individual predictors in model 104 + p
» p = number of predictors
Can also conduct a power analysis based on the effect size you desire to select your sample size
Other Considerations in Multiple Regression
– Multicollinearity or Singularity
• Singularity – when one predictor is a combination of other predictors included
• Multicollinearity - when other predictors can account for a high degree of variability in a predictor
Other Considerations in Multiple Regression
– Diagnostics for Multicollinearity or Singularity
Tolerance is used as diagnostic statistic
If other predictors used to predict a predictor, what variance is shared?
But reported as 1-R2, so closer to 1 is better -
less than .2 indicates a problem
Variance Inflation Factor (VIF) also used.
It is the reciprocal of Tolerance, so can range from 1 up. Reflects degree to which standard error of b is increased due to correlations among predictors.
value of 4 cause for some concern
value of 10 serious problem
Assessing the Outcome
Testing the Overall Model as a single outcome
How well do the set of predictors (X’s) predict the criterion (Y)
Ho: all b’s are = 0, all partial regression coefficients = 0 Or
Ho: R = 0, the Multiple Correlation Coefficient = 0
R = correlation of actual Y with weighted linear combination of predictors (X’s)
Or – since weighted linear combination leads to predicted scores
R = correlation of actual Y with predicted Yp
Reminder: Partitioning the Variability in Y
SSTotal = Sum (Y - Mean Y)2
variability of Y scores from the mean
Separated into
SSregression = Sum (Yp – Mean Y) 2 Improvement in predictions when using X (variability in Y explained by X),
rather than assuming everyone gets the Mean
SSresidual = Sum (Y - Yp) 2
Degree to which predictions do not match the actual scores
(prediction errors that have been minimized)
Linear Regression
2.50 3.00 3.50
gpa
90.00
100.00
110.00
120.00
iq
iq = 53.05 + 16.99 * gpaR-Square = 0.69
Mean IQ = 105
This would be your best ‘guess’ for every person if you had no useful predictor
Improvement in Prediction using GPA
Residual – distance from the
prediction line
Example from Simple Linear Regression
Residual much greater here
Mean GPA = 3.06
Test using F – similar to simple linear regression
Partition SST into • SSregression (explained by weighted combination)
• SSresidual (unexplained)
F = SSregression /df regression df = (p + a) - 1 SSresidual / df residual df = n – p - 1
F = MSregression = explained (systematic+unsystematic)
MSresidual unexplained (unsystematic)
Was R reliably different from 0 ? Yes, if F is significant
Recall: Standard Error of the Estimate = SQRT (MS residual)
Number of parameters in model (predictors + intercept) – df often indicated as p, since always only one a (df = p +1 -1)
R2 = SSregression = explained variability SST total variability
% of variance accounted for by the model(see next slide for ANOVA example)
Adjusted R2 for better estimate for population, adjusted based on number of predictors and sample size
Adjusted R2 = 1- ((1-R2) (n-1/n-p-1)) so lower if small sample, but many
predictors
Can use R2 for describing a sample
Tests of Between-Subjects Effects
Dependent Variable: Sensitive
83.200a 7 11.886 3.290 .003 .132
6451.600 1 6451.600 1785.585 .000 .922
14.400 1 14.400 3.985 .048 .026
44.600 3 14.867 4.115 .008 .075
24.200 3 8.067 2.233 .087 .042
549.200 152 3.613
7084.000 160
632.400 159
SourceCorrected Model
Intercept
GENDER
RELATE
GENDER * RELATE
Error
Total
Corrected Total
Type IIISum ofSquares df
MeanSquare F Sig.
EtaSquared
R Squared = .132 (Adjusted R Squared = .092)a.
Example from Handout Packet, Page approx. 47
Test of model in which there are 3 predictors used to predict the rating on “Sensitive”, the DV
Partial
In some cases, the purpose of the regression analysis is simply to see if the Model “works”.
Does it explain variance in the criterion? Can it be used to make predictions?
Thus, the overall test of the model is all you need, and you can interpret the R2 or R2
adj
and the SEE, if plan predictions
In other cases, you might want to know how the individual predictors contributed to the overall model.
Assessing the contribution of individual predictors
Dependent upon the set of predictors included!
Partial regression coefficient– can test to see if b = 0
Is b = 0 (slope = 0) when other predictors are held constant
Tested using a t test with df = n – p – 1
Beta – partial regression coefficient when all variables are standardized (standardized slope) If b is significant, so is beta
Test of Partial Regression Coefficient is like a typical statistical test of significance - it is or is not significant, and is influenced by sample size
Can also evaluate predictors based on “effect size” measures (practical significance)
These would be “significant” if b is significant
Partial correlation (pr) – as described in simple covariation section
correlation of predictor (X1) with DV (Y) after removing the variance in both explained by the other predictors
So both X1 and Y are adjusted before correlation is calculated
All other X’s are ‘partialed’ out of X1 and Y
pr2 – shared variance within context – what % of variability in Y does X1 explain after other
variables’ contributions to explaining both are removed
there is less than 100% of variability of Y left for X1 explain
Semi-partial (part) correlation (sr) –
correlation of predictor (X1) with DV (Y) after removing the variance of X1 shared with the other predictors
So X1 is adjusted by removing variance shared with other X’s
But all variability in Y is left to be explained
Assesses ‘unique’ contribution of X1 to explaining Y
There is 100% of variability in Y to explain for each X in model
sr2 is considered best measure of individual predictor importance (practical significance)
R2 will be lowered by sr2 for predictor when it is removed from model
(BOTH pr and sr ARE STILL DEPENDENT ON THE MODEL USED)
WHY?
Variability of DV (Y)
IV 1
IV 2
Shared variabilityDV and X1
Shared variabilityIV 1 and X2
Shared variabilityDV and X2
a
b
cd
Partial correlation (X1) = a/(a + d)
Semi-partial correlation (X1) = a/(a + b + c + d)
Types of Multiple Regression
Standard – all predictors entered together
contribution of each depends on others in the group
Assumes other variables would usually be there and/or are relevant
Four Humor StylesInvestment Model VariablesBig Five Personality Dimensions
Hierarchical Regression - enter in planned sequence
Can enter individual predictors one at a time
Or enter groups of variables at separate steps
As new predictors are added, each one can only explain variability that is left
Assess change in R2 at each step (increase significantly)and overall model when done
Predicting adult IQParental IQPrenatal experienceEarly infant experienceEducation
Statistical Methods –
Let the data determine inclusion in the model, not based on a logical or theoretical ‘plan’
Assess each step by evaluating change in R or R2
Usually an exploratory tool in possible model building
Requires a larger sample to have confidence
(40 cases per predictor)
Stepwise
Begins with single best predictor
Adds next best, and assesses if model is better
At each step, each variable is reassessed, and might be kept or removed
Stops when adding additional variables does not significantly improve model (R)
Forward inclusion
Begins with single best predictor
Adds next best, and assesses improvement
Once in, stay in, but only stay if improved model (R)
Backward exclusion
Begins with full model
Removes weakest contributor and assesses loss
Keeps removing unless significant drop in R
Research questions using Multiple Regression
Assess Overall Model
Assess individual predictors
Effects of adding or changing predictors
on overall model
on other individual predictors
Predictions in new sample
Other Multiple Regression issues/applications
Suppressor variables –
variables that improve the model due to correlations with other predictors, not criterion. They ‘suppress’ variance in another predictor that is ‘noise’
evident if simple r with criterion very low but contributes to model (sr is higher) can also produce a change in sign from r to b (i.e. positive r but negative b)
Other issues/applications
Mediation Models
Relationship of X to Y is mediated by some other variable
Positive use of Humor for self Perceived Stress
Positive Personality (optimistic, hopeful, happy)
Positive use of Humor for self Perceived Stress(High self-enhancing/Low self-defeating)
Humor use (H) predicts Perceived Stress (c) - the direct path
Humor use predicts Positive Personality (PP) (a)
Positive Personality predicts Perceived Stress, with Humor in model (b)
In Hierarchical Model, enter PP first, then H, if PP mediates H, H no longer ‘contributes’ to the model – the c’ path, indirect, not significant
a b
C’
C
Other issues/applications
Moderator Models – relationship of predictor with the criterion depends upon some other variable
(just like an interaction in ANOVA)
Yp = a + b1x1 + b2x2 + b3(x1x2) + residuals
Often requires some modifications of the data prior to the analysis
- centering variables to avoid multicollinearity (if predictors do not have true 0 scores)
Interaction term added to equation
Main effects
Best situation
CLEAR THEORY to be tested
Relationship Commitment (low 8 – 72 high)
satisfaction with outcomes (+) (low 3 – 21 high)
investments in relationship (+)
attractiveness of available alternatives (-) ( low 6 – high 48)
(Subj low 6 – 54 high)
(Obj none 0 - ?? Lots)
In Handout Packet
Begin by examining the individual variables for normality and outliers etc.
Can request Cook’s D to assess for outlier influence
Can check assumptions using plot from regression analysis
Scatterplot
Dependent Variable: Global Commitment
Regression Standardized Predicted Value
Regression Standardized Residual
Then look at the simple correlations (r)Expect predictors to correlate with criterion, but not a lot with each other
Correlations
1.000 .310 -.422 .395 .551
.310 1.000 -.237 .157 .339
-.422 -.237 1.000 -.257 -.440
.395 .157 -.257 1.000 .408
.551 .339 -.440 .408 1.000
. .003 .000 .000 .000
.003 . .020 .089 .001
.000 .020 . .013 .000
.000 .089 .013 . .000
.000 .001 .000 .000 .
75 75 75 75 75
75 75 75 75 75
75 75 75 75 75
75 75 75 75 75
75 75 75 75 75
Global Commitment
Global Satisfaction
Global alternatives
Objective Investments
Subjective Investments
Global Commitment
Global Satisfaction
Global alternatives
Objective Investments
Subjective Investments
Global Commitment
Global Satisfaction
Global alternatives
Objective Investments
Subjective Investments
Pearson Correlation
Sig. (1-tailed)
N
GlobalCommitment
GlobalSatisfaction
Globalalternatives
ObjectiveInvestments
SubjectiveInvestments
Check to see how well the model workedR and R2, and test of significanceStandard error of the estimate
ANOVAb
1751.967 4 437.992 10.889 .000a
2815.713 70 40.224
4567.680 74
Regression
Residual
Total
Model1
Sum ofSquares df
MeanSquare F Sig.
Predictors: (Constant), Subjective Investments, Global Satisfaction, ObjectiveInvestments, Global alternatives
a.
Dependent Variable: Global Commitmentb.
Model Summary
.619a .384 .348 6.34228Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), Global alternatives, GlobalSatisfaction, Objective Investments, SubjectiveInvestments
a.
To describe sample
To generalize to population
Typical residual
Coefficientsa
19.247 6.372 3.021 .004 6.539 31.955
.247 .214 .116 1.153 .253 -.180 .674 .310 .137 .108 .875 1.143
-.179 .098 -.192 -1.823 .073 -.376 .017 -.422 -.213 -.171 .791 1.264
3.390E-02 .019 .183 1.773 .081 -.004 .072 .395 .207 .166 .826 1.211
.346 .113 .353 3.071 .003 .121 .571 .551 .345 .288 .668 1.496
(Constant)
Global Satisfaction
Global alternatives
Objective Investments
Subjective Investments
Model1
B Std. Error
UnstandardizedCoefficients
Beta
Standardized
Coefficients
t Sig.LowerBound
UpperBound
95% Confidence Intervalfor B
Zero-order Partial Part
Correlations
Tolerance VIF
Collinearity Statistics
Dependent Variable: Global Commitmenta.
95
Now can look at individual predictors
check collinearity
see which predictors are individually significant
look at individual contributions (semi-partial or part r2)
• Go through example in SPSS
• Look at G*Power
• Stepwise example in Handouts