model=Y~XBoth Y and X can be matrices, in which case as many multiple regressions will be done as the number of variables in Y.
Consider the following (artificial) data set of ability tests, motivation tests, and performance measures. The Ability tests are (simulated) GRE Verbal, Quantitative, and Advanced, the motivation tests measure Need for Achievement and Anxiety, the Performance measures are graduate GPA, rated performance on the Prelims, and quality of the Masters Thesis.
Somehow we need to read the data and find the correlations.
datafilename="http://personality-project.org/R/datasets/psychometrics.prob2.txt"
dataset =read.table(datafilename,header=TRUE) #read the data file
names(dataset) #what are the variables?
dataset=dataset[,-1] #get rid of the ID
names(dataset) #check the names again
round(cor(dataset),2) #find the correlation matrix
dataset=scale(dataset) #convert to standardized scores
dataset=data.frame(dataset)
#gives this output
> datafilename="http://personality-project.org/R/datasets/psychometrics.prob2.txt"
> dataset =read.table(datafilename,header=TRUE) #read the data file
> names(dataset) #what are the variables?
[1] "ID" "GREV" "GREQ" "GREA" "Ach" "Anx" "Prelim" "GPA" "MA"
> dataset=dataset[,-1] #get rid of the ID
> names(dataset) #check the names again
[1] "GREV" "GREQ" "GREA" "Ach" "Anx" "Prelim" "GPA" "MA"
> round(cor(dataset),2) #find the correlation matrix
GREV GREQ GREA Ach Anx Prelim GPA MA
GREV 1.00 0.73 0.64 0.01 0.01 0.43 0.42 0.32
GREQ 0.73 1.00 0.60 0.01 0.01 0.38 0.37 0.29
GREA 0.64 0.60 1.00 0.45 -0.39 0.57 0.52 0.45
Ach 0.01 0.01 0.45 1.00 -0.56 0.30 0.28 0.26
Anx 0.01 0.01 -0.39 -0.56 1.00 -0.23 -0.22 -0.22
Prelim 0.43 0.38 0.57 0.30 -0.23 1.00 0.42 0.36
GPA 0.42 0.37 0.52 0.28 -0.22 0.42 1.00 0.31
MA 0.32 0.29 0.45 0.26 -0.22 0.36 0.31 1.00
> dataset=scale(dataset) #convert to standardized scores
> dataset=data.frame(dataset)
attach(dataset)
ability=cbind(GREV,GREQ,GREA) #form three different composites
motivation=cbind(Ach,Anx)
perform=cbind(Prelim,GPA,MA)
model.regress=lm(perform~ability+motivation)
model.regress #show the resulting regression
print(model.regress,digits=3) #another way of showing the result
summary(model.regress) #yet another way of showing the output
#produces this output
> attach(dataset)
ability=cbind(GREV,GREQ,GREA) #form three different composites
> motivation=cbind(Ach,Anx)
> perform=cbind(Prelim,GPA,MA)
> model=lm(perform~ability+motivation)
> model #show the resulting regression
Call:
lm(formula = perform ~ ability + motivation)
Coefficients:
Prelim GPA MA
(Intercept) -5.627e-16 1.786e-15 3.019e-15
abilityGREV 1.399e-01 1.987e-01 1.041e-01
abilityGREQ 3.944e-02 5.098e-02 2.555e-02
abilityGREA 4.041e-01 2.861e-01 3.111e-01
motivationAch 1.143e-01 1.190e-01 9.621e-02
motivationAnx -8.500e-03 -4.703e-02 -4.569e-02
> print(model,digits=3) #another way of showing the result
Call:
lm(formula = perform ~ ability + motivation)
Coefficients:
Prelim GPA MA
(Intercept) -5.63e-16 1.79e-15 3.02e-15
abilityGREV 1.40e-01 1.99e-01 1.04e-01
abilityGREQ 3.94e-02 5.10e-02 2.56e-02
abilityGREA 4.04e-01 2.86e-01 3.11e-01
motivationAch 1.14e-01 1.19e-01 9.62e-02
motivationAnx -8.50e-03 -4.70e-02 -4.57e-02
> summary(model) #yet another way of showing the output
Response Prelim :
Call:
lm(formula = Prelim ~ ability + motivation)
Residuals:
Min 1Q Median 3Q Max
-2.859621 -0.553213 -0.004029 0.503496 2.718487
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.627e-16 2.570e-02 -2.19e-14 1.000000
abilityGREV 1.399e-01 4.259e-02 3.283 0.001061 **
abilityGREQ 3.944e-02 3.954e-02 0.998 0.318722
abilityGREA 4.041e-01 4.539e-02 8.903 < 2e-16 ***
motivationAch 1.143e-01 3.441e-02 3.323 0.000925 ***
motivationAnx -8.500e-03 3.239e-02 -0.262 0.793050
---
Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Residual standard error: 0.8126 on 994 degrees of freedom
Multiple R-Squared: 0.3429, Adjusted R-squared: 0.3396
F-statistic: 103.8 on 5 and 994 DF, p-value: < 2.2e-16
Response GPA :
Call:
lm(formula = GPA ~ ability + motivation)
Residuals:
Min 1Q Median 3Q Max
-2.60408 -0.61369 0.02232 0.60416 2.53718
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.786e-15 2.665e-02 6.7e-14 1.000000
abilityGREV 1.987e-01 4.418e-02 4.497 7.72e-06 ***
abilityGREQ 5.098e-02 4.101e-02 1.243 0.214094
abilityGREA 2.861e-01 4.708e-02 6.078 1.73e-09 ***
motivationAch 1.190e-01 3.569e-02 3.334 0.000886 ***
motivationAnx -4.703e-02 3.360e-02 -1.400 0.161883
---
Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Residual standard error: 0.8429 on 994 degrees of freedom
Multiple R-Squared: 0.2931, Adjusted R-squared: 0.2895
F-statistic: 82.43 on 5 and 994 DF, p-value: < 2.2e-16
Response MA :
Call:
lm(formula = MA ~ ability + motivation)
Residuals:
Min 1Q Median 3Q Max
-2.66414 -0.61142 0.02422 0.61993 2.87736
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.019e-15 2.804e-02 1.08e-13 1.0000
abilityGREV 1.041e-01 4.648e-02 2.240 0.0253 *
abilityGREQ 2.555e-02 4.314e-02 0.592 0.5537
abilityGREA 3.111e-01 4.952e-02 6.282 4.98e-10 ***
motivationAch 9.621e-02 3.754e-02 2.563 0.0105 *
motivationAnx -4.569e-02 3.534e-02 -1.293 0.1964
---
Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Residual standard error: 0.8867 on 994 degrees of freedom
Multiple R-Squared: 0.2177, Adjusted R-squared: 0.2138
F-statistic: 55.32 on 5 and 994 DF, p-value: < 2.2e-16
These regressions were done with the raw data. This is very useful if we want to examine diagnostics for the regression, examine the residuals, look for badly fitted subjects, etc. But sometimes we want to do the regressions from the correlation matrix or covariance matrix without reverting to the raw data.
Using the same data set as before, we find the correlation matrix and then do some basic matrix algebra to do the multiple correlations.
r.matrix=cor(dataset) #find the correlations
round(r.matrix,2) #show themm to 2 decimal placeds
a.matrix=r.matrix[1:5,1:5] #the a.matrix are the predictors
b.matrix=r.matrix[1:5,6:8] #the b.matrix has the criteria
a.matrix #show the a.matrix
b.matrix #show the b.matrix
model.mat=solve(a.matrix,b.matrix) #solve the equation bY~aX
round(model.mat,2) #show the answer to 2 decimals
print(model.regression,digits=2) #compare this to the multiple regression using the raw data
#produces this output
> r.matrix=cor(dataset) #find the correlations
> round(r.matrix,2) #show themm to 2 decimal placeds
GREV GREQ GREA Ach Anx Prelim GPA MA
GREV 1.00 0.73 0.64 0.01 0.01 0.43 0.42 0.32
GREQ 0.73 1.00 0.60 0.01 0.01 0.38 0.37 0.29
GREA 0.64 0.60 1.00 0.45 -0.39 0.57 0.52 0.45
Ach 0.01 0.01 0.45 1.00 -0.56 0.30 0.28 0.26
Anx 0.01 0.01 -0.39 -0.56 1.00 -0.23 -0.22 -0.22
Prelim 0.43 0.38 0.57 0.30 -0.23 1.00 0.42 0.36
GPA 0.42 0.37 0.52 0.28 -0.22 0.42 1.00 0.31
MA 0.32 0.29 0.45 0.26 -0.22 0.36 0.31 1.00
> a.matrix=r.matrix[1:5,1:5] #the a.matrix are the predictors
> b.matrix=r.matrix[1:5,6:8] #the b.matrix has the criteria
> a.matrix #show the a.matrix
GREV GREQ GREA Ach Anx
GREV 1.000000000 0.728847119 0.6411517 0.005612846 0.010193145
GREQ 0.728847119 1.000000000 0.5963450 0.006846819 0.005469727
GREA 0.641151694 0.596345026 1.0000000 0.453464501 -0.389629680
Ach 0.005612846 0.006846819 0.4534645 1.000000000 -0.556186667
Anx 0.010193145 0.005469727 -0.3896297 -0.556186667 1.000000000
> b.matrix #show the b.matrix
Prelim GPA MA
GREV 0.4282415 0.4194686 0.3222975
GREQ 0.3830881 0.3669720 0.2873879
GREA 0.5724319 0.5162068 0.4545495
Ach 0.3033434 0.2763810 0.2634665
Anx -0.2278877 -0.2224047 -0.2192203
> model.mat=solve(a.matrix,b.matrix) #solve the equation bY~aX
> round(model.mat,2) #show the answer to 2 decimals
Prelim GPA MA
GREV 0.14 0.20 0.10
GREQ 0.04 0.05 0.03
GREA 0.40 0.29 0.31
Ach 0.11 0.12 0.10
Anx -0.01 -0.05 -0.05
> print(model.regression,digits=2) #compare this to the multiple regression using the raw data
Call:
lm(formula = perform ~ ability + motivation)
Coefficients:
Prelim GPA MA
(Intercept) -5.6e-16 1.8e-15 3.0e-15
abilityGREV 1.4e-01 2.0e-01 1.0e-01
abilityGREQ 3.9e-02 5.1e-02 2.6e-02
abilityGREA 4.0e-01 2.9e-01 3.1e-01
motivationAch 1.1e-01 1.2e-01 9.6e-02
motivationAnx -8.5e-03 -4.7e-02 -4.6e-02
Compare the results based upon the correlation matrix to that based upon the raw data. To find R^2, we need to multiply the beta weights times the zero order correlations