psych (version 2.3.3)

# lmCor: Multiple Regression and Set Correlation from matrix or raw input

## Description

A generalization of the lm function to correlation/covariance matrix input. Given a correlation matrix or a matrix or dataframe of raw data, find the multiple regressions and draw a path diagram relating a set of y variables as a function of a set of x variables. A set of covariates (z) can be partialled from the x and y sets. Regression diagrams are automatically included. Model can be specified in conventional formula form, or in terms of x variables and y variables. Multiplicative models (interactions) and quadratic terms may be specified in the formula mode if using raw data. By default, the data may be zero centered before finding the interactions. Will also find Cohen's Set Correlation between a predictor set of variables (x) and a criterion set (y). Also finds the canonical correlations between the x and y sets. Associated functions allow for cross validation of these regression models.

## Usage

lmCor(y,x,data,z=NULL,n.obs=NULL,use="pairwise",std=TRUE,square=FALSE,
main="Regression Models",plot=TRUE,show=FALSE,zero=TRUE, alpha = .05,part=FALSE)
#the prior name
setCor(y,x,data,z=NULL,n.obs=NULL,use="pairwise",std=TRUE,square=FALSE,
main="Regression Models",plot=TRUE,show=FALSE,zero=TRUE, alpha = .05,part=FALSE)

setCor.diagram(sc,main="Regression model",digits=2,show=FALSE,cex=1,l.cex=1,...)
#an alias to setCor or lmCorset.cor(y,x,data,z=NULL,n.obs=NULL,use="pairwise",std=TRUE,square=FALSE,
main="Regression Models",plot=TRUE,show=FALSE,zero=TRUE,part=FALSE)

mat.regress(y, x,data, z=NULL,n.obs=NULL,use="pairwise",square=FALSE) #the old form#does not handle formula input
matReg(x,y,C,m=NULL,z=NULL,n.obs=0,means=NULL,std=FALSE,raw=TRUE,part=FALSE)
crossValidation(model,data,options=NULL,select=NULL)
crossValidationBoot(y,x,data,n.iter=100)
matPlot(x, type = "b", minlength=6, xlas=0,legend=NULL,
lab=NULL,pch=16,col=1:6,lty=NULL,...)

## Value

beta

the beta weights for each variable in X for each variable in Y

R

The multiple R for each equation (the amount of change a unit in the predictor set leads to in the criterion set).

R2

The multiple R2 (% variance acounted for) for each equation

VIF

The Variance Inflation Factor which is just 1/(1-smc(x))

se

Standard errors of beta weights (if n.obs is specified)

t

t value of beta weights (if n.obs is specified)

Probability

Probability of beta = 0 (if n.obs is specified)

shrunkenR2

Estimated shrunken R2 (if n.obs is specified)

setR2

The multiple R2 of the set correlation between the x and y sets

itemresidualThe residual correlation matrix of Y with x and z removed

ruw

The unit weighted multiple correlation for each dependent variable

Ruw

The unit weighted set correlation

## Arguments

y

Three options: 'formula' form (similar to lm) or either the column numbers of the y set (e.g., c(2,4,6) or the column names of the y set (e.g., c("Flags","Addition"). See notes and examples for each.

x

either the column numbers of the x set (e.g., c(1,3,5) or the column names of the x set (e.g. c("Cubes","PaperFormBoard"). x and y may also be set by use of the formula style of lm.

data

A matrix or data.frame of correlations or, if not square, of raw data

C

A variance/covariance matrix, or a correlation matrix

m

The column name or numbers of the set of mediating variables (see mediate).

z

the column names or numbers of the set of covariates.

n.obs

If specified, then confidence intervals, etc. are calculated, not needed if raw data are given.

n.iter

Number of bootstrap resamples to take in crossValidationBoot

use

find the correlations using "pairwise" (default) or just use "complete" cases (to match the lm function)

std

Report standardized betas (based upon the correlations) or raw bs (based upon covariances)

part

z is specified should part (TRUE) or partial (default) correlations be found.

raw

Are data from a correlation matrix or data matrix?

means

A vector of means for the data in matReg if giving matrix input

main

The title for lmCor.diagram

square

if FALSE, then square matrices are treated as correlation matrices not as data matrices. In the rare case that one has as many cases as variables, then set square=TRUE.

sc

The output of lmCor or setCor may be used for drawing diagrams

digits

How many digits should be displayed in the setCor.diagram?

show

Show the unweighted matrix correlation between the x and y sets?

zero

zero center the x variables before finding the interaction terms.

alpha

p value of the confidence intervals for the beta coefficients

plot

By default, setCor makes a plot of the results, set to FALSE to suppress the plot

cex

Text size of boxes displaying the variables in the diagram

l.cex

Text size of numbers in arrows, defaults to cex

...

model

The resulting object from lmCor or bestScales

options

If using a bestScales object, which set of keys to use ("best.key","weights","optimal.key","optimal.weights")

select

Not yet implemented option to crossValidation

type

type="b" draws both lines and points

xlas

Orientation of the x labels (3 to make vertical)

minlength

When abbreviating x labels how many characters as a minimum

legend

If not NULL, the draw a legend at the appropriate location

pch

What plot character(s) to use in matPlot

col

colors to use in matPlot

lty

line types to use in matPlot

lab

What labels should be supplied to the lines in matPlot. (Defaults to colnames of the variables.)

## Author

William Revelle

Maintainer: William Revelle <revelle@northwestern.edu>

## Details

Although it is more common to calculate multiple regression and canonical correlations from the raw data, it is, of course, possible to do so from a matrix of correlations or covariances. In this case, the input to the function is a square covariance or correlation matrix, as well as the column numbers (or names) of the x (predictor), y (criterion) variables, and if desired z (covariates). The function will find the correlations if given raw data.

Input is either conventional formula mode, or the set of y variables and the set of x variables. Formula mode can be written in the standard formula style of lm (see last example). In this case, pairwise or higher interactions (product terms) may also be specified. By default, when finding product terms, the predictive variables are zero centered (Cohen, Cohen, West and Aiken, 2003), although this option can be turned off (zero=FALSE) to match the results of lm or the results discussed in Hayes (2013).

Covariates to be removed are specified by a negative sign in the formula input or by using the z variable. Note that when specifying covariates, the regressions are done as if the regressions were done on the partialled variables. This means that the degrees of freedom and the R2 reflect the regressions of the partialled variables. (See the examples.)

If using covariates, should they be removed from the dependent as well as the independent variables (part = FALSE, the default which means partial correlations or just the independent variables (part=TRUE or part aka semi-partial correlations). The difference between part correlations and partial correlations is whether the variance of the covariate is removed from both the DV and IVs (a partial correlation) or from just the IVs (a part correlation). The slopes of the regressions remain the same, but the amount of variance in the DV (and thus the standard errors) is larger when using part correlations. Using partial correlations for the covariates is equivalent to using the covariate in the regression when interpreting the effects of the other variables.

The output is a set of multiple correlations, one for each dependent variable in the y set, as well as the set of canonical correlations.

An additional output is the R2 found using Cohen's set correlation (Cohen, 1982). This is a measure of how much variance and the x and y set share.

Cohen (1982) introduced the set correlation, a multivariate generalization of the multiple correlation to measure the overall relationship between two sets of variables. It is an application of canoncial correlation (Hotelling, 1936) and $$1 - \prod(1-\rho_i^2)$$ where $$\rho_i^2$$ is the squared canonical correlation. Set correlation is the amount of shared variance (R2) between two sets of variables. With the addition of a third, covariate set, set correlation will find multivariate R2, as well as partial and semi partial R2. (The semi and bipartial options are not yet implemented.) Details on set correlation may be found in Cohen (1982), Cohen (1988) and Cohen, Cohen, Aiken and West (2003).

R2 between two sets is just $$R^2 = 1- \frac{\left | R_{yx} \right |}{\left | R_y \right | \left |R_x\right |} = 1 - \prod(1-\rho_i^2)$$ where R is the complete correlation matrix of the x and y variables and Rx and Ry are the two sets involved.

Unfortunately, the R2 is sensitive to one of the canonical correlations being very high. An alternative, T2, is the proportion of additive variance and is the average of the squared canonicals. (Cohen et al., 2003), see also Cramer and Nicewander (1979). This average, because it includes some very small canonical correlations, will tend to be too small. Cohen et al. admonition is appropriate: "In the final analysis, however, analysts must be guided by their substantive and methodological conceptions of the problem at hand in their choice of a measure of association." ( p613).

Yet another measure of the association between two sets is just the simple, unweighted correlation between the two sets. That is, $$R_{uw} =\frac{ 1 R_{xy} 1' }{(1R_{yy}1')^{.5} (1R_{xx}1')^{.5}}$$ where Rxy is the matrix of correlations between the two sets. This is just the simple (unweighted) sums of the correlations in each matrix. This technique exemplifies the robust beauty of linear models and is particularly appropriate in the case of one dimension in both x and y, and will be a drastic underestimate in the case of items where the betas differ in sign.

When finding the unweighted correlations, as is done in alpha, items are flipped so that they all are positively signed.

A typical use in the SAPA project is to form item composites by clustering or factoring (see fa,ICLUST, principal), extract the clusters from these results (factor2cluster), and then form the composite correlation matrix using cluster.cor. The variables in this reduced matrix may then be used in multiple R procedures using lmCor.

Although the overall matrix can have missing correlations, the correlations in the subset of the matrix used for prediction must exist.

If the number of observations is entered, then the conventional confidence intervals, statistical significance, and shrinkage estimates are reported.

If the input is rectangular (not square), correlations or covariances are found from the data.

The print function reports t and p values for the beta weights, the summary function just reports the beta weights.

The Variance Inflation Factor is reported but should be taken with the normal cautions of interpretation discussed by Guide and Ketokivi. That is to say, VIF > 10 is not a magic cuttoff to define colinearity. It is merely 1/(1-smc(R(x)).

The Guide and Ketokivi article is well worth reading for all who want to use various regression models.

crossValidation can be used to take the results from lmCor or bestScales and apply the weights to a different data set.

crossValidationBoot will bootstrap resample data and perform a lmCor and then return the crossValidation R values. Although it will handle covariates (z), it will not handle product terms.

matPlot can be used to plot the crossValidation values (just a call to matplot with the xaxis given labels). matPlot has been improved to draw legends and to allow for specifications of the line types.

setCorLookup will sort the beta weights and report them with item contents if given a dictionary. matReg is primarily a helper function for mediate but is a general multiple regression function given a covariance matrix and the specified x, y and z variables. Its output includes betas, se, t, p and R2. The call includes m for mediation variables, but these are only used to adjust the degrees of freedom. matReg does not work on data matrices, nor does it take formula input. It is really just a helper function for mediate

## References

J. Cohen (1982) Set correlation as a general multivariate data-analytic method. Multivariate Behavioral Research, 17(3):301-341.

J. Cohen, P. Cohen, S.G. West, and L.S. Aiken. (2003) Applied multiple regression/correlation analysis for the behavioral sciences. L. Erlbaum Associates, Mahwah, N.J., 3rd ed edition.

H. Hotelling. (1936) Relations between two sets of variates. Biometrika 28(3/4):321-377.

E.Cramer and W. A. Nicewander (1979) Some symmetric, invariant measures of multivariate association. Psychometrika, 44:43-54.

V. Daniel R. Guide Jr. and M. Ketokivim (2015) Notes from the Editors: Redefining some methodological criteria for the journal. Journal of Operations Management. 37. v-viii.

mediate for an alternative regression model with 'mediation'. predict to find predicted values given regression weights. cluster.cor, factor2cluster,principal,ICLUST, link{cancor} and cca in the yacca package. GSBE for further demonstrations of mediation and moderation.

## Examples

Run this code
#First compare to lm using data input
summary(lm(rating ~ complaints + privileges, data = attitude))
lmCor(rating ~ complaints + privileges, data = attitude, std=FALSE) #do not standardize
z.attitude <- data.frame(scale(attitude))  #standardize the data before doing lm
summary(lm(rating ~ complaints + privileges, data = z.attitude))  #regressions on z scores
lmCor(rating ~ complaints + privileges, data = attitude)  #by default we standardize and
# the results are the same as the standardized lm

R <- cor(attitude) #find the correlations
#Do the regression on the correlations
#Note that these match the regressions on the standard scores of the data
lmCor(rating ~ complaints + privileges, data =R, n.obs=30)

#now, partial out learning and critical
lmCor(rating ~ complaints + privileges - learning - critical, data =R, n.obs=30)
#compare with the full regression:
lmCor(rating ~ complaints + privileges + learning + critical, data =R, n.obs=30)

#Canonical correlations:

#The first Kelley data set from Hotelling
kelley1 <- structure(c(1, 0.6328, 0.2412, 0.0586, 0.6328, 1, -0.0553, 0.0655,
0.2412, -0.0553, 1, 0.4248, 0.0586, 0.0655, 0.4248, 1), .Dim = c(4L,
"math.power")))
lowerMat(kelley1)
data = kelley1, n.obs=140)
mod1$cancor #Hotelling reports .3945 and .0688 we get 0.39450592 0.06884787 #the second Kelley data from Hotelling kelley <- structure(list(speed = c(1, 0.4248, 0.042, 0.0215, 0.0573), power = c(0.4248, 1, 0.1487, 0.2489, 0.2843), words = c(0.042, 0.1487, 1, 0.6693, 0.4662), symbols = c(0.0215, 0.2489, 0.6693, 1, 0.6915), meaningless = c(0.0573, 0.2843, 0.4662, 0.6915, 1)), .Names = c("speed", "power", "words", "symbols", "meaningless"), class = "data.frame", row.names = c("speed", "power", "words", "symbols", "meaningless")) lowerMat(kelley) lmCor(power + speed ~ words + symbols + meaningless,data=kelley) #formula mode #lmCor(y= 1:2,x = 3:5,data = kelley) #order of variables input #Hotelling reports canonical correlations of .3073 and .0583 or squared correlations of # 0.09443329 and 0.00339889 vs. our values of cancor = 0.3076 0.0593 with squared values #of 0.0946 0.0035, lmCor(y=c(7:9),x=c(1:6),data=Thurstone,n.obs=213) #easier to just list variable #locations if we have long names #now try partialling out some variables lmCor(y=c(7:9),x=c(1:3),z=c(4:6),data=Thurstone) #compare with the previous #compare complete print out with summary printing sc <- lmCor(SATV + SATQ ~ gender + education,data=sat.act) # regression from raw data sc summary(sc) lmCor(Pedigrees ~ Sentences + Vocabulary - First.Letters - Four.Letter.Words , data=Thurstone) #showing formula input with two covariates #Do some regressions with real data (rather than correlation matrices) lmCor(reaction ~ cond + pmi + import, data = Tal.Or) #partial out importance lmCor(reaction ~ cond + pmi - import, data = Tal.Or, main="Partial out importance") #compare with using lm by partialling mod1 <- lm(reaction ~ cond + pmi + import, data = Tal.Or) reaction.import <- lm(reaction~import,data=Tal.Or)$resid
cond.import <-  lm(cond~import,data=Tal.Or)$resid pmi.import <- lm(pmi~import,data=Tal.Or)$resid
mod.partial <- lm(reaction.import ~ cond.import + pmi.import)
summary(mod.partial)
#lm uses raw scores, so set std = FALSE for lmCor
print(lmCor(y = reaction ~ cond + pmi - import, data = Tal.Or,std = FALSE,
main = "Partial out importance"),digits=4)
#notice that the dfs of the partial approach using lm are 1 more than the lmCor dfs

#Show how to find quadratic terms
sc <- lmCor(reaction ~ cond + pmi + I(import^2), data = Tal.Or)
sc
#pairs.panels(sc\$data) #show the SPLOM of the data

#Consider an example of a derivation and cross validation sample
set.seed(42)
ss <- sample(1:2800,1400)
model <- lmCor(y=26:28,x=1:25,data=bfi[ss,],plot=FALSE)
original.fit <- crossValidation(model,bfi[ss,]) #the derivation set
cross.fit <- crossValidation(model,bfi[-ss,])  #the cross validation set
summary(original.fit)
summary(cross.fit)
predicted <- predict(model,bfi[-ss,])
cor2(predicted,bfi[-ss,26:28])  #these are the correlations of the predicted with observed

#now do it for the correlations matrices - these should be the same
R1 <- lowerCor(bfi[ss,], show=FALSE)
R2 <- lowerCor(bfi[-ss,], show=FALSE)
mod1 <- lmCor(y=26:28,x=1:25,data=R1,plot=FALSE)
original <-  crossValidation(model,R1) #the derivation set capitalizes on chance
cross <- crossValidation(model,R2)  #the cross validation set -- values are lower
summary(original)  #inflated by chance
summary(cross)    #cross validated values are better estimates


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