In order to analyse the effect of bipolar vs. unipolar scales in the measurement of emotion we generated multiple sets of artificial data with known structure. Structural analyses of these data sets were done using exploratory factor analysis and the Very Simple Structure criterion (Revelle and Rocklin, 1979) to determine the most interpretable number of factors. These analyses were all done in the public domain statistical and data handling computer language R (R Development Core Team, 2004). To facilitate others who want to replicate or extend our analyses, we include in this appendix the original R code for all analyses reported.

The code in this appendix may be directly copied into R and executed. Electronic copies of the code in this appendix are also available at http://personality-project.org/r/measuringemotion.html . Very Simple Structure (Revelle and Rocklin, 1979) has been adapted for R and is available at http://personality-project.org/r/r.vss.html. 


#еееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееее 
#R Control sequence to examine the effect of skew in the measurement of affect
#
#Three functions are defined with default values:
#   simulate.items (nvar = 72 ,nsub = 500,	circum = TRUE, avloading =.6,  xbias=0,  ybias = -1) 
#   categorical.item <-function (item) 
#   truncate.item <- function(item,cutpoint=0)  
#
these functions are then called in a loop varying the number of subjects, the average loading, and type of structure
# output is evaluated in terms of the Very Simple Structure Criterion and chi square goodness of fit
#
#еееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееее 
#
#
simulate.items <- function (nvar = 72 ,nsub = 500,	circum = TRUE, avloading =.6,  xbias=0,  ybias = -1) 
#
# a function to generate  items with a particular structure
#True scores of items are assumed to be bivariate normal, bipolar and to lie in a two dimensional space
#Items loadings have either a simple structure (case 1) or to be uniformly distributed around a two dimensional space (case 2)
#Free parameters to be specified are:
#nvar		Number of variables    
#nsub		Number of subjects
#Circum		Simple Structure or Circumplex structure
#avloading	Average item reliability
#Amount of offset from 0 in the X and Y direction (xbias and ybias )
#
#default values are included in the function definition, other values may be specified when run
#


{ #begin function 
#
trueweight<- sqrt(avloading)    				#<---squared weights sum to 1 =>  true weight is sqrt(of reliability)			
errorweight <- sqrt(1-trueweight*trueweight)    #squared errors and true score weights add to 1

truex <- rnorm(nsub)  +xbias  			#generate normal true scores for x + xbias
truey <- rnorm(nsub)  + ybias			#generate normal true scores for y + ybias

if (circum) 
	{radia <- seq(0,2*pi,len=nvar+1)  #make a vector of radians (the whole way around the circle) if circumplex  
	rad <- radia[which(radia<2*pi)]                 #get rid of the last one
	} else rad <- rep(seq(0,3*pi/2,len=4),nvar/4)  #simple structure 
	
error<- matrix(rnorm(nsub*(nvar)),nsub)  					#create normal error scores
trueitem <- outer(truex, cos(rad)) + outer(truey,sin(rad))  #true score matrix for each item reflects structure in rad

item<- trueweight* trueitem  + errorweight*error    #observed item = true score + error score 
                                                    
return (item)  }   #the value of the function is the item matrix, ready for further analysis

#
#
# еееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееее
#
# Function to convert from continous variables to discrete (-3 <-> 3 ) categorical variables

categorical.item <-function (item) 
    {item = round(item)          #round all items to nearest integer value
	item[(item<=-3)] <- -3 	 #items < 3 become -3	
	item[(item>3) ] <-  3      #items >3 become 3 
	return(item) }             #the function returns these categorical items  and quits 
	
##	Function to convert a bipolar scale into a unipolar scale (i.e., to throw away information below a cutpoint)

truncate.item <- function(item,cutpoint=0)                      #truncate values less than cutpoint to zero
	{
		item[item < cutpoint] <- 0                               #item values < 0 are truncated  to zero (remove to not truncate) 
		return(item)
	}  
	
	
# еееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееееее
# simulate multiple sample sizes and examine the effect of unipolar vs. bipolar categorical items

samplesize=c(200,400,800,3200)                   #examine the effect of four sample sizes

nvar=18                                           #examine the effect of the number of variables
#vss.none <- list()                                #results will be stored here
vss.18 <- list()
for (i in 1:4)                                    #generate four data sets of varying sample size 
{ items=simulate.items(nsub=samplesize[i])
	catitem=categorical.item(items)
 	truncitem=truncate.item(catitem)
	#vss.none[[i]]=list(VSS(truncitem))             #examine the VSS criterion for the unrotated solution
	vss.18[[i]] <- list(VSS(truncitem,rotate="varimax"))  #examine VSS for the Varimax rotated solution
}

nvar=72
#vss.none <- list()                                #results will be stored here
vss.72 <- list()
for (i in 1:4)                                    #generate four data sets of varying sample size 
{ items=simulate.items(nsub=samplesize[i])
	catitem=categorical.item(items)
 	truncitem=truncate.item(catitem)
	#vss.none[[i]]=list(VSS(truncitem))             #examine the VSS criterion for the unrotated solution
	vss.18[[i]] <- list(VSS(truncitem,rotate="varimax"))  #examine VSS for the Varimax rotated solution
}
	
                   
nvar=72                          					 #results will be stored here
vss.bipolar <- list()                            #compare to bipolar scales

for (i in 1:4)                                    #generate four data sets of varying sample size 
{ items=simulate.items(nsub=samplesize[i])
	catitem=categorical.item(items)
 	#truncitem=truncate.item(catitem)
	#vss.none[[i]]=list(VSS(truncitem))             #examine the VSS criterion for the unrotated solution
	vss.bipolar[[i]] <- list(VSS(catitem,rotate="varimax"))  #examine VSS for the Varimax rotated solution
}


#now generate multiple plots 

plot.new()                   #set up a new plot page
par(mfrow=c(2,4))            #2 rows and 4 columns allow us to compare results
for (i in 1:4)                #for the 4 sample sizes show the VSS plots 

{ x=as.data.frame(vss.bipolar[[i]])
plotVSS(x,paste("N= ",samplesize[i], " 72 bipoloar variables")) }
for (i in 1:4) 
{ x=as.data.frame(vss.72[[i]])
plotVSS(x,paste("N= ",samplesize[i], " 72 unipolar variables")) }
vss.72bipolar=recordPlot()            #bipolar vs. unipolar VSS


plot.new()                   #set up a new plot page
par(mfrow=c(2,4))            #2 rows and 4 columns allow us to compare results
for (i in 1:4)                #for the 4 sample sizes show the VSS plots 

{ x=as.data.frame(vss.18[[i]])
plotVSS(x,paste("N= ",samplesize[i], " 18 unipoloar variables")) }
for (i in 1:4) 
{ x=as.data.frame(vss.72[[i]])
plotVSS(x,paste("N= ",samplesize[i], " 72 unipolar variables")) }
vss.18v72 = recordPlot()              #18 variable vs. 72 variable

The top four panels in Figure Appendix 1 shows the Very Simple Structure criterion applied for four sample sizes (N=200,400,800, and 3,200) to 72 bipolar items in a circumplex structure. The bottom four panels display the VSS criterion for 72 unipolar items also in a circumplex structure. Two things to note: as expected given that the bipolar data were generated to fit the circumplex, they do, and a two dimensional fits very well. In terms of conventional goodness of fit measures, the unipolar data require more than two dimensions, but the VSS criterion clearly indicates that a 2 dimensional solution is most interpretable if every item is thought to load on only one or at most two factors.

Figure 2 shows the Very Simple Structure criterion applied to four sample sizes (N = 200, 400 800, 3,200) for 18 versus 72 uni-polar variables in a circumplex structure.