Supplementary Material for Variants of Psychopathy
1
Supplementary Material
Variants of Psychopathy in Adult Male Offenders:
A Latent Profile Analysis
Andreas Mokros, Robert D. Hare, Craig S. Neumann, Pekka Santtila,
Elmar Habermeyer, and Joachim Nitschke
MPLUS
The following code is a script for Mplus (Muthén & Muthén, Los Angeles, CA) to
compute the latent class allocation for new data. Note that the order of latent classes in
the script differs from the ordering in the article (i.e., C#1 = LC2, C#2 = LC3, C#3 =
LC1).
TITLE:
This is an LCA with continuous latent class indicators using user-specified
parameter values without random starts.
DATA:
FILE IS input.dat;
VARIABLE: NAMES ARE f1 f2 f3 f4;
USEVARIABLES ARE f1-f4;
CLASSES = c (3);
ANALYSIS: TYPE = MIXTURE;
STARTS = 0;
LRTBOOTSTRAP=100;
LRTSTARTS = 2 1 50 15;
MODEL:
%OVERALL%
[ c#1@1.074 ];
[ c#2@-0.364 ];
%C#1%
[ f1@5.314 ];
[ f2@6.849 ];
[ f3@7.839 ];
[ f4@8.641 ];
Supplementary Material for Variants of Psychopathy
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f1@2.272 (5);
f2@0.857 (6);
f3@1.971 (7);
f4@1.300 (8);
%C#2%
[ f1@4.825 ];
[ f2@4.792 ];
[ f3@8.312 ];
[ f4@8.504 ];
f1@2.272 (5);
f2@0.857 (6);
f3@1.971 (7);
f4@1.300 (8);
%C#3%
[ f1@6.181 ];
[ f2@6.793 ];
[ f3@7.961 ];
[ f4@5.704 ];
f1@2.272 (5);
f2@0.857 (6);
f3@1.971 (7);
f4@1.300 (8);
OUTPUT: TECH1 TECH8 TECH11 TECH14;
SAVEDATA: FILE IS output.sav;
SAVE IS CPROB;
R
The following code is a script for R (R Core Team, 2012) to compute the Mahalanobis
distance of a case to the group centroids of manipulative psychopaths (LC1), aggressive
psychopaths (LC2), and sociopaths (LC3) based on the four PCL-R fators scores
(Interpersonal, Affective, Lifestyle, and Antisocial). Note that the allocation algorithm
is based on a fixed cluster specification. That is, cases from the present sample were
allocated to one of the three clusters if their posterior probability of class membership
was highest for this cluster.
Supplementary Material for Variants of Psychopathy
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In the classification routine, a new case will be allocated to a particular cluster if its
Mahalanobis distance toward that group is smaller than toward the other group. The
quadratic classification function is suitable for North American adult male offenders
with PCL-R total scores of at least 27 points. The data of the case are entered in the X
matrix. No other adjustments need to be made in the script. The example below contains
three cases (i.e., an X matrix with three rows).
Instead of using Mahalanobis distances, the prior probability of group allocation can be
acknowledged. For this purpose, the relative frequencies of LC1 and LC2 psychopaths
and LC3 sociopaths in the current sample (Study 1) served as estimates of the prior
probabilities. Then, p1, p2, and p3 give the posterior probabilities of group allocation
for LC1, LC2, and LC3, respectively. It should be noted, however, that the underlying
data were not multivariately normally distributed.
> # Reference: Tatsuoka, M. M. (1971). Multivariate analysis: Techniques for
educational and psychological research. New York: Wiley (pp. 218-232)
> # W := data matrix with subjects‘ subtotals on PCL-R facets Interpersonal, Affective,
Lifestyle, and Antisocial (in this order) in its four columns
> W <- matrix(c(4, 6.2439, 5.3091, 4.8657, 6, 6.7700, 6.9072, 4.5417, 9, 8.0209,
7.8165, 8.3519, 7, 5.4077, 8.6445, 8.5139),ncol=4,nrow=4)
>W
[,1] [,2] [,3] [,4]
[1,] 4.0000 6.0000 9.0000 7.0000
[2,] 6.2439 6.7700 8.0209 5.4077
[3,] 5.3091 6.9072 7.8165 8.6445
[4,] 4.8657 4.5417 8.3519 8.5139
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> M <- data.frame(matrix(rep(NA,length(W[,1])*6), nrow=length(W[,1])))
>M
X1 X2 X3 X4 X5 X6
1 NA NA NA NA NA NA
2 NA NA NA NA NA NA
3 NA NA NA NA NA NA
4 NA NA NA NA NA NA
> # y and z: data vectors with cluster means of manipulative and aggressive
psychopaths and of sociopaths, respectively
> x <- c(6.2439, 6.7700, 8.0209, 5.4077)
> y <- c(5.3091, 6.9072, 7.8165, 8.6445)
> z <- c(4.8657, 4.5417, 8.3519, 8.5139)
> # SSCP: sums-of-squares and cross-products matrices for manipulative psychopaths
(SSCP1), aggressive psychopaths (SSCP2), and sociopaths (SSCP3)
> SSCP1 <- matrix(c(520.9268,-19.9024,-108.4634,-10.5366,-19.9024,278.8223,57.6202,-24.0941,-108.4634,-57.6202,605.8746,-23.4460,-10.5366,-24.0941,23.4460,309.3031),nrow=4,ncol=4)
> SSCP1
[,1]
[,2]
[,3]
[,4]
[1,] 520.9268 -19.9024 -108.4634 -10.5366
[2,] -19.9024 278.8223 -57.6202 -24.0941
[3,] -108.4634 -57.6202 605.8746 -23.4460
[4,] -10.5366 -24.0941 -23.4460 309.3031
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> SSCP2 <- matrix(c(2318.4420,47.1983,-126.2215,54.8428,47.1983,617.8312,112.8481,2.7173,-126.2215,112.8481,1982.0633,54.1456,54.8428,2.7173,54.1456,1039.2015),nrow=4,ncol=4)
> SSCP2
[,1]
[,2]
[,3]
[,4]
[1,] 2318.4420 47.1983 -126.2215 -54.8428
[2,] 47.1983 617.8312 112.8481
2.7173
[3,] -126.2215 112.8481 1982.0633 54.1456
[4,] -54.8428 2.7173 54.1456 1039.2015
> SSCP3 <- matrix(c(453.1065,-17.2917,-37.7963,-131.0972,-17.2917,115.6250,11.1667,-8.1250,-37.7963,-11.1667,259.2593,-13.0556,-131.0972,-8.1250,13.0556,245.9583),nrow=4,ncol=4)
> SSCP3
[,1]
[,2]
[,3]
[,4]
[1,] 453.1065 -17.2917 -37.7963 -131.0972
[2,] -17.2917 115.6250 -11.1667 -8.1250
[3,] -37.7963 -11.1667 259.2593 -13.0556
[4,] -131.0972 -8.1250 -13.0556 245.9583
> # D: dispersion matrices for manipulative psychopaths (D1), aggressive psychopaths
(D2), and sociopaths (D3)
> D1 <- SSCP1*(1/(287-1))
> D1
[,1]
[,2]
[,3]
[,4]
[1,] 1.82142238 -0.06958881 -0.37924266 -0.03684126
[2,] -0.06958881 0.97490315 -0.20146923 -0.08424510
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[3,] -0.37924266 -0.20146923 2.11844266 -0.08197902
[4,] -0.03684126 -0.08424510 -0.08197902 1.08147937
> D2 <- SSCP2*(1/(948-1))
> D2
[,1]
[,2]
[,3]
[,4]
[1,] 2.44819641 0.049839810 -0.13328564 -0.057912144
[2,] 0.04983981 0.652408870 0.11916378 0.002869377
[3,] -0.13328564 0.119163780 2.09299187 0.057175924
[4,] -0.05791214 0.002869377 0.05717592 1.097361668
> D3 <- SSCP3*(1/(216-1))
> D3
[,1]
[,2]
[,3]
[,4]
[1,] 2.10747209 -0.08042651 -0.17579674 -0.60975442
[2,] -0.08042651 0.53779070 -0.05193814 -0.03779070
[3,] -0.17579674 -0.05193814 1.20585721 -0.06072372
[4,] -0.60975442 -0.03779070 -0.06072372 1.14399209
> # md: mahalanobis distances of subjects’ scores from cluster centroid of manipulative
psychopaths (md1), aggressive psychopaths (md2), and sociopaths (md3)
> for (i in 1:length(W[,1])) {M[i,1] <- sqrt(t(W[i,]-x)%*%solve(D1)%*%(W[i,]-x))}
> for (i in 1:length(W[,1])) {M[i,2] <- sqrt(t(W[i,]-y)%*%solve(D2)%*%(W[i,]-y))}
> for (i in 1:length(W[,1])) {M[i,3] <- sqrt(t(W[i,]-z)%*%solve(D3)%*%(W[i,]-z))}
> # p: posterior probabilities of group allocation toward manipulative psychopaths (p1),
aggressive psychopaths (p2), and sociopaths (p3)
> for (i in 1:length(W[,1])) {M[i,4] <- exp((-1/2)*(M[i,1]^2 + log(det(D1)) 2*log(287/1451)))/(exp((-1/2)*(M[i,1]^2 + log(det(D1)) - 2*log(287/1451))) + exp((-
Supplementary Material for Variants of Psychopathy
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1/2)*(M[i,2]^2 + log(det(D2)) - 2*log(948/1451))) + exp((-1/2)*(M[i,3]^2 +
log(det(D3)) - 2*log(216/1451))))}
> for (i in 1:length(W[,1])) {M[i,5] <- exp((-1/2)*(M[i,2]^2 + log(det(D2)) 2*log(948/1451)))/(exp((-1/2)*(M[i,1]^2 + log(det(D1)) - 2*log(287/1451))) + exp((1/2)*(M[i,2]^2 + log(det(D2)) - 2*log(948/1451))) + exp((-1/2)*(M[i,3]^2 +
log(det(D3)) - 2*log(216/1451))))}
> for (i in 1:length(W[,1])) {M[i,6] <- exp((-1/2)*(M[i,3]^2 + log(det(D3)) 2*log(216/1451)))/(exp((-1/2)*(M[i,1]^2 + log(det(D1)) - 2*log(287/1451))) + exp((1/2)*(M[i,2]^2 + log(det(D2)) - 2*log(948/1451))) + exp((-1/2)*(M[i,3]^2 +
log(det(D3)) - 2*log(216/1451))))}
> #The resulting data matrix M contains the classification statistics for each case in its
columns 1 to 6: Column 1 = Mahalanobis distance from cluster centroid of manipulative
psychopaths; Column 2 = Mahalanobis distance from cluster centroid of aggressive
psychopaths; Column 3 = Mahalanobis distance from cluster centroid of sociopaths;
Column 4 = posterior probability of group allocation toward manipulative psychopaths;
Column 5 = posterior probability of group allocation toward aggressive psychopaths;
and Column 6 = posterior probability of group allocation toward sociopaths;
>M
X1
X2
X3
X4
The data matrix can also be read into R using the "library(foreign)" subcommand. This
is based on the premise that the data file has the format tab-delimited and does not
contain a header. In the example, it is assumed that the file is stored in the directory
"/Users/Andreas/" under the file name "dataname.dat":
> library(foreign)
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> W <- as.matrix(read.table("/Users/Andreas/dataname.dat", header = FALSE))
Similarly, the results file can be saved as follows (by inserting the following as
the last line in the script):
> write.table(M, "/Users/Andreas/dataname_results.txt", sep = "\t")
Latent Profile Analysis with a PCL-R Threshold of 30
Several investigators have searched for latent classes in samples of offenders
selected on the basis of a PCL-R threshold higher than 27 (e.g., 29 by Skeem et al.,
2007; 30 by Hicks et al., 2004). For comparison with other studies we performed a
supplementary latent profile analysis (LPA) on a sample of offenders with a PCL-R
score of at least 30. This left us with 856 participants, representing about 15% of the
reference group.
Participants
The participants for this supplementary LPA were 856 offenders with a PCL-R
score of at least 30, a common research score for psychopathy (Hare, 2003). The mean
PCL-R total score in this restricted sample was 32.2 (SD = 2.1, minimum = 30,
maximum = 40). The results of a Mardia (1974) test spoke against multinormality of the
distribution of the PCL-R factor subtotals due to skewness (309.85, df = 20, p < .001)
and a platykurtic shape (-3.06, p < .001). The mean score for each factor was as follows:
Interpersonal, 5.8 (SD = 1.4); Affective, 6.8 (SD = 1.0); Lifestyle, 8.3 (SD = 1.3); and
Antisocial, 8.4 (SD = 1.4).
Results
A solution with two latent classes fit the data significantly better than did a unitary
solution without any latent classes. The modified likelihood-ratio test (Lo et al., 2001)
Supplementary Material for Variants of Psychopathy
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indicated, however, that up to five latent classes could be identified until a subsequent
(six-latent-classes solution) did not provide a significant improvement in model fit
anymore (see Table S-1).
--------------------------------------Insert Table S-1 about here
---------------------------------------Closer inspection revealed that the solutions with two and three latent classes were
nearly equivalent; the third latent class represented more or less a subdivision of the
larger latent class from the simpler solution with two latent classes. The adjusted Rand
Index comparing the solutions with two and three latent classes with each other (based
on allocations to single groups by virtue of maximum posterior probabilities) was .61.
Note that adjusted Rand Index values > .11 are considered significant and indicative of
strong agreement between clusters (Milligan & Cooper, 1986).
Settling on the most parsimonious model yielding a suitable fit (i.e., the solution
with two latent classes) yielded a replication of the manipulative (LC1) and aggressive
(LC2) variants described in the main analysis in the article. The mean probabilities for
allocation to the latent classes were high (> .85; see Table S-2 for details).
--------------------------------------Insert Table S-2 about here
---------------------------------------More specifically, offenders in LC1 clearly had a lower raw score average (scale of
0-10) on the Antisocial factor (M = 6.48) than did those in LC2 (M = 9.05; d = 2.85),
but still had a significantly higher mean raw score than offenders in the Reference
Group (5.7). Furthermore, offenders in LC1 and in the Reference Group certainly were
far more antisocial and aggressive than individuals in community samples (Coid, Yang,
Supplementary Material for Variants of Psychopathy
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Ullrich, Roberts, & Hare, 2009; Forth et al., 2013; Neumann & Hare, 2008; Weiler &
Widom, 1996). Conversely, offenders assigned to LC1 had a moderately (d = 0.53)
higher mean on the Interpersonal factor (M = 6.40) than the offenders in LC2 (M =
5.67). The differences between LC1 and LC2 on the Affective (d = 0.20) and the
Interpersonal factors (d = 0.31) were small (see Table S-3 for details; Figure S-1 shows
the mean profiles of the two latent classes.
--------------------------------------Insert Table S-3 about here
-----------------------------------------------------------------------------Insert Figure S-1 about here
---------------------------------------Discussion
The results of this supplementary LPA suggest that: (1) a “sociopathic” subgroup
did not emerge when the threshold for inclusion in the analysis was raised to 30; and (2)
the latent classes that did emerge (LC1 and LC2) were virtually the same as their
counterparts identified in the main LPA that used a PCL-R threshold of 27. That is, the
use of a strict or a relaxed PCL-R threshold for inclusion in the LPA yielded the same
manipulative (LC1) and aggressive (LC2) variants of psychopathy.
In statistical terms, the three-class solutions obtained with a PCL-R threshold of 27
was more robust than the two-class solution obtained in this supplementary analysis.
Only for the former did a likelihood-ratio test indicate that the solution in question fit
the data best. Furthermore, use of the lower threshold likely ameliorated some of the
issues associates with conducting LPA on more extreme portions of a distribution of
scores (Bauer & Curran, 2004).
Supplementary Material for Variants of Psychopathy
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Table S-1
Model Fit of the Latent Profile Analyses with up to Six Latent Classes (N =856).
Number of Latent Classes
1
2
3
4
5
6
Log-Likelihood
-5647.35
-5561.66
-5529.36
-5498.05
-5479.37
-5459.39
No. of Free Parameters
8
13
18
23
28
33
BIC
11348.71 11211.10 11180.27 11151.41 11147.80 11141.61
Adjusted BIC
11323.31 11169.82 11123.10 11078.37 11058.88 11036.81
AIC
11310.69 11149.32 11094.73 11042.11 11014.73 10984.79
a)
(-2)*Log-Likelihood Difference
171.38
64.60
62.62
37.36
39.96
LMR LRT, p value b) d)
.000
.000
.000
.006
.089
c) d)
Bootstrap LRT, p value
.000
.000
.000
.000
.000
1 - Entropy
.77
.66
.77
.76
.71
Note: AIC = Akaike’s Information Criterion. BIC = Bayesian Information Criterion. LRT = Likelihood Ratio Test. a Difference between models
of (k-1) and k classes. b Likelihood ratio test according to Lo et al. (2001). c LRT according to Nylund et al. (2007). d If < .05, a model with k
latent classes will fit significantly better than a model with (k-1) latent classes.
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Table S-2
Average Latent Class Probabilities for Most Likely Class Membership
Latent Class
n
Average Latent Class Probabilities
1
2
1
203
.88
.13
2
653
.04
.96
Table S-3
Means (SDs) and Effect Sizes (d) on Four Factors of the PCL-R in Two Latent Classes
PCL-R Factors
Cohen’s d
LC1
LC2
(n = 203)
(n = 653)
Interpersonal
6.40 (1.39)
5.67(1.39)
0.53
Affective
6.97 (0.99)
6.77 (0.99)
0.20
Lifestyle
8.60 (1.25)
8.21 (1.25)
0.31
Antisocial
6.48 (0.90)
9.05 (0.90)
2.85
Note. PCL-R = Psychopathy Checklist-Revised. LC1 and LC2 differed significantly
on each factor at p < .05, Bonferroni adjusted.
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Figure S-1
Supplementary analysis using a PCL-R cutscore of 30. Means of two latent classes referenced against
PCL-R factor scores of male offenders (N = 5,408; from Hare, 2003, p. 59). A z score of 0 represents
the mean of the reference group.