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					                                       Online material supplementary to Linscott et al. — page S1


                       ONLINE MATERIAL SUPPLEMENTARY TO:


       Seeking verisimilitude in a class: A systematic review of evidence that the
                  criterial clinical symptoms of schizophrenia are taxonic


                   Richard J. Linscott1-3, Judith Allardyce3, and Jim van Os3,4

           1
               To whom correspondence should be addressed; tel: +64 3 479 5689,
                     fax: +64 3 479 8335, e-mail: linscott@psy.otago.ac.nz
 2
     Department of Psychology, University of Otago, P. O. Box 56, Dunedin 9054, New
     Zealand; 3Department of Psychiatry and Neuropsychology, South Limburg Mental
    Health Research and Teaching Network, EURON, Maastricht University, P. O. Box
     616 (DRT 10), 6200 MD Maastricht, The Netherlands; 4Division of Psychological
     Medicine, Institute of Psychiatry, De Crespigny Park, Denmark Hill, London SE5
                                           8AF, UK.




Overview
          In this online supplement, we first describe two statistical approaches that may
yield evidence that is potentially relevant to questions about the latent1 structure of the
criterion symptoms of schizophrenia. Following these descriptions we present a pair
of simulations to illustrate their application. Finally, we illustrate reverse association
using evidence reported by Kendler et al.2


Coherent Cut Kinetic (CCK) Methods
          There are at least half a dozen CCK procedures, one of the most commonly
used of which is maximum covariance (MAXCOV) analysis. MAXCOV analysis is a
theorem-based bootstrapping procedure. It is unconventional in that it relies on visual

1
   ―Latent‖ is this context is statistical parlance for underlying or unobservable or
indirectly inferred, and is contrasted with ―manifest,‖ which means observable or
directly measurable.1 Thus, symptoms of schizophrenia, reflected in scores on
symptom rating scales, are manifest but underlying symptom factors (e.g., reality
distortion, disorganization, and negative) are latent. Equally, the endophenotypes of
schizophrenia, reflected in performance accuracy or reaction time or volume or
current and so on, are also manifest.
                                   Online material supplementary to Linscott et al. — page S2


analysis of descriptive statistics displayed graphically and is supplemented by
substantive tests of consistency among the bootstrapped parameters. CCK methods
can be understood by the following example. If a sample of men (n = 100) and
another of women (n = 100) are mixed and the correlation between the indicator
variables strength and muscle mass are plotted, a linear association will be visible in
the plot with no evidence for subgroups of men and women. If, however, the
covariance between strength and muscle mass on the y-axis is plotted against a ―cut‖
variable shoe size on the x-axis, the covariance will be noticeably high at the point of
shoe size that overlaps most between men and women, and much lower on the right
and left side, where groups are homogeneously male or female. (An example of this
pattern of covariance is illustrated in the top left panel of Figure S2). The principle
underlying these analyzes is that if two discrete groups exist that are discriminated by
an indicator variable, y, it follows that the two groups will differ in the mean of y.
This means that if the cases are sorted into taxon and complement groups, these
groups must differ in their means on y by some value, dy. If a second variable, x,
exists that also discriminates the taxon from the complement group but is not
correlated with y within either of the groups, then any cut of the distribution at a given
value of x will lead to a degree of separation between the means of y above and below
the cut.
           The MAXCOV procedure requires a minimum of three indicators, X, Y, and Z,
about which it can be assumed there is negligible within-class covariance. (This is
referred to as the assumption of conditional independence or local independence in
latent variable modeling.) Under this assumption, all observed covariance in a two-
class mix is determined by the class sizes (taxon base rate) and the separation
between taxon and complement groups. The commingled sample is ordered along one
indicator, X, cut into a sequence of sub-samples, and the covariance of the remaining
indicators, covYZ, within each sub-sample is calculated. If the latent structure of the
data is taxonic, the covariance coefficients will rise to a cusp or peak in the sub-
sample in which the base rate is closest to 50%. If the latent structure is dimensional,
covariance coefficients will be comparatively stable. The analysis is repeated by
ordering on Y (taking covXZ coefficients) and then on Z (covXY), and median or mean
covariance curves are obtained. If a peak or cusp is present, its location is used to
estimate the taxon base rate, and Bayesian posterior taxon membership probabilities
are obtained using the corresponding indicator cut scores. Case membership then
                                   Online material supplementary to Linscott et al. — page S3


serves as a basis for finding taxon-complement separation, within class covariance,
and estimating goodness of fit.
       As with all the methods considered here, indicators must be carefully selected.
In the case of MAXCOV and other CCK methods, the method should be used
iteratively to identify and eliminate inappropriate indicators.3 Indicator screening
should also be performed prior to analysis where possible. Specific considerations
include separation and conditional independence. For example, MAXCOV is not
recommended in situations where separation between classes is likely to be less than
1.2 SD2, or when the degree of correlation between indicators is so high that the
assumption of conditional independence (zero nuisance covariance) is substantially
violated.


Latent Variable Methods
       A latent variable is a random variable that is unobservable and that cannot be
expressed as a function of manifest or observable indicators alone.4 Instead, its
existence is inferred from patterns (correlation or covariation) among observed
indicators. Latent variables may be continuous (factors, traits, random effects) or
categorical (finite mixture or mixture distributions, latent class variables). Continuous
variables model unobserved dimensional constructs by explaining the observed
correlations, and are found using exploratory or confirmatory factor analysis (including
structural equation modeling / item response theory). Categorical variables model
unobserved typologies or classes of individuals and are found using latent class analysis
(LCA) or latent profile analysis (LPA). That is, these statistical procedures can be used
to provide competing (or complementary) dimensional and categorical representations,
respectively, of the same data, and interpretations of such representations are
constrained by the statistical procedures themselves. However, both approaches are used
simultaneously in hybrid models that explain covariance using both continuous and
categorical latent variables within a generalized latent variable framework.
       Exploratory and confirmatory factor analysis methods are based on the
common factor model. This model specifies that single or multiple continuous latent
variables (common factors) explain the correlations among the measured indicators,
whereas a unique factor (with a specific and an error component) influences only one
2
  In the class simulation used here, we are purposively overlooking this
recommendation. Use of lesser separation may generate unstable parameter estimates.
                                   Online material supplementary to Linscott et al. — page S4


indicator and does not account for the observed correlations. The common factor
model is used to estimate the pattern of association between the common factors and
each indicator, and indexes the common factor as loadings while residuals measure
the unique factor. Consequently, scores on one or more dimensions locate each
individual. Assumptions associated with exploratory and confirmatory factor analyzes
are that: (a) the sample is homogenous; (b) the latent variables are multivariate
normal; (c) residuals are normally distributed; (d) factors and residuals are
independent; (e) there is zero autocorrelation of residuals; and (f) relationships
between indicators and common factors are linear. Under these assumptions, the
distributions of the indicators are multivariate normal with no skewness or kurtosis.
       Finite mixture modeling identifies categorical latent variables representing two
or more subpopulations that differ qualitatively or quantitatively. Classical LCA and
LPA are special cases of mixture modeling. LCA uses categorical indicators as input
and LPA uses continuously scaled indicators. As in factor analysis, a single categorical
latent variable explains the covariance of the measured variables, and error is indexed
by residuals. In contrast to factor analysis, mixture models allow the assignment of
individuals to particular classes. Classical LCA and LPA have a number of
assumptions: (a) the sample is heterogeneous, that is, it is a mixture of two or more
subpopulations; (b) there is local independence, that is, within-class covariance is
zero; (c) the variance-covariance matrices are homogeneous across classes; (d) the
relationship between indicators and latent classes is linear. Given these assumptions,
the higher order moments (i.e., skewness and kurtosis) will deviate from zero as
differences between latent classes (i.e., of proportions, means, and variances) increase.
       A common assumption of latent continuous and categorical models is that the
indicators are independent, conditional on the class or common factor. In factor
analysis, this corresponds to the specification of uncorrelated residuals; in LCA and
LPA, to conditional independence. However, recently developed generalized latent
variable models take both categorical and continuous observed indicators as input.
This permits the relaxation of the conditional independence assumption.5-7 One such
hybrid approach, factor mixture modeling (FMM) allows complex structural
relationships by simultaneously modeling common factor models within two or more
latent classes. Thus, FMM is useful when there is reason to expect within class
correlations of observed variables. FMM assumptions include that: (a) the observed
sample is heterogeneous, that is, the joint distribution of the observed variables is a
                                    Online material supplementary to Linscott et al. — page S5


mixture distribution; and (b) there is multivariate normality within classes. Deviation
from the latter assumption may lead to over-extraction of classes. Recent simulation
studies suggest that hybrid modeling distinguishes correctly between simulations with
categorical and continuous latent structures, although the degree of class separation
and class base rates affect performance.5-7
        There are several important general considerations to bear in mind when
conducting or evaluating modeling research. Firstly, at the conceptual level, the
difference between categorical and continuous latent variables has important heuristic
significance. However, statistically, these are structurally equivalent; the K-class
model is structurally equivalent to a K – 1 factor model with continuous indicators.8, 9
Therefore, if one assumes sample heterogeneity but, in fact, indicators derive from a
latent continuous variable in a homogeneous population, LCA will extract (over-
extract) classes. Likewise, assuming sample homogeneity when, in fact, indicators
derive from a mixed distribution, exploratory factor analyzes will extract (over-
extract) factors.7
        Secondly, during model fitting, comparison is made among solutions from a
series of models with increasing numbers of classes, factors, or both. Interpretation of
the results—the choice of which heuristic model best fits data—cannot be made solely
on the basis of fit or other indices,1, 10-12 although some attempt to.13, 14 Rather,
interpretation depends on both statistical fit criteria and substantive reasoning.
        Several fit indices are used in latent variable modeling. If the maximum
likelihood algorithm is used to generate solutions, the log-likelihood estimation
indicates model fit. However, as log-likelihood increases with the number of parameters
used, fit is better determined using both log-likelihood and a log-likelihood-based
information criterion that favors parsimony. The most commonly encountered
information criteria are the Bayesian information criterion (BIC), the sample size
adjusted BIC (BIC´), and the Akaike information criterion (AIC). These three fit indices
differ in the ways they correct for sample size and free parameters. Higher log-
likelihood and lower information criteria scores indicate better fit. The likelihood ratio
test (LRT) allows comparison of a K – 1 class mixture model with a K class mixture
model. The mostly widely used LRT is the bootstrapped LRT.15 Finally, some software
packages provide estimates of overall classification quality, or entropy, for which unity
indicates perfect classification. Substantive considerations that may influence model
                                    Online material supplementary to Linscott et al. — page S6


selection involve analyzes of antecedents, covariates, and distal outcomes. Multiple
methodological and design variables also should not be overlooked.
       Thirdly, when the assumptions of the common factor model are met, unmixed
or single class distribution will have near-zero skewness and kurtosis. In contrast,
skewness and kurtosis deviate significantly from zero in mixed or commingled
distributions. Thus, maximum likelihood estimations of latent class and exploratory
factor models should be comparable using log-likelihood-based information criteria.
Here, again, better fit is indicated by larger log likelihood values.7


Simulations
Two Sample Simulations with Different Latent Structures
       Consider a sample of 700 patients from a population in which all members are
believed to be affected by Disorder A or Disorder B. For each individual, 8 normally
distributed continuous measures are available, representing scores on measures of
attributes that, although not pathognomonic, are known to be associated with the two
disorders. Indicators X1 to X5 are associated with Disorder A; and indicators X4 to X8
are associated with Disorder B. Thus, among these, indicators X4 and X5 are not
discriminating, measuring unitary features associated with both disorders. For each
measure, a higher score indicates more of the associated attribute.
       Two separate data sets were constructed, one representing a latent dimensional
structure in the population, the other representing a latent class structure where the
true prevalence of Disorder A is 25%. The latter was constructed first, using the Stata
random normal number generator, drawnorm, to generate n = 700 cases each with 8
normally distributed continuous scores. Subsequently, a latent structure was
introduced into the data by adding a constant to 25% of the data values, and
subtracting the same constant from the remaining data values. The constant was that
which resulted in a separation of 1 SD between groups’ mean scores on the
discriminating indices (i.e., X1 to X3 and X6 to X8) once the data were restandardized.
To simulate a dimensional data set, the correlation matrix for the class simulation was
obtained and used with the Stata drawnorm function to generate dimensional data
with the same correlation matrix. For all indicators in both simulations, M = 0.0 and
SD = 1.0. Other statistics for the data sets are presented in Table S1 and the
correlation matrices in Table S2. Univariate and bivariate density plots for two
example indicators from each set are illustrated in Figure S1.
                                   Online material supplementary to Linscott et al. — page S7


Table S1. Descriptive statistics for the class and dimension simulated data sets.

                                 Class                                            Dimension
Indicator
            Skewness    Kurtosis         XA - XB           Odds            Skewness     Kurtosis
X1           0.12        -0.19            1.00             6.53             0.05         0.26
X2           0.10        -0.06            0.94             5.15             -0.04       -0.01
X3           0.05        0.01             1.09             8.42             -0.07       -0.12
X4           -0.04       -0.01            0.08             1.10             -0.07       -0.41
X5           0.08        -0.18            0.09             1.26             -0.09       -0.21
X6           -0.20       -0.04           -1.06             7.17             0.15         0.20
X7           -0.07       0.25            -0.93             5.27             0.03        -0.02
X8           -0.11       -0.08           -0.98             6.40             0.05         0.34




Table S2. Covariance / correlation matrices for the class (beneath diagonal) and
dimension (above diagonal) simulated data. Parenthetical values along the diagonal
are the mean absolute differences in coefficients between the two simulations.

Indicator       X1        X2         X3             X4             X5        X6         X7          X8
  X1          (0.03)     0.14       0.25           0.07           -0.09     -0.20      -0.19       -0.20
  X2           0.17     (0.03)      0.23           0.01           0.03      -0.20      -0.14       -0.23
  X3           0.20      0.21      (0.03)          -0.02          -0.04     -0.25      -0.14       -0.28
  X4           0.02      0.03      -0.03           (0.03)         0.07      -0.03      -0.02       -0.08
  X5          -0.02      0.07       0.04           0.04           (0.05)    -0.03       0.00       0.09
  X6          -0.19     -0.14      -0.23           -0.02          0.00      (0.02)      0.22       0.18
  X7          -0.19     -0.14      -0.15           0.02           -0.01      0.19      (0.02)      0.14
  X8          -0.20     -0.21      -0.22           0.00           0.02       0.19       0.20       (0.04)
                                      Online material supplementary to Linscott et al. — page S8




                                       Class simulation




                                   Dimensional simulation




Figure S1. Illustrative bivariate and univariate density plots for indicators X1 and X2 from the
class (upper) and dimensional (lower) simulations.
                                   Online material supplementary to Linscott et al. — page S9


Indicator Selection and MAXCOV Analyzes of Simulations
       Two observations affect indicator selection. First, high scores on indicators X1
to X3 characterize Disorder A and high scores on X6 to X8 characterize Disorder B.
There is no a priori reason for including indicators X4 and X5 in an analysis because
these are not discriminative attributes. Secondly, and following from the first, low
scores on indicators X6 to X8 are possibly characteristic of Disorder A, given the
sampling population. So, as the six discriminating indicators must be consistent in the
direction of their discrimination, the scoring of X6 to X8 should be reversed.
       If we were not in a position to know as much about the population sample at
the outset, the same selection decisions could be made on the basis of scrutiny of
scatter plots and the correlation matrix (Table S2) and some preliminary MAXCOV
iterations. First, if indicators are sensitive, with separations of between 0.9 and 1.1
SD, and the base rate is assumed to range between 0.2 and 0.3, given conditional
independence, the pairwise correlation coefficients should fall between r = 0.13 and
r = 0.25.16, 17 Consequently, the near-zero pairwise correlations of X4 to X5 suggest
these measures cannot be used to separate the taxon from the complement and so
warrant exclusion. Second, the negative or parataxonic correlations among indicators
X1 to X3 and X6 to X8 imply that these indicators are not sensitive to the same latent
class.17 One might arrive at the conclusion that three scores should be reversed after
demonstrating taxon overlap when the two sets of three indicators are analyzed
separately.
       Thus, for both the class and dimensional simulations, the scores on X6 to X8
were reversed and these along with X1 to X3 were subjected to MAXCOV analyzes.
With six indicators, there are 60 indicator triplet combinations to be analyzed. The
principal graphical results are presented in Figure S2. The covariance curve for the
class simulation has a pronounced peak to the right of center, whereas for the
dimensional simulation, the covariance curve is both elevated and reveals no clear
peak, suggesting the latent structure is not taxonic. The observed base rates for the
class and dimensional simulations were M = 0.270 and M = 0.458, respectively.
                                                Online material supplementary to Linscott et al. — page S10




                                                  Class simulation

                  0.3


                                                                           300
     Covariance




                  0.2




                                                               Frequency
                                                                           200


                  0.1
                                                                           100



                  0.0
                                                                             0
                        -2             0              2                          0.0            0.5               1.0
                             Subsample z-coordinate                                Taxon membership probability



                                              Dimensional simulation

                  0.3


                                                                           300
     Covariance




                  0.2
                                                               Frequency




                                                                           200


                  0.1
                                                                           100



                  0.0
                                                                             0
                        -2             0              2                          0.0            0.5               1.0
                             Subsample z-coordinate                                Taxon membership probability

Figure S2. MAXCOV covariance curves and class membership probability histograms
obtained from analysis of six indicators for the class and dimensional simulations. The solid
line in the covariance curves provides a loess-smoothed covariance plot.




                  Several observations corroborate the findings from the class simulation: (a) the
mean within-class correlation coefficients for the taxon and complement were low,
r = -0.027 and r = 0.058, respectively, suggesting conditional independence (minimal
nuisance covariance); (b) the mean of indicator separations between the taxon and
complement classes identified from posterior probabilities was M = 0.95 (SD = 0.12);
(c) the Bayesian classification of individuals to classes resulted in a clear U-shaped
distribution of membership probabilities (Figure S2); (d) the mean residual pairwise
                                   Online material supplementary to Linscott et al. — page S11


covariance in the sample was covresidual = 0.036; and (e) the Jöreskog and Sörbom
goodness of fit index was GFI = 0.992. Base-rate variance is also used as a
consistency test, with large variance suggesting an unstable and therefore inconsistent
solution. In the class simulation, the observed standard deviation of base rates was
SD = 0.199, a value that is larger than desired. However, when the separation was
increased to 1.2 SD, the base rates were much more consistent, with SD = 0.109.
Stronger corroborative evidence would be obtained through the use of one or two
other coherent cut kinetic procedures, and such an approach is strongly recommended.
         Strictly speaking, the failure to observe a clear peak in the dimensional
simulation (Figure S2) means that there is no taxonicity to be corroborated by
consistency tests. Consistency tests are not dispositive given a flat or ambiguous
covariance curve. For comparison, however, the consistency indices from the
dimensional simulation were: (a) r = 0.050 and r = 0.114; (b) indicator separation
M = 0.74, SD = 0.16; (c) Figure 2; (d) covresidual = 0.066; and (e) GFI = 0.974. The
base-rate SD = 0.322; analyzes of a dimensional simulation corresponding to the 1.2
SD separation class simulation also yielded high base-rate variance, SD = 0.286.
         Given our omniscience with respect to true class membership in the class
simulation, it is possible to determine the classification accuracy of the MAXCOV
analysis for that simulation. There were 77 misclassifications: 31 false positives and
46 false negatives, giving sensitivity and specificity estimates of 0.74 and 0.94,
respectively, and a likelihood ratio of 12.4 (a likelihood ratio of 10 or greater is
generally considered diagnostic).


Latent Variable Modeling of Simulations
         We fitted a series of models to the dimensional and two-class simulated data
sets (Tables S3 and S4). For FMM fitted models, factor variance was fixed at zero and
factor loadings were constrained so that all the factor parameters are class specific.
That is, the analysis is fully exploratory. Log-likelihood ratios and information criteria
obtained for the dimensional and class simulations are shown in Tables S3 and S4,
respectively. Given the hypothetical nature of the simulation, we set aside
consideration of potential substantive issues; models were rejected on the basis of fit
alone.
                                       Online material supplementary to Linscott et al. — page S12


Table S3. Fit indices obtained for the dimensional simulation.

                            Log                                                  LRT
Model                                   AIC           BIC          BIC´
                         likelihood                                            p-valuea
LPA models
     1C                  -5956.54     11937.08    11991.69      11953.59          —
     2C                  -5830.11     11698.23    11784.70      11724.37         0.00
     3C                  -5803.86     11659.73    11778.06      11695.50         0.00
     4C                  -5795.65     11657.30    11807.50      11702.65         0.18
FMM models
     1F1Cb               -5806.76     11649.53    11731.45      11674.29          —
     2F1C                -5801.99     11649.99    11754.66      11681.63          —
     3F1C                -5800.06     11654.11    11776.99      11691.26          —
     1F2C                -5792.02     11634.04    11747.82      11668.44         1.00

Note. C = class; F = factor; 1F2C = 1-factor 2-class; AIC = Akaike information criterion; BIC
= Bayesian information criterion; BIC´ = sample size adjusted BIC; LRT = bootstrapped
likelihood ratio test.
a
    The LRT is not calculable for models with one class.
b
    The true latent structure.


Table S4. Fit indices obtained for the class simulation.

                            Log                                                  LRT
Model                                   AIC           BIC          BIC´
                         likelihood                                            p-valuea
LPA models
     1C                  -5956.54     11937.08     11991.69      11953.59         —
        b
     2C                  -5794.02     11626.03     11712.50      11652.18        0.00

     3C                  -5787.85     11627.70     11746.03      11663.47        1.00
FMM models
     1F1C                -5822.02     11680.04     11761.96      11704.81         —
     2F1C                -5820.25     11686.50     11791.16      11718.13         —
     3F1C                -5819.11     11692.23     11815.10      11729.37         —
     1F2C                -5800.13     11650.27     11764.05      11668.67        0.43

Note. See note to Table 3.
a
    The LRT is not calculable for models with one class.
b
    The true latent structure.
                                 Online material supplementary to Linscott et al. — page S13


        Analysis and model appraisal proceeded as follows for both simulations.
Beginning with one-class (1C) LPA, classes were added (i.e., 1C, 2C, 3C, . . . ) until
the bootstrapped LRT statistic was not significant, indicating the model should be
rejected in favor of one with fewer classes. Subsequently, increasing numbers of
factors (F) were added to one-class and then two-class FMM (i.e., 1F1C, 2F1C, 3F1C,
. . . , 1F2C, 2F2C, . . . ) until the model with the larger number of components was
rejected.
        To evaluate results for the dimensional simulation, consider Table S3. Among
the LPA models it can be seen that the information criteria deteriorate (decrease) as
the number of classes increase (Table S3). Critically, the bootstrapped LRT for the
four-class model is nonsignificant (p = 0.18), suggesting the three-class model is
preferable to the four-class model. Although neither the two- nor the three-class
models have good classification qualities, there were no obvious signs of inadmissible
parameter estimates (residuals) or extremely small classes to suggest over-extraction
of classes. Thus, had we only performed LPA, results from the dimensional
simulation would have likely been interpreted as supporting the two- or three-class
models rather than the correct 1F1C model. However, we went on to perform FMM.
Comparing only the exploratory factor analytic models (i.e., 1F1C, 2F1C, and 3F1C),
the correct model, 1F1C, would have been chosen on the basis of any of the
information criteria in Table S3. When the results of all the fitted models are
compared, the AIC and BIC´ favor the one-factor two-class (1F2C) solution.
However, 1F2C is not supported by the bootstrapped LRT, for which p = 1.0. That is,
1F2C is rejected because of the bootstrapped LRT and 1C1F, the correct model, has
the best fit.
        Turning to the class-simulation (Table S4), the information criteria improve
dramatically in the step from the one- to the two-class LPA model, but begin to
uniformly deteriorate in the subsequent transition to a three-class LPA model. If the
analyzes had been restricted to LPA only, the correct model (2C) would have been
chosen given any statistical criterion, including the bootstrapped LRT. In contrast, if
only FMM were performed, the erroneous 1F1C solution would have been selected on
the basis of a non-significant LRT for the 1F2C solution, although moderate factor
loadings obtained with this might have cast doubt on the solution. Comparison across
all LPA and FMM models shows the correct model, 2C, would be chosen using any
of the statistical indices.
                                                  Online material supplementary to Linscott et al. — page S14


                     As with MAXCOV, we compared true class membership and membership
assigned using the 2C LPA model. There were 49 misclassifications: 29 false
positives and 20 false negatives, giving sensitivity and specificity estimates of 0.89
and 0.94, respectively, and a likelihood ratio of 16.0.


Illustration of Reverse Association
                     To illustrate reverse association, we will consider data reported by Kendler et
al.2 that was used to test the validity of a six-class solution describing the latent
structure of features of schizophrenia and affective disorders. The six classes were
labeled classic schizophrenia, major depression, schizophreniform, bipolar
schizomania, schizodepression, and hebephrenia. The validating evidence included
differences in the frequencies of probands’ social and psychopathology outcomes and
relatives’ morbidity risk for psychosis and affective disorder among six classes. For
simplicity, we will restrict our discussion to the latter, which included rates for six
morbid outcomes: three schizophrenia-related outcomes (schizophrenia, nonaffective
psychoses, schizophrenia spectrum disorders) and three affective outcomes (bipolar
affective illness, unipolar affective illness, and affective illness). These outcome data
are shown in Figure S3.


                Schizophrenia-Related Outcomes                                       Affective Illness Outcomes

                                         Schizophrenia                                                       Bipola r illn ess
                50                       Nonaffective psychoses                 50                           Unipolar illness
                                         Schizophrenia spectrum                                              Affective illess


                40                                                              40
  Morbid risk




                                                                  Morbid risk




                30                                                              30



                20                                                              20



                10                                                              10



                 0                                                               0
                       I     II    III     IV      V      VI                           I    II    III   IV     V        VI


Figure S3. Relatives’ schizophrenia-related and affective illness morbidity risks for
six classes identified by Kendler et al.2 Class abbreviations: I = classic schizophrenia;
II = major depression; III = schizophreniform disorder; IV = bipolar-schizomania;
V = schizodepression; and VI = hebephrenia.
                                  Online material supplementary to Linscott et al. — page S15


        For the three schizophrenia-related outcomes, the differences in relatives’
morbidity risk across the six classes (Figure S3) are, in essence, a series of single
dissociations. If the same-process hypothesis were correct, plots of three or more
values of one parameter against the corresponding values of the other parameter must
yield monotonic-decreasing or monotonic-increasing curves. Thus, pairwise plots of
the schizophrenia-related outcomes are monotonic because, as can be seen in the top
panel of Figure S4, monotonic curves—specifically, curves that have no negatively
sloping segments—can be drawn over this panel to adequately represent the position
of all six classes. This suggests a single process could indeed account for the
differences among the six classes on these three variables. If such plots yielded
nonmonotonic curves (i.e., increases in one variable are sometimes associated with
decreases in the other variable and are sometimes associated with increases), this is
logically incompatible with the same-process hypothesis. Thus, the situation appears
to be different for the affective illness outcomes; one could not draw monotonic
curves to represent the groups shown in each of the plots in the bottom panel of
Figure S4. That is, monotonic curves do not adequately capture the relationships of
bipolar illness with the other two outcomes (affective illness, unipolar illness) because
curves that will adequately represent the position of all six classes must have
segments that are positively sloping (i.e., go up) as well as segments that are
negatively sloping (i.e., go down). Consequently, given the assumption that a
monotonic process leads to unipolar morbidity in relatives, these plots allow one to
reject the notion that the same process could possibly lead to bipolar morbidity in
relatives.
                                                                   Online material supplementary to Linscott et al. — page S16




                                                     Schizophrenia-Related Outcomes




                Nonaffective Psychosis
                                                                     VI



                                                   III
                                                     V
                                              IV I

                                         II
                Schizophrenia Spectrum




                                                                     VI                                               VI




                                                   V                                               V
                                                 III                                                III
                                              IV I                                           IIV
                                         II                                         II


                                                   Schizophrenia                    Nonaffective Psychosis


                                                           Affective Illness Outcomes
                Bipolar Illness




                                                     III
                                                                      II
                                                                     IV

                                                               V

                                                           I

                                              VI
                Affective Illness




                                                                      II
                                                                     IV                                    II
                                                                                                          IV

                                                               V                                   V
                                                           I                             I
                                                     III                                                        III


                                              VI                                   VI



                                                   Unipolar Illness                           Bipolar Illness


Figure S4. Pairwise associations among relatives’ morbidity risks for six classes
identified by Kendler et al.2 Error bars represent standard errors of risk. Class
abbreviations: I = classic schizophrenia; II = major depression; III = schizophreniform
disorder; IV = bipolar-schizomania; V = schizodepression; and VI = hebephrenia.
                                 Online material supplementary to Linscott et al. — page S17


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