Appendix 8. Shared environment assumptions
In this appendix, we discuss if the way we specified the shared environment might have led us to
draw the wrong conclusion regarding the general genetic factor. When modeling monozygotic
and dizygotic twins, it is reasonable to assume that both individuals within a given pair grow up
in the same family. We extended the ordinary twin model to pairs of different types of siblings,
and assumed that full siblings and maternal half-siblings shared all of the common environment
whereas paternal half-siblings shared none of it. It can be noted that other authors have used
other approaches that rest on other assumptions to test for shared environmental effects (38).
Results showed that regardless of how we modeled the shared environment, it played a
diminishingly small role for the covariation among the disorders and thus our conclusion that the
data support a general genetic factor of psychopathology when using national registers remains
valid.
How should the shared environment be parameterized based on annual tax records?
Although we assumed that the common environment was shared completely for full and
maternal half-siblings and not at all for paternal half-siblings, there may be other ways to
parameterize it. For example, to the extent that one assumes that the shared environment
consists of growing up in the same household, it could be quantified based on annual tax records,
similar to what have been used by Kendler and colleagues. (38) Specifically, these tax records
annually identify the building each person is registered to, such as a house or apartment complex.
To the extent that separated parents do not live in the same building, children who live with
different parents will have different values in these records. Of course, alike our original shared
environment assumption, this index is obviously not a description of reality; even though
children might be registered as living in the same or different households, this could be because
of tax and/or equality issues and is therefore not a perfect indicator for how much time they have
spent together. Nevertheless, it provides some insight into the siblings’ average living situation.
Because these records exist from 1968 and on, the respective pairs with complete information on
this variable included 267,714 pairs of full siblings, 28,434 pairs of maternal half-siblings, and
29,950 pairs of paternal half-siblings.
To estimate time spent in the same household for each sibling pair, we examined how many
years both individuals in each pair were registered to the same building according to the tax
records. We counted from the birth of the second sibling up till the year the first sibling turned
18. For example, if there was an age difference of four between two siblings, then they could
live in the same household for up the 14 years (i.e., 18 - 4). We then counted how many of those
years they were registered as living in the same building. For example, if two siblings with an
age difference of four were registered to the same building for 12 years, then we estimated that
they lived together 86 percent of the time (i.e., 12/14). Thus, this index ranged from zero (never
registered as living in the same building) to one (always registered to the same building).
As can be seen in the figures below outlining the respective distributions of time registered as
living in the same building, full and maternal half-siblings by and large grew up in the same
household, supporting our assumption that they shared all of the common environment. The
distribution based on paternal half-siblings, on the other hand, was largely bimodal, indicating
that about half lived in different buildings, but that the other half were registered as living in the
same building throughout childhood and adolescence. To the extent that the shared environment
is assumed to reflect time spent in the same household, this violates our assumption that they do
not share any of the common environment.
10000 15000
0
0
5000
Frequency
150000
Time living together:
Maternal half siblings
50000
Frequency
250000
Time living together:
Full siblings
0.0
0.2
0.4
0.6
0.8
1.0
Fraction together
0.0
0.2
0.4
0.6
0.8
1.0
Fraction together
8000
4000
0
Frequency
12000
Time living together:
Paternal half siblings
0.0
0.2
0.4
0.6
0.8
1.0
Fraction together
Although this parameterization of shared environment is based on a set of assumptions that
might not be entirely valid, as noted above, in the next section, we explored what happened to
the shared environment when we parameterized it in a number of different ways, including based
on the tax records. We provide arguments that the shared environment is of negligible
importance, regardless of what assumptions we make about it.
Does the original model that omitted the shared environment fit the data well?
Before fitting additional models, note that our original solution included two genetic and one
non-shared environment factor, as indicated by several dimensionality indices. To the extent that
the shared environment influenced the data, this model, which omitted the shared environment,
ought to fit poorly. However, by all measures, the original model fit the data very well (RMSEA
= .001; CFI = .99). This implies that the shared environment probably did not influence our
variables particularly much.
Fit statistics, however, pertain to the entire model. Another way to explore whether the shared
environment mattered is to examine the residual cross-sibling, cross-trait correlations separately
for maternal and paternal half-siblings. Regardless of whether our original parameterization or
the one based on tax records is the most optimal, we would expect the shared environment to
have a greater influence on maternal compared to paternal half-siblings. Thus, to the extent that
the shared environment influenced the observed correlations, omitting to model this component
ought to lead to larger residuals for the maternal half-siblings. In the figure below, we plotted
the absolute cross-sibling, cross-trait residuals (observed minus modeled correlations) separately
for maternal and paternal half-siblings. A visual inspection suggests that there is virtually no
difference in model misfit between maternal and paternal half-siblings. One way to quantify
differences between distributions is to compute Cohen’s d, which expresses the difference in
terms of standard deviations. Cohen’s d for the residuals between maternal and paternal halfsiblings equaled .03. Thus, even though we omitted to model the shared environment, model
misfit was about the same for maternal and paternal half-siblings, indicating that the shared
environment likely played a very small part.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Absolute residual cross-trait, cross-twin variances and covariances
30
20
10
Maternal half-siblings
0
10
20
30
Paternal half-siblings
Alternate parameterizations of the shared environment and its subsequent influence
Although our original model that omitted the shared environment appeared to fit the data very
well, one may argue that our shared environment assumptions were problematic in light of the
observation that a sizable proportion of paternal half-siblings were registered as living in the
same building. To remedy this, we ran an additional set of Cholesky decompositions in which
we parameterized the shared environment in several different ways. For each parameterization,
we examined the magnitude of the first shared environment Eigen value (summarized in Table 1
below). To the extent that this value did not exceed unity, it implies that the shared environment
had a diminishingly small influence on the observed overlap among the disorders.
Table 1. Alternate parameterizations of the shared environment and the corresponding first Eigen value.
Shared environment parameterizations
Models
Full C = 1 Mat C = 1 Pat C = 1 Pat = .5 Pat = 0 Pat = 1/0
Original model
x
x
x
Alternate model 1
x
x
Alternate model 2
x
x
x
Alternate model 3
x
x
x
Alternate model 4
x
x
x
Note. C = Shared environment.
Full = full siblings; Mat = Maternal half-siblings; Pat = Paternal half-siblings.
Alternate model 4 is limited to participants who lived together more than 80% (C = 1)
or less than 20% (C = 0) of their childhood according to tax records.
First C Eigen value
0.17
0.70
0.56
0.61
0.66
First, we ran a model (alternate model 1) based on only full and maternal half-siblings because
these groups both evidenced highly similar distributions of time registered as living in the same
building (i.e., the vast majority of siblings from both groups were registered as living in the same
building throughout childhood and adolescence). The first shared environment Eigen value
based on a Cholesky decomposition was .70, that is, it did not exceed the commonly used cutoff
of one (meaning that it accounted for less than one variable). This indicates that even when we
relied on only full and maternal-half siblings (who by and large appeared to grow up in the same
household), there was very little meaningful covariance in the shared environment matrix,
supporting our decision to exclude it.
Second, we ran two additional models in which we specified that the shared environment was
shared at unity (alternate model 2) or at .5 (alternate model 3) among paternal half-siblings. The
first shared environment Eigen values were .56 and .61, respectively. This indicates that
regardless of how the shared environment was parameterized among paternal half-siblings, it did
not appear to influence the observed overlap among the disorders.
Third, we ran a Cholesky decomposition in which the shared environment parameterization was
based on the living situation according to annual tax records (alternate model 4). We selected
the full siblings (257,575 pairs) and maternal half-siblings (22,522 pairs) who were registered as
living in the same building at least 80 percent of the time. We split the paternal half-siblings into
two groups: those who had lived together more than 80 percent of the time (15,821 pairs), and
those who had lived together less than 20 percent of the time (11,258 pairs). For the former
group, we fixed the shared environment parameter at unity, and for the latter group, we fixed it at
zero. Results demonstrated that the first shared environment Eigen value equaled .66. Thus,
when we analyzed the subsample for which there was annual housing information and
parameterized the shared environment accordingly, the shared environment still did not have a
meaningful influence on the observed overlap among the disorders.
Summary
In this appendix, we explored a number of different ways to parameterize the shared
environment. Regardless of how we analyzed the data, we did not find evidence for an
appreciable effect of the shared environment on the correlational structure. Therefore, even
though none of the shared environment assumptions used in the models above are probably
correct, it does not seem to matter because the shared environment did not appear to influence
the covariances among our variables. Thus, we believe that our conclusion that a general genetic
factor underlies a large amount of the overlap among disorders in national registers is supported.