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MANKIND QUARTERLY 2021 61:3 578-598
578
Are there Complex Assortative Mating Patterns for
Humans? Analysis of 340 Spanish Couples
Emil O. W. Kirkegaard*
Ulster Institute for Social Research, London, UK
* Corresponding author: the.dfx@gmail.com
Assortative mating for both physical and psychological traits is
well-established in many animal species, including humans. Most
studies, however, only compute linear measures of mate similarity,
typically Pearson correlations. However, it is possible that trait
similarity, or dissimilarity, has complex patterns missed by the
correlation metric. We investigated a dataset of 340 Spanish couples
for evidence of relationships across 7 traits: age, educational
attainment, intelligence, and the scales of the Eysenck Personality
Questionnaire: Extroversion, Psychoticism, Neuroticism, and the Lie
scale. We replicated well known linear assortative mating for age,
intelligence and education. Like most studies, we find weak to no
assortative mating for the personality traits. Analysis of nonlinear
patterns using regression splines failed to reveal anything beyond
the linear relations. Finally, we examined cross-trait variation for
couples but we found little of note. Overall, it does not appear that
there are complex patterns for traits in human couples.
Key Words: Assortative mating, Mate similarity, Nonlinear, Trade-
off, Human, Cross-trait association
Large meta-analyses show that in nature, mates tend to be more
phenotypically similar to each other than random pairs of individuals. This is
referred to as assortative mating (Janicke et al., 2019; Vandenberg, 1972). It
contrasts with disassortative or negatively assortative mating, where mates are
less similar than chance for some phenotype (e.g. coloration in wolves, Hedrick
et al., 2016). Both patterns are found in nature, but the former is more common.
Evidence is abundant that humans also mate assortatively (Luo, 2017). Indeed,
most human languages contain sayings about this phenomenon such as “birds
KIRKEGAARD, E.O.W. COMPLEX ASSORTATIVE MATING PATTERNS IN HUMANS
579
of a feather flock together”, “krage søger mage” (crow seeks mate, in Danish), or
“rybak rybaka vidit izdaleka” (fisherman sees another fisherman from afar, in
Russian).
1
Assortative mating is a special case of the more general phenomenon of
social similarity. In other words, it’s not just mates but also neighbors, friends,
colleagues, and any other network connections, that tend to be similar. Many or
most of these network links are genetically unrelated, at least for recent family
history, and thus the phenotypic similarity cannot be entirely explained with genes
shared by recent descent. Instead, social similarity mainly reflects self-selection
and various assortment processes in human networking behavior. For instance,
people who are similar in vocational interests often end up studying the same
topic in college, and end up friends. Similarly, people who share an interest in
dancing may meet at a club or dance and become friends. Unsurprisingly,
research also indicates that people are more friendly disposed to people who are
like themselves, called social homophily (love of the same, in Greek) (Aiello et al.,
2012; Huber & Malhotra, 2017; McPherson et al., 2001).
Social similarity, including assortative mating, differs importantly in strength
by phenotype. Age is probably the strongest sorter, as evidenced by the existence
of widespread terms used to refer to the ‘age-outgroup’ (“old people”, “boomers”,
“kids”, etc.). Most countries enforce age-groupings for children during childhood
— children are largely segregated by year of birth in the first 10 or so years of
school systems — so to some extent, the strong age similarity in networks is
socially imposed not merely self-selected. Some typically found values for
assortative mating correlations in humans are (findings summarized from Luo,
2017): age .70s to .90s, educational attainment or general social status .40s to
.60s, attitudes (e.g. political or religious opinions) .40s to .70s, and .10s to .40s
for personal values (e.g. risk taking), general intelligence .40s, mental health or
well-being .20s to .50s (depending on disorder), habitual behaviors (e.g. smoking,
exercise) .20s to .50s. Personality (mostly self-rated OCEAN/Big Five
2
) shows
surprisingly weak assortative mating, with typical coefficients from 0 to .30s. Even
various physical traits show some association (e.g. hand length r = .18;
Vandenberg, 1972), typically .10s to .20s for specific traits, and .30s to .40s for
physical attractiveness. Height is a particularly well-studied trait (k = 154, Stulp et
al., 2017), probably due to ease of accurate measurement and strong role in
1
A collection of these can be found at https://about.imtranslator.net/birds-of-a-feather-
flock-together/
2
Openness, Conscientiousness, Extroversion, Agreeableness, and Neuroticism/
Emotional Stability.
MANKIND QUARTERLY 2021 61:3
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human social perception, and shows an overall association of r = .23. There did
not seem to be any historical changes in the degree of similarity. There are also
strong patterns of assortative mating for race and ethnicity, decreasing in recent
decades. However, because these are measured as nominal traits in most
studies, Pearson correlations depend strongly on the sample proportions, so the
effect sizes are not comparable to the other traits listed, which are based on
continuous or quasi-continuous traits.
The existing literature suffers from some limitations. First, most studies only
examine the strength of a linear relationship, usually using Pearson correlations,
or similar effect size metrics appropriate for ordinal or nominal data (such as race
or religion). Thus, it is possible that they have missed nonlinear and especially
non-monotonic or threshold relationships. For instance, the review of effect sizes
summarized above (Luo, 2017) does not mention the word “linear” at all. There is
however at least one study that examined nonlinear assortative mating and its
importance for marital satisfaction (Luo & Klohnen, 2005; see also
Nikoloulopoulos & Moffatt, 2019). This study found mixed results, and thus there
is a need for more and larger studies.
Second, while assortative mating for height is well-established, it is also very
common to see women advertise in their dating profiles that the man must be
taller (so called male-taller norm). This female demand should result in a
particular nonlinear pattern for height relationships such that there is a curve
break close to the height parity line.
3
Not many studies have actually examined
this, but one that did found that this effect was relatively weak compared to what
might have been expected from the extensive talk about this in the online dating
literature (Stulp et al., 2013). Height was also found to be positively related to
fertility (and thus fitness) for men, but not women (Stulp et al., 2015). Both sexes
were subject to stabilizing selection (i.e. deviation from an optimal height had
lower fertility), but men were also subject to some directional selection, with
highest fertility at approximately 1 standard deviation in height above the mean.
Patterns such as these might exist for other traits as well, and should be explored
in detail to understand ongoing human sexual selection. Similarly, a recent study
using self-reported preferences found that very high levels of a trait appear
intimidating to potential mates and may be selected against. This could potentially
induce some nonlinear patterns, depending on the existence of assortative
3
Specifically, there should be a relative lack of males with just below female height
compared to what would be expected based on the linear measure of association (e.g.
Pearson correlation). This pattern would result if there was a tendency specific female
(or male or both) aversion to female > male height pairs.
KIRKEGAARD, E.O.W. COMPLEX ASSORTATIVE MATING PATTERNS IN HUMANS
581
mating for the trait in question and how much is too much (Gignac & Starbuck,
2019).
Third, phenotypes are usually examined one by one, with scant research on
cross-trait similarity or trade-offs. Trade-offs are expected based on some models
of human mate preferences. There is indeed some evidence of this for height and
education (Ponzo & Scoppa, 2015). Such models posit a composite score,
termed sexual market value or mate value (Fisher et al., 2008), for an individual
based on their phenotypes (and assumed genotypes due to genetic influence on
traits). Hence, a partner who is relatively poor in one phenotype can make up for
it by having a good ranking in another phenotype. Thus, we would expect to see
selection-induced negative correlations across differences on traits in pairs, such
that partners who are well matched on phenotype A show larger gaps on
phenotype B, where both are phenotypes that are sought.
4
This is an example of
the more general phenomenon of collider bias (Rohrer, 2018). Figure 1 illustrates
this concept.
4
Net negative correlations only result if the collider bias is strong enough to offset any
positive association between the traits that might exist in the general population.
MANKIND QUARTERLY 2021 61:3
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Figure 1. Illustrative example of collider bias for two components of a composite
mate value score. Black line shows total sample fit (no association), while the
blue lines show the fit for the two subsamples (negative associations). Smoothing
lines by general additive model.
In the figure we see things from the perspective of a woman who is unwilling
to date a partner below composite mate value of 1, perhaps because she
perceives herself to be at this value and prefers hypergamy (Esteve et al., 2012;
Harinam & Henderson, 2020; Tuckfield, 2019). In this case, the men in the upper
right part (turquoise color) are in her potential dating pool. Among these men, her
two favored traits of attractiveness and niceness are negatively associated (r = -
.66, top blue line) despite being unassociated in the total male population (black
line, r = .00). There is also a weaker induced negative association among the men
with too low composite scores (r = -.22, bottom blue line).
Thus, bringing the above together, the purpose of the current study was to
examine possible nonlinear cross-mate similarity or dissimilarity across
phenotypes in a suitably large dataset.
Data and methods
We use archival data from Colom et al. (2002). This study was published in
Spanish. First author Roberto Colom kindly shared the anonymized data with me
(personal communication, 2015). The dataset contains trio data on 342 parent
duos and one child. The child was a student enrolled in a class at university, so
there was some indirect selection for parental education and intelligence levels.
The dataset includes measures of personality measured using Eysenck’s
Personality Questionnaire-Revised (Spanish adaptation, Aguilar et al., 1990),
which contains scales for Extroversion, Neuroticism, Psychoticism, and the Lie
scale. The first two are similar to the existing scales from other batteries such as
NEO-PI-R, IPIP and HEXACO tests among others. Psychoticism is unique to
Eysenck’s test and is plausibly seen as a combination of the OCEAN traits of
Agreeableness and Conscientiousness. It has been found to be related to
creative achievement and various forms of socially undesirable behavior
(Eysenck, 1995). The Lie scale is somewhat unusual. It consists of items that
relate to overly positive self-presentation, intended to measure social desirability
bias (Eysenck & Eysenck, 2011; Ferrando et al., 1997). Aside from personality,
the dataset includes age, educational attainment (coded on a 1-4 scale, 1 =
primary education level, 2 = secondary education level, 3 = technical studies, and
4 = university level), and intelligence. The latter was measured using the TIG-2
(Test De Inteligencia General, test of general intelligence, 2nd edition) (Gough &
KIRKEGAARD, E.O.W. COMPLEX ASSORTATIVE MATING PATTERNS IN HUMANS
583
Domino, 1963; publisher site TEA, 2020). This is a Spanish developed dominos-
based abstract reasoning test, similar to the Raven Progressive Matrices test.
The appendix contains some example items from the test manual.
We used regression splines to detect and estimate the effect size of nonlinear
associations. Since the reader may be unfamiliar with this approach, and question
its utility, we carried out a simple simulation to test whether it worked as expected.
Specifically, we simulated data based on a sine function in the region 0 to 3.14
(π), which has an inverted U-shaped pattern, and a correlation of 0.00 with a
uniform variable. We simulated datasets that varied in sample size and amount
of random normal error added to see how this affected results. For each simulated
dataset, we fit a linear and a natural spline model and computed the model
adjusted R2 values. From these we computed the gain in adj. R2 as the measure
of nonlinearity in the data, i.e., the extra ability of the natural spline to predict or
explain the data, adjusted for overfitting (James et al., 2013, Chapter 7). For each
parameter combination, we repeated the simulation 10 times. Figure 2 shows the
results.
Figure 2. Results of simulation study for detection of linear pattern in regression
model. The R² adjusted gain is based on comparison of linear model fit with a
natural spline model. The grey ribbon shows the variation across each of the 10
replicates.
MANKIND QUARTERLY 2021 61:3
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We see that without noise, the spline model always fits near-perfectly. The
R2 gain is slightly above 1.00, showing that there is some bias in this metric, or at
least, bias in the correction when the pattern in the data is perfect.
5
For the
realistic cases with noise, we see that the spline is able to tell the pattern from the
noise, at least after about n > 100. When the noise amount is very large (e.g. SD
= 2), no amount of data helps (R2 gain ≃ 0).
Figure 3 shows the simulation data when n = 340, as in this study, and noise
is 0.5. In this figure, the R2 gain is 0.25 (R2 adj. linear = -.003, R2 adj. spline =
.245). Thus, if there are any nonlinear patterns in the data that are similar to those
in our simulation, we should be able to find them with this approach.
Figure 3. Sine function pattern with n = 340 and noise standard deviation = 0.5.
The red line shows the linear model fit, and the blue line shows the spline fit.
5
The theoretically expected value here is R2 = 1.00 and R2 gain of 1.00 since the linear
model has r = .00 and thus R2 = .00.
KIRKEGAARD, E.O.W. COMPLEX ASSORTATIVE MATING PATTERNS IN HUMANS
585
Results
Figure 4 shows the heatmap of the correlation matrix for the included
quantitative variables. We used latent correlations for the ordinal educational
attainment variables, based on the assumption of an underlying continuous
educational attainment trait expressed as one of the four ordinal categories.
Figure 4. Heat map of correlation matrix of study variables. Correlations are
Pearson correlations except for education (ed) where latent (polychoric)
correlations are used. Correlations of |r| > .10 have p < .05. E = Extroversion, ed =
educational attainment, intel = intelligence, Lies = Lie scale (social desirability), N
= Neuroticism, P = Psychoticism.
We see evidence of assortative mating, replicating the findings of the original
Spanish study: .81 for age, .65 for education, .48 for intelligence, .26 for the Lie
scale (socially desirable responding). The main personality dimensions of the
Eysenck test, however, show scant evidence of assortative mating: -.01 for
Extroversion, .07 for Neuroticism, and .13 for Psychoticism. These findings are in
line with most of the literature on self-reported personality data. We also see some
MANKIND QUARTERLY 2021 61:3
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expected cross-trait correlations within person, and cross-trait assortative mating
correlations, chiefly for education and intelligence. The within-person correlations
between education and intelligence are relatively weak compared to typical
values in the literature, which may be due to the student sampling procedure.
Generally speaking, the cross-trait assortative mating correlations are similar to
the within-self correlations. For instance, father educational attainment correlates
-.20 with mother’s Neuroticism, and so does mother’s own educational
attainment.
To further explore possible nonlinear relationships, we plotted the six (quasi)-
continuous phenotypes across the couples. Prior to this, they were standardized
to allow plotting using the same x and y axis scales. The results are shown in
Figure 5.
Figure 5. Scatterplots with smoothing fits for 6 phenotypes (standardized within
sex). Smooth fit via LOESS. E = Extroversion, ed = educational attainment, intel
= intelligence, Lies = Lie scale (social desirability), N = Neuroticism, P =
Psychoticism.
KIRKEGAARD, E.O.W. COMPLEX ASSORTATIVE MATING PATTERNS IN HUMANS
587
Aside from a slight indication of a nonlinear pattern for Psychoticism (P),
there appears to be no evidence of nonlinearity in these figures. To quantitatively
confirm this result, we fit regression models for each phenotype. There were 4
models for each, 2 linear models in each direction (i.e., predict male phenotype
from female and vice versa), as well as 2 nonlinear models with a natural spline.
For the resulting models, we compared the adjusted R2 values to look for
incremental validity. Table 1 shows the results. We see that there is essentially
no evidence of any nonlinearity, the gain columns are near-zero, as was expected
from inspection of the scatterplots.
Table 1. Model fit results (adjusted R2) for models estimating nonlinear patterns.
Trait
Mom
linear
Mom
nonlinear
Dad
linear
Dad
nonlinear
Mom
Dad
Age
0.66
0.66
0.66
0.66
0.00
0.00
Neuroticism
0.00
-0.01
0.00
0.00
-0.01
0.00
Extroversion
0.00
0.00
0.00
-0.01
0.01
0.00
Psychoticism
0.01
0.03
0.01
0.02
0.02
0.00
Lie scale
0.07
0.07
0.07
0.06
0.00
0.00
Intelligence
0.23
0.26
0.23
0.23
0.03
0.00
To examine the possibility of cross-trait trade-offs, we computed the distance
between each couple based on their sex-normed standardized values. A value of
0 thus means partners have the same relative phenotypic standing (e.g., both are
average, or both are 1 standard deviation above average). A value of 1 means
that the female is 1 standard deviation higher in the phenotype, though not
necessarily in the natural unit (e.g. females who are +1 SD above female average
height are still below the average male norm). Figure 6 shows the correlation
matrix of the resulting distance scores.
The correlations among distance scores revealed a few results. Chiefly, we
see that couples more distant in Neuroticism and Psychoticism were closer in the
Lie scale. Reversely, couples more distant in education were also more distant in
intelligence, and the same were seen for Psychoticism and Neuroticism,
Extroversion and Psychoticism, as well as Lie scale and intelligence.
MANKIND QUARTERLY 2021 61:3
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Figure 6. Heat map of correlation matrix of distance scores. Correlations of |r| >
.10 have p < .05. E = Extroversion, ed = educational attainment, intel =
intelligence, Lies = Lie scale (social desirability), N = Neuroticism, P =
psychoticism.
Variance effects
A reviewer suggested looking into possible variance effects. The idea is here
that though trait levels might not predict the mean values of trait levels in mates,
they might nonetheless predict the variance. For instance, it is well known that
older people’s partners have a wider age span (and thus variance) than younger
people. In particular, older men may sometimes date women several decades
younger than themselves. Statistically, this would then result in heteroscedasticity
(HS), i.e., unequal variance of the residuals across the levels of the predictor
variable (in this case, male age). To test for this, we examined popular methods
for detecting and quantifying HS. However, we found them lacking robustness
against outliers and in ability to detect nonlinear HS. Figure 7 illustrates HS using
simulated data.
KIRKEGAARD, E.O.W. COMPLEX ASSORTATIVE MATING PATTERNS IN HUMANS
589
Figure 7. Illustrations of heteroscedasticity.
We follow the popular approach of using the squared residuals for testing for
HS. We add the twist that we employ rank transformed data to increase the
robustness, and we employ natural splines to detect nonlinear HS (Harrell, 2015).
We carried out a simulation study to test the properties of our approach and the
results indicated they work approximately as expected with regards to rates of
false positives, but that the test sometimes had troubles distinguishing between
linear and nonlinear HS. The details of this are given in the appendix.
Applying the method to the present dataset produces 48 results since there
are 6 traits, 2 directions (male to female trait, and reversely), and 4 tests for each
(linear vs. nonlinear, raw vs. rank transformation). Nominally, 38% of these found
HS (p < .05). However, most of these concern the raw data approach which is not
robust (42% for raw, 33% for ranks). HS for age is the most common finding, but
least interesting (75% for age but 30% for other traits). The supplementary
materials concern detailed output for some of these tests, though all in all, the
findings were not of much interest. Figure 8 shows an example. The results show
that while we detect some HS, it seems to be of either little importance
(intelligence, Extroversion), dubious reality (Psychoticism), or unsurprising (age).
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Figure 8. Observed heteroscedasticity in cross-mate trait association for 4 traits.
Discussion
The present study examined assortative mating in a relatively large Spanish
dataset of parents (npairs = 340). We find evidence of assortative mating of
approximately typical magnitudes as reported in the literature (Luo, 2017). We
examined the data for possible nonlinear mate trait associations but found not
much. Simulations indicated that we probably had good power to detect nonlinear
patterns if they had existed and were large enough to care about. Thus, nonlinear
mating patterns do not seem to exist for the traits examined in this study. We
furthermore examined the data for evidence of trade-offs between traits. We did
find some correlations that are larger than chance, but no particular model was
suggested by the overall pattern of results. Generally speaking, the associations
were weak across traits that were not correlated within persons, thus suggesting
a general lack of trade-offs. At a reviewer’s request, we examined the data for
heteroscedasticity (i.e. non-constant residual variance across predictor’s range).
We find evidence of such heteroscedasticity, but it was generally not interesting.
The most robust finding was that for age, mothers’ age varied more with
increasing fathers’ age, a well-known phenomenon (Buss, 2019).
KIRKEGAARD, E.O.W. COMPLEX ASSORTATIVE MATING PATTERNS IN HUMANS
591
The primary limitations of the study are as follows. First, the personality data
were based on self-report measures. Self-report measures have been found to
be problematic in multiple domains. In psychiatry, other-reports (and combined
self + other reports) produce higher heritability estimates (Faraone & Larsson,
2019), as do other-reports in personality psychology, with heritabilities around
80% compared to the usual 40-50% (Riemann et al., 1997). Multi-informant data
of anti-social behavior reached a heritability of 96% whereas typical single-
measure data only find about 30-50% (Baker et al., 2007). Other-reports of
personality also show stronger correlations to outcomes such as academic
achievement and job performance (Connelly & Ones, 2010). Other-reported job
performance (compared to self-reported) shows stronger correlations to
objectively measured job performance (Jaramillo et al., 2005). Taken together,
this casts doubt on some of the low or null findings in the present study. It may
be that self-rating biases are (negatively) confounded such that spouses rate
themselves unrelated in personality, perhaps using each other as reference
groups (Heine et al., 2002). Future studies should attempt to get third party-
reported, or at least, spouse-reported personality data for further analysis.
Second, the sample was recruited from students’ parents at the university,
causing some restriction in range in parental education and intelligence. This
seemed to cause some lower than expected levels of correlations among
education and intelligence, but was unlikely to cause notable range issues for the
other traits examined.
Third, the collection of traits examined was less than ideal. It would be
preferable to have a broader personality scale (e.g. based on item data from IPIP)
in future studies, as well as measures of attractiveness, BMI/obesity, humor
ability, psychopathology, political & religious views, and other traits known to
show strong relations between mates. Future studies should attempt to collect a
broad set of traits for analysis.
Supplementary materials
Data, R code, code output (R notebook), and figures are available at
https://osf.io/qjcp6/. The R notebook is also available at
https://rpubs.com/EmilOWK/complex_assortative_mating. For the study of
heteroscedasticity, see https://rpubs.com/EmilOWK/heteroscedasticity.
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Appendix
TIG domino test
Example items from the TIG domino test. Copied from the Spanish test manual.
Heteroscedasticity testing
We implemented an approach using the popular squared residuals approach
to detect and quantify heteroscedasticity (HS) in model residuals. The novel part
about our approach is the use of both raw and rank transformed data for
robustness, and the use of natural splines (restricted cubic splines) to test for
nonlinear HS. Specifically, our approach consists of:
1. Fitting a linear model with the predictor and outcome variable.
2. Squaring the residuals from the fitted model.
3. Create rank transformed variables.
4. Fit linear and natural spline models to predict the transformed residuals
from the predictor. These are done using the rms package (Harrell, 2019).
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5. Derive p values for these comparisons, as well as the effect sizes in
adjusted R2. The p values come from the model comparisons, either against the
null (linear model) or the linear model (spline model). These are based on the
likelihood ratio as recommended by Frank Harrell.
To see if the approach worked as expected, we simulated 30,000 datasets
at n = 200 with either no HS, linear HS, or nonlinear HS (10,000 for each). This is
the same approach as that for Figure 2 in the main text except that used n = 1000
instead of n = 200. The strength of the HS was set to 5. This is done by multiplying
the standard normal residuals by the HS strength as well as a factor based on the
X value. Thus the standard deviations of the residuals are up to 5 times larger at
the designated points on the X axis. For the linear HS, the max value is at x=10,
and for the nonlinear HS case, it is at both x=0 and x=10.
For each of the simulated datasets, we fit the 4 regression discussed above,
and saved the p values and adjusted R2 values. We then plotted the distribution
of the p values by simulation scenario, shown in Figure A1.
Figure A1. Distribution of p values from simulation tests.
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597
Quantitatively, we can also examine how many p values are below a certain
threshold to examine the false positive and negative rates. These are shown in
Table A1.
Table A1. Quantitative results of p values from simulations.
Condition
Test
p mean
Fraction p<05
Fraction p<01
no HS
linear rank
0.501
0.053
0.011
no HS
linear raw
0.493
0.053
0.011
no HS
spline rank
0.494
0.048
0.009
no HS
spline raw
0.497
0.047
0.009
linear HS
linear rank
0.000
1.000
1.000
linear HS
linear raw
0.000
1.000
1.000
linear HS
spline rank
0.381
0.132
0.038
linear HS
spline raw
0.436
0.123
0.048
nonlinear HS
linear rank
0.500
0.049
0.009
nonlinear HS
linear raw
0.402
0.139
0.046
nonlinear HS
spline rank
0.000
1.000
1.000
nonlinear HS
spline raw
0.000
1.000
1.000
The results show that multiple things of interest:
1. For the case with no HS (row 1 in Figure A1), the distribution of p values is
roughly uniform. The table results show the false positive rates are close to the
theoretically expected ones (rows 1-4 in Table A1).
2. For the case with linear HS (row 2 in Figure A1; rows 5-8 in Table A1), the
distribution of p values for linear tests is almost entirely at 0, indicating ~100%
power to detect the HS. For the spline tests, the p values are skewed towards
0, but not overwhelmingly so (12-13%).
3. For the case with nonlinear HS (row 3 in Figure A1; rows 9-12 in Table A1),
the distribution of p values from spline tests is almost entirely at 0 (i.e. ~100%
power), while the linear tests using raw data (but not ranks) are skewed
towards 0, but again not overwhelmingly so (14%).
Thus, to summarize, the false positive rates are close to the expected rates
(i.e., p = .05 means 5% false positives given null HS). For the true cases of linear
and nonlinear HS, the detections have good power, but sometimes detect the
wrong kind of HS, detecting as linear when nonlinear or reversely about 13% of
the time at this sample size and HS strength. It is strange that the spline false
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positive is only seen for the raw data, but not the rank transformed. Further
research might clarify this anomaly.
