# Analysis of Pleiotropy for Testosterone and Lipid Profiles in Males and
Females

Srijan Chattopadhyay<EMAIL_ADDRESS>[ Swapnaneel
Bhattacharyya<EMAIL_ADDRESS>[ Sevantee Basu
<EMAIL_ADDRESS>[ Indian Statistical Institute, 203
Barrackpore Trunk Rd, Kolkata, WB 700108, India.

###### Abstract

In modern scientific studies, it is often imperative to determine whether a
set of phenotypes is affected by a single factor. If such an influence is
identified, it becomes essential to discern whether this effect is contingent
upon categories such as sex or age group, and importantly, to understand
whether this dependence is rooted in purely non-environmental reasons. The
exploration of such dependencies often involves studying pleiotropy, a
phenomenon wherein a single genetic locus impacts multiple traits. This
heightened interest in uncovering dependencies by pleiotropy is fueled by the
growing accessibility of summary statistics from genome-wide association
studies (GWAS) and the establishment of thoroughly phenotyped sample
collections. This advancement enables a systematic and comprehensive
exploration of the genetic connections among various traits and diseases.
additive genetic correlation illuminates the genetic connection between two
traits, providing valuable insights into the shared biological pathways and
underlying causal relationships between them. In this paper, we present a
novel method to analyze such dependencies by studying additive genetic
correlations between pairs of traits under consideration. Subsequently, we
employ matrix comparison techniques to discern and elucidate sex-specific or
age-group-specific associations, contributing to a deeper understanding of the
nuanced dependencies within the studied traits. Our proposed method is
computationally handy and requires only GWAS summary statistics. We validate
our method by applying it to the UK Biobank data and present the results.

GWAS, LDSC, Heritability, Pleiotropy, Testosterone,

###### keywords:

, ,

##  1 Introduction

The term ’pleiotropy’ was introduced by German scientist Ludwig Plate over a
century ago to characterize the phenomenon wherein a hereditary unit
influences more than one trait within an organism (Stearns (2010)).
Subsequently, pleiotropy has remained a subject of extensive research and
discussion. Prior to the rise of human genetics as a prominent field,
investigations into pleiotropy primarily occurred within model organisms and,
at a more theoretical level, within the realm of evolutionary biology (Wagner
and Zhang (2011)). Throughout the past few decades, numerous proposals have
emerged regarding the classification of various forms of pleiotropy, and
Solovieff et al. (2013) outlined three main categories of pleiotropy in the
realm of complex traits. Besides that, based on the extent to which genetic
overlap is assessed, pleiotropy analysis is classified into genome-wide,
regional, and single variants. Genome-wide methods mostly seek the extent to
which genetic factors contribute to the covariance between two traits which is
captured by additive genetic correlation (Hackinger and Zeggini (2017)).

The additive genetic correlation is a quantitative genetic parameter, which is
usually defined as the correlation of breeding values of traits and has been
anticipated to signify the pleiotropic influence of genes or the correlation
between causal loci in two traits. The predominant source of the correlation
stems from the pleiotropy of genes, although Linkage Disequilibrium (LD) can
also play a contributory role. If two traits are influenced by genetic
variants that are physically close to each other on a chromosome, LD can lead
to a correlation in the inheritance of these traits. In other words, if there
is LD between the genetic variants associated with two traits, it means that
these variants tend to be inherited together. As a result, the traits they
influence may also show a correlation. Studies on genetically correlated
traits enhance our comprehension of complex characteristics by unveiling
genetic variations contributing to diseases, refining genetic prediction, and
providing valuable insights for informing therapeutic interventions.

Genome-wide association studies (GWAS) have introduced a novel paradigm for
assessing additive genetic correlation s among various disease traits. The
primary objective of Genome-wide association studies (GWAS) is to discern
associations between genotypes and phenotypes by examining disparities in the
allele frequency of genetic variants among individuals who share similar
ancestry but exhibit phenotypic differences. To date, more than 5,700 Genome-
wide association studies (GWAS) have been undertaken, investigating a spectrum
of more than 3,300 traits (Watanabe et al. (2019)). An impetus for enhanced
statistical power has propelled Genome-wide association studies (GWAS) sample
sizes significantly beyond the threshold of one million participants (Lee et
al. (2018), Jansen et al. (2019)). Using modern sophisticated statistical
tools, besides individual-level genetic data, additive genetic correlation can
also be estimated using GWAS summary statistics (Van Rheenen et al. (2019)).
Linkage disequilibrium score regression (LDSC) (Bulik-Sullivan et al. (2015a))
was the first method introduced to estimate the additive genetic correlation
from GWAS summary statistics.

It relies on the observation, anticipated under conditions of polygenicity,
that the extent of linkage disequilibrium (LD) a Single Nucleotide
Polymorphism (SNP) shares with other genetic variants directly influences its
likelihood of being correlated with causal variants. Consequently, a higher
degree of LD corresponds to an increased expected association test statistic.
Due to its exclusive dependence on summary statistics obtained from the
entirety of a Genome-wide association study (GWAS), the LDSC approach proves
to be highly efficient, making it particularly well-suited for analyses
involving very large sample sizes (Warrington et al. ). Also since there is no
bias introduced by sample overlap, LDSC can explore additive genetic
correlations among hundreds of traits with precision (Bulik-Sullivan et al.
(2015b), Brainstorm et al. (2018)).

In this paper, we present a novel approach designed to discern whether the
influence of specific factors on a collection of phenotypes is contingent upon
sex or age group. It is imperative to underscore that a substantial
dissimilarity in the effects across diverse categories, such as sex or age
group, often suggests the involvement of environmental factors. Our proposed
methodology seeks to alleviate the confounding impact of environmental
variables, focusing instead on investigating the potential genetic
underpinnings through an examination of pleiotropy.

Crucially, our method operates seamlessly using only Genome-Wide Association
Study (GWAS) summary statistics, rendering it computationally efficient. By
applying our methodology to the investigation of the impact of testosterone on
the lipid profile (Cholesterol, HDL, LDL, and Triglyceride), we aim to analyze
whether sex disparities in these effects are attributable to purely non-
environmental reasons. Leveraging GWAS Summary Statistics derived from the UK
Biobank Data, our findings align with a well-established and widely recognized
observation: the heightened testosterone production observed in males compared
to females.

This work not only contributes a methodological advancement for dissecting sex
and age group-specific effects but also provides empirical insights into the
genetic factors influencing the observed disparities in the impact of
testosterone on lipid profiles between males and females. The utilization of
GWAS summary statistics from a large-scale dataset like the UK Biobank
enhances the robustness and applicability of our findings in understanding the
intricacies of genetic contributions to phenotypic variations.

The paper is organized as follows. In Section 2.1, we discuss the basic notion
of additive genetic correlation and LD Score Regression and explain how the
former can be calculated and used to measure pleiotropy. We also discuss some
matrix comparison methods that will be used in our work. In Section 2.2, we
present our whole work in detail. Section 3 contains the outputs of our work
and discussion on our findings and observations. Finally, in Section 4, we
brief up the conclusions from the whole study.

##  2 Methods and Materials

###  2.1 Overview of existing methods

Pleiotropy is evident when a genetic locus influences multiple traits.
Pleiotropy is ascribed to a genetic locus, indicating the potential for
pleiotropic effects at either the DNA variant or genetic level. Within the
framework of discussing additive genetic correlation, our focus is
specifically directed toward DNA variants. A significant divergence arises in
whether the two phenotypes form an integral part of a causal cascade, termed
as vertical or mediated pleiotropy, or if they lack a direct linkage,
characterized as horizontal or biological pleiotropy. A comprehensive
understanding of horizontal pleiotropy holds the promise of enriching our
insights into shared biological processes between traits. Conversely, vertical
pleiotropy assumes importance in unraveling causality, presenting invaluable
insights for formulating intervention strategies in disease prevention. It is
noteworthy that, (Grüneberg (1938),Wagner and Zhang (2011)) in earlier
investigative endeavors, which laid the foundational groundwork for pleiotropy
understanding, vertical pleiotropy was termed ’spurious pleiotropy’—a term now
utilized to describe spurious additive genetic correlation estimates resulting
from biases, misclassification, or linkage disequilibrium. Techniques designed
to discriminate between vertical and horizontal pleiotropy suggest that, more
often than not, a combination of both contributes to the perceived additive
genetic correlation (Zhu et al. (2018), Verbanck et al. (2018)). Genetic
correlation describes the average effect of pleiotropy across all causal loci.

In the quantitative model, for each individual, two traits ($a$ and $b$) are
each decomposed into a genetic value($g$) and a residual value($e$):

$\begin{split}a&=g_{a}+e_{a}\\\ b&=g_{b}+e_{b},\end{split}$ (2.1)

the additive genetic correlation ($\rho_{g}$) of the trait is,

$\rho_{g}=\frac{\sigma_{g_{a},g_{b}}}{\sqrt{\sigma_{g_{a}}^{2}\sigma_{g_{b}}^{2}}},$
(2.2)

where $\sigma_{g_{a},g_{b}}$ is the population covariance of the genetic
values $g_{a},g_{b}$ and $\sigma_{g_{a}}^{2},\sigma_{g_{b}}^{2}$ are the
genetic variance of the two traits in the population. Since, $\rho_{g}$ is a
correlation, $\rho_{g}\in[-1,1]$. From the expression, it is also clear that
$\rho_{g}$ is the proportion of variance that two traits share due to genetic
causes (Falconer (1996), Lynch et al. (1998), Maes (2000)). We denote the
estimated additive genetic correlation by $r_{g}$.

If the traits are standardized (that is, phenotypic variance is 1) and the
genetic values consider only the additive genetic effects, then the genetic
variances are narrow-sense heritabilities and the additive genetic correlation
can be expressed in terms of coheritability ($h_{ab}$) :

$\rho_{g}=\frac{h_{ab}}{\sqrt{h_{a}^{2}h_{b}^{2}}}.$ (2.3)

Thus from the estimates of heritability and coheritability, one can obtain the
estimate of additive genetic covariance. However, during such estimates,
sufficient care needs to be taken for the choice of latent model to avoid
potential bias of the estimates since one of the most common issues of
ambiguity arises in differentiating the conceptual parameters from the
estimates derived from the available data (Zuk et al. (2012)).

To estimate $\rho_{g}$ using LDSC, GWAS Summary Statistic is used as input. In
GWAS, conventional practice involves the utilization of linear or logistic
regression models, depending on whether the phenotype is continuous (such as
height, blood pressure or body mass index) or binary (such as the presence or
absence of disease), respectively. An additional individual-specific random
effect term is also considered in the model to account for genetic relatedness
among individuals (Loh et al. (2018)). The linear regression model for GWAS is
written as,

$\begin{split}Y&=W\alpha+X_{s}\beta_{s}+g+e\\\ g&\sim
N(0,\sigma_{A}^{2}\psi)\\\ e&\sim N(0,\sigma_{e}^{2}I)\end{split}$ (2.4)

where, for each individual, $Y$ is a vector of phenotype values, $W$ is a
matrix of covariates including an intercept term, $\alpha$ is a corresponding
vector of effect sizes, $X_{s}$ is a vector of genotype values for all
individuals at SNP $s$, $\beta_{s}$ is the corresponding fixed effect size of
genetic variant $s$ (also known as the SNP effect size), $g$ is a random
effect that captures the polygenic effect of other SNPs, $e$ is a random
effect of residual errors, $\sigma_{A}^{2}$ measures the additive genetic
variation of the phenotype, $\psi$ is the standard genetic relationship
matrix, $\sigma_{e}^{2}$ measures residual variance and $I$ is an identity
matrix. In logistic regression models, a logit link function is used for
binomially distributed case–control phenotypes to model outcome odds
(Uffelmann et al. (2021)).

In bivariate LDSC, coheritability ($h_{ab}$) can be estimated from GWAS
association statistics for SNP $j$ of both traits ($z_{aj}$ and $z_{bj})$ and
the SNP LD scores ($l_{j}$). The regression relationship is also influenced by
the sample size for two traits ($N_{a},N_{b}$) and the total number of SNPs
($M$). The intercept term estimates sample overlap ($N_{s})$ and hence
reflects the proportion of shared individuals (
$\frac{N_{s}}{\sqrt{N_{a}N_{b}}}$) and their phenotypic correlation
($\rho_{p}$):

$E[z_{aj}z_{bj}|l_{j}]=\frac{\rho_{p}N_{s}}{\sqrt{N_{a}N_{b}}}+\frac{h_{ab}}{M}\sqrt{N_{a}N_{b}}l_{j}$
(2.5)

In the same way, heritability can be computed by taking $a=b$ (Van Rheenen et
al. (2019)). It is to be noted that in the latter case, an additional
intercept term is often incorporated to accommodate residual confounding, such
as population stratification, within a given dataset.

It is to be noted that, the precision of the above estimates are reflected in
the corresponding p-values and standard errors. While the estimates are
derived separately, it is noteworthy that they commonly originate from a
singular dataset. Therefore, necessary multiple testing corrections such as
the Bonferroni multiple testing method (Holm (1979)), Benjamini-Hochberg
method (Benjamini and Hochberg (1995)) are applied to the p-values to keep the
overall error rate (or false positive rate) to less than or equal to the user-
specified p-value cutoff or error rate. However, the estimates are kept
unchanged.

The additive genetic covariance matrix ($\mathbf{G}$) plays a crucial role in
modeling the evolutionary dynamics and figuring the principle of adaptation
for the vector of mean phenotypes
($\mathbf{\bar{z}}=(\bar{z}_{1},\cdots,\bar{z}_{m})^{T}$ with T denoting
transpose) in a population (Lande (1979)). If the mean fitness in the
population is denoted by $\bar{W}$, the evolution of the mean phenotype vector
($\Delta\mathbf{\bar{z}}$) satisfies the following relation, known as
multivariate evolutionary response equation for natural selection

$\Delta\mathbf{\bar{z}}=\mathbf{G}\hskip 2.84544pt\nabla\text{In}\hskip
2.84544pt\bar{W}$ (2.6)

where $\text{In}\hskip 2.84544pt\bar{W}$ is the selection gradient. Thus, the
additive genetic covariance matrix ($\mathbf{G}$) along with the selection
gradient ($\text{In}\hskip 2.84544pt\bar{W}$) jointly determines the
trajectory and speed of the evolution.

To find sex-specific or age-specific dependencies we compute the additive
genetic covariance matrices for each sex or age group and implement matrix
comparison methods to compare the additive genetic covariance matrices
calculated for each category. Numerous approaches have been suggested and
employed for the comparison of covariance and correlation matrices. Shaw et
al. (1995), Roff (2000), Bégin and Roff (2001) introduces several methods of
matrix comparison. When a population is exposed to random linear selection
gradients population-level consequences of matrix differences can be compared
using Random skewer analysis introduced in Cheverud (1996), Cheverud and
Marroig (2007), Roff et al. (2012), Aguirre et al. (2014). An alternative
approach due to Krzanowski (Krzanowski (1979), Aguirre et al. (2014)) involves
the comparison of the genetic covariance matrix based on the subpart of the
matrix which contains most of the genetic variance and this method is known as
Krzanowski’s common subspace analysis. Another method that serves as a
comprehensive assessment of matrix similarity through a sequence of
hierarchical tests is Flury hierarchy analysis (Flury (1988), Arnold and
Phillips (1999), Phillips and Arnold (1999), Steppan, Phillips and Houle
(2002), Roff et al. (2012)). In this paper, for our purpose, we use
Krzanowski’s common subspace analysis and random skewer analysis.

In Krzanowski’s common subspace analysis to find the similarity among
subspaces of $G$, this method starts by finding the common subspace
$\mathbf{H}$ among $p=2$ species as

$\mathbf{H}=\sum_{t=1}^{p}\mathbf{A_{t}A_{t}^{T}}$ (2.7)

where $t=1,\cdots,p$ are index species and $\mathbf{A_{t}}$ contains the
subset $k_{t}$ eigenvectors of $G_{t}$ as columns and $k$ is an integer
smaller than the dimension of $G$, which is chosen initially (in our work, we
choose $k$ to be 2). These $k$ vectors define the dominant subspace of the $G$
matrices and only a $k$ dimensional comparison is interpretable as adding any
further dimension will make that orthogonal to one of the species subspaces
$(k=\text{min}(k_{i}),i=1,\cdots,p)$ (Krzanowski (1979)). Henceforth, an
eigendecomposition of $\mathbf{H}$ is obtained to find the $k$ eigenvectors
$\mathbf{h_{i}}$ of $\mathbf{H}$ that contain the genetic variation of the
linear trait combination that is shared across the $p$ species. The
eigenvalues of matrix H can attain a maximum value of $p=2$. In the limit,
when an eigenvector is associated with an eigenvalue of 2, it indicates that a
linear combination of traits completely elucidates the genetic variance shared
across the three species (Gosden and Chenoweth (2014)).

If an eigenvalue deviates from this maximum value of $p=2$, it implies that
the trait combination corresponding to the eigenvector of $\mathbf{H}$ cannot
be entirely reconstructed from the $k=2$ eigenvectors of matrix G in at least
one of the species. This departure signals that the dominant aspect of the G
matrix in at least one species is not perfectly aligned with the eigenvector
of $\mathbf{H}$ (Gosden and Chenoweth (2014), Aguirre et al. (2014)). The
dissimilarity in alignment can be quantified by evaluating the angle
$\delta_{t}$ between each eigenvector($h_{i}$) of matrix $\mathbf{H}$ and the
subspaces of species $t$ ($\mathbf{A_{t}}$) as(Krzanowski (1979), Gosden and
Chenoweth (2014))

$\delta_{t}=\cos^{-1}\left(\sqrt{h_{i}^{T}\mathbf{A_{t}A_{t}^{T}}h_{i}}\right)$
(2.8)

The random skewer analysis is grounded in the multivariate breeder’s equation,
$\Delta z=G\beta$, wherein $\Delta z$ represents the vector of trait changes
and $\beta$ denotes a vector of selection gradients. This analytical approach
derives its significance from the biological interpretation inherent in the
equation. Selection vectors $\beta$ are stochastically generated and projected
through the G matrices, thereby yielding the predicted response to selection,
$\Delta z$. The angle $\theta$ between $\Delta z$ and $\beta$ serves to
quantify the extent of deflection. Moreover, if we want to draw comparisons
between two matrices $G_{1},G_{2}$ then for the same vector $\beta$, the angle
($\phi$) between $\Delta z_{1},\Delta z_{2}$ where $\Delta z_{1}=G_{1}\beta$
and $\Delta z_{2}=G_{2}\beta$ is computed. It is noteworthy that to compute
the angle ($\phi$) between two vectors $v_{1},v_{2}$ the following formula is
used,

$\phi=\cos^{-1}\left(\frac{{v_{1}^{T}v_{2}}}{{\sqrt{v_{1}^{T}v_{1}\cdot
v_{2}^{T}v_{2}}}}\right)$ (2.9)

The computed angle measures the disparity between the two matrices under
consideration which in our context are the additive genetic covariance and
correlation matrix for males and females.

###  2.2 Our work

In this section, we elucidate our approach to scrutinizing sex-specific or
age-specific dependencies among a set of phenotypes influenced by a singular
factor. This methodology is applied to the UK Biobank dataset, aiming to
discern the impact of testosterone on cholesterol, HDL, LDL, and triglyceride
levels for males and females.

####  2.2.1 The data

Our data is taken from GWAS round 2 dataset from UK Biobank - Neale lab
(released publicly on 1st August 2018). GWAS analysis of 7,221 phenotypes
across 6 continental ancestry groups are in the UK Biobank dataset. We took
Cholesterol(mmol/L), HDL Cholesterol (mmol/L), LDL direct (mmol/L),
Triglycerides (mmol/L), and Testosterone (nmol/L), each for males, females and
combined. In each of the dataframe, there were 13791468 number of subjects,
and the columns were ”variant”, ”minor_allele”, ”minor_AF”,
”low_confidence_variant”, ”n_complete_samples”, ”AC”, ”ytx”, ”beta”, ”se”,
”tstat”, ”pval”. ”variant” is Variant identifier in the form
”chr:pos:ref:alt”, where ”ref” is aligned to the forward strand of GRCh37 and
”alt” is the effect allele. ”minor_allele” is the minor allele (alt allele is
not always minor). ”minor_AF” is frequency of the minor allele in the
n_complete_samples defined for the corresponding phenotype.
”low_confidence_variant” contains flag indicating low confidence results based
on the following heuristics: Case/control phenotypes: minor_AF ¡ 0.001,
Categorical phenotypes with less than 5 categories: minor_AF ¡ 0.001,
Quantitative phenotypes: minor_AF ¡ 0.001. ”n_complete_samples” represents
number of samples defined for this phenotype. ”AC” is allele count of alt
allele calculated on dosages within n_complete_samples. ”ytx” is dot product
of phenotype vector y and genotype vector x (alt allele count in cases for
case/control phenotypes). ”beta” is estimated effect size of alt allele. ”se”
is estimated standard error of beta. ”tstat” is t-statistic of beta estimate
(= beta/se). Finally, ”pval” is p-value of beta significance Wald’s test.

####  2.2.2 Finding the additive genetic correlation using LDSC

The first step is to obtain the additive genetic correlation between pairs of
traits. We use LDSC to find the additive genetic correlation of the traits
under consideration. We used the summary statistic from GWAS for the two
traits of interest which include information such as SNP (Single Nucleotide
Polymorphism) identifiers, effect sizes, and standard errors. We first add
RSID (Reference SNP IDentifier) in the original dataset to ensure consistency
and clarity in identifying genetic variations.

After the initial dataset is prepared, we undergo the process of munging for
organizing and formatting the data to make it suitable for analysis with LDSC.
The goals involve ensuring the integrity and consistency of genetic data
files, particularly in the context of Genome-Wide Association Studies (GWAS).
The process includes validating file formats, identifying and addressing
issues with data delimiters, confirming header name consistency, and checking
for multiple models or traits in GWAS. Additionally, there is a focus on
maintaining uniformity in SNP ID formats, handling missing or incomplete
information, and verifying the presence of essential columns. Quality checks
cover various aspects, such as statistical values, allele frequencies, and
potential issues like allele flipping or strand ambiguity. The ultimate aim is
to prepare the data for downstream analyses, ensuring accuracy and adherence
to standards compatible with tools like LDSC and IEU OpenGWAS, with the
flexibility to perform liftover to a desired reference genome if needed. The
necessary codes for our work are taken from Bulik-Sullivan et al. (2015b),
Zheng et al. (2016) available publicly${}^{\href
https://github.com/bulik/ldsc,\href
https://neurogenomics.github.io/MungeSumstats/articles/MungeSumstats.html}$.

Finally, the LDSC software is used to run LD Score Regression on the munged
data. This step involves calculating LD Scores for each SNP in the reference
panel, obtaining SNP-based heritability estimates and finally calculating the
additive genetic correlation using Equation 2.3. Thus we have the estimate of
the additive genetic correlation for a pair of traits. Applying the method
described in this section individually for males and females, for all pairs of
traits under consideration, we have the additive genetic covariance and
correlation matrix for males and females.

####  2.2.3 Comparing sex-specific additive genetic correlation matrices

After getting the whole variance-covariance matrix for Males and Females, we
compare the whole multivariate structure for the mentioned traits and even
after removing some of the traits as long as we get an increasing similarity
between the two sex. We did the comparison by different methods of comparing
matrices (some are specially designed and used for covariance matrices). We
used Krzanowski’s common subspaces analysis (Chakrabarty and Schielzeth
(2020), Aguirre et al. (2014)). For this, we used evolqg (Melo et al. (2015))
package in R and the function named KrzSubspace for subspace analysis and
noted down the outputs in Section 3. We also did a principal component
analysis on those covariance matrices and compared the direction for maximum
variance explained between males and females and attached those plots with
discussion in the next section. For that, we have used eigen function from the
base package of R. Furthermore, we have used Random skewers method (Cheverud
(1996), Aguirre et al. (2014)) on different subsets of the whole covariance
structure and discussed those results in Section 3. For that, we used phytools
(Revell (2012)) package in R and used the skewers function. Finally, we use a
probabilistic approach to measure the disparity between the additive genetic
covariance matrices for males and females. Let $\hat{G}_{m}$ and $\hat{G}_{f}$
be the estimates of the additive genetic covariance matrix for males ($G_{m}$)
and females ($G_{f}$). We take the Bhattacharyya distance between normal
distributions with mean $0$ and covariance matrices $\hat{G}_{m},\hat{G}_{f}$
to be our measure of disparity of $G_{m}$ and $G_{f}$ i.e. if
$d\left(G_{m},G_{f}\right)$ denotes the disparity between the additive genetic
covariance matrices for males and females and for two probability
distributions $P,Q$ $D_{B}\left(P,Q\right)$ denotes the Bhattacharyya distance
(Bhattacharyya (1943)) between $P,Q$ then,

$d\left(G_{m},G_{f}\right)=D_{B}\left(N\left(0,\hat{G}_{m}\right),N\left(0,\hat{G}_{f}\right)\right)$
(2.10)

where $N\left(0,\hat{G}_{m}\right),N\left(0,\hat{G}_{f}\right)$ denote the
multivariate normal distributions with mean $0$ and covariance matrices
$\hat{G}_{m},\hat{G}_{f}$ respectively. For the computation of the
Bhattacharyya distance, we use bhattacharyya.dist function of fpc(Hennig and
Imports (2015)) package in R. We compare the results obtained using these
three methods in Section 3.

##  3 Results and Discussion

In this section, we denote the estimated additive genetic correlation by
$r_{g}$ while $p$ denotes the p-value of the corresponding estimates after
necessary multiple testing corrections.

Figure 1: additive genetic correlation of Cholesterol and Testosterone, HDL
and Testosterone(Sex-specific)

Figure 2: additive genetic correlation of LDL and Testosterone, Triglyceride
and Testosterone (Sex-specific)

Figure 3: Additive genetic correlations for Males and Females

From Figure 1, we can see that for males, there is a positive additive genetic
correlation between cholesterol and testosterone ($r_{g}=0.1242,p=0.0212$),
i.e. with a significant amount of confidence against the null. This is also
checked by the results of Gårevik et al. (2012), where they concluded that
total cholesterol level was significantly increased by 15% two days after the
testosterone injection (p = 0.007). The results are in concordance with that
of Freedman et al. (1991), where they also acknowledged that levels of HDL
cholesterol are positively associated with endogenous levels of testosterone
in men, but they also made a cautious comment about the causality behind this
correlation, but by additive genetic correlation study through LDSC, we
confirm that the correlation of phenotypic effects of these by genetic
variants across the genome on testosterone and cholesterol is significant and
positive, i.e. due to genetic link between these, the correlation remains
positive for males. Whereas for females we note the exact opposite thing,
i.e., we see that the additive genetic correlation between these for females
is negative with both confidence coefficients being negative. Also, we note an
interesting thing in fig. 3 that the confidence intervals for males are way
more shorter than females for each of the additive genetic correlation s. That
is an indicator of the fact that due to a lower amount of testosterone being
produced in females, the effects with the other phenotypes are not that much
stronger as of males. That’s why we notice less evidence against the zero
additive genetic correlation s as the p-value being
$0.0586,0.147,0.282,0.04245$. Three of them are bigger than the standard
$0.05$ norm and two are very close, while only one being less but not in a
significant amount. That clearly indicates that for females, the additive
genetic correlation s between testosterone with the others is not that much
strong. On the other hand, we collect strong evidence against zero correlation
for most of the cases for males, and with an almost nonzero correlation in
each of the cases. That signifies the importance of the amount of testosterone
to keep body hormones at their level very much. Leinonen et al. (2023) study
shows that increasing testosterone in males overall promoted a favorable
metabolic profile. It was positively correlated with HDL, cholesterol
($r_{g}>0.20,p<0.0011$), while negatively correlated with triglycerides
($r_{g}=-0.18,p=4.23-05$), that is concordant with our findings (Figure 1, 2,
3) using only the UK Biobank GWAS, whereas they used some other resources and
constructed polygenic scores (PGS) for total testosterone, sex-hormone binding
globulin (SHBG) and free testosterone. In contrast, they observed a negative
correlation of HDL cholesterol with testosterone in females
($r_{g}=-0.18,p=4.2e-05$). But our study got the same results but at a lower
significance level, that is for females, the linear correlation between those
is not very significant, i.e. quite ambiguous. Ivanova et al. (2023) also
obtained the same types of results for subjects in Russia. Testosterone is the
most important circulating testicular androgen in men. Prospective cohort
studies have shown that low serum testosterone levels are associated with a
number of common disorders like cardiovascular disease (Vandenput and Ohlsson
(2014)), which is caused by lipid profiles, though previous research are
diverse in this context. According to Rubinow and Page (2012) even when
reductions in HDL-cholesterol are observed as a consequence of androgen
therapy, the implications for cardiovascular risk modification remain highly
uncertain. whereas Laughlin, Barrett-Connor and Bergstrom (2008) claims that
declining testosterone levels in elderly men are thought to underlie many of
the symptoms and diseases of aging and Vikan et al. (2009) also consciously
reported the results obtained by their research about this.

Trait1 | Trait2 | $r_{g}$ | SE($r_{g}$) | CI($r_{g}$) | $\text{cov}_{g}$ | SE($\text{cov}_{g}$) | CI($\text{cov}_{g}$) | p-value | $p_{\text{adjusted}}$
---|---|---|---|---|---|---|---|---|---
Triglyceride(Both) | Testosterone(Both) | -0.37 | 0.03 | [-0.43, -0.30] | -0.04 | 0.01 | [-0.24, -0.18] | 0.00 | 0.00
Cholesterol(Both) | Testosterone(Both) | 0.12 | 0.05 | [0.03, 0.21] | 0.01 | 0.00 | [0.00, 0.05] | 0.01 | 0.02
HDL(Both) | Testosterone(Both) | 0.30 | 0.02 | [0.26, 0.35] | 0.04 | 0.00 | [0.10, 0.13] | 0.00 | 0.00
LDL(Both) | Testosterone(Both) | 0.07 | 0.04 | [-0.01, 0.15] | 0.01 | 0.00 | [0.01, 0.06] | 0.11 | 0.12
HDL(Female) | Testosterone(Female) | -0.05 | 0.03 | [-0.11, 0.02] | -0.01 | 0.00 | [-0.02, 0.00] | 0.15 | 0.16
Cholesterol(Female) | Testosterone(Female) | -0.08 | 0.04 | [-0.16, 0.00] | -0.01 | 0.00 | [0.00, 0.03] | 0.06 | 0.07
LDL(Female) | Testosterone(Female) | -0.05 | 0.05 | [-0.14, 0.04] | -0.00 | 0.00 | [0.01, 0.03] | 0.28 | 0.29
Triglyceride(Female) | Testosterone(Female) | -0.08 | 0.04 | [-0.17, -0.00] | -0.01 | 0.01 | [-0.02, 0.00] | 0.04 | 0.05
Cholesterol(Female) | HDL(Female) | 0.38 | 0.08 | [0.23, 0.53] | 0.06 | 0.01 | [0.04, 0.08] | 0.00 | 0.00
LDL(Female) | HDL(Female) | 0.05 | 0.07 | [-0.09, 0.18] | 0.01 | 0.01 | [-0.01, 0.02] | 0.49 | 0.49
Triglyceride(Female) | HDL(Female) | -0.65 | 0.06 | [-0.77, -0.53] | -0.13 | 0.02 | [-0.17, -0.09] | 0.00 | 0.00
Triglyceride(Female) | LDL(Female) | 0.44 | 0.09 | [0.27, 0.61] | 0.06 | 0.01 | [0.04, 0.08] | 0.00 | 0.00
Cholesterol(Female) | LDL(Female) | 0.94 | 0.01 | [0.91, 0.96] | 0.10 | 0.03 | [0.05, 0.15] | 0.00 | 0.00
Cholesterol(Female) | Triglyceride(Female) | 0.28 | 0.07 | [0.14, 0.42] | 0.04 | 0.01 | [0.02, 0.07] | 0.00 | 0.00
Cholesterol(Male) | Testosterone(Male) | 0.12 | 0.05 | [0.02, 0.23] | 0.02 | 0.01 | [0.00, 0.03] | 0.03 | 0.03
HDL(Male) | Testosterone(Male) | 0.29 | 0.03 | [0.23, 0.36] | 0.05 | 0.01 | [0.04, 0.06] | 0.00 | 0.00
LDL(Male) | Testosterone(Male) | 0.14 | 0.06 | [0.02, 0.26] | 0.01 | 0.01 | [0.00, 0.03] | 0.02 | 0.03
Triglyceride(Male) | Testosterone(Male) | -0.37 | 0.04 | [-0.45, -0.28] | -0.07 | 0.01 | [-0.08, -0.05] | 0.00 | 0.00
Cholesterol(Male) | HDL(Male) | 0.51 | 0.10 | [0.31, 0.71] | 0.07 | 0.01 | [0.05, 0.09] | 0.00 | 0.00
Triglyceride(Male) | HDL(Male) | -0.61 | 0.07 | [-0.76, -0.47] | -0.12 | 0.03 | [-0.17, -0.06] | 0.00 | 0.00
LDL(Male) | HDL(Male) | 0.34 | 0.11 | [0.13, 0.56] | 0.04 | 0.01 | [0.02, 0.06] | 0.00 | 0.00
Triglyceride(Male) | LDL(Male) | 0.23 | 0.09 | [0.05, 0.41] | 0.03 | 0.01 | [0.01, 0.04] | 0.01 | 0.02
Cholesterol(Male) | LDL(Male) | 0.96 | 0.01 | [0.94, 0.98] | 0.08 | 0.02 | [0.03, 0.12] | 0.00 | 0.00
Cholesterol(Male) | Triglyceride(Male) | 0.25 | 0.10 | [0.06, 0.44] | 0.03 | 0.01 | [0.01, 0.06] | 0.01 | 0.02
Table 1: Additive Genetic Correlations and Covariances between the traits with
respective Standard errors and 95$\%$ approximate confidence Intervals.
”$r_{g}$” represents the estimated additive genetic correlation, ”SE($r_{g}$)”
is the estimated standard errors, ”CI” represents the 95$\%$ confidence
interval, ”$\text{cov}_{g}$” represents estimated Genetic Covariance,
”SE($\text{cov}_{g}$)” denotes estimated standard error of the estimated
genetic covariance and ”$p_{\text{adjusted}}$” denotes the adjusted p-value
after multiple testing error.

Zhang et al. (2014) did a retrospective study on 11000 subjects to evaluate
the relationship between serum total testosterone level and lipid profile
after adjusting for some traditional confounding factors, free thyroid
hormones and TSH in Chinese men, but they only included middle aged and
elderly people (though, finally only 4114 male subjects from the general
population were evaluated). In the bivariate correlation analysis, the
correlation between Cholesterol and Testosterone was $r_{g}=-0.077,p=0.00$,
between $logTG$ and Testosterone, $r_{g}=-0.374,p=0.00$, between LDL and
Testosterone, $r_{g}=-0.069,p=0.00$. So, they didn’t get a strong correlation
for LDL and Total Cholesterol. Firstly, this correlation was not purely
genetic, also number of subjects was much much less in this study compared to
our case, that can introduce a bias (also bias of age in their study) which
can cause the amount of correlation significantly less. However, we got
positive correlation of LDL, HDL and Total Cholesterol with Testosterone with
a very significant p value ($\leq 0.03$) and less amount of SE, whereas
significant negative correlation with Triglyceride which matches with that of
Zhang et al. (2014). Also, Shamim et al. (2019) reported the correlations of
Triglyceride ($p=0.0$), LDL ($p=0.5$), Cholesterol ($p=0.114$) with
Teststerone were negative, and with HDL it was positive ($p=0.001$). However,
the study was only on 200 people and results were highly non-significant.
However none of these studies considered only the additive genetic
correlations from their data, they only estimated the sample total correlation
from the collected data.

Traits | $r_{g}$ | SE($r_{g}$) | p-value | $cov_{g}$ | SE($cov_{g}$) | $p_{adjusted}$
---|---|---|---|---|---|---
Testosterone | 0.01 | 0.05 | 0.87 | 0.00 | 0.01 | 0.87
Cholesterol | 0.85 | 0.04 | 0.00 | 0.09 | 0.02 | 0.00
LDL | 0.81 | 0.06 | 0.00 | 0.07 | 0.02 | 0.00
HDL | 0.91 | 0.02 | 0.00 | 0.18 | 0.03 | 0.00
Triglyceride | 0.90 | 0.03 | 0.00 | 0.17 | 0.03 | 0.00
Table 2: Additive Genetic Correlation between Male and Female traits.
”$r_{g}$” means ”Estimated Additive Genetic Correlation”, ”SE” means estimated
standard errors of the estimates, ”p” is the actual p-value and
”$p_{\text{adjusted}}$” is the adjusted p-value for multiple testing error.
”$cov_{g}$” means the estimated genetic covariance

Figure 4: Heatmap for Estimated Additive Genetic Correlation matrix Gender | Trait | $h(or,var_{g})$ | SE($h(or,var_{g})$) | CI($h(or,var_{g})$)
---|---|---|---|---
M | Testosterone | 0.17 | 0.02 | [0.13, 0.21]
M | Triglyceride | 0.18 | 0.04 | [0.11, 0.25]
M | LDL | 0.07 | 0.03 | [0.01, 0.12]
M | HDL | 0.20 | 0.03 | [0.13, 0.27]
M | Cholesterol | 0.09 | 0.02 | [0.04, 0.14]
F | Testosterone | 0.07 | 0.03 | [0.01, 0.13]
F | Triglyceride | 0.20 | 0.03 | [0.14, 0.26]
F | LDL | 0.09 | 0.03 | [0.04, 0.15]
F | HDL | 0.21 | 0.02 | [0.16, 0.26]
F | Cholesterol | 0.11 | 0.03 | [0.06, 0.16]
Table 3: Heritibility of different traits, ”M” means Male, ”F” means Female.
”h” is the estimated scaled heritibility or genetic variance, ”hse” is the
standard error of the estimate, CI is the approximate 95% Confidence Interval
of the estimates

We observe from Table 3, the heritibilities of the selected lipid profiles and
testosterone are ¡ 0.22 (with SE $\leqslant 0.4$). From Figure 4, the
similarity between the additive genetic correlation matrix for males and
females restricted to only lipid profile (i.e. HDL, LDL, Cholesterol,
Triglycerides) is noteworthy. In particular, the additive genetic correlation
matrices restricted to HDL, LDL, and Cholesterol have high similarity between
males and females. Triglyceride levels and functioning changes during
pregnancy time in Females(Ghio et al. (2011)). That can lead to significant
difference in the distribution structures and sparsity of the underlying
functioning of those.(Herrera et al. (2006)). Zhang et al. (2018) studied the
triglyceride distributions for the people of North China. They performed a
chi-squared test for the distributions of triglyceride sex wise and the
distribution of triglyceride levels was different between males and females
($\chi^{2}=155.77,p<0.001$). Seidell et al. (1991) studied the sex effects of
lipid profiles in European (The Netherlands, Sweden, Italy, and Poland) men
and women, and their study concluded that fat distribution may be responsible
for male/female differences in serum triglycerides but that such conclusions
are less clear for HDL, total and LDL cholesterol. However, our study shows
that the covariance variance structure for HDL, LDL and Cholesterol almost
remains the same for females and males. The additive genetic
variance–covariance matrix (G) summarizes the multivariate genetic
relationships among a set of traits. So, we did a matrix comparison between
females and males using Krzanowski’s common subspaces analysis (Aguirre et al.
(2014), Chakrabarty and Schielzeth (2020)), Random Skewer’s Method (Cheverud
and Marroig (2007), Aguirre et al. (2014)) and also using Bhattacharyya’s
distance assuming gaussianity (Bhattacharyya (1943), Hennig (2010)). All of
them lead to the same result, i.e., the distribution of the lipid profile
concatenated with testosterone is similar for Males and Females. However, the
amount of similarity increases when we remove the testosterone from the
matrix. The shared space matrix is

$\begin{bmatrix}0.19054899&0.2499404&-0.03439405&-0.02583797&-0.3068839\\\
0.24994042&1.1479944&0.31350653&0.52330668&-0.6512001\\\
-0.03439405&0.3135065&0.62441997&0.73292240&0.4520197\\\
-0.02583797&0.5233067&0.73292240&0.89170573&0.4072541\\\
-0.30688389&-0.6512001&0.45201973&0.40725406&1.1453309\\\ \end{bmatrix}$

The largest two eigen values of the shared space matrix are 1.992132 and
1.896234. And, the corresponding eigen vectors are

$\begin{bmatrix}0.05550021&-0.21930879\\\ -0.26871768&-0.72488680\\\
-0.55657367&-0.01659567\\\ -0.65778482&-0.12427234\\\
-0.42691382&0.64087733\\\ \end{bmatrix}$

PC1 vs PC2 plot for the variance-covariance matrices of males and females are
in Figure 6. From Figure 6, we can see that the direction of maximum variance
for males and females are very close to each other. However, a difference in
Triglyceride and Testosterone are noted, which can be explained by the
different functions of those in different sexes as described earlier. However,
when testosterone is removed from the picture, the similarity between the
directions of maximum explained variance even increases (Figure 7) and after
removing triglycerides also, that direction almost becomes the same (Figure
7). That clearly indicates that except the complexities and functionalities
created by Testosterone or Triglycerides in different sex, the lipid profiles
of Males and Females matches very significantly. Keenly looking into Figure 7,
we observe that the first maximum explainable direction is the most similar
for all than the second. That means that most of the noises in the pattern of
these traits in both the sexes is almost the same. Using Random Skewers Method
(with seed 2023 in R), we get $r=0.9119407$ with $p=0.01$. After removing
testosterone, $r=0.9698086$, $p=0.0$, and after removing triglyceride also,
$r=0.9815491,p=0.0$. Using bhattacharrya’s distance under gaussianity
assumption, the calculated measure for the whole matrix is 0.02, which is
pretty small. So, using all the measures we can see that the underlying
variance-covariance structure for the lipid profiles even with testerone is
very much similar for males and females, and the similarity increases as we
remove testosterone and triglyceride from the structure.

Figure 5: Heritibility of the traits Figure 6: PC1 vs PC2 of the full
variannce covariance matrix (including Testosterone)

Figure 7: PC1 vs PC2 plot of the variance covariance matrix (first one,
removing testosterone only and second one, removin triglyceride too)

##  4 Conclusions

Significant Positive and Negative correlations were found between testosterone
and other lipid profiles for male, whereas for females the correlations are
not much significant and almost close to 0. By the multivariate covariance
matrix analysis, we confirm that for males and females, there is not much
difference in the multivariate lipid profile structure.

##  5 Acknowledgements

The research conducted was graciously supported by the National Institute of
Biomedical Genomics, Kalyani, India. We extend our sincere gratitude to our
supervisors, Dr. Analabha Basu, Dr. Anasuya Chakrabarty, and Dr. Diptarup
Nandi, for their invaluable guidance and support throughout this study. We
would like to express a special acknowledgment to Dr. Anasuya Chakrabarty,
whose guidance played a pivotal role in shaping the trajectory of our
research. We are truly appreciative of her continuous involvement and
insights.

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