# Magnetic properties of binary alloys Ni1-xMox and Ni1-yCuy close to critical
concentrations

R.-Z. Lin  These authors contributed equally to this work  Department of
Physics, National Cheng Kung University, Tainan 701, Taiwan Center for Quantum
Frontiers of Research & Technology (QFort), National Cheng Kung University,
Tainan 701, Taiwan Taiwan Consortium of Emergent Crystalline Materials,
National Science and Technology Council, Taipei 10622, Taiwan J.-X. Hsu  These
authors contributed equally to this work  Department of Physics, National
Cheng Kung University, Tainan 701, Taiwan Center for Quantum Frontiers of
Research & Technology (QFort), National Cheng Kung University, Tainan 701,
Taiwan Taiwan Consortium of Emergent Crystalline Materials, National Science
and Technology Council, Taipei 10622, Taiwan E.-P. Liu  Center for Condensed
Matter Sciences, National Taiwan University, Taipei 10617, Taiwan Department
of Physics, National Taiwan University, Taipei 10617, Taiwan W.-T. Chen
Center for Condensed Matter Sciences, National Taiwan University, Taipei
10617, Taiwan Center of Atomic Initiative for New Materials, National Taiwan
University, Taipei 10617, Taiwan Taiwan Consortium of Emergent Crystalline
Materials, National Science and Technology Council, Taipei 10622, Taiwan C.-L.
Huang<EMAIL_ADDRESS>Department of Physics, National Cheng Kung
University, Tainan 701, Taiwan Center for Quantum Frontiers of Research &
Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan Taiwan
Consortium of Emergent Crystalline Materials, National Science and Technology
Council, Taipei 10622, Taiwan

###### Abstract

The search for the ferromagnetic quantum critical point (FM QCP) has always
been a captivating research topic in the scientific community. In pursuit of
this goal, we introduced nonmagnetic transition metals to alloy with elemental
nickel, studying the magnetic properties of nickel binary alloys Ni1-xMox and
Ni1-yCuy as a function of $x$ and $y$ up to the critical concentrations
$x_{cr}$ and $y_{cr}$ at which the FM transition $T_{\rm C}$ disappears.
$T_{\rm C}-x(y)$ phase diagrams were constructed via Arrott-Noakes scaling of
magnetization data. An enhanced specific heat coefficient $C/T$ is observed
near $y_{cr}$, manifesting the effect of quantum fluctuations near a FM-to-
paramagnetic quantum phase transition. This implies the plausible FM QCP in
Ni1-yCuy. However, in the case of Ni1-xMox, although the enhancement of the
specific heat coefficient also occurs as $x$ approaches $x_{cr}$, spin glass
behavior is observed through ac magnetic susceptibility. This observation
rules out the possibility of the existence of the FM QCP in Ni1-xMox.

##  I INTRODUCTION

A ferromagnetic quantum critical point (FM QCP) is one of the most exotic
quantum states in condensed matter physics. Such a state has a FM long-range
ordered ground state before tuning. When some temperature-irrelevant tuning
parameter, such as, physical pressure or chemical substitution, is applied,
the transition temperature $T_{\rm C}$ can be continuously suppressed to
absolute zero at which the FM order is destroyed by quantum fluctuations of
the order parameter [1]. The increasing attention drawn to search for the FM
QCP is not only due to its rarity compared the antiferromagnetic counterpart
but also because the quantum fluctuations near the QCP often lead to emergent
physics phenomena, e.g., unconventional superconductivity and non-Fermi-liquid
(NFL) behavior [2, 3, 1].

Tuning a ferromagnet to the QCP is not trivial [1]. The only controlled method
that has been experimentally [4, 5, 6, 7, 8] and theoretically [9, 10, 11, 12]
verified involves introducing an appropriate amount of chemical disorder. To
meet this condition, a binary alloy of a nonmagnetic transition metal $A$ and
an FM element, such as Fe, Co, or Ni, is arguably the simplest one. It is
natural to expect that the $T_{\rm C}$ of the binary FM alloy can be
suppressed as the concentration of $A$ increases. This is indeed observed in
several Ni${}_{1-x}A_{x}$ alloys [13, 14, 15, 16, 17, 18]. However, for all
the Ni${}_{1-x}A_{x}$ alloys, only one FM QCP exists: Ni1-xRhx with $x=0.375$
where NFL behavior is reported [8, 19]. Therefore, it is worth to further
investigate the magnetic properties of other Ni${}_{1-x}A_{x}$ alloys to
explore other possible FM QCPs. In this work, we report a detailed study on
Ni1-xMox and Ni1-yCuy binary alloys. Structure analysis confirmed the face-
centered-cubic structure of nickel in all samples. The degree of chemical
disorder can be identified through electrical resistivity measurements. The
$T_{\rm C}-x(y)$ phase diagrams were constructed with the aid of magnetic
susceptibility measurements and scaling analysis on isothermal magnetization
curves. Enhanced low-temperature specific heat was observed near the critical
concentrations ($x_{cr}$ and $y_{cr}$) of both alloys at which the $T_{\rm C}$
vanishes, indicating the effect of quantum fluctuations. Finally, the ac
magnetic susceptibility measurement on Ni1-xMox at $x_{cr}$ revealed spin
glass behavior, most probably due to the random alignment of magnetic moments
in the system. This result rules out the existence of the FM QCP in the
Ni1-xMox alloy. In terms of Ni1-yCuy, the current data shed light on the
existence of the FM QCP near $y_{cr}$, and more experiments at lower
temperatures in the future will be needed to confirm this matter.

##  II RESEARCH METHODS

Polycrystalline Ni1-xMox and Ni1-yCuy samples were prepared by arcmelting
high-purity Ni, Mo, and Cu elements. Samples were sealed under vacuum within a
quartz tube. Subsequently, Ni1-xMox underwent annealing at 1,000∘ C for a
duration of five days, and was cooled in air. Ni1-yCuy was subjected to
annealing at 700∘ C for a week, and was quenched in water to avoid the phase
separation below 400∘ C [20]. X-ray measurements were carried out using a
Bruker D2 Phaser diffractometer. The chemical composition was determined by
electron probe microanalysis (EPMA) using a JEOL JXA 8530F Hyperprobe.
Resistivity and specific heat were measured using a Quantum Design (QD)
Dynacool physical property measurement system. Magnetization and magnetic
susceptibility measurements were carried out using a QD magnetic property
measurement system.

The phase purity of samples was confirmed by room temperature powder x-ray
diffraction measurements, as shown in Fig. 1(a) and (c). Because of the
extreme hardness of the arcmelted samples, x-ray diffraction data were
collected on cut and polished specimens. The x-ray patterns were refined using
GSAS II. The resulting lattice parameters vary linearly with nominal Mo and Cu
concentrations, as shown in Fig. 1(b) and (d), consistent with the Vegard’s
law. The EPMA determined compositions also vary linearly with the nominal
composition values, as shown in Fig. 9. Therefore, nominal composition is
adopted throughout the main text.

Figure 1: (a)(c) Representative x-ray diffraction patterns of Ni1-xMox and
Ni1-yCuy at room temperature. (b)(d) Refined lattice constants as a function
of Mo and Cu concentrations.

##  III RESULTS and DISCUSSIONS

###  III.1 Electrical resistivity

Figure 2 shows the temperature dependence of resistivity normalized to its
value at 300 K, $\rho(T)/\rho_{\rm{300K}}$. For Ni1-xMox,
$\rho(T)/\rho_{\rm{300K}}$ decreases as the temperature decreases, as expected
for a metallic sample. The fact that $\rho(T)/\rho_{\rm{300K}}$ at 2 K and
$\rho_{\rm{300K}}$ are both the smallest for $x=0.05$ (Fig. 2(a) and the
inset), the largest for $x=0.13$, and increase with $x$, indicates that the
disorder effect increases with the amount of Mo in Ni1-xMox. For Ni1-yCuy,
metallic behavior, i.e., $\rho(T)/\rho_{\rm{300K}}$ decreases upon cooling, is
seen in $y$ = 0.20-0.56, as shown in Fig. 2(b). As $y>0.56$,
$\rho(T)/\rho_{\rm{300K}}$ slightly increases as the temperature decreases
from 300 to 2 K. This fact and a negative transverse magnetoresistance (not
shown) in $y>0.56$ samples are considered evidence of charge transport
involving weak Anderson localization [21].

In contrast to the monotonic increase of $\rho_{\rm{300K}}$ with $x$ in
Ni1-xMox, $\rho_{\rm{300K}}$ of Ni1-yCuy fluctuates greatly and does not show
a clear relationship with $y$. The $\rho_{\rm{300K}}$ values for $y=0.40-0.60$
is two-three times larger than those for $y=0.20,0.30$, and 0.80, as shown in
the inset of Fig. 2(b). The difference in the evolution of transport behavior
between Ni1-xMox and Ni1-yCuy, as the concentration of nonmagnetic elements
increases, is most probably due to the lower sample inhomogeneity in Ni1-yCuy,
particularly close to $y=0.5$, which is evidenced by a larger error bar in
EPMA results as compared to Ni1-xMox (see Fig. 9).

Figure 2: Normalized resistivity $\rho/\rho_{\rm{300K}}$ for (a) Ni1-xMox and
(b) Ni1-yCuy. The insets show the absolute value of $\rho$ at 300 K for each
concentration.

###  III.2 Magnetic susceptibility

The magnetic susceptibility $\chi=M/H$ under the magnetic field $H=1000$ Oe of
Ni1-xMox and under the magnetic field $H=100$ Oe of Ni1-yCuy is shown in Fig.
3(a) and (c), respectively. When $T$ is lowered through a paramagnetic-to-FM
transition, $\chi$ sharply increases. The $T_{\rm C}$ decreases with
increasing $x$ and $y$, as expected in diluted magnetic alloys, and is
consistent with the previous report [22].

For itinerant magnetic systems, the temperature dependence of the inverse
magnetic susceptibility is analogous to the Curie-Weiss law developed for
local moment systems, i.e.,

$\chi(T)=\frac{D}{T-\theta^{*}_{\rm CW}},$ (1)

where $D$ is a constant and $\theta^{*}_{\rm CW}$ is Curie-Weiss-like (CW-
like) temperature [23, 24]. In other itinerant ferromagnets as the magnetic
order is tuned by chemical substitution, the positive $\theta^{*}_{\rm CW}$
has been found to trace $T_{\rm C}$ well, decrease with increasing
substitutions, and change sign from positive to negative around the critical
concentration [25]. We fit the high temperature $\chi^{-1}(T)$ using Eq. (1),
as shown by a dashed line in Fig. 3(b), and obtained $\theta^{*}_{\rm CW}$ vs.
$x$ and $y$ in Fig. 4(a) and (b), respectively. The data shows that
$\theta^{*}_{\rm CW}$ changes sign at $x=0.10$ and $y=0.56$. Both values are
very close to the critical concentrations $x_{cr}$ and $y_{cr}$ determined
from the $T_{\rm C}$-$x(y)$ phase diagrams, which will be discussed later. The
coincidence of the concentration where the $\theta^{*}_{\rm CW}$ changes sign
and the critical concentration has been observed in other ferromagnets where
QCPs exist at the same tuning parameter [25, 26, 27].

Figure 3: (a)(c) Magnetic susceptibility $\chi$ of Ni1-xMox and Ni1-yCuy
measured with $H=1000$ Oe. (b)(d) Inverse $\chi$ vs. $T$. (a) and (b); (c) and
(d) use the same legends. Figure 4: The Curie-Weiss-like temperature
$\theta^{*}_{\rm CW}$ as a function of (a) $x$ in Ni1-xMox and (b) $y$ in
Ni1-yCuy. Dashed lines are guides to the eye.

###  III.3 Arrott-Noakes scaling of magnetization

In the vicinity of $T_{\rm C}$, the isothermal magnetization curves can be
scaled based on an Arrott-Noakes equation of state, from which critical
scaling exponents $\beta,\gamma,$ and $\delta$ are extracted [28]. The Arrott-
Noakes equation of state, i.e.,

$M^{1/\beta}=a^{*}+b^{*}(\frac{H}{M})^{1/\gamma}$ (2)

leads to the scaling of $M$ since the critical fluctuations are the dominant
energy scale. $a^{*}$ and $b^{*}$ are scaling constants. Equation (2) yields
parallel lines, forming the so-called modified Arrott plot, with $M\sim
H^{1/\delta}$ at $T_{\rm C}$, and giving rise to the critical exponents
$\beta$, $\gamma$, and $\delta$. To ensure the analysis is within the regime
where critical fluctuations dominate, magnetic isotherms were measured in
evenly spaced temperature steps in the range $(1-3\%)T_{\rm C}\leq
T\leq(1+3\%)T_{\rm C}$, where $T_{\rm C}$ corresponds to the temperature at
which the modified Arrott plot intersects at the origin. The left columns of
Figs. 10 and 11 show the scaling results of Ni1-xMox and Ni1-yCuy,
respectively. Critical exponents $\beta$ and $\gamma$ can likewise be
determined from the scaling theory which describes the reduced $M$ versus the
reduced $\mu_{0}H$ in the form [29],

$M/|T-T_{\rm{C}}|^{\beta}=f_{\pm}(\mu_{0}H/|T-T_{\rm{C}}|^{\beta+\gamma}),$
(3)

which makes $M$ to fall on two universal curves: one for the FM state,
$T<T_{\rm C}$, and the other for the paramagnetic state, $T>T_{\rm C}$. The
right columns of Figs. 10 and 11 display the scaling plots for the $M$ shown
in the left columns. It is evident that the branches for $T<T_{\rm C}$ and
$T>T_{\rm C}$ each collapse onto one curve. The scaling tendencies are
apparent, showcasing the congruence between the modified Arrott plot and the
Arrott-Noakes scaling analysis.

The results of scaling analysis are summarized in Fig. 5(a) and (b) where we
show the evolution of the critical exponents $\beta$, $\gamma$, and $\delta$
as a function of $x$ and $y$. These exponents are not independent from each
other and should follow the Widom relation $\delta=\gamma/\beta+1$ [30]. To
test if our scaling results are consistent with the relation, we compare
$\delta_{scaling}$ values derived from the scaling analysis with
$\delta_{\rm{Widom}}$ values deduced from the Widom relation after introducing
$\gamma_{scaling}$ and $\beta_{scaling}$. The deviation =
$(\delta_{scaling}-\delta_{\rm{Widom}})/\delta_{\rm{Widom}}\times$ 100% is
shown in Fig. 5(c) and (d). The deviation is less than 4% in Ni1-xMox and 8%
in Ni1-yCuy. For Ni1-xMox, the critical exponents deviate from the values of
pure nickel ($\beta=0.39$, $\gamma=1.315$, and $\delta=4.37$ [31]), and
slightly increase as $x$ increases. For Ni1-yCuy, the critical exponents do
not show a continuous trend as a function of $y$, but fluctuate around the
values $\beta\approx 0.56$, $\gamma\approx 1.77$, and $\delta\approx 4.44$.
The evolution of $\beta$, $\gamma$, and $\delta$ cannot be explained by
currently theoretical models. As the concentration of the nonmagnetic
substitution increases, one would expect the magnetic interactions to become
more anisotropic compared to pure nickel due to disorder. Subsequently, the
critical dimension is reduced so that critical fluctuations become important.
One would expect $\delta$ and $\gamma$ increase, and $\beta$ decreases [32].
One potential explanation is that quantum fluctuations cause substantial
renormalizations of the low-energy theory underlying the classical critical
point, and thus drive the system into a strong-coupling regime that is beyond
the established theoretical models [25].

Figure 5: Critical exponents as a function of (a) $x$ in Ni1-xMox and (b) $y$
in Ni1-yCuy. Enlarged scale for the exponent $\beta$ is shown in the inset of
(a). (c,d) Deviation from the Widom relation. See text for detail. Figure 6:
(a) $T_{\rm C}-x$ phase diagram of Ni1-xMox. (b) $T_{\rm C}-y$ phase diagram
of Ni1-yCuy. FM = ferromagnetic state, PM = paramagnetic state, and SG = spin
glass.

###  III.4 Low temperature specific heat

As the $T_{\rm C}$ is suppressed and approaches zero, the influence of thermal
fluctuations on the order parameter fluctuations diminishes, while the
influence of quantum fluctuations becomes dominant. This is best captured by
low-temperature specific heat measurement as the specific heat can directly
probe the low-energy magnetic excitations. Figures 7(a) and (c) show the
temperature dependence of zero-field specific heat $C/T$ for Ni1-xMox and
Ni1-yCuy, respectively. For the concentrations significantly below $x_{cr}$
and $y_{cr}$, $C/T$ saturates at constant values as the temperature is
decreased to 1.8 K, indicating Fermi-liquid behavior for metals. As $x$ and
$y$ approach $x_{cr}$ and $y_{cr}$, $C/T$ elevates and increases as the
temperature decreases. We plot the $C/T$ value at 2 K as a function of $x$ and
$y$ in Fig. 7(b) and (d), respectively. It is evident that $C/T$ peaks at
$x_{cr}\sim 0.095$ and $y_{cr}\sim 0.54$. In several magnetic systems tuned to
the QCP, the similar phenomenon has been observed where the maximal $C/T$
centers around the critical concentration, reflecting the effect of enhanced
quantum fluctuations [33, 7].

Figure 7: Specific heat of (a) Ni1-xMox and (c) Ni1-yCuy plotted as $C/T$ vs.
log$T$. $C/T$ at 2 K as a function of (b) $x$ and (d) $y$.

###  III.5 Spin glass in Ni1-xMox

It is known that the chemical replacement in a magnetic system can lead to
strong disorder, subsequently causing spin glass behavior. This phenomenon has
been observed in several ferromagnets as the chemical substitution is
introduced to suppress the $T_{\rm C}$ [1]. We have examined Ni1-xMox and
Ni1-yCuy down to 2 K and indeed found spin glass behavior in the former with
$x=0.095$ via the ac magnetic susceptibility measurements. Figure 8(a) shows
the temperature dependence of the real part ($\chi^{\prime}$) of the ac
magnetic susceptibility where an ac field of 2.5 Oe with the frequency $f$
varies between 0.1 and 1000 Hz was applied. A peak in $\chi^{\prime}(T)$
centered around $T_{f}=6.3$ K for $f=0.1$ Hz is observed. As $f$ increases,
$T_{f}$ increases, and the overall intensity of $\chi^{\prime}(T)$ decreases.
This phenomenon is exactly how a spin glass system manifests in
$\chi^{\prime}(T)$ [34]. We can quantify the extent of peak shift by utilizing
the Mydosh parameter ($S$) and categorize the class of the spin glass
accordingly,

$S=\Delta T_{f}/T_{f}\cdot\Delta(\rm{log}_{10}\mathit{f}),$ (4)

where $\Delta T_{f}=T_{f}(f_{1})-T_{f}(f_{2})$ and
$\Delta(\rm{log}_{10}\mathit{f})=\rm{log}_{10}\mathit{(f_{1})}-\rm{log}_{10}\mathit{(f_{2})}$
with $f_{1}$ = 1000 Hz and $f_{2}$ = 0.1 Hz. The calculated $S=0.0133$ falls
within the range of 0.005-0.08, and thus imply Ni1-xMox with $x=0.095$ belongs
to the canonical spin-glass system [34].

To understand the dynamical properties of the spin glass phenomenon in
Ni1-xMox, we applied the standard theory for dynamical scaling near $T_{\rm
C}$. The conventional result of the dynamical scaling establishes a
relationship between the critical relaxation time $\tau$ and the correlation
length $\xi$ as $\tau\sim\xi^{z}$, and $\xi$ diverges with the temperature as
$\xi\sim(T/(T-T_{\rm C}))^{\nu}$. In the context of a spin glass, the above
equation can be expressed as:

$\tau=\tau_{0}(\frac{T_{f}}{T_{g}}-1)^{-z\nu},$ (5)

where $\tau$ is the relaxation time with dependence on the measurement
frequency ($\tau=1/f$), $\tau_{0}$ is the characteristic relaxation time of a
single spin flip, $T_{f}$ is the freezing temperature below which the moments
freeze, $T_{g}$ is the freezing temperature as $f$ approaches zero, and $z\nu$
is the dynamical exponent. As indicated by Eq. 5, a divergence occurs as $f$
decreases, resulting in multiple local minima that make the fitting
challenging. Therefore, it is common to rewrite the Eq. 5 into a linear
relationship [35, 36],

$\textup{ln}(\tau)=\textup{ln}(\tau_{0})-z\nu\textup{ln}(\frac{T_{f}}{T_{g}}-1).$
(6)

We decide $T_{g}=5.97$ K by comparing the $R^{2}$ values, as shown in the
inset of Fig. 8(b). The obtained value for $\tau_{0}$ is $3.63\times 10^{-17}$
s, and the value of $z\nu$ is 15.28$\pm$0.15, as shown in Fig. 8(b). For
canonical spin glass systems, $\tau_{0}$ is typically between $10^{-12}$ and
$10^{-14}$ s, while for cluster glass systems, $\tau_{0}$ falls within the
range of $10^{-7}$ to $10^{-10}$ s [34]. Our result is three orders of the
magnitude smaller than the lower-bound value for canonical spin glass systems.
Besides, the exponent $z\nu$ in different systems lie between 4 and 12 [37].
Our result $z\nu\sim 15$ is outside the range. Based on the scaling relation
$\nu d=2\beta+\gamma$, the $\nu$ is of the order of 1, and hence the $z$ value
is as large as 15, which cannot be explained by any known theories. One
potential scenario involves the amplification of quantum fluctuations, which
strengthens the influence of strong disorder in Ni1-xMox in proximity to
$x_{cr}$. This phenomenon propels the system into a novel strong-coupling
regime that is challenging to reach through conventional calculations.

Figure 8: (a) Temperature dependence of the real part ($\chi^{\prime}$) of the
ac magnetic susceptibility of Ni1-xMox with $x=0.095$. (b) Liner dynamical
scaling of the relaxation time vs. the reduced temperature. The inset shows
$R^{2}$ values obtained from linear fits in the main panel with different
$T_{g}$’s.

##  IV SUMMARY

We have studied the magnetic properties of binary nickel alloys Ni1-xMox and
Ni1-yCuy via the structural analysis, electrical resistivity, magnetic
susceptibility, critical exponents analysis based on isothermal magnetization
curves, specific heat, and the ac magnetic susceptibility measurements. Our
aim is to search for FM QCPs among binary nickel alloys. In Ni1-xMox, the FM
QCP is prohibited due to the spin glass behavior near $x=x_{cr}$. The same
outcome has been observed in Ni1-xVx [18], Ni1-xPdx [17, 38], and other Ce-
and U-based ferromagnets with composition tuning [39, 40, 41, 42].

For Ni1-yCuy, we have observed the effect of quantum fluctuations from
enhanced specific heat around the critical contribution. While that implies
the existence of the FM QCP, making the decisive call requires more
measurements at lower temperatures. Examining the $\rho(T)$ plot in Fig.2(b)
and the larger error bars in EPMA (Fig.9(b) in APPENDIX A), it seems that
Ni1-yCuy exhibits a much stronger disorder effect and sample inhomogeneity
compared to Ni1-xMox. However, spin glass or short-range order is not observed
down to 1.8 K in Ni1-yCuy. To address this concern, conducting ac magnetic
susceptibility measurements at lower temperatures would provide further
clarification. It will also be important to check whether the divergence of
$C/T$ persists at lower temperatures. Muon spin relaxation measurements for
various concentrations up to $y_{cr}$ can help determine whether the
ferromagnetic transition remains second order, which is the prerequisite for a
QCP [8].

## Acknowledgements

We thank C.-C. Yang at SQUID VSM Lab, Instrumentation Center, National Cheng
Kung University (NCKU), and Ms. J. Kang at Center for Condensed Matter
Sciences (CCMS), National Taiwan University (NTU) for technical support. We
also thank C.-W. Wang and C.-M. Wu at NSRRC for the preliminary neutron
experiments at ANSTO. This work is supported by the National Science and
Technology Council in Taiwan (grant numbers NSTC 109-2112-M-006-026-MY3,
110-2124-M-006-009) and the Higher Education Sprout Project, Ministry of
Education to the Headquarters of University Advancement at NCKU. WTC
acknowledges the National Science and Technology Council of Taiwan, for
funding 111-2112-M-002-044-MY3, 112-2124-M-002-012, the Featured Areas
Research Center Program within the framework of the Higher Education Sprout
Project by the Ministry of Education of Taiwan 112L900802, and Academia Sinica
project number AS-iMATE-111-12.

## APPENDIX

###  IV.1 Electron microscope microanalysis

For EPMA, we measured 20 data points across a flat surface of each sample and
the standard deviation in composition is $\leq$2% for Ni1-xMox and $\leq$10%
for Ni1-yCuy. The largest standard deviation in Ni1-yCuy is found around
$y=0.50$, reflecting samples’ inhomogeneity.

Figure 9: (a) Mo concentrations in Ni1-xMox and (b) Cu concentrations in
Ni1-yCuy determined by EPMA vs. nominal concentrations.

###  IV.2 Modified Arrott plots and Arrott-Noakes scaling plots

Left columns of Fig. 10 and Fig. 11 display the modified Arrott plots for
Ni1-xMox with $x=0.05-0.09$ and Ni1-yCuy with $y=0.40-0.50$, respectively.
Right columns of Fig. 10 and Fig. 11 display the Arrott-Noakes scaling plots
of the same data on the left columns.

Figure 10: The Left column shows modified Arrott plots of Ni1-xMox with
$x=0.05-0.09$. The right column shows Arrott-Noakes scaling plots of the same
data displayed in the left column. Figure 11: The Left column shows modified
Arrott plots of Ni1-yCuy with $y=0.40-0.50$. The right column shows Arrott-
Noakes scaling plots of the same data displayed in the left column.

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# Multiple State Analysis, a Multidimentional Approach to Multiple Time-to-
Event Data and
Life Course Health Trajectories:
Application to Patients with Myocardial Infarction

Marc Delord
School of Life Course & Population Sciences
Department of Population Health Sciences
King’s College London
London, United kingdom
& Annastazia Learoyd
School of Life Course & Population Sciences
Department of Population Health Sciences
King’s College London
London, United kingdom
& Abdel Douiri
School of Life Course & Population Sciences
Department of Population Health Sciences
King’s College London
London, United kingdom
Correspondence<EMAIL_ADDRESS>
###### Abstract

Life course epidemiology of chronic diseases has been dominated so far by the
environmental approach. Whether it focuses on early life exposures / events or
later lifestyle behaviors, this approach assumes that previous life
experiences interact at each stage of life and shape subsequent health
trajectories. Inspired by the analysis of social trajectories, focusing on
transitions between multiple states, in multiple dimensions of the social
experience, we propose here a novel empirical approach to multiple time-to-
event health data, denoted as _multiple state analysis_. Alike the so-called
state sequence analysis, the aim of _multiple state analysis_ is to create
typologies of the main life course trajectories. This approach is applied to
real world data from a south London general practice electronic health record
system from which multiple long term conditions associated with myocardial
infarction were considered. Among expected results such as the recurrent role
of hypertension, _multiple state analysis_ shows that different patterns of
long term conditions including physical and mental health conditions, are
associated with the onset of myocardial infarction but also with socio-
demographics such as sex and ethnicity.

_Keywords_ Jaccard Dissimilarity Index $\cdot$ Multimorbidity $\cdot$ Multiple
Sequence Analysis $\cdot$ Multiple State Analysis $\cdot$ Myocardial
Infarction $\cdot$ Ward clustering

##  1 Introduction

The notion that life course health trajectories are influenced by early life
events and the ever-changing historical context (Elder,, 1998), or by later
_lifestyle_ associated exposure (Hill and Doll,, 1957) have dominated the
epidemiological field so far. Conversely, we can assume that patterns induced
by the main combinations of early life events and specific socio demographic
contexts, would produce tractable patterns of the main life-course health
trajectories. Progress in information technologies and the availability to
academic research of publicly funded health databases including daily life
general practice records represent an unprecedented opportunity to better
evaluate this hypothesis. Along with the main trajectories which are well
documented such as the association of hypertension with cardio-vascular
diseases (Carter et al.,, 2015), this research is expected to better identify
critical trajectories and potentially uncover unexpected patterns and at-risk
populations allowing better informed healthcare policies.

Whereas life course epidemiology has developed the environmental model-based
approach to chronic diseases, focusing on lifestyles and exposures, the social
sciences have proposed a sequential approach of life experience trajectories.
This empirical approach focuses on transitions between distinct states and
aims at creating typologies of trajectories by identifying patterns of
recurrent sequences. In subsequent developments of the so-called state
sequence analysis, Pollock (Pollock,, 2007) proposed an extended approach to
life course trajectories denoted as _multiple sequence analysis_ where
multiple dimensions of the social experience are considered when building a
typology.

In this paper, we propose a generalization of Pollock’s approach, to multiple
time-to-event endpoints through the analysis of multiple long term conditions
associated with myocardial infarction. Myocardial Infarction, leading to
“heart attack," is one of the leading causes of death in high-income countries
(Roth et al.,, 2015). Myocardial infarction is caused by decreased or complete
cessation of blood flow in the myocardium and results in irreversible damage
to the heart muscle (Jaffe,, 2013). Most of the time, myocardial infarction is
due to underlying coronary artery disease (Libby,, 2013). Well-known
modifiable risk factors associated with coronary artery disease and myocardial
infarction are smoking, abnormal blood lipid profile, hypertension, diabetes
mellitus, abdominal obesity, psycho-social factors, diet, physical activity
and alcohol consumption (protective) (Yusuf et al.,, 2004, Anand et al.,,
2008). Some non-modifiable risk factors for myocardial infarction include
advanced age, male gender (males tend to have myocardial infarction earlier in
life) and genetics (Anand et al.,, 2008, Nielsen et al.,, 2013).

This paper is organized as follows: section 1 presents the background of the
analysis of state sequences, a motivating example justifying the use of the
proposed approach and ends by the presentation of the _multiple state
analysis_ method. In section 2 the analysis of multiple long term conditions
associated with myocardial infarction is presented. Finally in section 3 the
_multiple state analysis_ method is briefly discussed in light of the study
results.

###  1.1  State Sequence Analysis

State sequence analysis is the analysis of sequences describing individual
life courses with the objective to develop exploratory typologies of the main
trajectories at the population level. As this empirical approach would always
produce some results (Levine,, 2000), and bearing in mind that state sequence
analysis is an exploratory exercise, the sound interpretation of resulting
typologies, grounded by relevant theories (Shalizi,, 2009) remains an
important step of the state sequence analysis workflow. In the social sciences
and epidemiological setting, beside the interest of a typology by itself
(Pollock,, 2007), the use of auxiliary interpretative variables such as socio-
demographic covariates or known risk factors, is of particular interest to
evaluate the relevance of typologies insofar as public health policies depends
on identifiable modifiable risk factors and associated populations. First
established in the area of social sciences, state sequence analysis is
applicable to domains where temporal or spatial transitions are meaningful,
such as in the social sciences, in biology or in healthcare research (Vanasse
et al.,, 2020).

Although state sequence analysis is an empirical method, it is important to
state that in the settings mentioned earlier, analysed sequences require some
$good$ properties to produce meaningful results. Especially, transitions
between states need to be meaningful and successive states should be quasi-
exclusive from one another: hospital discharge in healthcare research, disease
recovery in epidemiology, or change in marital status in social sciences are
examples of meaningful state transitions. In other situations however, the
simple record of successive non exclusive events may not fully render
subjects’ experiences and the analysis of such sequences may be misleading.

For instance, in the medical field, the onset of some communicable or non
communicable diseases such as long term conditions, the record of which may
not always be fully accurate or the diagnosis based on heterogeneous criteria,
will produce such sequences. Importantly, if the sequence of acute health
events and long term conditions is informative for single time-to-event
analysis, the concept of multimorbidity, that is a state characterized by the
accumulation of 2 or more long term conditions, that may occur simultaneously,
represents a relevant information in term of life course health trajectories
(Barnett et al.,, 2012). However, this dimension may not be fully captured by
the simple sequence of events further analysed by state sequence analysis.

To illustrate this situation let’s consider health record histories of a
homogeneous group of patients presenting a diagnosis of diabetes and
hypertension at about the age of fifty and a record of stroke at about the age
of 60. Though we intuitively understand that these patients share similar
experiences in terms of health trajectories, the simple sequences derived from
their historical records will present a significant divergence in the critical
ten years preceding the onset of stroke, according to whether hypertension or
diabetes was first recorded. As a consequence, the simple analysis of these
sequences will produce a typology dominated by the nature of the first
recorded long term condition (diabetes or hypertension). This conclusion would
be arguably misleading as, by hypothesis, analysed patients show similar
health trajectories.

One way to overcome this situation could be to produce some ad hoc sequences
for these patients such as:  (i) diabetes or hypertension (ii) $\\{$diabetes
and hypertension$\\}$ or $\\{$ hypertension and diabetes $\\}$ (iii)
$\\{$diabetes and hypertension and stroke$\\}$ or
$\\{$hypertension and diabetes and stroke$\\}$, where
$\\{\>\>.\>\>and\>\>.\>\>and\>\>...\\}$ represent a new states of
multimorbidity. However, in this situation, characterized by different states
of multimorbidities, the definition of simple sequences may not be
straightforward and would require simplifying rules produced by experts of the
relevant field in order to obtain a reduced set of workable and meaningful
life course health sequences.

###  1.2 Multiple State Analysis

In a similar situation considering social trajectories Pollock (Pollock,,
2007) proposed the multiple-sequence analysis where sequences associated with
distinct dimensions of the social experience are analysed in conjunction with
one another. In order to deal with limitations associated with the improper
use of state sequence analysis when analyzing non excluding / non competing
events, we propose a novel approach related to Pollock’s method denoted
hereafter as _multiple state analysis_ – or multiple time to event analysis –
applied to the analysis of multimorbidity associated with myocardial
infarction as an example.

From a statistical perspective, the proposed method requires a
multidimentional representation of patients’ health records. In subsequent
steps, the classical state sequence analysis principles are applied: given an
appropriate metrics, pairwise dissimilarity between patients are computed
before a clustering method is applied to create a typology.

####  1.2.1 Individual patient’s state matrix and group summaries

_Multiple state analysis_ requires individual health patients’ records to be
formatted into multiple time-to-event indicator tracks, stacked in ${t\times
k}$ state matrices, $t$ and $k$ being the maximum age observed in the cohort
and the total number of conditions considered in the analysis, respectively.

If onset of disease $l$ for patient $j$ is $t_{j}$, we note:
$\textrm{M}_{.l}^{j}=\textrm{I}_{[t\geq t_{j}]}$, where $\textrm{M}_{.l}^{j}$
stands for column $l$ of matrix $\textrm{M}^{j}$, and I is the positive
integer indicator function. For instance in the following example,
$\textrm{M}^{j}$ shows that the analysis is conducted on long term conditions
$a$, $b$, $c$, $d$ and $e$ from 0 to 99 years old for patient $j$, onset of
diseases $a$, $b$, $c$, and $e$ being 2, 46, 47 and 48 respectively. Patients
may be censored or experience a competing event such as death or any other
event whose occurrence either precludes the occurrence of diseases under
examination or fundamentally alters the probability of their occurrence
(Gooley et al.,, 1999).

$\tau$ being patients’ censoring times vector,
$\textrm{c}^{j}=\textrm{I}_{[t\geq\tau_{j}]}$ represents patient’s $j$
censoring indicator and $\bar{\textrm{c}}^{j}=\textrm{I}_{[t<\tau_{j}]}$, the
follow up period for patient $j$. In the proposed example, patient $j$ is
censored at age 83.

$\mathbf{M^{j}}=\blockarray{cccccc}&abcde\\\ \block{c(ccccc)}000000\\\
100000\\\ 210000\\\ \vdots\vdots\vdots\vdots\vdots\vdots\\\ 4510000\\\
4611000\\\ 4711100\\\ 4811101\\\ \vdots\vdots\vdots\vdots\vdots\vdots\\\
8111101\\\ 8211101\\\ 8311101\\\ \vdots\vdots\vdots\vdots\vdots\vdots\\\
9911101\\\ ,\>\>\mathbf{\bar{c}^{j}}=\blockarray{cc}&\\\ \block{c(c)}01\\\
11\\\ 21\\\ \vdots\vdots\\\ 451\\\ 461\\\ 471\\\ 481\\\ \vdots\vdots\\\ 811\\\
821\\\ 830\\\ \vdots\vdots\\\ 990\\\ $ (1)

An appealing feature of the proposed individual patients’ data representation
is its straightforward interpretation in terms of the classical time-to-event
indicators such as overall survival and cumulative incidence functions, and
associated graphical representations, when interested in statistical summaries
associated to the entire cohort or any subgroup of patients/diseases. This is
of particular interest in the medical setting where these statistics are
routinely interpreted.

####  1.2.2 Dissimilarity index matrix

The objective pursued when creating a typology of life course health
trajectories is to categorise patients such that patients belonging to the
same group share similar trajectories relatively to patients belonging to
other groups. This process implies therefore the use of a
dissimilarity/distance metric to obtain pairwise patients’ profile
dissimilarities to be stored in a distance matrix or _dissimilarity index
matrix_. Based on this matrix, a clustering method can be used to finally
create a typology.

As _multiple state analysis_ deals with state indicators, the relationship
between two patients can be summarised using a 2 $\times$ 2 contingency table
$[(p,r),(s,q)]$,

  1. (i)

$q$ being the number of matching time units where both patients are affected
by a given long term condition,

  2. (ii)

$p$ being the number of matching time units where both patients are free from
the considered long term condition,

  3. (iii)

$s$ and $r$ being the number of matching time units where both patients are in
different states of health regarding the specified long term condition and,

  4. (iv)

$t$ being the length of the sequence.

In this setting, the dissimilarity between profiles $x^{i}$ and $x^{j}$ can be
expressed as:

$\begin{cases}d(x^{i},x^{j})=\frac{r+s}{q+r+s+p},\\\ q+r+s+p=t\end{cases}$

Considering states of illness as more informative than healthy states,
omitting $p$, the number of negative matches in the denominator, leads to the
Jaccard dissimilarity index (Jaccard,, 1901):

$d(x^{i},x^{j})=\frac{r+s}{q+r+s}=1-\frac{q}{q+r+s}.$

Since $r+s=t-p-q$, we can rewrite the above relation as follows:

$d(x^{i},x^{j})=1-\frac{q}{t-p}.$

A composite analogue to the Jaccard dissimilarity index can be derived as:

$\begin{cases*}d(M^{i},M^{j})=1-\frac{Q}{t^{*}-P},\\\
Q=\sum_{l=1}^{k}q_{l},\\\ P=\sum_{l=1}^{k}p_{l},\\\ t^{*}=kt\end{cases*}$ (2)

i.e. $Q$ and $P$ are the sum of $q$ and $p$ over all considered long term
conditions.

Using the matrix notations (1) from 1.2.1 and setting $\bar{\textrm{C}^{i}}$
as the $t\times t$ diagonal matrix with $\bar{\textrm{c}^{i}}$ as diagonal
entries, the censored quantities defined above can be computed as:

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\textrm{tr}\left(\left(\textrm{M}^{i}{{}^{\prime}}\bar{\textrm{C}^{i}}\bar{\textrm{C}^{j}}\right)\left(\textrm{M}^{j}{{}^{\prime}}\bar{\textrm{C}^{i}}\bar{\textrm{C}^{j}}\right)^{\prime}\right)$
$\displaystyle P$ $\displaystyle=$
$\displaystyle\textrm{tr}\left(\left(\left(\textrm{M}^{i}-\mathds{1}\right)^{\prime}\bar{\textrm{C}^{i}}\bar{\textrm{C}^{j}}\right)\left(\left(\textrm{M}^{j}-\mathds{1}\right)^{\prime}\bar{\textrm{C}^{i}}\bar{\textrm{C}^{j}}\right)^{\prime}\right)$
$\displaystyle t^{*}$ $\displaystyle=$
$\displaystyle\textrm{tr}\left(\left(\mathds{1}^{\prime}\bar{\textrm{C}^{i}}\bar{\textrm{C}^{j}}\right)\left(\mathds{1}^{\prime}\bar{\textrm{C}^{i}}\bar{\textrm{C}^{j}}\right)^{\prime}\right),$

where tr denotes the trace operator and $\mathds{1}$ is the $t\times k$ matrix
with all entries set to 1.

Using this notation, we can translate the cases of two patients as described
above. Patient $i$ has a record of hypertension ($hyp$) at 51 and a record of
diabetes ($dm$) at 52, whereas patient $j$ has a record of diabetes at 51 and
a record of hypertension at 52.

Simple sequences of patients $i$ and $j$ would be:

${x^{i}}=\blockarray{cc}&\\\ \block{c(c)}500\\\ 51hyp\\\ 52dm\\\ 53dm\\\
\vdots\vdots\\\ 60dm\\\ \;,\>\>{x^{j}}=\blockarray{cc}&\\\ \block{c(c)}500\\\
51dm\\\ 52hyp\\\ 53hyp\\\ \vdots\vdots\\\ 60hyp\\\ \;,$

whereas the state matrices of patients $i$ and $j$ would be:

$\mathbf{m^{i}}=\blockarray{cccccc}&dmhyp\\\ \block{c(ccccc)}5000\\\ 5101\\\
5211\\\ 5311\\\ \vdots\vdots\\\ 6011\\\
\;,\>\>\mathbf{m^{j}}=\blockarray{cccccc}&dmhyp\\\ \block{c(ccccc)}5000\\\
5110\\\ 5211\\\ 5311\\\ \vdots\vdots\\\ 6011\\\ \;.$

The Jaccard dissimilarity index computed from the simple sequences of patients
$i$ and $j$ equals 1 (its maximum). It equals $\frac{1}{10}$ using the state
matrix notation and the composite Jaccard index proposed in (2), which is what
we expect for these patients with similar health trajectories.

####  1.2.3 Clustering method

Although _multiple state analysis_ is not restricted to a specific clustering
procedure, we have used in this paper the Ward’s hierarchical clustering
method (Ward Jr,, 1963). At the starting point of this procedure, each
instance is considered as a cluster of its own, then clusters are recursively
merged such that the resulting cluster structure presents the minimum cost in
terms of the _within clusters_ variance. This objective is reached by
minimizing

$W_{i,j}=\frac{n_{i}n_{j}}{n_{i}+n_{j}}\;d(c_{i},c_{j})^{2}$

over $i$ and $j$, $N_{i}$ being the cardinality of cluster $i$ and
$d(c_{i},c_{j})$ the Euclidean distance between centroids of clusters $i$ and
$j$. Finally, dissimilarities are updated at each step following the Lance-
Williams formula (Lance and Williams,, 1967):

$\displaystyle d(c_{i}\cup c_{j},c_{k})$ $\displaystyle=$
$\displaystyle\alpha_{i}\;d(c_{i},c_{k})+\alpha_{j}\;d(c_{j},c_{k})+\beta\;d(c_{i},c_{j}),$
$\displaystyle\begin{cases}\alpha_{i}=\frac{n_{i}+n_{k}}{n_{i}+n_{j}+n_{k}}\\\
\alpha_{j}=\frac{n_{j}+n_{k}}{n_{i}+n_{j}+n_{k}}\\\
\beta\>\>=\frac{n_{k}}{n_{i}+n_{j}+n_{k}}\;.\end{cases}$

This formula allows the use of the Ward method in the general setting where
$d(.)$ does not denote a _proper_ Euclidean distance. Following the Lance-
Williams formula, the updated dissimilarity between merged clusters $i$ and
$j$ ($c_{i}\cup c_{j}$) and any other clusters $k$ ($c_{k}$) is the simple
linear combination of individual dissimilarities between involved clusters.

The result of this process is a dendrogram, representing the nested groups of
patients and similarity levels at which clusters merged. Clustering of the
data is obtained by cutting the dendrogram to get a workable number of
clusters.

In the next section, we present an application of _multiple state analysis_
using electronic health records of patients with myocardial infarction and
other comorbidity. We show subsequently how multivariate logistic regressions
can be used to evaluate associations between clusters and socio-demographic
variables and other risk factors.

##  2 Multiple State Analysis of multimorbidity associated to myocardial
infarction

###  2.1  Patients

####  2.1.1  Primary care registry

In an application of the proposed method to real world data, we considered
electronic health records 33 diseases and long term conditions in adult
patients aged over 18 and registered in 41 general practices in south London
between April 2005 and April 2021. The data were provided by the Lambeth
DataNet. Recorded long term conditions were: cancers and neoplasms (1),
cardiovascular diseases (stroke (2), heart failure (3), peripheral vascular
disease (4), atrial fibrillation (5), hypertension (6), myocardial infarction
(7) and transient ischemic attack (8)), infectious diseases (viral hepatitis B
& C (9), Hiv/aids (10)), inflammatory diseases / diseases of the immune system
(rheumatoid arthritis (11) and inflamatory bowel disease (IBD) (12), lupus
(13)), kidney diseases (chronic kidney disease grade 3 to 5 (CKD 3-5) (14)),
liver diseases (15), mental health conditions (anxiety disorders (16),
depression (17), alcohol dependence (18), serious mental illness (19) and
substance dependency (20), learning disabilities (21)), metabolic / endocrine
diseases (diabetes (22) and morbid obesity (23)), musculoskeletal conditions
(osteoarthritis (24) and osteoporosis (25)), Sickle-Cell Anaemia (26),
neurological conditions (parkinson’s (27), multiple sclerosis (28), epilepsy
(29) and dementia (30)), respiratory diseases (asthma (31) and COPD (32)) and
chronic pain (33). Patient electronic files includes the date at which any of
the above long term conditions was ever recorded. Incident cases were defined
as the first ever diagnostic record.

Of note some early diseases such as asthma were traced back and added to
individual files outside patients follow-up interval.

###  2.2  Method

The analysis was conducted on patients showing a record of myocardial
infarction, according to the 3 steps described in 1.2: i) arrange patients
individual records into multiple time-to-event endpoints stacked in individual
patients’ state matrices and censoring indicators, ii) compute pairwise
patients’ dissimilarities on individual state matrices (and censoring
indicators) and apply a clustering method, and iii) define a typology.

State matrices were computed considering the records of the 33 diseases and
long term conditions enumerated above. Figure 1 represents example of state
matrices from 4 patients randomly samples from the cohort.

Figure 1: Example of life course patients’ history (state matrices): status of
33 long term conditions are considered from 0 to 104 years old

Pairwise dissimilarities between patients were computed using the composite
Jaccard dissimilarity index (Jaccard,, 1901) as described in 1.2.2. The
Jaccard coefficient, also referred to as the binary metric, is a measure of
dissimilarity between binary samples, such as state indicator. It can be
interpreted, for a given long term condition, as the amount of time two
patients are simultaneously in a the same state (in term of patient’s age)
divided by the the amount of time these patients are in different states.
Finally, agglomerative hierarchical clustering was computed using the Ward’s
method and Hubert’s C index was used as an indicator in choosing a relevant
size of typology in a convenient and workable range (Hubert and Levin,, 1976).

####  2.2.1  Graph representation

To better visualize and understand the link between onset of myocardial
infarction and other associated long term conditions in defined clusters, we
used a graph where conditions are represented by dots whose x-coordinate
represents the median age at onset. On the y axis, dots are conveniently
assigned to layers (levels), such that significant transitions are represented
by edges oriented from a higher levels to a lower levels (Sugiyama et al.,,
1981). The number of layers in the graph can therefore be interpreted as the
maximum number of significant transitions from the first event to the last
event represented on the graph according to some threshold.

####  2.2.2  Statistical analysis of cluster trajectories

In order to describe patients characteristics with respect to the proposed
typology, socio-demographic variables, risk factors and diseases status
indicators were crossed with cluster status indicators. All variables were
binary or categorical except age at onset of myocardial infarction and the
multimorbidity index.

Socio-demographics, risk factors and long term condition indicators, were
displayed as frequencies and percentage or median and interquartile range as
appropriate. Associations between variables and clusters was tested using chi-
squared test or Fisher exact test as appropriate, or using Kruskal–Wallis test
for numeric variables (tables LABEL:tab:t1 and LABEL:tab:t2).

Multivariate association between clusters and socio-demographic factors / risk
factors, and long term conditions was estimated using logistic regressions
where cluster indicators were explained by tested variables. Results were
displayed using heatmaps were non significant result, at the 5% level, were
omitted.

###  2.3  Results

From 856342 single patients for which electronic records were available, 5093
(0.6%) had a record of myocardial infarction which corresponds to an incidence
rate of 94.4 cases per 100,000 person-years. The median age at onset of
myocardial infarction was 61. Among patients with a record of myocardial
infarction, 68% were male, 68% were white and the majority of the population
belonged to IMD quintile 1-2 (most deprived) (vs quintile 3-5 (less
deprived)). The main comorbidities associated with myocardial infarction were
hypertension (67%) and diabetes (40%) and the median number of comorbidities
associated with myocardial infarction was 5. The median follow-up was 9.6
years, 61% of patients were censored or died at end of follow-up (39%) (table
LABEL:tab:t1).

After pairwise patients’ dissimilarity computation and the clustering,
Hubert’s $C$ index was computed for a range of partition sizes. A 10 group
partition, corresponding local minimum the $C$ index between 6 and 15 clusters
was retained for this exploratory analysis.

####  2.3.1  Typology annotation

_Demographics and risk factors_

The 10 clusters were ordered by decreasing frequency and numbered from 1 to
10. Cluster 1 represents 1175 patients (23.1%) and clusters 9 and 10 represent
218 (4.3%) and 103 (2.0%) patients respectively.

Table LABEL:tab:t1 displays basic demographics and risk factors associated
with clusters. Briefly, the median age at onset of myocardial infarction
ranged from 47-48 in clusters 8 and 4 respectively, to 78 in cluster 2. The
proportion of female patients ranged from 15.7% in cluster 4 to 47.8% in
cluster 2. The proportion of White patients varied from 45.3% in clusters 6
and 9 to 82.0% in cluster 7, whereas the proportion of Black patients was 8.8%
in cluster 7 and reached 30.8% in cluster 9, and the share of the Asian
population ranged from 5.0% in cluster 7 to 25.8% in cluster 6. IMD-quintile
index was not significantly different from cluster to cluster. Former or
current smokers were over represented in cluster 7 (85%), and polymedication
was more frequently observed in clusters 2, 3 and 6, and finally, the highest
median number of comorbidities associated with myocardial infarction was
observed in cluster 3 (median of 7).

Alternatively, figure 2 displays the multivariate log-odds ratio of
demographic variables and risk factors associated with the different clusters.

Figure 2: Log-odds ratio of socio-demographic variables and risk factors
associated with clusters: values are derived from multivariate logistic
regressions where cluster indicators are explained using displayed variables.
Positive log-odds ratio show over representation of the corresponding
population traits in a given cluster as compared to other patients showing a
record of myocardial infarction. Only significant coefficients are displayed
(at the 5% level).

_Associated long term conditions_

The distribution of myocardial infarction comorbidities according to the
different clusters is displayed in table LABEL:tab:t2: Among the most frequent
comorbidities associated with myocardial infarction, including hypertension
(67.1%), diabetes (39.2%), Osteoarthritis (29.8%), depression (26.2%) and
asthma (15.5%)), the proportion of hypertensive patients ranged from 40.8% in
cluster 10 to 100% in cluster 9, diabetes proportion ranged from 23.8% in
clusters 2 to 99.5% in cluster 6, osteoarthritis proportion ranged from 17.5%
in cluster 10 to 60.2% in cluster 3, depression proportion ranged from 10.1 in
cluster 5 and 6 / 10.2 in cluster 2 to 86.2% in cluster 7, and asthma
proportion ranged from 5.0% in cluster 5 to 100% in cluster 10.

Figure 3 displays the multivariate log-odds ratio of long term conditions
associated with clusters, and finally a graphical representations of the onset
of myocardial infarction and associated long term conditions in proposed in
figure 4.

Figure 3: Log-odds ratio of the 32 long term conditions associated with
clusters: values are derived from multivariate logistic regressions where
cluster indicators are explained by displayed variables. Positive log-odds
ratio show over representation of the corresponding population traits in a
given cluster as compared to other patients showing a record of myocardial
infarction. Only significant coefficients are displayed (at the 5% level).

Figure 4: Graphical representation of clusters: long term conditions are
represented by colored dots. Dots x-coordinate represents the median age at
onset of the given long term condition. Dots are presented on the the y axis
according to layers, such that no significant transitions exist between dots
from the same layer. Significant transitions are represented by edges oriented
from a higher level to a lower levels.

###  2.4  Clusters examination

From tables LABEL:tab:t1 and LABEL:tab:t2, and results of the above
multivariate statistical analysis on patients’ socio-demographic variables,
risk factors and associated long term conditions, it is now possible to better
characterise the resulting typology of myocardial infarction.

Hypertension is the most frequent long term condition associated with
myocardial infarction in this study. It precedes onset of myocardial
infarction in 2/3 of hypertensive patients. This pattern is the main
characteristic of cluster 1 and 2 (accounting for 23.1% and 15.6% of analyzed
patients respectively). Median age at onset of myocardial infarction
significantly differs in each cluster, from 57.6 year in cluster 1 to 78.2
year in cluster 2 . Cluster 1 is also characterized by a high multimorbidity
index with a median of 6. Comorbidities observed in cluster 1 include
infectious diseases (HIV, B & C viral hepatitis), inflammatory conditions
(IBD, rheumatoid arthritis), mental illness, substance dependency (including
alcohol and smoking) and stroke. Comorbidities observed in cluster 2 are
linked to higher age, they include dementia and osteoporosis but also
cardiovascular conditions such as atrial fibrillation and transient ischemic
attack.

Cluster 3 (11.2% of patients) is also characterized by a high multimorbidity
index (median of 7), and a history of asthma, COPD and osteoarthritis.

Cluster 4 (10.9% of patients) is singular: most of patients in this group are
younger male patients with a median of 48.1 year at onset of myocardial
infarction. Patients in this cluster have also the lowest multimorbidity index
(median of 4).

The structure of cluster 5 (10.2% of patients) is similar to cluster 4 with a
low multimorbidity index (median of 5). However, patients in this cluster are
significantly older at onset of myocardial infarction (median age of 62.2) and
significantly more affected by hypertension (75.6% vs 58.7%) and COPD.

Cluster 6 and 8, accounting for 8.1% and 6.8% of analyzed patients
respectively, are characterised by diverse metabolic disorders including
diabetes (99.5% and 71.2% of patients respectively). Cluster 8 has a higher
mulimorbidity index (median of 6). Along with diabetes, the main conditions
associated with myocardial infarction in this cluster are cardio-vascular
conditions (hypertension (83.7%) and peripheral vascular disease), anxiety
disorders and morbid obesity. Importantly, if both clusters are also
characterised by a higher proportion of non-White patients, cluster 8 is the
youngest group of patients with a median age at onset of myocardial infarction
of 47.2 (vs 65.2 in cluster 6).

White patients are over represented in cluster 7 (82.0%). This cluster,
accounting for 8.0% of analyzed patients, is also characterised by mental
health conditions (including depression (86.2%) and anxiety disorders), COPD,
and the highest proportion of smokers or former smoker (85.4%).

Like clusters 6 and 8, non-White patients are over represented in cluster 9
(accounting for 4.3% of analyzed patients). This cluster is also characterised
by a high rate of morbid obese patients and more strikingly by hypertension as
100% of patient in this cluster suffer from this long term condition.

Finally, cluster 10 is characterized by a young population at onset of
myocardial infarction (49.2 year) with a high multimorbidity index (median of
6). Along with asthma which affects all patients in this cluster, the other
comorbidities associated with myocardial infarction in this cluster are:
morbid obesity, liver disease and anxiety disorder.

##  3  Discussion and concluding remarks

We have presented in this paper a novel method aimed to analysing multiple
time-to-event data denoted as _multiple state analysis_ with application to
the analysis of life-course health trajectories. This method relates to the
sequence analysis developed in the field of the social sciences, and more
particularly to Pollock’s multivariate extension of sequence analysis, where
multiple aspects of the social experience, such as marital and employment
status, are jointly analysed using a _multiple sequences analysis_ strategy.
Before pairwise subject dissimilarities computation and its subsequent
clustering phase, the proposed _multiple state analysis_ requires only a
binary coding of individual data into states matrices and associated follow up
indicator. From this, a typology of the main life course trajectories is
derived.

In an application of _multiple state analysis_ to multimorbidity associated
with myocardial infarction, we analysed 5093 unique individual patients
experiencing up to 32 different commodities, aged from 0 to 104, with various
follow-up times. Following our strategy, patients could be mapped back to a
limited number of clusters associated with major life-long health trajectories
that is, dominant sequences of long term conditions associated with myocardial
infarction showing distinct demographic characteristics and levels of the main
known myocardial infarction risk factors:

  1. (a)

hypertension, which is globally the main driver of myocardial infarction, is
not evenly distributed across clusters. Examination of cluster 9 reveals that
myocardial infarction occurs in this cluster in the context of a longer and
systematic background of hypertension, which is diagnosed with a median of 10
years before the onset of myocardial infarction in all patients in this
cluster,

  2. (b)

a similar observation can be made in clusters 6, 7 and 10 with diabetes,
depression and asthma respectively. Of note, all patients in cluster 10 have a
history of asthma, which was diagnosed with a median of more than 26 years
before the onset of myocardial infarction. This pattern is also seen in
cluster 3, where 80.2% of patients with myocardial infarction have a history
of osteoarthritis and/or asthma,

  3. (c)

in clusters 4 and 5, which are characterized by low multimorbidity indexes,
myocardial infarction arises in a non-specific background. However, median age
at onset of myocardial infarction is lower in cluster 4 as compared to
clusters 5 (48.1 vs 62.2 respectively),

  4. (d)

finally, our results highlight the link between ethnicity and driving long
term conditions such as diabetes which is associated with Asian, Black and
Mixed ethnicity in cluster 6 and 8, and hypertension which is associated with
Asian and Black ethnicity in cluster 9.

The ability to produce a limited number of distinct and meaningful profiles
from arbitrary large populations, as presented in the proposed application of
_multiple state analysis_ , has been denoted by Pollock as a powerful trait of
his technique (_multiple sequence analysis_) which may therefore be used as an
end in itself (Pollock,, 2007).

Without loss of generality, the approach proposed in this paper can be seen as
an extension of Pollock’s multiple sequence analysis when patients (or any
subject of interest) may experience an arbitrary number of events for which
time of occurrence is of interest. Of note, the method is not constrained by
the record of rare events and it allows for right or interval censoring.
Finally, its implementation is straightforward using the basic libraries of
popular statistical software.

_Multiple state analysis_ is particularly appropriate to the analysis of
multiple comorbidities as this phenomenon is characterized by the accumulation
over time of potentially many long term conditions such as diabetes or
hypertension as well as acute events such as myocardial infarction or stroke.
In contrast to situations where transition between exclusive states is of
interest, such as changes in the marital status in the analysis of social
trajectories, in the epidemiological setting of multimorbidity, patients’
health trajectories are characterised by different levels of multimorbidity
over life courses. In this setting, the simple record of events, and its
resulting sequences, analysed as sequences of exclusive events, may lead to
irrelevant conclusions.

Examination of the typology created using _multiple state analysis_ shows that
resulting groups of patients are discriminated in terms of the main myocardial
infarction driving long term conditions but also with respect to major
demographics variables including age, sex and ethnicity, and common risk
factors such as cholesterol or smoking status. Assuming that early life
exposures as well as some latter behaviors / exposures are implicitly
apprehended through the continuous record of acute diseases and long term
conditions, theses results support the environmental approach hypothesis of
life course epidemiology whether it focuses on early life exposure or later
behaviors / exposures. Our results also implicitly support a genotype /
environment hypothesis to myocardial infarction, depending on the extent to
which long term conditions driving the occurrence of myocardial infarction are
influenced by gene / environment exposure.

From a public health perspective however, the genotype / environment balance
origin of medical records history is not of prime importance as public health
policies would rather focus on modifiable risk factors of diseases. To this
end, _multiple state analysis_ is a particularly relevant tool as it allows
stratification of at-risk populations in terms of the main demographics
variable as well as the main driving long term conditions allowing tailored
communication according to target population.

To conclude, the proposed approach to multiple time-to-event data denoted here
as _multiple state analysis_ is a versatile tool, comparable in its objective
and results to the simple or multiple analysis of sequence data, but adapted
to the situation where the times to potentially many non exclusive events need
to be analysed in a multidimensional manner as opposed to the simple analysis
of the resulting sequences of events. Alike the simple or multiple sequence
analysis, the aim of _multiple state analysis_ is the development of
exploratory typologies allowing to uncover broad distinctions between majors
life course trajectories, without focusing on a single event or point in time.
As such, results of _multiple state analysis_ may be valuable for further
modeling, to guide public health decision making or as an end in itself
(Pollock,, 2007).

##  Acknowledgements

We would like to thank Mark Ashworth, Stevo Durbaba, Xiaohui Sun, and Mohammad
Yadegarfar for their help in data extraction and approvals. Approval for the
analysis of fully anonymized data was granted by Lambeth DataNet Clinical
Commissioning Group and Information Governance Steering Group. This project is
funded by King’s Health Partners / Guy’s and St Thomas Charity ‘MLTC Challenge
Fund’ (grant number EIC180702) and support from the National Institute for
Health and Care Research (NIHR) under its Programme Grants for Applied
Research (NIHR202339) and the NIHR Applied Research Collaboration (ARC) South
London at King’s College Hospital NHS Foundation Trust. The views expressed
are those of the authors and not necessarily those of the NIHR or the
Department of Health and Social Care.

Table 1:  Distribution of socio-demographic variables and risk factors associated with myocardial infarction according the clusters Cluster # |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total | p
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Sex | Female | 320 (27.2) | 379 (47.8) | 257 (45.2) | 87 (15.7) | 136 (26.3) | 135 (32.5) | 135 (33.3) | 94 (27.3) | 73 (33.5) | 24 (23.3) | 1640 (32.2) | <0.001
Age at onset | Median (IQR) | 57.6 (14.6) | 78.2 (12.4) | 70.2 (12.2) | 48.1 (7.9) | 62.2 (8.3) | 65.2 (12.8) | 57.2 (15.4) | 47.2 (13.0) | 59.7 (12.6) | 49.2 (14.6) | 61.2 (19.9) | <0.001
Ethnicity | White | 617 (71.2) | 416 (76.8) | 282 (68.4) | 286 (68.4) | 270 (73.2) | 135 (45.3) | 260 (82.0) | 123 (47.7) | 72 (45.3) | 49 (62.0) | 2510 (67.5) | <0.001
| Asian | 77 (8.9) | 46 (8.5) | 55 (13.3) | 51 (12.2) | 38 (10.3) | 77 (25.8) | 16 (5.0) | 39 (15.1) | 31 (19.5) | 9 (11.4) | 439 (11.8) |
| Black | 133 (15.3) | 59 (10.9) | 66 (16.0) | 58 (13.9) | 52 (14.1) | 65 (21.8) | 28 (8.8) | 69 (26.7) | 49 (30.8) | 14 (17.7) | 593 (15.9) |
| Other | 40 (4.6) | 21 (3.9) | 9 (2.2) | 23 (5.5) | 9 (2.4) | 21 (7.0) | 13 (4.1) | 27 (10.5) | 7 (4.4) | 7 (8.9) | 177 (4.8) |
IMD quintile $\dagger$ | 1-2 | 786 (66.9) | 527 (66.5) | 389 (68.5) | 344 (62.0) | 321 (62.1) | 269 (64.8) | 281 (69.4) | 243 (70.6) | 138 (63.3) | 70 (68.0) | 3368 (66.1) | 0.065
Cholesterol | yes | 860 (75.0) | 528 (69.9) | 445 (79.0) | 385 (72.1) | 379 (74.3) | 292 (72.5) | 318 (78.9) | 271 (79.5) | 157 (72.0) | 68 (67.3) | 3703 (74.4) | 0.001
Smoking ever | yes | 941 (82.0) | 474 (62.8) | 405 (71.9) | 430 (80.5) | 377 (73.9) | 280 (69.5) | 344 (85.4) | 271 (79.5) | 144 (66.1) | 79 (78.2) | 3745 (75.3) | <0.001
Polymedication | yes | 426 (36.8) | 337 (44.2) | 244 (43.3) | 149 (27.6) | 170 (33.9) | 177 (44.0) | 143 (35.9) | 122 (35.7) | 69 (31.9) | 30 (29.1) | 1867 (37.4) | <0.001
Multimorbidity index | Median (IQR) | 6.0 (3.0) | 5.0 (3.0) | 7.0 (3.0) | 4.0 (3.0) | 5.0 (4.0) | 5.0 (3.0) | 6.0 (4.0) | 6.0 (4.0) | 5.0 (4.0) | 5.0 (3.0) | 5.0 (3.0) | <0.001
Status | Dead | 389 (33.1) | 518 (65.3) | 244 (43.0) | 170 (30.6) | 222 (42.9) | 187 (45.1) | 101 (24.9) | 90 (26.2) | 59 (27.1) | 21 (20.4) | 2001 (39.3) | <0.001
Total N (%) |  | 1175 (23.1) | 793 (15.6) | 568 (11.2) | 555 (10.9) | 517 (10.2) | 415 (8.1) | 405 (8.0) | 344 (6.8) | 218 (4.3) | 103 (2.0) | 5093 |
$\dagger$ IMD quitile 1 (most deprived)-2 vs 3-5 (less deprived)
Table 1: Distribution of socio-demographic variables and risk factors associated with myocardial infarction according the clusters (continued) Table 2: Distribution of the 32 comorbidities associated with myocardial infarction according the clusters Cluster # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total | p
---|---|---|---|---|---|---|---|---|---|---|---|---
Cancer | 273 (23.2) | 141 (17.8) | 107 (18.8) | 71 (12.8) | 96 (18.6) | 69 (16.6) | 61 (15.1) | 28 (8.1) | 23 (10.6) | 11 (10.7) | 880 (17.3) | <0.001
Stroke | 235 (20.0) | 149 (18.8) | 112 (19.7) | 65 (11.7) | 71 (13.7) | 72 (17.3) | 57 (14.1) | 55 (16.0) | 26 (11.9) | 14 (13.6) | 856 (16.8) | <0.001
Heart Failure | 292 (24.9) | 271 (34.2) | 162 (28.5) | 157 (28.3) | 170 (32.9) | 128 (30.8) | 79 (19.5) | 109 (31.7) | 65 (29.8) | 23 (22.3) | 1456 (28.6) | <0.001
Peripheral Vascular Disease | 141 (12.0) | 92 (11.6) | 68 (12.0) | 74 (13.3) | 38 (7.4) | 70 (16.9) | 45 (11.1) | 58 (16.9) | 26 (11.9) | 5 (4.9) | 617 (12.1) | <0.001
Atrial Fibrillation | 166 (14.1) | 231 (29.1) | 133 (23.4) | 86 (15.5) | 133 (25.7) | 65 (15.7) | 42 (10.4) | 38 (11.0) | 33 (15.1) | 11 (10.7) | 938 (18.4) | <0.001
Hypertension | 706 (60.1) | 562 (70.9) | 389 (68.5) | 326 (58.7) | 391 (75.6) | 290 (69.9) | 205 (50.6) | 288 (83.7) | 218 (100.0) | 42 (40.8) | 3417 (67.1) | <0.001
Transient Ischemic Attack | 78 (6.6) | 89 (11.2) | 32 (5.6) | 30 (5.4) | 35 (6.8) | 26 (6.3) | 17 (4.2) | 21 (6.1) | 19 (8.7) | 4 (3.9) | 351 (6.9) | <0.001
Viral Hepatitis (B & C) | 41 (3.5) | 1 (0.1) | 6 (1.1) | 8 (1.4) | 1 (0.2) | 4 (1.0) | 4 (1.0) | 5 (1.5) | 1 (0.5) | 1 (1.0) | 72 (1.4) | <0.001
Hiv/aids | 52 (4.4) | 0 (0.0) | 0 (0.0) | 3 (0.5) | 2 (0.4) | 0 (0.0) | 8 (2.0) | 4 (1.2) | 1 (0.5) | 5 (4.9) | 75 (1.5) | <0.001
Rheumatoid Arthritis | 46 (3.9) | 12 (1.5) | 28 (4.9) | 4 (0.7) | 12 (2.3) | 3 (0.7) | 11 (2.7) | 7 (2.0) | 4 (1.8) | 5 (4.9) | 132 (2.6) | <0.001
IBD $\dagger$ | 45 (3.8) | 6 (0.8) | 8 (1.4) | 9 (1.6) | 9 (1.7) | 1 (0.2) | 4 (1.0) | 3 (0.9) | 1 (0.5) | 4 (3.9) | 90 (1.8) | <0.001
Lupus | 6 (0.5) | 0 (0.0) | 2 (0.4) | 0 (0.0) | 0 (0.0) | 0 (0.0) | 2 (0.5) | 4 (1.2) | 2 (0.9) | 0 (0.0) | 16 (0.3) | 0.016
CKD 3-5 $\ddagger$ | 270 (23.0) | 318 (40.1) | 169 (29.8) | 105 (18.9) | 167 (32.3) | 155 (37.3) | 67 (16.5) | 94 (27.3) | 64 (29.4) | 13 (12.6) | 1422 (27.9) | <0.001
Liver Disease | 40 (3.4) | 6 (0.8) | 5 (0.9) | 5 (0.9) | 5 (1.0) | 5 (1.2) | 4 (1.0) | 12 (3.5) | 3 (1.4) | 3 (2.9) | 88 (1.7) | <0.001
Anxiety Disorders | 299 (25.4) | 64 (8.1) | 149 (26.2) | 81 (14.6) | 58 (11.2) | 33 (8.0) | 239 (59.0) | 100 (29.1) | 24 (11.0) | 37 (35.9) | 1084 (21.3) | <0.001
Depression | 393 (33.4) | 81 (10.2) | 160 (28.2) | 78 (14.1) | 52 (10.1) | 42 (10.1) | 349 (86.2) | 114 (33.1) | 31 (14.2) | 35 (34.0) | 1335 (26.2) | <0.001
Alcohol Dependence | 246 (20.9) | 10 (1.3) | 25 (4.4) | 37 (6.7) | 8 (1.5) | 5 (1.2) | 49 (12.1) | 23 (6.7) | 4 (1.8) | 8 (7.8) | 415 (8.1) | <0.001
Serious Mental Illness | 113 (9.6) | 8 (1.0) | 10 (1.8) | 9 (1.6) | 5 (1.0) | 4 (1.0) | 25 (6.2) | 13 (3.8) | 2 (0.9) | 2 (1.9) | 191 (3.8) | <0.001
Substance Dependency | 105 (8.9) | 11 (1.4) | 9 (1.6) | 12 (2.2) | 1 (0.2) | 3 (0.7) | 18 (4.4) | 13 (3.8) | 2 (0.9) | 7 (6.8) | 181 (3.6) | <0.001
Learning Disabilities | 14 (1.2) | 0 (0.0) | 0 (0.0) | 4 (0.7) | 1 (0.2) | 3 (0.7) | 4 (1.0) | 0 (0.0) | 0 (0.0) | 0 (0.0) | 26 (0.5) | 0.002
Diabetes | 378 (32.2) | 189 (23.8) | 201 (35.4) | 178 (32.1) | 133 (25.7) | 413 (99.5) | 129 (31.9) | 245 (71.2) | 101 (46.3) | 27 (26.2) | 1994 (39.2) | <0.001
Obesity (morbidly obese) | 106 (9.0) | 22 (2.8) | 46 (8.1) | 50 (9.0) | 15 (2.9) | 33 (8.0) | 22 (5.4) | 56 (16.3) | 28 (12.8) | 13 (12.6) | 391 (7.7) | <0.001
Osteoarthritis | 301 (25.6) | 250 (31.5) | 359 (63.2) | 107 (19.3) | 135 (26.1) | 106 (25.5) | 118 (29.1) | 68 (19.8) | 54 (24.8) | 18 (17.5) | 1516 (29.8) | <0.001
Osteoporosis | 34 (2.9) | 91 (11.5) | 46 (8.1) | 11 (2.0) | 34 (6.6) | 16 (3.9) | 18 (4.4) | 4 (1.2) | 5 (2.3) | 1 (1.0) | 260 (5.1) | <0.001
Sickle-Cell Anaemia | 1 (0.1) | 0 (0.0) | 0 (0.0) | 1 (0.2) | 0 (0.0) | 1 (0.2) | 0 (0.0) | 0 (0.0) | 0 (0.0) | 0 (0.0) | 3 (0.1) | 0.693
Parkinson’s | 23 (2.0) | 17 (2.1) | 11 (1.9) | 4 (0.7) | 8 (1.5) | 10 (2.4) | 5 (1.2) | 4 (1.2) | 0 (0.0) | 0 (0.0) | 82 (1.6) | 0.122
Multiple Sclerosis | 3 (0.3) | 0 (0.0) | 1 (0.2) | 0 (0.0) | 0 (0.0) | 0 (0.0) | 2 (0.5) | 1 (0.3) | 0 (0.0) | 0 (0.0) | 7 (0.1) | 0.409
Epilepsy | 105 (8.9) | 4 (0.5) | 13 (2.3) | 8 (1.4) | 12 (2.3) | 5 (1.2) | 6 (1.5) | 10 (2.9) | 5 (2.3) | 4 (3.9) | 172 (3.4) | <0.001
Dementia | 85 (7.2) | 155 (19.5) | 74 (13.0) | 37 (6.7) | 64 (12.4) | 42 (10.1) | 31 (7.7) | 5 (1.5) | 13 (6.0) | 4 (3.9) | 510 (10.0) | <0.001
Asthma | 161 (13.7) | 72 (9.1) | 211 (37.1) | 45 (8.1) | 26 (5.0) | 29 (7.0) | 78 (19.3) | 47 (13.7) | 16 (7.3) | 103 (100.0) | 788 (15.5) | <0.001
COPD $\S$ | 227 (19.3) | 102 (12.9) | 176 (31.0) | 63 (11.4) | 79 (15.3) | 34 (8.2) | 97 (24.0) | 23 (6.7) | 13 (6.0) | 25 (24.3) | 839 (16.5) | <0.001
Chronic Pain | 769 (65.4) | 479 (60.4) | 475 (83.6) | 254 (45.8) | 264 (51.1) | 222 (53.5) | 264 (65.2) | 219 (63.7) | 110 (50.5) | 57 (55.3) | 3113 (61.1) | <0.001
Total N (%) | 1175 (23.1) | 793 (15.6) | 568 (11.2) | 555 (10.9) | 517 (10.2) | 415 (8.1) | 405 (8.0) | 344 (6.8) | 218 (4.3) | 103 (2.0) | 5093 |
$\dagger$ Inflammatory Bowel Disease
$\ddagger$ Chronic Obstructive Pulmonary Disease
$\S$ Chronic Kidney Disease stage 3 to stage 5
Table 2: Distribution of the 32 comorbidities associated with myocardial
infarction according the clusters (continued)

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