UDK 523.92+51.71

The Structure of the Solar Cycle and of the Activity Cycles of Late-Type Stars

V.N. Obridko1, D.D. Sokoloff1,2, M.M.Katsova3

1IZMIRAN, Troitsk, Moscow, Russia

2Lomonosov Moscow State University, Moscow, Russia

3Sternberg Astronomical Institute of the Lomonosov Moscow State University,
Moscow, Russiaя

Abstract

It is shown that the description of the solar cycle that takes into account
the odd zonal harmonic of the solar magnetic field allows us to deepen our
knowledge of two important aspects of the solar activity. First, to clarify
and expand predictions of the evolution of the cyclic activity of the Sun in
the near future. Second, to develop a program for monitoring the
spectrophotometric characteristics of radiation of the solar-type stars aimed
at obtaining new information about their magnetic fields.

Keywords: solar cycle, late-type stars

## Introduction

The classical description states that the solar activity cycle is the
interaction between the main dipole component of the solar poloidal magnetic
field and the toroidal magnetic field. Here, the physical nature of these
components is assumed to be intuitively clear and is not subjected to special
analysis. However, the real extended solar cycle is much more complicated than
the models and requires a detailed description. In this work, we develop the
concept of an extended solar cycle by finding out what zonal harmonics are
responsible for the features of the equatorial and polar propagation in the
activity tracers on the surface. As a result, we arrive at the conclusion that
the zonal harmonics with $l=5$ play a decisive role in separating the
phenomena of both types associated with odd zonal harmonics.

According to the original scheme, the maximum of the equatorial field
coincides with the inversion of the polar field, and both hemispheres are
antisymmetric in the field sign. However, this simple and clear concept of the
mean field does not fully explain a number of features of the solar activity.
In particular, why are there cycles of different heights, how do spots arise
from the mean field, and what is the nature of active longitudes? This scheme
and its generalizations deal with zonal characteristics; therefore,
supersynoptic maps are analyzed, on which any dependence on longitude
disappears.

In our research, we applied this description to the concept of an extended
solar cycle. This concept was first formulated in 1988 [1, 2]. Shortly before
this, observations appeared showing that the magnetic activity of one cycle
could coexist with the magnetic activity of the previous cycle, and the
overlapping interval could be up to seven years [3]. As seen from observations
of the solar surface, the extended solar cycle begins at high latitudes during
the sunspot maximum and consists of a relatively short poleward branch called
’’Rush-to-the-Poles’’ (RTTP) and a long branch directed towards the equator
and passing through the solar minimum and the following solar cycle ([4], see
also [5] and references therein).

The idea of an extended solar cycle as a single entity formed by the observed
space-time structures was the result of many years of observations and a hard
work on compiling catalogues carried out by the generation of outstanding
observers of the past.

Some of these structures, such as prominences and filaments (e.g., see [6-9]),
ephemeral active regions [10], and most features observed in the solar coronal
green line [11], are naturally interpreted as magnetic phenomena. The origin
of the other, such as the zonal structures of the torsion oscillations
[12-14], is not yet completely clear. We should also mention long-term series
of geomagnetic data, in which features are found that are consistent with the
pattern of solar magnetic cycles overlapping in time [15, 16].

In the recent work of our team [17], we made the following additions to the
concept of the extended solar cycle:

1\. The initial paradigm of the solar dynamo is reasonably consistent with the
description of the 11-year cycle of solar activity using the first two odd
zonal harmonics of the large-scale solar magnetic field.

2\. On the other hand, this description needs a modification and refinement
within the concept of the extended solar cycle. During the overlapping phase,
three waves of activity coexist on the solar surface leading to the appearance
and intensification of the odd zonal harmonic with $l=5$.

3\. The maximum amplitude of the harmonic with $l=5$ was decreasing strongly
during the past four cycles of solar activity, similar to the cycle amplitude
obtained from sunspots. The magnitude of the harmonic corresponding to the
Gaussian coefficient $g_{50}$ is observed to fall especially strongly after
2000, which agrees with low Cycle 24. Of course, four cycles are hardly
sufficient for convincing statistics, but it seems reasonable to expect that
Cycle 25 will not be much higher than Cycle 24.

These conclusions were supported and refined by new observations. Such
refinements, which appeared after the publication of [17], are given below and
constitute one of the goals of this work.

Another goal of this work is to form an idea of the activity cycles on stars
of late spectral types using the solar cycle as an example. Naturally, the
observations of the magnetic activity on these stars are much less detailed
than the scheme of the extended solar cycle described above. Therefore, it is
important to understand to what extent this complex scheme can be compared
with data on the stellar magnetic activity that are available now or can be
obtained in the foreseeable future.

##  1 The role and meaning of the fifth harmonic

The central role in the proposed description of the phenomenon of the solar
cycle belongs to the zonal harmonic $l=5$. Let’s consider its behavior in more
detail.

Figure 1 shows a synoptic map of the magnetic field. One can see the magnetic
flux propagate from the latitudes $\theta=-30^{\circ}$ and
$\theta=+30^{\circ}$ to the poles. This phenomenon, commonly referred to as
Rush-to-the-Poles (RTTP), is mainly associated with large-scale fields. Upon
reaching the poles, RTTP replaces the previous wave, and a reversal of the
polar field occurs. Besides that, there is a well-known wave moving from mid
latitudes to the equator. The polar and equatorial waves appear almost
simultaneously, and have opposite predominant polarities of the magnetic
field. At some moments, two Rush-to-the-Poles waves with opposite field signs
coexist on the Sun. When one of them almost reaches the pole, the other only
appears at mid latitudes. In such periods, three zones of alternating
magnetic-field polarity exist in each hemisphere on the Sun.

Figure 1: Supersynoptic map based on Kitt Pick Observatory data (positive
polarity - white, negative polarity - dark). The vertical lines indicate the
positions of the maxima of the fifth harmonic. The ordinate axis is the sine
of latitude; the horizontal bars mark the latitudes of $30^{\circ}$,
$0^{\circ}$, and $-30^{\circ}$.

At first glance, it is not quite clear why we emphasize the role of one of the
zonal harmonics. As an example, let’s take a closer look at the supersynoptic
map in the vicinity of 1999 (see Fig. 1). At the north pole, we can still see
the wave of positive polarity, which formed in 1989 (i.e., in the growth phase
of Cycle 22), and was moving to the pole during 13 years until 2002 (i.e., in
fact, till the maximum of Cycle 23). At the same time, another wave of
negative polarity existed at mid latitudes since 1999 and was also drifting
poleward for 13 years until 2012 (i.e., until the ascending branch of cycle
24). In the same year (1999), the third wave appeared, which, unlike the first
two waves, was not strictly unipolar, but nevertheless it had predominant
positive polarity, corresponding to the polarity of the leading spots in
active regions. This wave was propagating towards the equator.

As a result, we have three zones of alternating polarity in each hemisphere.
As a whole, the interaction of these three magnetic systems continues from
1989 to 2005, i.e., 16 years. Thus, there is a period when the waves of all
three types are present simultaneously. At this time, six intermittent zones
are observed in general on the visible solar disk, which exactly corresponds
to the zonal harmonic with $l=5$. We call this time interval the overlapping
phase. We will see below that the overlapping phase can be quantitatively
described by the fifth zonal harmonic, which, in this context, can be called
the height of the overlapping zone.

Figure 1 was plotted using WSO (John Wilcox Stanford Observatory) data for the
period from May 1976 (Carrington rotation 1641) up to the present
(http://wso.stanford.edu/forms/prsyn.html). The observed solar surface
magnetic field was expanded into harmonic coefficients, and the time evolution
of the corresponding coefficients was represented over the last four cycles of
solar activity. We analyzed WSO synoptic maps of the line-of-sight component
of the photospheric magnetic field expanded into the series of associated
Legendre polynomials $P_{ml}$ [18].

All components of the magnetic field at a given point of a spherical layer
between the photosphere and the so-called source surface can be reconstructed
under the potential approximation from the line-of-sight component. The source
surface is defined as a spherical surface, on which all magnetic lines become
radial. It is assumed that it lies at a distance $R_{s}=2.5R_{o}$ from the
center of the Sun ($R_{0}$ is the solar radius). In this case, the classical
scheme was used, which assumes that the potential approximation can also be
applied to the lower boundary surface, i.e., the photosphere level. Figure 2
shows the actual field structure of the fifth harmonic on June 1, 1999. All
six zones are clearly visible.

Figure 2: The field structure on the photosphere with the fifth harmonic alone
taken into account. One rotation is shown, centered on June 1, 1999.

Next, following [19] (see also [20]), we calculate the mean square radial
component of the magnetic field on a sphere of radius $R$. Using the
orthogonality of the Legendre polynomials, we calculate integrals over a
spherical surface and obtain the result in closed form. Below, we give as an
example the result of calculations on the photosphere $i(B_{r})|_{R_{0}}$ and
on the source surface $i(B_{r})|_{R_{s}}$:

$i(B_{r})|_{R_{0}}=\sum_{lm}\frac{(l+1+l\zeta^{2l+1})^{2}}{2l+1}(g^{2}_{lm}+h^{2}_{lm}),$
(1)

$i(B_{r})|_{R_{s}}=\sum_{lm}(2l+1)\zeta^{2l+4}(g^{2}_{lm}+h^{2}_{lm}),$ (2)

where $\zeta=R_{0}/R_{s}$. Hence, the contribution of the $l^{th}$ mode to the
mean solar magnetic field includes all coefficients with $l$. We assume
$R_{s}=2.5R_{0}$; i.e., $\zeta=0.4$. Choosing appropriate values of $l$ and
$m$ in these relations, we obtain the contribution of the corresponding
harmonics to the mean square magnetic field on the corresponding surface. The
calculation results are shown in Figure 3.

Figure 3: Time evolution of the mean square magnetic field associated with the
first odd axisymmetric harmonics up to $l=7$ [21]. The lowest panel shows the
time variation in sunspot numbers (SSN) during the solar cycle; the top panel
represents the time evolution of the polar magnetic field.

The harmonics with $l=1$ and $l=3$ behave in accordance with the classical
dynamo theory; i.e., they follow the evolution of the polar magnetic field and
change in anti-phase with the sunspot numbers. On the contrary, the harmonic
with $l=5$ behaves in essentially different way and is maximum just at the
beginning of the minimum of the polar magnetic field [17].

To sum up, we can say that shortly before the sign reversal of the polar
magnetic field, both the large-scale and the relatively small-scale magnetic
fields appear almost simultaneously on the photosphere surface (the
overlapping phase). Then, however, these fields behave differently. The large-
scale component as a unipolar feature moves to the poles (Rush-to-the-Poles
phenomenon). As a result, two waves of opposite polarities appear, propagating
on the solar surface. On the contrary, the local magnetic field is represented
by bipolar features propagating towards the equator. Formally, the separation
of both types of waves is quantitatively related to the harmonic with $l=5$.

The maximum of the fifth harmonic is displayed as a rather sharp peak
coinciding in time with a relatively short period of coexistence of all three
waves in each hemisphere.

At the time when the nearest wave reaches the pole and disappears, a reversal
of the polar field occurs. From that moment on, only two waves are observed on
the Sun in each hemisphere, and the harmonic with $l=5$ virtually disappears.

Note that the maxima of the zonal harmonic with $l=5$ are clearly defined in
Cycles 21, 22, and 23, but are almost invisible in Cycle 24, when we are
dealing with an extended minimum and a rather indistinct overall structure
after 2012 (see Fig.1). One can see that for the fifth harmonic, $B^{2}$ does
not exceed 0.1 $G^{2}$ during the entire period after 2000. It should be noted
that Cycle 24 was anomalous in many respects. In particular, it was strongly
distorted by asymmetry. The maxima in the northern and southern hemispheres
were so much separated in time that the total number of sunspots was much
lower than in other cycles.

The behavior of the fifth harmonic until 2022 strongly resembled the situation
before Сycle 24; therefore, we suggested in [17] that Cycle 25 cannot be
significantly higher than Cycle 24.

##  2 Estimates of the height and date of Cycle 25

The prospects for Cycle 25 have become somewhat more definite lately, after
our work [17] was already published. First of all, we can use data on the
behaviour of the polar field. The connection between the intensity of the
polar field and the height of the following solar cycle is best substantiated
physically and relies on the basic concepts of the solar dynamo (for details,
see extensive reviews [22, 23].

The linear relationship between the maximum value of the polar field $B_{\rm
pol}$ at the end of the previous cycle (usually near the SSN minimum) and the
height of the following sunspot cycle based on data for the last four cycles
is shown in Fig. 4 and is approximated by the relation

$SSN_{max}=36.405+1.3666B_{\rm pol}.$ (3)

The observed maximum value of $B_{\rm pol}$ was reached in the summer of 2019
and was equal to 65 $\mu T$ (http://wso.stanford.edu/Polar.html). Hence, the
predicted value of SSN in Cycle 25 is 125.2. This is only 8.8 units higher
than in Cycle 24 (116.4).

Figure 4: Relationship between the maximum value of the polar field, $B_{\rm
pol}$, before the cycle and the height of the coming sunspot cycle. The
expected value for Cycle 25 is indicated by the arrow. Figure 5: Variations in
the squared amplitude of the fifth zonal harmonic over the past two years.

It is to be noted that the variation of the fifth harmonic also became more
regular (see Fig. 5). At the same time, its maximum value reached in 2022.5
was 0.98 $G^{2}$, which is almost an order of magnitude lower than the maximum
values of this harmonic in Cycles 21 and 22. This probably indicates that the
mechanism of generation of the three waves mentioned above does not work
efficiently enough.

Figure 6: The lag (difference between $T_{\rm max}$ and $T_{P}$,): the
positive values indicate that the polarity reversal occurred prior to the
maximum of the sunspot cycle; the negative values show that the polarity
reversal was delayed. The lag values were calculated separately for the north
(open circles) and the south pole (triangles). The average reversal dates
(avg) are shown with squares.

As the dates of the maximum, we can use the dates of reversal of the polar
field. According to the classical model, the dates of the polarity reversals
and cycle maxima must coincide or at least be close in time. Figure 6
illustrates the time delay between the polarity reversals $T_{P}$ and the
$SSN(T_{\rm max})$ dates. The lag was calculated as the difference between
$T_{\rm max}$ and $T_{P}$. The positive values indicate that the polarity
reversal occurred prior to the maximum of the sunspot cycle; the negative
values show that the polarity reversal was delayed. The lag values were
calculated separately for the north (open circles) and the south pole
(triangles). The average reversal dates (avg) are shown with squares.

It turned out that in the relatively high (or at least, medium) Cycles 21 and
22, the SSN maximum either coincided with or took place slightly ahead of the
polarity reversal, while in the low Cycles 23 and, especially, 24, the
polarity reversal occurred 1.0 - 1.5 years earlier. Note that the very concept
of the exact date of the cycle maximum is rather conventional. Usually, the
date of the maximum is the month when the curve of the sunspot numbers
smoothed over 13 months reaches its maximum value.

However, there is a long time interval when the values on the curve differ
from the maximum value rather weakly, by only a few units. Therefore, it would
be more correct to speak of a maximum phase. Now that we already know the
polarity reversal date at the beginning of Cycle 25 in the north hemisphere
(March 2023), we can expect the cycle maximum between the end of 2023 and the
second half of 2024, which is much earlier than predicted from a simple
11-year periodicity (2025 or early 2026).

Thus, the maximum may come earlier than predicted (already in 2023), and its
height is expected to be 125.2, i.e., only 9 units more than in Cycle 24. The
earliest date of the onset of the maximum phase is November-December 2023.

Another question the future observations may answer is whether the Gnevyshev-
Ohl rule remains valid for this cycle. The point is that there are several
formulations of this rule [24]. The original rule states that the sum of
sunspot numbers per cycle in even cycles is less than in subsequent odd cycles
[25]. Later, it was extended to the monthly maximum sunspot numbers per cycle.
The rule in the latter formulation was broken in the pair of Cycles 22-23.

If our calculations are correct, the Gnevyshev-Ohl rule in this formulation
will hold in the pair of Cycles 24-25, although Cycle 25 will only slightly
exceed Cycle 24 in height. However, it follows from our calculations that the
growth branch in Cycle 25 will be very short (4.0–4.5 years), i.e., much
shorter than the growth branch of Cycle 24 (about 5.5 years). Therefore, there
is no certainty that the rule for the integral sum of sunspot numbers per
cycle in its classical form will hold in the pair of Cycles 24-25. Here, a
very long maximum phase in Cycle 24 (almost 3 years) can play a special role.

The results of this section are briefly summarized as express information in
[26].

##  3 A possible scenario for observing the structure of activity on late-type
stars

The structure of the solar cycle described above can be used not only to
forecast the periodic solar activity. It can also indicate what exactly should
be observed to better understand the activity on stars more or less similar to
the Sun, i.e., the late-type stars. We know from spectrophotometric data and,
first of all, from the data of the H-K Project [27] and conceptually similar
projects [28] that a number of late-type stars exhibit activity cycles similar
to the solar ones, in particular, their duration is comparable to the length
of the solar cycle. Although the periodicity of solar type on the stars with
similar physical characteristics seems natural, the discovery of these cycles
was a fundamental achievement of astronomy. Note that on many stars observed
within the framework of the mentioned projects, the detected time variations
of the studied spectral lines do not have a pronounced periodicity. On some
stars, secondary cycles were detected along with the main ones (see, for
example, [29, 30]). The analysis of archival time series available for
individual stars (e.g., [31, 32]) shows that instead of a cyclic behavior,
fluctuations can form a continuous spectrum of variations.

It is believed that the stellar activity cycles and other similar variations
are associated with inductive effects and rank among the phenomena studied in
the framework of the hydromagnetic dynamo theory (although, there are stars
with a relic magnetic field, as, apparently, are Ap stars (e.g., [33])). The
variable activity of late-type stars is usually discussed within the paradigm
that the magnetic field excited by the stellar dynamo has a configuration
similar to that of the solar magnetic field. This point of view was
characteristic of the first (successful) attempts to detect stellar cycles. In
a few cases, the time-latitude pattern of the cyclic activity of particular
stars could be reconstructed (at least roughly) from the available data, and
it really turned out to be similar to the solar activity (see, for example,
[34]). However, there are many stars of late spectral types whose magnetic-
field structure and cyclic activity may differ very much from those of the
Sun. For example, it is widely believed that the magnetic activity of M-type
dwarfs differs markedly from that of the Sun (e.g., [35]), although the
question of what exactly is this difference remains open. Another example is
the discovery of stars comparable to the Sun, but having kilogaussian magnetic
fields (e.g., [36]) or producing much stronger flares than the Sun [37]. All
said above makes us believe that the next step in the study of the stellar
magnetism should comprise observations that will either show the possibility
of excitation of essentially different magnetic configurations on stars or
prove that all really existing configurations are similar to those of the Sun.

The existing theoretical models of the stellar dynamo allow various magnetic
field configurations excited by dynamo mechanisms similar to that acting on
the Sun. Virtually all scientists involved in modeling spherical dynamos have
noted this under various circumstances (see [38] for one of the first
references), although, of course, it is difficult to prove by numerical
simulations that such configurations actually occur on stars. On the other
hand, theoretical arguments can be provided in favor of the statement that all
stars close to the Sun in their physical properties must have a magnetic field
similar to that of the Sun (cf. [39]). Obviously, the described problem should
be solved by observational methods, i.e., by constructing latitude-time
diagrams of stellar activity based on a representative sample of observational
data for stars of late spectral types. Since the problem is rather general,
this sample does not have to contain only stars that are very similar in their
properties to the Sun.

Fortunately, the available observational capabilities, first of all, the
inverse Doppler imaging known since the 80s of the past century [40], allow
the solution of such problems, especially taking into account that we are only
interested in the latitudinal distribution of star spots, and there is no need
to determine their sidereal longitude.

The proposed program involves long-term monitoring of stellar activity
comparable in duration to the the H-K Project. (It is remarkable that this
Project had started before the inverse Doppler imaging technique was
developed). Such a program requires that we clearly define the particular
features in the described scenario of a solar cycle that are to be compared
with the stellar activity data, since the discovery of their entire variety is
apparently the matter of a distant future. This is precisely the purpose of
the present section.

The idea of the proposed monitoring is that, relying only on temperature data
and restoring the latitude-time distribution of star spots over two or three
cycles of stellar activity, we can to establish more or less confidently the
similarity or dissimilarity between the solar activity and the activity of a
given star. This idea, is based on the fundamental zonal model of star spot
distribution developed by Alekseev and Gershberg [41]. But our goal is not to
describe the distribution of spots using a limited number of parameters, but
to find out what magnetic configurations they are associated with and what can
this tell us about the operation of the stellar dynamo.

Instead of reconstructing the positions of individual star spots on the basis
of the reverse Doppler images, it seems enough to restore the position of the
latitude band, $\Theta$, in which these spots arise, its half-width $\theta$,
its contrast with the surrounding surface, $K$, and the evolution of these
parameters over a stellar activity cycle. It is also necessary to find out
whether such a band is observed separately in each hemisphere or whether a
single band of activity common to both hemispheres appears in the vicinity of
the stellar equator.

Let us explain how the behavior of these parameters helps us understand
whether the cyclic activity of a given star resembles that of the Sun. The
presence of a single activity band near the stellar equator, which does not
split into two individual bands in each hemisphere, would indicate that we are
dealing with a magnetic configuration with quadrupole rather than dipole
symmetry. This test is based on the fact that the toroidal component of the
mean magnetic field with dipole symmetry vanishes at the stellar equator,
while in the case of the quadrupole symmetry it need not vanish. Similar
configurations appear in stellar dynamo models along with ordinary solar-type
configurations, but have not yet been detected in observational practice. They
are discussed in literature (e.g., [42]). In archival observations of solar
activity, a similar episode was reported in the 18th century [43], although
the quality of observations of that time does not allow us to confidently
insist on its reality.

Estimating the half-width of the magnetic activity band, $\theta$, we can
determine what zonal harmonic of the magnetic field is excited by the stellar
dynamo mechanism. For example, as shown by observations of the mean magnetic
field on the solar surface, the harmonic $l=5$ dominates the Sun, which
corresponds to the width of the activity band of about $30^{\circ}$. Indeed, a
zonal harmonic with index $l$ divides the sphere into $l+1$ bands. The dipole
has two bands (two hemispheres), the quadrupole has three bands, etc. For
$l=5$, we have six bands of width $180^{\circ}/6=30^{\circ}$ (Fig. 2) (cf. [
17]). This is the generally recognized zone of local activity. The Carrington
rotation rate corresponds exactly to the middle of this zone - $16^{\circ}$.
The sunspot cycle usually begins approximately at the upper boundary of this
zone — $30^{\circ}$.

Changes in the position of the central latitude of the activity band,
$\Theta(t)$, can help us understand whether the activity waves propagate
towards the stellar equator, as happens on the Sun, or towards the poles of a
star, which is also quite possible within the concept of the stellar dynamo at
a certain distribution its sources (see, for example, [44]).

And finally, the contrast of the activity band against the surrounding stellar
surface can show how much the flare activity of a star may differ from that of
the Sun (cf. [45]).

It is natural to reconstruct the magnetic configuration on stars from
observations of light curves in different spectral ranges on the basis of the
reverse Doppler imaging technique. In other words, instead of choosing the
initial set of the spot formation zones with the parameters corresponding to
the observed light curves, it is proposed to consider the probability density
of the spot formation at a given latitude (or a similar quantity) as a
continuous function of latitude. This function is found from the solution of
the corresponding inverse problem. In the process, we propose that the
residual minimization procedure applied to determine the discrepancy between
the model and observations include terms that depend on the number and size of
the desired distribution by using the Tikhonov regularization or maximum
entropy methods. Experience shows that the use of such methods for mapping the
temperature distribution on the stellar surface allows us to avoid uneven
detalization of the obtained pattern generated by observational errors rather
than real observations.

Of course, we describe a minimal observation program. One can supplement
temperature observations with magnetic data, i.e. support the inverse Doppler
images with inverse Zeeman-Doppler images. This makes it possible to
significantly expand the capabilities of monitoring, e.g., to distinguish
between the quadrupole and dipole magnetic fields much more confidently, since
the polarity rule similar to the Hale’s one sounds different for these
configurations. For quadrupole configurations, the dominant polarity in both
hemispheres of the star coincides, and the polarity reversal occurs at the
transition from one cycle to the other. Recall that, originally, the Hale
polarity rule was formulated shortly after the first successful measurements
of the sunspot magnetic field, without years of preliminary observations.

A representative sample for such a study can be formed in different ways.
There is no need to focus only on solar-type stars; however, it is important
that these stars have convective zones and exhibit noticeable differential
rotation, since otherwise the question of a solar-type dynamo becomes
meaningless. Implementation of such a project is a very difficult task. The
difficulty lies not so much in developing new expensive equipment, but, as
seen from the experience of the H-K Project, in recruiting a small but well-
motivated team of researchers and ensuring a succession of scientific
generations in the team.

Let us briefly formulate the difference between the proposed observation
program and the programs currently being developed.

1\. We propose to focus on a long-term monitoring of temperature variations on
a large but limited number of selected stars similar to the Sun in their
physical properties rather than expanding the list of stars under
investigation and focusing on the variety of methods available to characterize
the magnetic field of each star at a given moment in time.

2\. Shift the main attention from studying the field distribution over the
stellar surface at a given time to studying the latitude-time distribution of
the magnetic field over time intervals comparable to several cycles of stellar
activity.

## Conclusions

Let us summarize the potential capabilities of the updated description of the
solar cycle proposed in the recent works of our team and discussed above.

Firstly, this description may allow a more accurate and reliable prediction of
the evolution of the cyclic activity of the Sun. Such a forecast may be tested
in the near future.

Secondly, this description can serve as the basis of a program for studying
the magnetic activity of stars more or less similar to the Sun in their
physical characteristics. Despite the significant effort required to implement
such a program (similar to the H-K Project), it seems feasible within a
limited time frame and will allow us to radically improve our knowledge of the
magnetic activity of stars.

## Acknowledgements

The authors are grateful to Dr. T.Hoeksema for open access to data on the site
http://wso.stanford.edu. We also express our gratitude to the Ministry of
Science and Higher Education for support in the frames of grants
075-15-2020-780 (VNO and MMK) and 075-15-2022-284 (DDS).

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# Supporting information for "Optimal risk-assessment scheduling for primary
prevention of Cardiovascular disease"

Francesca Gasperoni
MRC Biostatistics Unit
University of Cambridge
Cambridge, U.K.
<EMAIL_ADDRESS>
&Christopher H. Jackson
MRC Biostatistics Unit
University of Cambridge
Cambridge, U.K.
<EMAIL_ADDRESS>
&Angela M. Wood
Department of Public Health and Primary Care
University of Cambridge
Cambridge, U.K.
<EMAIL_ADDRESS>
&Michael J. Sweeting
Department of Health Sciences
University of Leicester
Leicester, U.K.
<EMAIL_ADDRESS>
&Paul J. Newcombe
MRC Biostatistics Unit
University of Cambridge
Cambridge, U.K.
<EMAIL_ADDRESS>
&David Stevens
Liverpool Centre for Cardiovascular Science
University of Liverpool
Liverpool Heart & Chest Hospital
Department of Cardiovascular and Metabolic Medicine
Institute of Life Course and Medical Sciences
University of Liverpool
Liverpool, U.K.
<EMAIL_ADDRESS>
&Jessica K. Barrett
MRC Biostatistics Unit
University of Cambridge
Cambridge, U.K.
<EMAIL_ADDRESS>

##  1 Cohort selection, risk factors and outcome definitions

In this section we report details related to cohort selection and variables
included in the proposed model.

In Figure 1, we represent the scheme of the cohort selection. The final
derivation dataset is composed of 1,774,220 people distributed across 270
practices in the UK; while the validation dataset is composed of 836,044
people distributed in 136 practices.

Figure 1: Flow chart of the selection process of the analysed cohort.

Following Xu et al. (2021), we define the study entry for each person as the
latest of the following four dates: the date of 6 months after registration at
the general practice; the date the individual turned 30 years of age; the date
that the data for the practice were up to standard (Tate et al., 2017); or
April 01, 2004, the date that the Quality and Outcomes Framework (QOF) was
introduced (National Health Service, 2011). We define the study exit for each
person as the earliest of the following dates: the date of deregistration at
the practice; the individual’s death; the date that the individual turned 95
years of age; the last contact date for the practice with CPRD; or the
administration end date (November 2017).

The Read codes (used to identify outcomes in CPRD) and International
Classification of Diseases, Tenth Revision, codes (used to identify outcomes
in primary or secondary diagnosis fields from Hospital Episode Statistics and
in underlying or subordinate cause of death fields from the Office for
National Statistics) are provided in the Web Appendix 1, Web Tables 1 and 2 of
Xu et al. (2021).

Previous diagnosis of diabetes, renal disease, depression, migraine, severe
mental illness, rheumatoid arthritis and atrial fibrillation are ascertained
from CPRD Read codes. Blood pressure medication (yes/no) is ascertained from
CPRD prescription information and it is defined as the date of first
prescription. Statin initiation is defined as the date of first CPRD
prescription (code list for CPRD prescription provided in Web Appendix 2, Web
Table 3 of Xu et al. (2021)). Finally, Townsend deprivation index ranges from
1 to 20. This index presents a total of 1979 missing values (0.08% of the
whole cohort), that are imputed through the mean value per each landmark.

We set to missing biologically implausible values: BMI $>80$ kg/m2 ; SBP
$>250$ mmHg or $<60$ mmHg; total cholesterol level $>20$ mmol/L or $<1.75$
mmol/L; HDL cholesterol level $>3.1$ mmol/L or $<0.3$ mmol/L.

Furthermore, we consider two different sets of risk factors, if we are
performing the analysis before or after landmark age $60$. At all landmark
ages, we include the BLUPs of BMI, HDL, SBP, total cholesterol and smoking. At
all landmark ages, we include blood pressure medication, Townsend deprivation
index, previous diagnosis of diabetes, depression, migraine and severe mental
illness. Previous diagnosis of renal disease, rheumatoid arthritis and atrial
fibrillation are included only after landmark age $60$. This choice is
motivated by the fact that these specific conditions are extremely rare at
younger ages and the estimates of Cox model in Eq. 12 of the main manuscript
are unfeasible.

##  2 Details of the Incremental Net Benefit function

###  2.1 Derivation of EFLY before and after statins initiation

We evaluate as benefit the restricted event free life years (EFLY), i.e. we
investigate the time to CVD diagnosis, $T$, restricted to the observable time
window $[L_{a},L_{a}+10]$.

To define the restricted EFLY, $min(T,L_{a}+10)$, we assume that the time to
CVD can be written as $T=T^{NS}+T^{S}$, where $T^{NS}$ is event-free time
elapsed before statin initiation and $T^{S}$ is the event-free time elapsed
after statin initiation. The definition of the distribution of the time to
CVD, $T$, should reflect the fact that statin usage has a positive impact on
CVD-free life expectancy (Ferket et al., 2012). In order to express this gain
in EFLY, we quantify the effect of statins via the hazard ratio $\theta$. In
this article, we set $\theta=0.8$, following previous meta-analysis of statin
trials (Unit Epidemiological Studies, 2005).

The discontinuity point in the definition of $T$, that separates the time to
CVD without statins $T^{NS}$ and the time with statins $T^{S}$, is defined as
$\tau_{k^{*}_{i,L_{a}}}$, the first visit among the scheduled ones
$\boldsymbol{\tau}$ that happens after $t^{*}_{i,L_{a}}$, the predicted time
when the 5-year CVD risk of person $i$ exceeds the 5% threshold at landmark
age $L_{a}$. $\tau_{k^{*}_{i,L_{a}}}$ is landmark and person-specific and
depends on the risk-assessment strategy under evaluation, $\boldsymbol{\tau}$.

We can define the hazard rate of CVD onset, given the personal covariates
known at time $L_{a}$ and $\tau_{k^{*}_{i}}$ as reported in Eq. (1).

$\displaystyle\lambda(t;\textbf{x}_{i}(L_{a}),L_{a},\tau_{k^{*}_{i,L_{a}}})$
$\displaystyle=\lambda^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a})\cdot\mathbbm{1}{\\{t\leq\tau_{k^{*}_{i,L_{a}}}\\}}+\lambda^{S}(t;\textbf{x}_{i}(L_{a}),L_{a})\cdot\mathbbm{1}{\\{t>\tau_{k^{*}_{i,L_{a}}}\\}}$
$\displaystyle=\lambda_{0}(t;L_{a})\cdot\exp\\{\textbf{x}_{i}(L_{a})^{T}\boldsymbol{\beta}(L_{a})\\}\cdot\left[\mathbbm{1}{\\{t\leq\tau_{k^{*}_{i,L_{a}}}\\}}+\theta\cdot\mathbbm{1}{\\{t>\tau_{k^{*}_{i,L_{a}}}\\}}\right],\quad
t\geq L_{a}.$ (1)

We can compute the cumulative hazard function
$\Lambda(t;\textbf{x}_{i}(L_{a}),L_{a},\tau_{k^{*}_{i,L_{a}}})$ as explained
in Eq. (2).

$\displaystyle\Lambda(t;\textbf{x}_{i}(L_{a}),L_{a},\tau_{k^{*}_{i,L_{a}}})$
$\displaystyle=\int_{L_{a}}^{t}\lambda_{0}(u)\cdot\exp\\{\textbf{x}_{i}(L_{a})^{T}\boldsymbol{\beta}(L_{a})\\}\cdot\left[\mathbbm{1}{\\{u\leq\tau_{k^{*}_{i,L_{a}}}\\}}+\theta\cdot\mathbbm{1}{\\{u>\tau_{k^{*}_{i,L_{a}}}\\}}\right]\,du$
$\displaystyle=\begin{cases}\Lambda_{0}(t;L_{a})\exp\\{\textbf{x}_{i}(L_{a})^{T}\boldsymbol{\beta}(L_{a})\\},&t\leq\tau_{k^{*}_{i,L_{a}}}\\\
\Lambda_{0}(\tau_{k^{*}_{i,L_{a}}})\exp\\{\textbf{x}_{i}(L_{a})^{T}\boldsymbol{\beta}(L_{a})\\}+(\Lambda_{0}(t;L_{a})-\Lambda_{0}(\tau_{k^{*}_{i,L_{a}}}))\cdot\theta\cdot\exp\\{\textbf{x}_{i}(L_{a})^{T}\boldsymbol{\beta}(L_{a})\\},&t>\tau_{k^{*}_{i,L_{a}}}\end{cases}$
(2)

Defining $\Lambda^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a})$ as
$\Lambda_{0}(t;L_{a})\exp\\{\textbf{x}_{i}(L_{a})^{T}\boldsymbol{\beta}(L_{a})\\}$,
we can rewrite Eq. (2) as follows:

$\displaystyle\Lambda(t;\textbf{x}_{i}(L_{a}),L_{a},\tau_{k^{*}_{i,L_{a}}})$
$\displaystyle=\begin{cases}\Lambda^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a}),&t\leq\tau_{k^{*}_{i,L_{a}}}\\\
\Lambda^{NS}(\tau_{k^{*}_{i,L_{a}}};\textbf{x}_{i}(L_{a}),L_{a})+(\Lambda^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a})-\Lambda^{NS}(\tau_{k^{*}_{i,L_{a}}};\textbf{x}_{i}(L_{a}),L_{a}))\cdot\theta,&t>\tau_{k^{*}_{i,L_{a}}}\end{cases}$
(3)

Finally, we the survival function
$S(t;\textbf{x}_{i}(L_{a}),\tau_{k^{*}_{i,L_{a}}})$ in Eq. (4).

$\displaystyle S(t;\textbf{x}_{i}(L_{a}),L_{a},\tau_{k^{*}_{i,L_{a}}})$
$\displaystyle=\exp\\{-\Lambda(t;\textbf{x}_{i}(L_{a}),\tau_{k^{*}_{i,L_{a}}},L_{a})\\}=\begin{cases}S^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a}),&t\leq\tau_{k^{*}_{i,L_{a}}}\\\
S^{S}(t;\textbf{x}_{i}(L_{a}),L_{a}),&t>\tau_{k^{*}_{i,L_{a}}}\end{cases}$ (4)

where:

$S^{S}(t;\textbf{x}_{i}(L_{a}),L_{a})=S^{NS}(\tau_{k^{*}_{i,L_{a}}};\textbf{x}_{i}(L_{a}),L_{a})\cdot\left(\frac{S^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a})}{S^{NS}(\tau_{k^{*}_{i,L_{a}}};\textbf{x}_{i}(L_{a}),L_{a})}\right)^{\theta}$
(5)

Note that if a person is never prescribed statins, then we have
$S(t;\textbf{x}_{i}(L_{a}),L_{a})=S^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a})$.

Conditioning on $\tau_{k^{*}_{i,L_{a}}}$, $\tau_{k^{*}_{i,L_{a}}}\leq
L_{a}+10$, we are able to define the restricted EFLY in Eq. (6).

$\displaystyle EFLY$
$\displaystyle=EFLY_{NS}(\tau_{k^{*}_{i,L_{a}}})+EFLY_{S}(\tau_{k^{*}_{i,L_{a}}})$
$\displaystyle=\int_{L_{a}}^{\tau_{k^{*}_{i,L_{a}}}}S^{NS}(t;\textbf{x}_{i}(L_{a}),L_{a})\,dt+\int_{\tau_{k^{*}_{i,L_{a}}}}^{L_{a}+10}S^{S}(t;\textbf{x}_{i}(L_{a}),L_{a})\,dt.$
(6)

###  2.2 Expected number of risk assessments

The expected number of risk assessment,
$\mathbb{E}_{\boldsymbol{\tau}}[N_{i}]$, is part of the costs associated with
a specific risk assesment strategy $\boldsymbol{\tau}$. We assume that the
CVD-risk assessment of a person is performed up to time
$\tau_{k^{*}_{i,L_{a}}}$ (i.e., no more visits after statins initiation).

In Fig. 2, we represent an illustrative example to show how the expected
number of visits is computed for two different risk-assessment schedules
($\bar{\boldsymbol{\tau}}$ in the top row and $\tilde{\boldsymbol{\tau}}$ in
the bottom row) for a generic person whose 5-year CVD risk exceeds the 5%
threshold at $t^{*}_{i}$ (dashed black line). According to both risk-
assessment schedules, person $i$ should start taking statins from the third
visit, which means $\bar{k}^{*}_{i}=\tilde{k}^{*}_{i}=3$ and
$\mathbb{E}_{\bar{\boldsymbol{\tau}}}[N_{i}]=\mathbb{E}_{\tilde{\boldsymbol{\tau}}}[N_{i}]=3$.

If a person never crosses the $5\%$ threshold, they will never start taking
statins and the expected number of visits including the baseline visit is
$\mathbb{E}_{\boldsymbol{\tau}}[N_{i}]=1+10/\Delta\tau$, where $\Delta\tau$ is
the time between visits according to visit schedule $\boldsymbol{\tau}$. From
Fig. 2, the expected number of visits for a person whose 5-year CVD risk never
crosses the 5% threshold are 6, according to $\bar{\boldsymbol{\tau}}$ and
4.33, according to $\tilde{\boldsymbol{\tau}}$.

Figure 2: Example figure for describing the procedure for computing
$k^{*}_{i}$, $\tau_{k^{*}_{i}}$, given $t_{i}^{*}$ and two specific risk-
assessment strategies. In the upper part of the figure, we consider risk-
assessment schedule $\bar{\boldsymbol{\tau}}$ (visits every 2 years), while in
the lower part, we consider another risk-assessment schedule
$\tilde{\boldsymbol{\tau}}$ (visits every 3 years). This person is expected to
cross the $5\%$ threshold at $t_{i}^{*}$, which implies that this person is
going to start taking statins at the third visit
($\bar{k}_{i}^{*}=\tilde{k}_{i}^{*}=3$) in both cases
($\mathbb{E}_{\bar{\boldsymbol{\tau}}}[N_{i}]=\mathbb{E}_{\tilde{\boldsymbol{\tau}}}[N_{i}]=3$).
However, $\bar{\tau}_{3}$ and $\tilde{\tau}_{3}$ are different,
$\bar{\tau}_{3}<\tilde{\tau}_{3}$, which means that time spent under statins
blue boxes) is longer if we focus on $\bar{\boldsymbol{\tau}}$ (risk-
assessment every 2 years). The statin-free time is represented through red
boxes.

##  3 Optimal risk-assessment scheduling: CVD free life years

In this section, we report the recommended risk-assessment strategies across
different landmark ages for women in Table 1 and for men in Table 2. In each
table, we describe the results based on the landmark ages (values in columns)
and on the 5-year CVD risk categories estimated at each landmark age (values
in rows). The percentage associated to each number of the table is computed
with respect to the landmark cohort. For example, if we focus on women at high
risk at landmark age 40 (third column from the left in Table 1), we note that
76 (0.05% of 155497) women are recommended to have a risk-assessment every
year. Furthermore, we report the total number of people belonging to each risk
category and the total number of people among them, whose 5-year CVD risk is
not expected to cross the 5% threshold in the next 10 years. These two numbers
are at the bottom of each risk category block. For example, if we focus on
women at high risk at landmark age 40 (third column from the left in Table 1),
we note that 485 are labelled as High risk and 50 (10.31%) of them are not
expected to cross the 5% threshold. At the bottom of both tables we record the
landmark cohort size and total number of people whose 5-year CVD risk is not
expected to cross the 5% threshold in the next 10 years. Looking at these
rows, we note that the landmark cohort size decreases over landmark ages. The
total number of people whose 5% CVD risk is not expected to cross the
threshold decreases over landmark ages. This line should be read in pair with
the top one where the total number of people labelled as very high risk is
reported (note that these numbers increase over landmark ages). These
observations hold for both women and men.

Relevant differences between women in Table 1 and men in Table 2 are related
to the time when the number of people whose 5-year CVD risk is not expected to
cross the threshold starts dropping (last row) and the time when the
percentage of people labelled as very high risk starts increasing (top row).
Indeed, for women the first number start dropping at landmark age 65 (from
34.79% to 8.18%), while for men at landmark age 55 (from 43.01% to 5.6%).
Analogously, almost a quarter of the age 65 landmark cohort of women (24.62%)
is labelled as very high risk, similar number is reached by men already at
landmark age 55 (28.1%).

Table 1:
Optimal CVD-risk assessment frequency across landmark ages (in columns),
stratified by baseline risk category (in rows). This table refers to women and
the outcome of interest is 10-year CVD. In the two bottom lines we report the
landmark cohort size and the total number of people whose 5-year CVD risk is
not expected to cross the 5% threshold in the next 10 years.

Risk class | Optimal | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80
---|---|---|---|---|---|---|---|---|---|---
Very high | - | 359 (0.23%) | 1119 (0.7%) | 3586 (2.47%) | 5068 (4.12%) | 11953 (10.95%) | 21274 (24.62%) | 46926 (74.18%) | 46361 (99.09%) | 37178 (100%)
High | 1 | 76 (0.05%) | 405 (0.25%) | 876 (0.6%) | 1498 (1.22%) | 1995 (1.83%) | 5795 (6.71%) | 7928 (12.53%) | 303 (0.65%) | -
2 | 88 (0.06%) | 465 (0.29%) | 969 (0.67%) | 1746 (1.42%) | 1963 (1.8%) | 7044 (8.15%) | 4798 (7.58%) | 98 (0.21%) | -
3 | 67 (0.04%) | 96 (0.06%) | 585 (0.4%) | 1004 (0.82%) | 2000 (1.83%) | 4434 (5.13%) | 1431 (2.26%) | 11 (0.02%) | -
4 | 5 (0%) | 144 (0.09%) | 163 (0.11%) | 876 (0.71%) | 1759 (1.61%) | 463 (0.54%) | 946 (1.5%) | - | -
5 | 76 (0.05%) | 69 (0.04%) | 387 (0.27%) | 217 (0.18%) | 2108 (1.93%) | 351 (0.41%) | - | 2 (0%) | -
6 | 69 (0.04%) | 128 (0.08%) | 263 (0.18%) | 491 (0.4%) | 1871 (1.71%) | 10 (0.01%) | - | - | -
7 | - | 7 (0%) | 17 (0.01%) | 99 (0.08%) | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 104 (0.07%) | 84 (0.05%) | 1000 (0.69%) | 522 (0.42%) | 1632 (1.49%) | 617 (0.71%) | 113 (0.18%) | - | -
Total | 485 | 1398 | 4260 | 6453 | 13328 | 18714 | 15216 | 414 | 0
Never cross | 50 (10.31%) | 8 (0.57%) | 438 (10.28%) | 82 (1.27%) | 3 (0.02%) | 123 (0.66%) | 2 (0.01%) | - | -
Med-high | 1 | 5 (0%) | 6 (0%) | 1 (0%) | 10 (0.01%) | - | - | - | - | -
2 | 51 (0.03%) | 235 (0.15%) | 118 (0.08%) | 123 (0.1%) | 58 (0.05%) | 1432 (1.66%) | 111 (0.18%) | 1 (0%) | -
3 | 65 (0.04%) | 448 (0.28%) | 210 (0.14%) | 950 (0.77%) | 342 (0.31%) | 4915 (5.69%) | 151 (0.24%) | 6 (0.01%) | -
4 | 32 (0.02%) | 762 (0.48%) | 529 (0.37%) | 1827 (1.48%) | 772 (0.71%) | 5868 (6.79%) | 445 (0.7%) | 3 (0.01%) | -
5 | 102 (0.07%) | 543 (0.34%) | 1178 (0.81%) | 1651 (1.34%) | 1864 (1.71%) | 7755 (8.98%) | - | - | -
6 | 6 (0%) | 382 (0.24%) | 433 (0.3%) | 807 (0.66%) | 2 (0%) | - | - | - | -
7 | - | - | - | - | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 1315 (0.85%) | 1911 (1.19%) | 8943 (6.17%) | 12354 (10.03%) | 27238 (24.95%) | 11430 (13.23%) | 408 (0.64%) | 2 (0%) | -
Total | 1576 | 4287 | 11412 | 17722 | 30276 | 31400 | 1115 | 12 | 0
Never cross | 821 (52.09%) | 627 (14.63%) | 4870 (42.67%) | 7729 (43.61%) | 1138 (3.76%) | 1437 (4.58%) | 2 (0.18%) | - | -
Med-Low | 1 | - | - | - | - | - | - | - | - | -
2 | 4 (0%) | - | - | - | - | - | - | - | -
3 | 12 (0.01%) | 52 (0.03%) | 1 (0%) | 35 (0.03%) | - | 14 (0.02%) | - | - | -
4 | 27 (0.02%) | 237 (0.15%) | 20 (0.01%) | 177 (0.14%) | 12 (0.01%) | 123 (0.14%) | - | - | -
5 | - | 174 (0.11%) | 12 (0.01%) | 132 (0.11%) | - | - | - | - | -
6 | - | - | - | - | - | - | - | - | -
7 | - | - | - | - | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 8618 (5.54%) | 20071 (12.55%) | 40276 (27.79%) | 56987 (46.28%) | 50104 (45.89%) | 14861 (17.2%) | - | - | -
Total | 8661 | 20534 | 40309 | 57331 | 50116 | 14998 | 0 | 0 | 0
Never cross | 7692 (88.81%) | 16253 (79.15%) | 32028 (79.46%) | 54396 (94.88%) | 33458 (66.76%) | 5493 (36.62%) | - | - | -
Low | 1 | - | - | - | - | - | - | - | - | -
2 | - | - | - | - | - | - | - | - | -
3 | - | - | - | - | - | - | - | - | -
4 | - | - | - | - | - | - | - | - | -
5 | - | - | - | - | - | - | - | - | -
6 | - | - | - | - | - | - | - | - | -
7 | - | - | - | - | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 144416 (92.87%) | 132637 (82.91%) | 85340 (58.89%) | 36561 (29.69%) | 3500 (3.21%) | 17 (0.02%) | - | - | -
Total | 144416 | 132637 | 85340 | 36561 | 3500 | 17 | 0 | 0 | 0
Never cross | 143864 (99.62%) | 131791 (99.36%) | 81119 (95.05%) | 36540 (99.94%) | 3384 (96.69%) | 17 (100%) | - | - | -
| Total | 155497 | 159975 | 144907 | 123135 | 109173 | 86403 | 63257 | 46787 | 37178
| Never cross | 152427 (98.03%) | 148679 (92.94%) | 118455 (81.75%) | 98747 (80.19%) | 37983 (34.79%) | 7070 (8.18%) | 4 (0.01%) | - | -

Table 2:
Optimal CVD-risk assessment frequency across landmark ages (in columns),
stratified by baseline risk category (in rows). This table refers to men and
the outcome of interest is 10-year CVD. In the two bottom lines we report the
landmark cohort size and the total number of people whose 5-year CVD risk is
not expected to cross the 5% threshold in the next 10 years.

Risk class | Optimal | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80
---|---|---|---|---|---|---|---|---|---|---
Very high | - | 761 (0.58%) | 4077 (2.89%) | 13792 (10.73%) | 30009 (28.1%) | 56178 (60.93%) | 63144 (93.15%) | 45486 (99.94%) | 30978 (100%) | 23189 (100%)
High | 1 | 165 (0.13%) | 1385 (0.98%) | 3956 (3.08%) | 7634 (7.15%) | 10305 (11.18%) | 2816 (4.15%) | 22 (0.05%) | - | -
2 | 288 (0.22%) | 1388 (0.98%) | 4891 (3.8%) | 7441 (6.97%) | 7357 (7.98%) | 1350 (1.99%) | 4 (0.01%) | - | -
3 | 252 (0.19%) | 606 (0.43%) | 4893 (3.81%) | 9968 (9.33%) | 3500 (3.8%) | 32 (0.05%) | - | - | -
4 | 156 (0.12%) | 331 (0.23%) | 669 (0.52%) | 4094 (3.83%) | 5584 (6.06%) | 24 (0.04%) | - | - | -
5 | 58 (0.04%) | 386 (0.27%) | 161 (0.13%) | 968 (0.91%) | 211 (0.23%) | - | - | - | -
6 | 40 (0.03%) | 82 (0.06%) | 205 (0.16%) | 9 (0.01%) | - | - | - | - | -
7 | 10 (0.01%) | 105 (0.07%) | 85 (0.07%) | - | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 8 (0.01%) | 222 (0.16%) | 98 (0.08%) | 117 (0.11%) | 909 (0.99%) | 2 (0%) | - | - | -
Total | 977 | 4505 | 14958 | 30231 | 27866 | 4224 | 26 | 0 | 0
Never cross | 1 (0.1%) | 139 (3.09%) | 70 (0.47%) | 12 (0.04%) | 61 (0.22%) | - | - | - | -
Med-high | 1 | 3 (0%) | 24 (0.02%) | - | 1 (0%) | - | - | - | - | -
2 | 99 (0.08%) | 228 (0.16%) | 792 (0.62%) | 60 (0.06%) | 9 (0.01%) | 81 (0.12%) | - | - | -
3 | 199 (0.15%) | 1211 (0.86%) | 4038 (3.14%) | 418 (0.39%) | 76 (0.08%) | 70 (0.1%) | - | - | -
4 | 363 (0.28%) | 2820 (2%) | 7097 (5.52%) | 6711 (6.28%) | 991 (1.07%) | 144 (0.21%) | - | - | -
5 | 766 (0.58%) | 3935 (2.78%) | 10357 (8.06%) | 9012 (8.44%) | - | - | - | - | -
6 | 637 (0.48%) | 962 (0.68%) | 7175 (5.58%) | - | - | - | - | - | -
7 | 236 (0.18%) | 150 (0.11%) | 50 (0.04%) | - | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 767 (0.58%) | 4954 (3.51%) | 8638 (6.72%) | 22640 (21.2%) | 6485 (7.03%) | 125 (0.18%) | - | - | -
Total | 3070 | 14284 | 38147 | 38842 | 7561 | 420 | 0 | 0 | 0
Never cross | 116 (3.78%) | 1839 (12.87%) | 2287 (6%) | 2258 (5.81%) | 937 (12.39%) | - | - | - | -
Medium-low | 1 | - | - | - | - | - | - | - | - | -
2 | 3 (0%) | - | - | - | - | - | - | - | -
3 | 10 (0.01%) | - | 13 (0.01%) | 1 (0%) | - | - | - | - | -
4 | 261 (0.2%) | 132 (0.09%) | 221 (0.17%) | 8 (0.01%) | - | - | - | - | -
5 | 583 (0.44%) | 2191 (1.55%) | 501 (0.39%) | 1 (0%) | - | - | - | - | -
6 | 775 (0.59%) | 237 (0.17%) | 38 (0.03%) | - | - | - | - | - | -
7 | - | - | - | - | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 22119 (16.81%) | 71349 (50.49%) | 55336 (43.05%) | 7547 (7.07%) | 592 (0.64%) | 2 (0%) | - | - | -
Total | 23751 | 73909 | 56109 | 7557 | 592 | 2 | 0 | 0 | 0
Never cross | 14996 (63.14%) | 59809 (80.92%) | 47403 (84.48%) | 3593 (47.55%) | 507 (85.64%) | - | - | - | -
Low | 1 | - | - | - | - | - | - | - | - | -
2 | - | - | - | - | - | - | - | - | -
3 | - | - | - | - | - | - | - | - | -
4 | - | - | - | - | - | - | - | - | -
5 | - | - | - | - | - | - | - | - | -
6 | - | - | - | - | - | - | - | - | -
7 | - | - | - | - | - | - | - | - | -
8 | - | - | - | - | - | - | - | - | -
9 | - | - | - | - | - | - | - | - | -
10 | 102989 (78.29%) | 44529 (31.51%) | 5541 (4.31%) | 149 (0.14%) | 3 (0%) | - | - | - | -
Total | 102989 | 44529 | 5541 | 149 | 3 | 0 | 0 | 0 | 0
Never cross | 100361 (97.45%) | 44111 (99.06%) | 5533 (99.86%) | 114 (76.51%) | 3 (100%) | - | - | - | -
| Total | 131548 | 141304 | 128547 | 106788 | 92200 | 67790 | 45512 | 30978 | 23189
| Never cross | 115474 (87.78%) | 105898 (74.94%) | 55293 (43.01%) | 5977 (5.6%) | 1508 (1.64%) | - | - | - | -

In Figure 3 we report a detailed representation of risk profiles,
$\hat{r}_{i}(s+5;\textbf{x}_{i}(s),s)$ $s\in\\{40,\dots,50\\}$, for women at
$L_{a}=40$, whose 5-year CVD risk at $L_{a}=40$ is classified as high. It is
immediate to notice that the median $\tau_{k^{*}_{i,40}}$ (black solid lines)
increases according to the optimal frequency recommendation. Indeed, people
whose 5-year CVD risk is expected to exceed the 5% threshold later in time are
more likely to be recommended a lower frequency risk-assessment strategy.
Furthermore, it is interesting to notice that the 5-year CVD risk estimated at
the baseline is relevant for the risk-assessment recommendation, but the
information given by the risk profile is fundamental for the risk-assessment
recommendation. Indeed, we observe that different risk-profile trends (more
steep or more flat) can lead to opposite risk-assessment recommendation even
for people at high risk of CVD.

Figure 3: Risk profile representation for women at $L_{a}=40$. Each panel
represents an optimal risk-assessment frequency. Each line is the estimated
risk profile for a specific person. The solid black line represents the median
expected time of crossing the 5% threshold (represented via the horizontal red
line). The dashed lines represent the first and third quartiles of the
$t_{k^{*}_{i},40}$ distribution.

###  3.1 Descriptive statistics of the landmark cohorts

In this Section, we report the descriptive characteristics of each landmark
cohort, stratified by the 5-year CVD risk classification at the landmark age.
Descriptive statistics are reported in table 3 and 4 for women and men
respectively. We observe a higher risk for people under blood pressure
medication, with diagnoses of depression, diabetes, migraine, renal disease,
rheumatoid arthritis, severe mental illness and systemic lupus eritematosus.
Higher values of SBP, total cholesterol (TCHOL), BMI are associated with
people at higher risk. The Townsend 20 index and smoking are also associated
with higher risk.

Table 3:
Descriptive characteristics at each landmark age for the women cohorts.

variable | Risk class | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80
---|---|---|---|---|---|---|---|---|---|---
AF | Very high risk | 0 % | 0.27 % | 0.47 % | 0.67 % | 5.53 % | 4.37 % | 2.8 % | 3.4 % | 5.63 %
High risk | 0.41 % | 0.14 % | 0.56 % | 0.67 % | 0.65 % | 0.14 % | 0 % | 0 % |
Med-high risk | 0.19 % | 0.26 % | 0.39 % | 0.63 % | 0.1 % | 0 % | 0 % | 0 % |
Med-low risk | 0.16 % | 0.23 % | 0.36 % | 0.39 % | 0 % | 0 % |  |  |
Low risk | 0.12 % | 0.15 % | 0.2 % | 0.29 % | 0 % | 0 % |  |  |
BP med | Very high risk | 95.82 % | 91.6 % | 92.58 % | 83.88 % | 80.16 % | 77.39 % | 57.62 % | 52.06 % | 62.15 %
High risk | 87.01 % | 85.62 % | 81.34 % | 64.71 % | 62.07 % | 52.84 % | 1.27 % | 0.48 % |
Med-high risk | 78.43 % | 72.31 % | 65.12 % | 53.63 % | 39.6 % | 16.35 % | 0 % | 0 % |
Med-low risk | 55.01 % | 44.93 % | 31.13 % | 22.5 % | 8.73 % | 1.63 % |  |  |
Low risk | 7.61 % | 7.47 % | 3.43 % | 3.38 % | 0.2 % | 0 % |  |  |
Depression | Very high risk | 83.29 % | 73.19 % | 82.12 % | 72.97 % | 64.78 % | 43.67 % | 28.51 % | 20.35 % | 19.39 %
High risk | 75.05 % | 64.02 % | 71.53 % | 64.7 % | 47.21 % | 29.05 % | 4.3 % | 2.17 % |
Med-high risk | 75.19 % | 63.24 % | 61.72 % | 49.15 % | 32.24 % | 18.46 % | 0.63 % | 0 % |
Med-low risk | 66.92 % | 52.9 % | 44.86 % | 28.02 % | 11.96 % | 7 % |  |  |
Low risk | 26.21 % | 25.03 % | 14.42 % | 7.79 % | 0.97 % | 0 % |  |  |
Diabetes | Very high risk | 22.84 % | 21.18 % | 11.88 % | 13.12 % | 6.22 % | 4.31 % | 2.07 % | 1.96 % | 2.11 %
High risk | 15.67 % | 11.52 % | 4.91 % | 3.47 % | 1.41 % | 0.37 % | 0 % | 0 % |
Med-high risk | 6.92 % | 5.09 % | 2.28 % | 1.06 % | 0.43 % | 0.09 % | 0 % | 0 % |
Med-low risk | 3.54 % | 1.91 % | 0.76 % | 0.11 % | 0.07 % | 0 % |  |  |
Low risk | 0.31 % | 0.15 % | 0.06 % | 0.01 % | 0 % | 0 % |  |  |
Migraine | Very high risk | 62.95 % | 52.37 % | 41.27 % | 26.2 % | 20.06 % | 15.24 % | 7.61 % | 6.7 % | 5.19 %
High risk | 52.58 % | 46.28 % | 33.12 % | 21.57 % | 16.63 % | 11.44 % | 8.68 % | 0.97 % |
Med-high risk | 54.57 % | 41.99 % | 27.26 % | 18.33 % | 12.51 % | 8.08 % | 25.83 % | 0 % |
Med-low risk | 42.54 % | 30.5 % | 16.82 % | 12 % | 6.93 % | 3.47 % |  |  |
Low risk | 9.99 % | 8 % | 6.11 % | 4.97 % | 2.8 % | 0 % |  |  |
Renal dis. | Very high risk | 3.06 % | 2.59 % | 3.04 % | 3.18 % | 5.09 % | 9.32 % | 6.86 % | 7.86 % | 12.18 %
High risk | 1.86 % | 1.79 % | 2.68 % | 2.76 % | 3.4 % | 3.03 % | 0.97 % | 0 % |
Med-high risk | 1.46 % | 2.19 % | 2.22 % | 2.06 % | 2.13 % | 1.15 % | 0.09 % | 0 % |
Med-low risk | 1.2 % | 1.24 % | 1.29 % | 1.3 % | 1.18 % | 0.32 % |  |  |
Low risk | 0.36 % | 0.5 % | 0.67 % | 0.93 % | 1.26 % | 0 % |  |  |
RA | Very high risk | 2.23 % | 1.52 % | 1.78 % | 2.41 % | 8.52 % | 6.64 % | 3.53 % | 2.89 % | 3.03 %
High risk | 1.03 % | 1.14 % | 1.62 % | 2.08 % | 3.3 % | 1.94 % | 0.03 % | 0.48 % |
Med-high risk | 0.7 % | 1.52 % | 1.64 % | 1.81 % | 1.51 % | 0.43 % | 0 % | 0 % |
Med-low risk | 0.89 % | 1.01 % | 1.27 % | 1.44 % | 0.2 % | 0.03 % |  |  |
Low risk | 0.66 % | 0.86 % | 1.05 % | 1.42 % | 0 % | 0 % |  |  |
SMI | Very high risk | 9.75 % | 6.43 % | 9.82 % | 8.07 % | 5.19 % | 4.52 % | 1.99 % | 1.61 % | 1.73 %
High risk | 7.01 % | 4.94 % | 5.92 % | 3.89 % | 2.54 % | 0.96 % | 0.02 % | 0 % |
Med-high risk | 8.38 % | 4.25 % | 3.93 % | 2.43 % | 1.31 % | 0.22 % | 0 % | 0 % |
Med-low risk | 5.05 % | 2.9 % | 1.72 % | 0.96 % | 0.32 % | 0.01 % |  |  |
Low risk | 0.78 % | 0.81 % | 0.23 % | 0.2 % | 0.03 % | 0 % |  |  |
SLE | Very high risk | 1.11 % | 0.89 % | 0.45 % | 0.41 % | 0.42 % | 0.44 % | 0.3 % | 0.32 % | 0.23 %
High risk | 0 % | 0.36 % | 0.35 % | 0.46 % | 0.38 % | 0.32 % | 0.2 % | 0.24 % |
Med-high risk | 0.57 % | 0.44 % | 0.41 % | 0.37 % | 0.29 % | 0.21 % | 0.18 % | 0 % |
Med-low risk | 0.38 % | 0.37 % | 0.34 % | 0.27 % | 0.23 % | 0.27 % |  |  |
Low risk | 0.19 % | 0.22 % | 0.23 % | 0.25 % | 0.4 % | 0 % |  |  |
BMI | Very high risk | 1.22 (1.13) | 0.89 (1.04) | 0.33 (0.86) | 0.08 (0.77) | 0.01 (0.75) | -0.14 (0.63) | -0.27 (0.53) | -0.37 (0.49) | -0.46 (0.44)
High risk | 0.88 (1.03) | 0.58 (0.94) | 0.16 (0.78) | -0.04 (0.68) | -0.11 (0.6) | -0.24 (0.52) | -0.45 (0.4) | -0.78 (0.46) |
Med-high risk | 0.54 (0.94) | 0.36 (0.86) | 0.04 (0.72) | -0.1 (0.62) | -0.21 (0.51) | -0.32 (0.44) | -0.74 (0.4) | -1.18 (0.52) |
Med-low risk | 0.23 (0.83) | 0.05 (0.71) | -0.14 (0.59) | -0.21 (0.52) | -0.34 (0.41) | -0.48 (0.42) |  |  |
Low risk | -0.33 (0.48) | -0.32 (0.47) | -0.34 (0.45) | -0.39 (0.44) | -0.66 (0.39) | -0.97 (0.3) |  |  |
HDL | Very high risk | -1.08 (0.44) | -0.7 (0.57) | -0.46 (0.57) | -0.32 (0.57) | -0.03 (0.61) | 0.01 (0.59) | 0.2 (0.59) | 0.29 (0.59) | 0.3 (0.58)
High risk | -0.86 (0.48) | -0.57 (0.54) | -0.28 (0.55) | -0.14 (0.52) | 0.09 (0.56) | 0.19 (0.54) | 0.44 (0.56) | 1.77 (0.68) |
Med-high risk | -0.68 (0.47) | -0.43 (0.54) | -0.16 (0.53) | -0.01 (0.49) | 0.19 (0.53) | 0.3 (0.52) | 1.12 (0.82) | 2.52 (0.28) |
Med-low risk | -0.48 (0.42) | -0.26 (0.46) | -0.02 (0.47) | 0.17 (0.45) | 0.33 (0.51) | 0.57 (0.65) |  |  |
Low risk | -0.12 (0.33) | 0.01 (0.4) | 0.2 (0.46) | 0.46 (0.57) | 0.72 (0.71) | 1.91 (0.76) |  |  |
SBP | Very high risk | 0.19 (0.67) | 0.23 (0.64) | 0.04 (0.6) | 0.05 (0.55) | 0.15 (0.51) | 0.18 (0.46) | 0.2 (0.4) | 0.26 (0.39) | 0.33 (0.42)
High risk | 0 (0.57) | 0.01 (0.58) | -0.1 (0.54) | -0.07 (0.5) | 0.01 (0.43) | 0.06 (0.37) | 0.02 (0.3) | -0.02 (0.36) |
Med-high risk | -0.24 (0.55) | -0.17 (0.53) | -0.22 (0.51) | -0.15 (0.46) | -0.07 (0.37) | -0.02 (0.33) | -0.35 (0.32) | -0.32 (0.24) |
Med-low risk | -0.47 (0.47) | -0.4 (0.45) | -0.37 (0.42) | -0.26 (0.38) | -0.2 (0.34) | -0.2 (0.34) |  |  |
Low risk | -0.85 (0.33) | -0.69 (0.34) | -0.54 (0.35) | -0.44 (0.36) | -0.58 (0.33) | -0.78 (0.23) |  |  |
Smoke | Very high risk | 0.71 (0.24) | 0.71 (0.24) | 0.68 (0.26) | 0.72 (0.24) | 0.51 (0.3) | 0.4 (0.29) | 0.26 (0.23) | 0.2 (0.19) | 0.18 (0.19)
High risk | 0.66 (0.25) | 0.68 (0.24) | 0.6 (0.26) | 0.61 (0.25) | 0.42 (0.26) | 0.29 (0.22) | 0.17 (0.1) | 0.08 (0.08) |
Med-high risk | 0.64 (0.24) | 0.65 (0.24) | 0.56 (0.26) | 0.51 (0.25) | 0.35 (0.21) | 0.24 (0.14) | 0.07 (0.08) | 0.07 (0.08) |
Med-low risk | 0.61 (0.24) | 0.6 (0.24) | 0.5 (0.24) | 0.38 (0.18) | 0.28 (0.15) | 0.18 (0.12) |  |  |
Low risk | 0.46 (0.2) | 0.43 (0.2) | 0.38 (0.18) | 0.27 (0.16) | 0.14 (0.13) | 0.04 (0.06) |  |  |
TCHOL | Very high risk | 0.07 (0.56) | 0.34 (0.67) | 0.3 (0.6) | 0.42 (0.6) | 0.44 (0.53) | 0.43 (0.51) | 0.39 (0.46) | 0.36 (0.46) | 0.3 (0.48)
High risk | 0.01 (0.56) | 0.15 (0.57) | 0.22 (0.51) | 0.38 (0.5) | 0.43 (0.44) | 0.44 (0.43) | 0.44 (0.39) | 0.31 (0.55) |
Med-high risk | -0.1 (0.48) | 0.09 (0.5) | 0.16 (0.44) | 0.33 (0.42) | 0.42 (0.4) | 0.43 (0.38) | 0.43 (0.48) | 0.12 (0.38) |
Med-low risk | -0.22 (0.4) | -0.05 (0.38) | 0.09 (0.38) | 0.28 (0.36) | 0.39 (0.35) | 0.39 (0.39) |  |  |
Low risk | -0.38 (0.24) | -0.21 (0.27) | 0.01 (0.31) | 0.21 (0.35) | 0.24 (0.42) | 0.04 (0.57) |  |  |
Townsend20 | Very high risk | 15.64 (3.66) | 15.01 (4.15) | 14.75 (4.16) | 13.64 (4.73) | 12.97 (4.8) | 11.68 (5.05) | 9.44 (5.19) | 8.69 (5.19) | 9.01 (5.2)
High risk | 15.45 (3.74) | 14.35 (4.46) | 13.28 (4.82) | 12.71 (4.96) | 11.32 (5.1) | 9.22 (5.17) | 5.58 (3.79) | 4.28 (2.97) |
Med-high risk | 14.97 (4.2) | 13.91 (4.69) | 12.4 (5.15) | 11.29 (5.31) | 9.59 (5.13) | 7.61 (4.41) | 3.65 (2.64) | 2 (1.28) |
Med-low risk | 14.09 (4.6) | 12.93 (5.09) | 11.05 (5.4) | 9.18 (5.06) | 6.32 (4.12) | 4.24 (3.1) |  |  |
Low risk | 9.02 (5.52) | 8.31 (5.31) | 7.04 (4.75) | 5.38 (3.97) | 3.1 (2.24) | 2.24 (1.48) |  |  |

Table 4:
Descriptive characteristics at each landmark age for the men cohorts.

variable | Risk class | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80
---|---|---|---|---|---|---|---|---|---|---
AF | Very high risk | 1.18 % | 0.78 % | 1.01 % | 1.4 % | 2.42 % | 2.35 % | 3.42 % | 5.45 % | 8.27 %
High risk | 0.61 % | 1.09 % | 0.96 % | 0.81 % | 0.01 % | 0 % | 0 % |  |
Med-high risk | 1.04 % | 0.73 % | 0.54 % | 0.63 % | 0.01 % | 0 % |  |  |
Med-low risk | 0.45 % | 0.33 % | 0.41 % | 0.69 % | 0 % | 0 % |  |  |
Low risk | 0.2 % | 0.29 % | 0.32 % | 0.67 % | 0 % |  |  |  |
BP med | Very high risk | 84.23 % | 80.23 % | 59.76 % | 50.48 % | 36.44 % | 31.47 % | 36.19 % | 44.84 % | 55.08 %
High risk | 78.51 % | 56.12 % | 30.35 % | 11.88 % | 5.5 % | 2.37 % | 0 % |  |
Med-high risk | 61.79 % | 29.84 % | 9.59 % | 3.13 % | 3.85 % | 1.43 % |  |  |
Med-low risk | 18.26 % | 4.46 % | 2.22 % | 2.17 % | 2.53 % | 0 % |  |  |
Low risk | 1.13 % | 0.84 % | 1.08 % | 0 % | 0 % |  |  |  |
Depression | Very high risk | 55.98 % | 44.94 % | 39.69 % | 31.7 % | 21.12 % | 14.4 % | 11.64 % | 10.35 % | 9.81 %
High risk | 56.29 % | 39.13 % | 27 % | 14.88 % | 4.52 % | 2.94 % | 3.85 % |  |
Med-high risk | 47.88 % | 33.09 % | 17.43 % | 4.83 % | 3.08 % | 1.67 % |  |  |
Med-low risk | 42.5 % | 15.17 % | 6.74 % | 3.71 % | 1.86 % | 0 % |  |  |
Low risk | 7.2 % | 5.39 % | 3.83 % | 2.68 % | 0 % |  |  |  |
Diabetes | Very high risk | 25.23 % | 4.34 % | 6.81 % | 4.17 % | 2.92 % | 2.45 % | 2.94 % | 3.7 % | 3.82 %
High risk | 8.7 % | 3.44 % | 1.68 % | 0.78 % | 0.16 % | 0.28 % | 0 % |  |
Med-high risk | 5.64 % | 1.79 % | 0.61 % | 0.29 % | 0.09 % | 1.19 % |  |  |
Med-low risk | 1.1 % | 0.59 % | 0.16 % | 0.17 % | 0.51 % | 0 % |  |  |
Low risk | 0.08 % | 0.46 % | 0.18 % | 1.34 % | 0 % |  |  |  |
Migraine | Very high risk | 8.67 % | 17.41 % | 10.62 % | 7.6 % | 4.75 % | 3.45 % | 3.16 % | 2.63 % | 2.27 %
High risk | 10.95 % | 14.85 % | 6.32 % | 4.15 % | 2.22 % | 3.41 % | 0 % |  |
Med-high risk | 9.15 % | 10.21 % | 4.54 % | 1.57 % | 1.93 % | 2.38 % |  |  |
Med-low risk | 6.18 % | 3.66 % | 1.86 % | 1.01 % | 2.03 % | 0 % |  |  |
Low risk | 3.84 % | 1.06 % | 1.3 % | 0 % | 0 % |  |  |  |
Renal dis | Very high risk | 3.42 % | 1.91 % | 1.89 % | 1.76 % | 2.19 % | 2.26 % | 3.66 % | 6.29 % | 10.26 %
High risk | 1.74 % | 1.95 % | 1.38 % | 0.86 % | 0.27 % | 2.72 % | 0 % |  |
Med-high risk | 1.47 % | 1.04 % | 0.64 % | 0.62 % | 0.2 % | 3.1 % |  |  |
Med-low risk | 0.7 % | 0.42 % | 0.5 % | 0.98 % | 0.34 % | 50 % |  |  |
Low risk | 0.25 % | 0.38 % | 0.83 % | 0 % | 0 % |  |  |  |
RA | Very high risk | 0.66 % | 0.69 % | 0.77 % | 0.79 % | 1.3 % | 1.09 % | 1.24 % | 1.38 % | 1.44 %
High risk | 0.51 % | 0.55 % | 0.66 % | 0.59 % | 0.03 % | 0.09 % | 0 % |  |
Med-high risk | 0.55 % | 0.55 % | 0.5 % | 0.52 % | 0.04 % | 0 % |  |  |
Med-low risk | 0.39 % | 0.41 % | 0.47 % | 0.91 % | 0 % | 0 % |  |  |
Low risk | 0.31 % | 0.42 % | 0.7 % | 0.67 % | 0 % |  |  |  |
SMI | Very high risk | 6.44 % | 5.96 % | 3.73 % | 2.3 % | 1.92 % | 1.22 % | 1.2 % | 1.14 % | 1.11 %
High risk | 4.09 % | 5.44 % | 2.17 % | 1.15 % | 0.14 % | 0.14 % | 0 % |  |
Med-high risk | 3.78 % | 3.4 % | 1.33 % | 0.6 % | 0.08 % | 0.24 % |  |  |
Med-low risk | 2.74 % | 1.16 % | 0.63 % | 0.81 % | 0.17 % | 0 % |  |  |
Low risk | 1.17 % | 0.49 % | 0.76 % | 0.67 % | 0 % |  |  |  |
SLE | Very high risk | 0 % | 0.1 % | 0.09 % | 0.09 % | 0.07 % | 0.08 % | 0.09 % | 0.09 % | 0.09 %
High risk | 0.2 % | 0.02 % | 0.1 % | 0.05 % | 0.04 % | 0.14 % | 0 % |  |
Med-high risk | 0 % | 0.09 % | 0.05 % | 0.06 % | 0.07 % | 0.24 % |  |  |
Med-low risk | 0.05 % | 0.05 % | 0.04 % | 0.08 % | 0 % | 0 % |  |  |
Low risk | 0.04 % | 0.02 % | 0.07 % | 0 % | 0 % |  |  |  |
BMI | Very high risk | 0.78 (1.09) | 0.45 (0.94) | 0.09 (0.73) | -0.1 (0.62) | -0.22 (0.53) | -0.31 (0.49) | -0.4 (0.49) | -0.46 (0.47) | -0.56 (0.43)
High risk | 0.38 (0.91) | 0.16 (0.7) | -0.14 (0.56) | -0.22 (0.44) | -0.32 (0.37) | -0.62 (0.44) | -1.15 (0.5) |  |
Med-high risk | 0.21 (0.77) | -0.05 (0.56) | -0.18 (0.44) | -0.26 (0.39) | -0.5 (0.45) | -0.89 (0.43) |  |  |
Med-low risk | -0.11 (0.53) | -0.19 (0.37) | -0.25 (0.37) | -0.49 (0.46) | -0.82 (0.43) | -1.02 (0.78) |  |  |
Low risk | -0.28 (0.32) | -0.32 (0.38) | -0.52 (0.45) | -0.94 (0.43) | -1.47 (0.18) |  |  |  |
HDL | Very high risk | -0.7 (0.51) | -0.41 (0.52) | -0.23 (0.51) | -0.07 (0.5) | 0.08 (0.5) | 0.2 (0.56) | 0.28 (0.63) | 0.28 (0.64) | 0.3 (0.62)
High risk | -0.51 (0.46) | -0.28 (0.48) | -0.07 (0.43) | 0.08 (0.41) | 0.25 (0.46) | 0.79 (0.82) | 1.91 (1.3) |  |
Med-high risk | -0.39 (0.44) | -0.18 (0.41) | -0.01 (0.37) | 0.17 (0.43) | 0.63 (0.76) | 1.78 (1.09) |  |  |
Med-low risk | -0.22 (0.32) | -0.07 (0.3) | 0.1 (0.41) | 0.68 (0.81) | 1.73 (1.01) | 2.52 (0.98) |  |  |
Low risk | -0.07 (0.31) | 0.11 (0.48) | 0.66 (0.83) | 2.36 (0.93) | 4.04 (0.08) |  |  |  |
SBP | Very high risk | 0.17 (0.65) | 0.13 (0.59) | -0.08 (0.54) | -0.08 (0.5) | -0.04 (0.44) | -0.03 (0.41) | -0.01 (0.44) | 0.03 (0.45) | 0.05 (0.46)
High risk | -0.11 (0.54) | -0.15 (0.47) | -0.28 (0.41) | -0.21 (0.34) | -0.17 (0.32) | -0.41 (0.37) | -0.66 (0.42) |  |
Med-high risk | -0.23 (0.5) | -0.33 (0.38) | -0.33 (0.33) | -0.29 (0.32) | -0.39 (0.38) | -0.59 (0.42) |  |  |
Med-low risk | -0.46 (0.34) | -0.44 (0.27) | -0.41 (0.31) | -0.49 (0.37) | -0.57 (0.42) | -0.56 (0.37) |  |  |
Low risk | -0.58 (0.25) | -0.58 (0.29) | -0.62 (0.35) | -0.66 (0.37) | -0.58 (0.59) |  |  |  |
Smoke | Very high risk | 0.7 (0.22) | 0.68 (0.23) | 0.65 (0.25) | 0.54 (0.28) | 0.4 (0.26) | 0.29 (0.25) | 0.22 (0.23) | 0.17 (0.22) | 0.14 (0.21)
High risk | 0.68 (0.22) | 0.64 (0.23) | 0.61 (0.25) | 0.42 (0.2) | 0.28 (0.15) | 0.15 (0.15) | 0.06 (0.08) |  |
Med-high risk | 0.65 (0.21) | 0.65 (0.23) | 0.5 (0.19) | 0.34 (0.17) | 0.17 (0.16) | 0.11 (0.12) |  |  |
Med-low risk | 0.66 (0.19) | 0.54 (0.17) | 0.39 (0.17) | 0.2 (0.19) | 0.14 (0.15) | 0.04 (0.02) |  |  |
Low risk | 0.52 (0.18) | 0.38 (0.2) | 0.18 (0.18) | 0.15 (0.15) | 0.13 (0.15) |  |  |  |
TCHOL | Very high risk | 0.68 (0.75) | 0.59 (0.59) | 0.52 (0.52) | 0.45 (0.48) | 0.39 (0.42) | 0.3 (0.41) | 0.19 (0.43) | 0.08 (0.44) | 0.02 (0.45)
High risk | 0.5 (0.56) | 0.5 (0.48) | 0.44 (0.42) | 0.41 (0.36) | 0.36 (0.32) | 0.16 (0.46) | -0.45 (0.64) |  |
Med-high risk | 0.47 (0.51) | 0.43 (0.4) | 0.42 (0.33) | 0.37 (0.33) | 0.23 (0.44) | 0.02 (0.57) |  |  |
Med-low risk | 0.33 (0.33) | 0.37 (0.28) | 0.36 (0.32) | 0.21 (0.46) | 0.03 (0.53) | -1.1 (0.51) |  |  |
Low risk | 0.24 (0.24) | 0.27 (0.33) | 0.17 (0.45) | 0.03 (0.61) | 0.09 (0.25) |  |  |  |
Townsend 20 | Very high risk | 13.97 (4.83) | 12.85 (5.17) | 12.81 (5.16) | 11.26 (5.35) | 10.34 (5.25) | 8.7 (5.22) | 8.39 (5.17) | 8.53 (5.2) | 8.73 (5.21)
High risk | 13.63 (4.75) | 12.2 (5.38) | 11.61 (5.34) | 10.22 (5.31) | 5.96 (4.02) | 4.8 (3.7) | 3.23 (2.41) |  |
Med-high risk | 12.9 (5.18) | 12.2 (5.36) | 10.92 (5.25) | 6.42 (4.41) | 4.98 (3.86) | 4.4 (3.47) |  |  |
Med-low risk | 12.9 (5.09) | 10.29 (5.16) | 6.48 (4.44) | 5.35 (4.11) | 4.55 (3.61) | 6.5 (0.71) |  |  |
Low risk | 8.65 (5.41) | 5.98 (4.81) | 4.96 (4) | 4.72 (4.23) | 5 (1.73) |  |  |  |

##  4 Validation: C-index and Brier score

###  4.1 Validation

We validate the 2-stage landmark model for estimating the probability of not
being diagnosed with CVD before statins initiation, described in Section 3.2
of the main manuscript. The estimated c-indices are represented via dots in
Figure 4, while the Brier scores are represented in Figure 5.

Figure 4: Estimated c-indices for the second landmark model for men (blue
dots) and women (red dots) for different values of starting time $t$. Each
point represents a c-index computed for a specific
$s=L_{a}\in\\{40,45,..,80\\}$ and $w=10$, since we are interested in the
discrimination accuracy of the 10-year CVD risk. Points at $L_{a}=40$,
represents the c-indices estimated with $s=40$ and $w=10$. The solid black
lines represent the overall c-index across landmark ages and gender (dashed
lines represent $95\%$ confidence interval). The dashed red line at 0.5
represents the minimum sensible value of the c-index. Figure 5: Estimates of
Brier Score, $BS_{s}(w)$ where $s=L_{a}\in\\{40,45,..,80\\}$ and $w=10$. Each
$BS_{L_{a}}(10)$ is represented through a colored dot (blue dots for men and
red dots for women).

In Figure 4, we found good overall discrimination (overall c-index equal to
0.77, represented via a solid black line).

Secondly, we validate the extended 2-stage landmarking approach described in
Section 3.3 of the main manuscript and we report the estimated c-indices in
Figure 6 and the estimated Brier Scores in Figure 7.

Figure 6: Estimated c-indices for the first landmark model for women (left
panel) and men (right panels) for different values of starting time $s$. Each
point represents a c-index computed for a specific $s\in\mathcal{P}_{L_{a}}$
and $w=5$, since we are interested in the discrimination accuracy of the
5-year CVD risk. We associate a specific color to each landmark set. All
points in light blue are associated with $L_{a}=40$, and from the first point
from the left we have $s\in\\{40,41,42,..,50\\}$.The dashed red line at 0.5
represents the minimum sensible value of the c-index. Values lower than 0.5
are recorded at older ages, for the latest time-windows (i.e., 83-88, 84-89,
85-90 in orange for men, 88-93, 89-94, 90-95 in violet for men). Figure 7:
Estimates of Brier Score, $BS_{s}(w)$ where $s\in\mathcal{P}_{L_{a}}$ and
$w=5$. Each $BS_{s}(5)$ is represented through a colored dot (each landmark
age is associated to a specific color). BS associated to women are reported in
the left panel, while the BS associated to men are reported in the right
panel.

Note that the extended 2-stage landmarking approach for estimating
$t_{i,L_{a}}^{*}$ in general has good discriminatory power and good prediction
accuracy (the lower the Brier Score, the higher the predictive accuracy of the
model). The model performs better for women than men (at each time $s$ the
c-index for women is higher than the c-index for men, and the inverse for the
Brier score). Furthermore, the model performance tends to decrease for higher
landmark ages. Low c-indices with high standard deviations are found for older
landmark ages ($75$, $80$), for later prediction times $s\in\\{83,84,85\\}$
and $s\in\\{88,89,90\\}$ respectively. This could be due to the fact that the
mean follow-up time is lower at higher landmark ages and very few people are
observed after 83 years at landmark age 75 and after 88 years at landmark age
80.

We observe similar trends between Figure 4, Figure 5 and Figure 6, Figure 7:
lower discrimination and prediction accuracy for the men landmark cohorts and
a decline in model performance as the landmark age increases.

##  5 Sensitivity analysis: exploring the effect of NB parameters

We consider three sensitivity analyses by varying key parameters as follows:

  * •

$\lambda\in[20,000;30,000]$ £/year, while $u_{s}=0.997$, $c_{s}=150$ £/year,
and $c_{\nu}=18.39$ £/visit. The results are reported in panel A of Figure 8.

  * •

$u_{s}\in[0.997;1]$, while $\lambda=25,000$ £/year, $c_{s}=150$ £/year, and
$c_{\nu}=18.39$ £/visit. The results are reported in panel B of Figure 8.

  * •

$c_{s}\in[4;320]$ £/year, while $\lambda=25,000$ £/year, $u_{s}=0.997$, and
$c_{\nu}=18.39$ £/visit. The results are reported in panel C of Figure 8.

Figure 8: Sensitivity analysis of $\lambda$ (panel A), $c_{s}$ (panel B),
$u_{s}$ (panel C). In the y-axis of all panels we represent the proportion of
the cohort. In panel A, the value of $\lambda$ is reported in the x-axis and
it ranges in $[20,000;30,000]$ £/year. $u_{s}=0.997$, $c_{s}=150$ £/year, and
$c_{\nu}=18.39$ £/visit. In panel B, the value of $u_{s}$ is reported in the
x-axis and it ranges in $[0.997;1]$. $\lambda=25,000$ £/year, $c_{s}=150$
£/year, and $c_{\nu}=18.39$ £/visit. In panel C, the value of $c_{s}$ is
reported in the x-axis and it ranges in $[4;320]$ £/year. $\lambda=25,000$
£/year, $u_{s}=0.997$, and $c_{\nu}=18.39$ £/visit. This figure appears in
color in the electronic version of this article.

In general, we observe that results are robust with respect to the parameter
choice.

In panel A of Figure 8, we vary the value of $\lambda$ from 20,000 £/year to
30,000 £/year, as $\lambda$ increases, the 10-year frequency is optimal for
fewer people, while intermediate frequency (such as 4-7 years) becomes optimal
for a larger proportion of people. Visits every 1, 2, 3 years are optimal for
a constant proportion of the cohort. An increasing $\lambda$ can be
interpreted as a stronger willingness to pay for increased expectancy of CVD-
free life years, so intermediate frequencies tend to be preferred over
10-yearly risk-assessments.

We see an analogous behaviour for $u_{s}$ (panel B of Figure 8). We vary the
utility factor from 0.997 (which implies a decrease of quality of life equal
to 0.003) to 1 (which implies that taking statins has no effect at all on the
quality of life). We notice that 1 to 4-year risk-assessment strategies are
the optima for a constant number of people. We note when the impact of statins
on quality of life is low ($u_{s}$ tends to 1), the 10-year frequency schedule
is less preferred, while intermediate frequencies (4-8 years) are preferred.
If no burden is associated with taking statins ($u_{s}=1$), then the optimal
strategy is to initiate statins immediately.

In contrast, the higher the price of statins, $c_{s}$, the more risk-
assessment strategies associated with less frequent visits are to be preferred
(panel C of Figure 8). This is expected because higher costs imply decreased
net benefit of statin usage.

We investigated also $c_{\nu}$ varying between 15 £/visit to 1000 £/visit
(results not shown). Despite the broad range explored, the optimal schedule
proportions are unchanging across all values of $c_{\nu}$. This result is
expected because this term of the NB is not comparable in scale with the terms
associated with expected event free life years in Eq. (2) of the manuscript.

It is also immediate to notice from the range reported in the y-axis of Figure
8 that the greatest part of the whole cohort ($>$70%) is recommended to be
assessed every 10 years. This is due to the fact that we are considering the
stacked landmark cohorts and the biggest landmark cohorts are those ones
collected at $L_{a}=40$, $L_{a}=45$, that are composed of younger and
healthier people.

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