# Using Small Domain Estimation to obtain better retrospective Age Period
Cohort insights

Koen Simons
Epidemiology and Biostatistics
Melbourne School of Population and Global Health
University of Melbourne, AU
<EMAIL_ADDRESS>&Rebecca Bentley
Centre for Health Equity
Melbourne School of Population and Global Health
University of Melbourne, AU
<EMAIL_ADDRESS>&Lyle Gurrin
Epidemiology and Biostatistics
Melbourne School of Population and Global Health
University of Melbourne, AU
<EMAIL_ADDRESS>0000-0002-6534-2277 0000-0003-3334-7353
0000-0001-7052-1969

###### Abstract

Recent changes in housing costs relative to income are likely to affect
people’s propensity to Housing Affordability Stress (HAS), which is known to
have a detrimental effect on a range of health outcomes. The magnitude of
these effects may vary between subgroups of the population, in particular
across age groups. Estimating these effect sizes from longitudinal data
requires Small Domain Estimation (SDE) as available data is generally limited
to small sample sizes. In this paper we develop the rationale for smoothing-
based SDE using two case studies: (1) transitions into and out of HAS and (2)
the mental health effect associated with HAS. We apply cross-validation to
assess the relative performance of multiple SDE methods and discuss how SDE
can be embedded into g-computation for causal inference.

_K_ eywords Small Domain Estimation; Small Area Estimation; Smoothing; Age
Period Cohort; Mental Health; Housing Affordability Stress; g-computation

##  1 Introduction

The number of people experiencing housing affordability stress (HAS) is
increasing in many high-income countries due to the rapid rise in the past 10
years of the cost of housing relative to incomes (Taylor and Tyers,, 2017). At
the same time, a growing number of studies in a variety of settings report
associations between HAS and mental health (Bentley et al.,, 2011; Mason et
al.,, 2013; Bentley et al.,, 2016; Marí-Dell’Olmo et al.,, 2017). Many of
these studies have considered changes in people’s housing situation over time
to assess concordant changes in their health status. On the whole, there is
evidence that HAS has a short term negative effect on mental health. The
increasing prevalence of HAS combined with the known relationship between
housing affordability and the risk of poor mental health is, potentially, an
important determinant of population health. It may provide an indicator at the
population level of the “health” of housing within a given society or country,
in this case Australia. Here we attempt a big-picture view of this
relationship by looking across time and tracking age-based cohorts, seeking to
describe the impact of the cost of housing over time on population well-being.

The experience of being unable to afford the cost of housing is an obvious
source of stress. Despite this, housing affordability stress is difficult to
define and multiple operational definitions exist (Nepal et al.,, 2010). A
definition that has been widely used in research and by national data
collection agencies is the so-called 30/40 rule, where a household is
considered to be in housing affordability stress if it spends at least 30% of
its gross income on housing costs and is in the bottom 40% of the national
equivalized disposable income distribution (Trewin,, 2002). For a household to
“exit” from unaffordable housing requires an opportunity to increase income
and/or an opportunity to decrease housing-related expenses, such as through
improved employment status, moving house or receiving government subsidies.
Clearly, the abundance or scarcity of such opportunities depends on both
personal and market characteristics. For example, education and experience are
personal factors that influence employment opportunities, whereas house prices
differ strongly by location. Financial transactions within families (such as
gifts or bequests) may alter these probabilities on the individual level, and
expectations about wealth and attitudes to the management of finances will be
influenced by parents and the wider family environment (Cigdem and Whelan,,
2017). While transfers are generally within-family, expectations reflect
comparisons with peers and/or previous generations; they are not limited to
the individual’s circumstances. The probability of exiting from HAS is,
therefore, likely to be dependent on region, age and social stratum. Since
demographics, policy and market conditions naturally change over time, there
is possibly temporal variation of the exit probability. Most of these factors
would also be expected to influence the probability of exposure to or “entry”
into HAS. At the population level, these combined influences result in age-,
time-, region- and social stratum-specific differences in the incidence of and
emergence from HAS. Identifying and understanding these differences may lead
to more efficient policies by targeting interventions to lift people out of
HAS. Similarly, identifying differences in the rates of entry into HAS enables
targetted preventive measures.

There are complexities in estimating effects of HAS on health outcomes.
Ideally stratum-specific entry and exit rates would be estimated directly from
longitudinal data where HAS status is recorded at two or more times. Census
information may also provide a large or exhaustive sample that could be used
to estimate the prevalence of HAS. Such data are, however, unlikely to be set
up for longitudinal analysis, so entry and exit probabilities cannot be
obtained without further assumptions. Similarly, estimation of stratum-
specific effects of HAS requires that information on health outcomes is linked
to exposure data. While the frequency of the creation of record-linked data
resources continue to improve, concerns for privacy may limit access to the
resulting data.

One feature that these data platforms share is that sufficiently rich data for
the analyses we propose are likely to be limited to small sample sizes.
Moreover, small sample sizes lead to even smaller sample sizes per stratum.
These very small sample sizes provide little information on stratum-specific
rates and effects. Small Domain Estimation (SDE) or Small Area Estimation
(SAE) aims to overcome this problem and can be used to obtain efficient
estimates of HAS incidence and prevalence. Here, small refers to the sample
size of an individual stratum, as opposed to the geographical area or the
total population size of a stratum. Domain refers to the level of aggregation
that is the target of estimation; a domain may be composed of multiple strata.
Efficiency for domain-specific estimates is gained by including information
about the structure of the data or through a trade-off between the bias and
the variance of estimation. Rao and Molina, (2015) and Pfeffermann et al.,
(2013) describe various statistical methods to obtain small domain estimates.
Rahman et al., (2013) evaluated the use of SAE to obtain efficient estimates
of HAS prevalence from survey data. Their findings indicate that there is
large geographical variation in HAS: “Almost two-thirds of these households
are located in statistical local areas (SLAs) in eight capital cities, and a
large number of them are in Sydney and Melbourne.”

In this paper, we focus primarily on variation in entry and exit rates by age,
birth-year and calendar-year and do not consider explicitly spatial variation
or produce sex-specific estimates. First, we introduce three traditional
estimators to discuss the bias-variance trade-off: direct estimation, complete
pooling and partial pooling. Second, we define an estimator based on tensor
smoothing splines that exploits the relative similarity between adjacent
domains, and we discuss the strengths and weaknesses by applying Cross-
Validation on an example dataset. Third, we consider the effect of HAS on
changes in Mental Health (MH) and the modification of these changes by age,
birth-year and calendar year. We illustrate how the different Small Domain
Estimators can be used to estimate both the main effect of HAS on Mental
Health as well as for estimating and visualising the heterogeneity of effect.
We discuss the limitations of these methods in the context of Age-Period-
Cohort data where, for identification, an arbitrary constraint must be
imposed. We argue, however, that this does not limit interpretability of the
findings. We conclude that tensor splines and other SDE approaches are useful
for detecting trends and/or hot spots as well as for generating hypotheses. If
backed by appropriately specified Directed Acyclic Graphs (DAG), the ability
to explicitly allow for heterogeneity of effect may be advantageous when
employing a potential outcomes approach to causal inference.

##  2 Data

Since 2001, the Melbourne Institute of Applied Economic and Social Research is
annually collecting data for the longitudinal Household, Income and Labour
Dynamics in Australia Survey (HILDA). At baseline, 7682 households were
interviewed and 19914 individuals were included. As is common in such studies,
participants drop out for various reasons. In 2011, a ‘top-up’ sample was
added, consisting of 2153 new households or 5451 new individuals.

HILDA data has previously been used to investigate HAS (Borrowman et al.,,
2017) and the effects of housing conditions on mental health (Bentley et al.,,
2018; Singh et al.,, 2019). For this study, we included, for each calendar-
year, all individuals who were between 25 and 64 years old and who had moved
out of their parents house. Individuals with missing data on household
equivalised income or housing-related costs were discarded for the affected
time periods. Similarly, analysis of changes in mental health, SF36, was
restricted to available data.

##  3 Entry and Exit Rates

###  3.1 SDE, partial pooling and smoothing splines

Entry into and exit from Housing Affordability Stress can be seen as a two
compartment model: the population not in HAS, who are at risk of entering into
HAS, and the population in HAS, who may exit. We assume that the two
transition processes are first order Markov processes, specifically:

$\displaystyle p_{\mathrm{entry}}$
$\displaystyle=p\left(Y_{it}=1|Y_{it-1}=0,Y_{it-2},Y_{it-3},\ldots{},A_{it},A_{it-1},A_{it-2},\ldots{}\right)$
$\displaystyle=p\left(Y_{it}=1|Y_{it-1}=0,A_{it}\right)$ $\displaystyle
p_{\mathrm{exit}}$ $\displaystyle=p\left(Y_{it}=0|Y_{it-1}=1,A_{it}\right)$

Wherein $Y_{it}=1$ indicates that individual $i$ experienced HAS in year $t$
and $A_{it}$ represent covariates such as age.

Given a sample $S$ of observations $(t,a_{t},y_{t},y_{t-1})$, unbiased
estimates of the conditional entry and exit probabilities can be obtained
directly from the sample averages:

$\displaystyle\hat{p}_{\mathrm{exit}}^{d}(a,t)$
$\displaystyle=\sum_{i:Y_{it-1}=1,A_{it}=a}Y_{it}/n_{at}$ $\displaystyle
n_{at}=\\#\\{i\in S:Y_{iT-1}=1,A_{iT}=a,T=t\\}$

However, the variance of the _direct_ estimators $\hat{p}^{d}(a,t)$ is
$O(1/n_{at})$ and will be large for small $n$ as is illustrated in Figure 1
for the exit probability, which suffers smaller denominators as the number of
individuals with HAS is a small fraction of the population as shown in Figure
8 in appendix.

Figure 1: Probability of exit from unaffordable housing estimated using direct
estimation from the sample, without pooling. For ease of visualisation only
ages 30, 40, 50 and 60 are shown.

A straightforward way to reduce the variance is to combine the observations
from multiple domains and calculate a _pooled_ estimate. The _completely
pooled_ estimator is defined as:

$\displaystyle\hat{p}_{\mathrm{exit}}^{c}(a,t)$
$\displaystyle=\sum_{i,T:Y_{iT-1}=1}Y_{iT}/n$ $\displaystyle
n=\sum_{A,T}n_{AT}$

This estimate has lower variance, however, the estimator is biased unless the
transition probability is homogeneous over all domains. Moreover, the
magnitude of the bias depends on the heterogeneity and is unknown. It is
possible to define alternative estimators with a different trade-off between
bias and variance, e.g. by combining (aggregating) domains into a set of
super-domains. Considering the previously introduced estimators, a compromise
between the unbiased direct estimator and the biased completely pooled
estimator can be obtained by weighting the estimators:

$\displaystyle\hat{p}_{\mathrm{exit}}^{w}(a,t)$
$\displaystyle=w_{at}\hat{p}_{\mathrm{exit}}^{c}(a,t)+(1-w_{at})\hat{p}_{\mathrm{exit}}^{d}(a,t)$

For each domain $(a,t)$, the weighted estimate is between the direct and
completely pooled estimates. Hence, they can be seen as shrinkage estimators
that pull the direct estimators towards an overall mean and reduce the
heterogeneity of the estimates without requiring it to be zero. This produces
estimates that have a variance between the completely pooled and the direct
estimates and bias must also be between the two. Naturally, the amount of bias
and variance depend on the choice of the domain-specific weights. There are
multiple ways to obtain suitable weights $w_{at}$. One option is to rewrite
the problem as a (Bayesian) hierarchical model such that there is an overall
mean odds ($p_{o}$) but domain-specific odds vary around this mean. Working on
the log-odds scale, a typical model is:

$\displaystyle Y_{it}|a$ $\displaystyle\sim\mathrm{Binomial}(p_{a,t})$
$\displaystyle\mathrm{logit}(p_{a,t})$ $\displaystyle=\pi_{at}$ $\displaystyle
u_{at}\sim N(0,\sigma)$ $\displaystyle\pi_{at}$ $\displaystyle=\pi_{o}+u_{at}$

In this approach, the amount of shrinkage can be controlled indirectly by
choosing a prior for $\sigma$ that puts more or less weight on values close to
0. Together with the data, this results in a posterior distribution for
$\sigma$ and for both the overall mean probability $p_{o}$ and the domain-
specific transition probabilities $p_{at}$. The estimates for domains with
lower sample sizes, relative to the average sample size, will be shrunk more
than those with more observations. In addition, the weighted estimates
wherefore the respective direct estimates are relatively further away from the
overall mean, will be shrunk more than those that are closer to the overall
mean. Besides shrinking, this property is also referred to as
‘regularization’, as it shrinks outlying values towards the mean; ‘borrowing
strength’ or _‘partial pooling’_ as it augments the domain-specific sample
with observations from other domains in such a way that the amount of pooling
or borrowing depends on the relative sample size.

Throughout the remainder or this manuscript we will refer to such estimates as
partially pooled estimates. Results of partial pooling for the exit
probabilities are visualized in the left panel of Figure 2. As expected,
partial pooling results in estimates that are less heterogeneous than the
direct estimates presented earlier. Comparison with Figure 1 reveals gains in
_stated_ precision, however, credible intervals remain wide and no patterns
across age or time are revealed.

Figure 2: Probability of exit from unaffordable housing estimated using small
domain estimation. Left: estimates obtained from a partial pooling model with
iid error. Right: estimates obtained using a tensor spline basis. For ease of
visualisation only ages 30, 40, 50 and 60 are shown.

Nonetheless, such patterns are intuitively expected: aggregate changes in
market conditions and in demographics are often slow. Furthermore, conditions
for a 30-year-old person will be more similar to conditions for a 31-year-old
person than those for a 60 year old person. It is sensible to expect that the
entry and exit probabilities vary smoothly over time and age:

$\displaystyle Y_{it}|a$ $\displaystyle\sim\mathrm{Binomial}(p_{a,t})$
$\displaystyle\mathrm{logit}(p_{a,t})$ $\displaystyle=s(a,t)$

Wherein $s$ is a _smooth_ 2d function.

Options for estimating $s$ include 2d kernels, thin plate smoothing splines
and tensor splines. To distinguish the smooth estimators from partially pooled
estimators, consider the naive kernel that assigns weight $w_{at}=1$ if
$|A-a|<5$ & $|t-T|<5$ and $w_{at}=0$ elsewhere. Define the naive smooth
estimator as:

$\displaystyle\hat{p}_{\mathrm{exit}}^{n}(a,t)$
$\displaystyle=\sum_{i,T:Y_{iT-1}=1}^{|A_{iT}-a|<5,|T-t|<5}Y_{iT}/n_{at}$
$\displaystyle n_{at}$ $\displaystyle=\\#\\{(T,A_{iT},Y_{iT},Y_{iT-1})\in
S:Y_{iT-1}=1,|A_{iT}-a|<5,|T-t|<5\\}$

This naive smooth estimator uses overlapping age-period windows of ten years
by ten years and completely pools all the data within each window while
ignores all data outside of the window. Using a moving window yields an
average absolute difference between estimates for two adjacent domains that
will be lower than when direct estimation is used. In other words, the
estimates are less _wiggly_ 111For twice differentiable functions $f(x)$,
wiggle can be defined as $\int|\partial^{2}f(x)/\partial x^{2}|dx$, ie a line
with constant slope has zero wiggle. This can be further improved upon by
using a different kernel such that more weight is given to observations closer
to the centre of the window, optimising the width of the window, allowing the
window’s width and height to be different, etc. Indeed, the naive kernel has
multiple deficiencies, including poor performance at the edges of the data:
the estimate for the youngest age $a_{\mathrm{min}}$ will include information
from observed ages $a_{\mathrm{min}}$ to $a_{\mathrm{min}}+5$ and if there is
a monotonous trend in the exit probability over age, then the estimate will be
closer than the estimate for age $a_{\mathrm{min}}+2.5$; the estimate is
biased. Complete and partially pooled estimates will also be biased when a
trend exists as they pool all data towards the global mean regardless of the
existence of trends.

Instead, a linear model could be adopted to account for the possibility of
linear trends, quadratic and higher order terms can be included to allow for
more complex patterns. It is well known that using piecewise polynomials such
as cubic b-splines can be used to perfectly fit data as long as enough knots
are included in the basis functions; a saturated tensor b-spline model
provides the same estimates a the direct estimator. By applying a penalty to
the amount of wiggle, p-splines (Eilers and Marx,, 1996) shrink the
coefficients that describe the response surface towards a flatter but not
necessarily more horizontal surface. Hence, tensor p-splines are a useful
method for estimating 2d smooth functions such as $s(a,t)$. Other options
include thin plate smoothing splines, radial base splines and methods that
implicitly estimate the correlation between parameters for adjacent cells.
Opsomer et al., (2008) specifically used radial bases of p-splines to create
small domain estimates; the Besag-York-Mollie (BYM) model includes random
effects that are spatially correlated, relying either on an adjacency matrix
or on a distance metric (Besag et al.,, 1991). Assuming there is zero spatial
correlation reduces the BYM model to partial pooling.

Despite similarities in aims, these options are not equivalent and the
relative merits are often studied by simulations, e.g. Chambers et al.,
(2009). For this application, we recommend tensor smoothing splines as there
is no reason to expect the response surface to be isotropic: there is no prior
reason to enforce that age 30 is equally similar to age 31 as 2002 is similar
to 2003. Tensor splines are invariant to the choice of scales (Wood et al.,,
2013), whereas thin plates are isotropic on the chosen scale and conditional
autoregressive models are similarly dependent on the relative scaling of age
and calendar time.

###  3.2 Evaluating performance

We used the R (R Core Team,, 2018) package brms (Bürkner,, 2017) to estimate a
smoothing tensor spline response surface for the exit probabilities. The right
panel of Figure 2 shows that the (stated) precision is greatly increased and
two trends are revealed: the probability of exit from HAS decreases with age
and has, for the most part, been declining over the study period. This figure
is restricted to four one-year age-groups for visual clarity but has the
advantage that the credible intervals are included. The companion Figure 3
shows the patterns across all age-calendaryear combinations but without
indication of uncertainty.

Figure 3: Heatmaps of posterior means for the probability of exit by APC.
Left: map obtained from a partial pooling model with iid error. Right: map
obtained using a tensor spline basis.

In Figures 2 and 3, based on the HILDA data, show that only tensor splines
yield visually clear gradients. Such gradients facilitate the generation of
hypotheses that explain the apparent differences across domains. Moreover,
Figure 2 demonstrates that tensor smoothing splines can result in higher
_stated_ precision compared to direct and partially pooled estimates. However,
complete pooling provides even more optimistic _stated_ precision: the pooled
estimate is 46% with 95% confidence interval ranging 45% to 47%. Figure 2
shows that the other methods produce point-wise intervals that are at least
four times wider. Considering only stated precision, tensor splines remain
second to complete pooling. However, it is known that complete pooling is
biased if effects are indeed heterogeneous. Instead of relying on the width of
confidence or credible intervals, model fit can be assessed more formally by
calculating the _expected log posterior predictive density_ of the observed
data, given model $m$:

$\displaystyle\mathrm{elpd}^{(m)}$
$\displaystyle=\sum_{i}\mathrm{log}p(y_{i}|\boldsymbol{y},m)$
$\displaystyle=\sum_{i}\int\mathrm{log}\left(p(y_{i}|\hat{\pi}_{i}^{(m)})\right)p(\hat{\boldsymbol{\pi}}^{(m)}|\boldsymbol{y},m)d\hat{\boldsymbol{\pi}}^{(m)}$
$\displaystyle\sim\sum_{i}\int\mathrm{log}\left(p(y_{i}|\hat{\pi}_{i}^{(m)})\right)p(\boldsymbol{y}|\hat{\boldsymbol{\pi}}^{(m)},m)p(\boldsymbol{\hat{\pi}}^{(m)})d\hat{\boldsymbol{\pi}}^{(m)}$

Through the rest of the paper we will omit the dependency on the model $m$.
The estimated variability over the domains and the average stated precision
are calculated using:

$\displaystyle V(E\left[\hat{\pi}_{at}\right])$
$\displaystyle\sim\sum_{at}\int\left(\hat{\pi}_{at}-\hat{\pi}_{o}\right)^{2}p(\boldsymbol{y}|\boldsymbol{\hat{\pi}})p(\boldsymbol{\hat{\pi}})d\boldsymbol{\hat{\pi}}$
$\displaystyle E\left[V(\hat{\pi}_{at})\right]$
$\displaystyle\sim\sum_{at}\int\left(\hat{\pi}_{at}-\bar{\pi}_{at}\right)^{2}p(\boldsymbol{y}|\boldsymbol{\hat{\pi}})p(\boldsymbol{\hat{\pi}})d\boldsymbol{\hat{\pi}}$

Wherein $\pi_{o}$ represents the mean log odds over all observations and
$\bar{\pi}_{at}$ is the domain-specific posterior mean:

$\displaystyle\hat{\pi}_{o}$ $\displaystyle=\frac{1}{n}\sum_{i}\hat{\pi}_{i}$
$\displaystyle\bar{\pi}_{at}$
$\displaystyle\sim\int\hat{\pi}_{at}p(\boldsymbol{y}|\boldsymbol{\hat{\pi}})p(\boldsymbol{\hat{\pi}})d\boldsymbol{\hat{\pi}}$

As defined above, elpd provides a measure of fit and does not penalise the
complexity of the model. Therefore, it is likely to favour models that over-
fit the data. In order to more objectively evaluate the performance of the
estimators, we used 5-fold Cross-Validation (CV) stratified by age-calendar-
year; we repeatedly held out approximately222Due to the small sample sizes it
is not always possible to withhold exactly 20% of the data in each domain,
e.g. if there are 6 observations in a age-period combination, one of the folds
must include two observations and the other four folds contain one observation
each. 20% of the sample data from each domain and generated posterior
predictions for these test datasets based on the remaining 80% of the data as
training data. The above definition of expected log posterior density is
trivially extended to new data. Using the same folds for each model, it is
possible to calculate the pairwise differences in expected log posterior
predictive density:

$\displaystyle\mathrm{elpd}_{i}^{(m)}$
$\displaystyle\sim\int\mathrm{log}p(y_{i}|\boldsymbol{\hat{\pi}})p(\boldsymbol{y}^{\mathrm{train}}|\boldsymbol{\hat{\pi}})p(\boldsymbol{\hat{\pi}})d\boldsymbol{\hat{\pi}}$
$\displaystyle\Delta^{(m,w)}\mathrm{elpd}$
$\displaystyle=\sum_{i}(\mathrm{elpd}_{i}^{(m)}-\mathrm{elpd}_{i}^{(w)})$

Table 1:  Comparison of out of sample predictive performance of estimates of
age-calendar-year-specific probabilities of exit from unaffordable housing,
using 5-fold Cross-validation stratified by age-calendar-year.

Estimator | Whole sample | 5-fold CV
---|---|---
| elpd |  $\Delta$ elpd |  $V(E\left[\hat{\pi}_{at}\right])$* |  $E\left[V(\hat{\pi}_{at})\right]$* | elpd | sd |  $\Delta$ elpd | sd
Complete pooling: $\pi_{at}=\pi_{c}$ | -6209.7 | -112.2 | 0.0 | 0.4 | -6210.0 | 7.2 | -29.7 | 7.4
Partial pooling: $\pi_{at}=\pi_{o}+u_{at}$ | -6097.4 |  | 31.2 | 74.6 | -6180.4 | 10.3 |  |
Tensor spline: $\pi_{at}=\pi_{o}+s(at)$ | -6125.0 | -27.5 | 74.1 | 4.5 | -6129.8 | 14.5 | 50.5 | 11.8

Complete pooling**: $\pi_{at}=\pi_{c}$ -6209.7 -112.3 0.0 0.4 -6210.7 7.2
-30.3 7.4 Direct estimator**: $\pi_{at}=\pi_{at}$ -6052.6 44.9 369.9 279.8
-6775.4 32.7 -595.0 25.2

* Rescaled by 1000 * Partial pooling and tensor spline estimators were obtained using the package brms, however, the direct estimator was not. To account for possible numeric deficiencies, the completely pooled estimator was fit both with and without brms.

###  3.3 Results

Table 3.2 lists the expected log posterior predictive density calculated both
with the whole sample and with the cross-validation procedure. Looking at
within-sample assessment, the direct estimator has an expected log posterior
density of -6052.6, which is higher than the elpd of the other methods. The
within-sample assessment of performance is often too optimistic. In
particular, the direct estimator performs substantially worse in a “new”
sample: the elpd drops to -6775.4. In contrast, the completely pooled
estimator has near identical performance within and out of sample, elpd
-6209.7 respectively -6210. A small decrease is to be expected in cross-
validation as the procedure uses less data to fit the model. As expected,
partial pooling is a favourable compromise between complete pooling and direct
estimation; with elpd -6180.4 under CV it performs better than the two
estimators it is based on. However, the largest elpd under CV for this
dataset, -6129.8, is obtained by the tensor spline-based estimates. Its
within-sample elpd is close to that obtained under CV, whereas partial pooling
is more optimistic within-sample.

Partial pooling and tensor splines outperform direct estimation and complete
pooling on out of sample assessment. This improvement can be attributed to a
favourable bias-variance trade. This trade is possible due to variation across
domains in true transition probabilities; if there is no such variation then
the completely pooled estimator is unbiased and has minimal variance. However,
Table 3.2 shows that the direct, partially pooled and tensor spline estimates
vary across domains. For this dataset, tensor splines net twice as much
variation in estimated probabilities as partial pooling (0.0741 respectively
0.0312) but not nearly as much variation as the direct estimator (0.3699).
Simultaneously, tensor splines have the second highest precision: the
variability of the posterior, averaged over all domains, is 0.0045 for tensor
splines, 0.0004 for complete pooling and much higher for partial pooling and
direct estimation. If there truly is heterogeneity, the reduced heterogeneity
in the estimates implies bias. This bias is traded for reduced variability of
the posterior compared to the direct estimator. In this sample, the trade-off
provides a net benefit for both partial pooling and tensor splines but the
exchange is much more favourable for tensor splines: less bias is taken for a
larger reduction in variance.333For partial pooling the trade-off can be
further optimised, by manipulating the weights $w_{at}$ directly or through
the prior on $\sigma$. For tensor splines the trade-off can also be optimised
by changing the penalty parameter (or by altering the number of knots used to
construct the b-spline bases). For both methods, a set of solutions can be
defined and indexed by one or more ‘tuning parameters’. Choosing the optimal
tuning parameter is a direct trade of bias for variance within a method- and
data specific set of solutions. Clearly, tensor splines have a favourable
exchange rate here.

To understand how tensor splines attain higher efficiency, we repeat the
Cross-Validation with a different set-up. Instead of sampling within each
domain, it is possible to perform cross-validation by repeatedly holding out
entire domains, e.g. excluding ages (corresponding to rows in Figure 3),
calendar-years (columns) or birth-years (diagonals). Table 3.3 shows the
results for leaving out calendar-years: the performance of complete pooling
remains the same as under the stratified CV scenario, however, partial pooling
now performs on par with complete pooling, while it was superior previously.
Tensor spline smoothing remains superior in the new scenario but the
difference in elpd within-sample and under cross-validation has changed from
-4.8 (= -6129.8 + 6125) to -13.7 (= -6138.7 + 6125). These results illustrate
that tensor spline smoothing combines information from the target domain with
neighbouring domains to efficiently estimate domain-specific exit
probabilities. If only neighbour domains are available, the procedure still
works but the accuracy is affected. In contrast, partial pooling combines
information from the target domain and the overall mean. If no observations
from the target domain are available, the predictions effectively use
information from the overall mean only and hence the predicted values are very
similar to those derived by complete pooling. If there are trends in over the
domains, shrinking the estimate towards the overall mean may generate more
bias than shrinking the estimate towards the nearest neighbours’ mean.

Table 2:  Comparison of out of sample predictive performance of estimates of
age-calendaryear-specific probabilities of exit from unaffordable housing,
using 5-fold Cross-validation wherein entire calendaryears were left out.

* Rescaled by 1000

##  4 Effect of HAS on Mental Health

###  4.1 Modelling choices

The combination of demographic change with changing market conditions may lead
to changes in expectations and modify the mental health effects associated
with HAS. To investigate this, we again assume that mental health scores are
the result of a first order Markov Process, specifically:

$\displaystyle
p\left(Y_{it}=y_{it}|Y_{it-1},Y_{it-2},\ldots{},A_{it},A_{it-1}\ldots{}\right)$
$\displaystyle=p\left(Y_{it}=y_{it}|Y_{it-1},A_{it}\right)$

The distribution of SF36 is skewed and it is reasonable to expect that,
without external influences, there is no change in Mental health:
$E[Y_{it}|Y_{it-1}]=Y_{it-1}$. Hence, we opt to model the first differences:

$\displaystyle\Delta Y_{it}$ $\displaystyle=Y_{it}-Y_{it-1}$

The distribution of first differences is symmetrical, however, there is
evidence of conditional heterogeneity and regression to the mean (Figure 3.2
in appendix). To accommodate this we fitted the following tensor spline model:

$\displaystyle\left\\{\begin{array}[]{ll}\Delta Y_{it}&\sim
N\left(\mu_{it},\sigma_{it}\right)\\\ \mu_{it}&=\alpha+\beta
Y_{it-1}+s_{1}(A_{it},t)\\\
\mathrm{log}\sigma_{it}&=s_{h}(Y_{it-1})\end{array}\right.$

Wherein $s_{1}$ is a smooth 2d function modelled using tensor splines and
$s_{h}$ is a 1d smooth function. For convenience, we use fully-parametric
cubic b-splines with 5 degrees of freedom for $s_{h}$. Similar to the models
for entry and exit, a comparison partial pooling model is defined by replacing
$s_{1}$ by $u_{at}\sim N(0,\sigma_{d})$.

These models allow for an age and calendar-year effect on the change in Mental
Health but do not include HAS. We contrast these models with two alternatives.
The first includes an effect of HAS ($M$) on changes in Mental Health ($\Delta
Y$), the second allows for the effect of HAS to be modified by age and
calendar-year:

$\displaystyle\left\\{\begin{array}[]{ll}\Delta Y_{it}&\sim
N\left(\mu_{it},\sigma_{it}\right)\\\
\mu_{it}&=\alpha+\beta_{1}Y_{it-1}+s_{1}(A_{it-1},t)+\beta_{2}M_{it-1}\\\
\mathrm{log}\sigma_{it}&=s_{h}(Y_{it-1})\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{ll}\Delta Y_{it}&\sim
N\left(\mu_{it},\sigma_{it}\right)\\\
\mu_{it}&=\alpha+\beta_{1}Y_{it-1}+s_{1}(A_{it},t)+\beta_{2}M_{it-1}s_{2}(A_{it},t)\\\
&=\alpha+\beta_{1}Y_{it-1}+(1-M_{it-1})s_{1}(A_{it},t)+\beta_{2}M_{it-1}+M_{it-1}s_{2}^{\star}(A_{it},t)\\\
\mathrm{log}\sigma_{it}&=s_{h}(Y_{it-1})\end{array}\right.$

These models correspond to the Directed Acyclic Graph shown in Figure 4. By
including both a main effect for HAS and a smooth interaction between HAS, Age
and Period, these models are not identifiable unless additional constraints
are imposed. For $s_{1}$ and $s_{2}^{\star}$ the smooth functions are
constrained such that their mean is zero. $s_{2}$ must be constrained to have
mean equal to one as is typical for a Varying Coefficient Model. Albeit both
formula containing two two-dimensional tensor splines are mathematically
equivalent, they may result in different estimates depending on how the knots
are chosen and how penalties are implemented. In this paper we used the second
form only.

$\mathrm{SF36}_{t}-\mathrm{SF36}_{t-1}$$\mathrm{SF36}_{t-1}$$\mathrm{HAS}_{t}$$\mathrm{Age-
Period-Cohort}_{t}$ Figure 4: Direct Acyclic Graph for the evolution of Mental
Health influenced by Housing Affordability Stress and an age-period-cohort
effect.

We expect that models including smooth terms will result in patterns that may
be of interest. Unlike the entry and exit models, additional covariates are
included in models. To calculate an average effect of exposure, we use the
_potential outcomes_ framework (Rubin,, 1974; Holland,, 1986; Hernan and
Robins,, 2010) to estimate the expected change in MH for each individual using
their observed covariates and comparing these with the predicted outcomes for
the counterfactual state wherein HAS is altered but all other covariates are
retained. The average effect of exposure can be obtained by averaging the
difference between predictions for HAS=1 minus the predictions for HAS=0.
Performing this calculation, we must make the strong causal assumptions (1)
that there are no (un)measured variables that affect the outcome other than
the variables included in the model: age, calendar-year, previous Mental
Health and HAS, (2) that exposure could have been different, and (3) that
changing the exposure for one person does not affect exposure for other
people. These assumptions are also referred to as exchangeability,
manipulability and the stable unit treatment value assumption. If any of these
assumptions is violated, the calculations do not provide a meaningful answer
to a causal question.

Averaging the differences in counterfactual outcomes over all observed
covariates provides an estimate of the average effect of HAS. We define the
domain-specific average effect of HAS by as the domain-specific mean over all
remaining variables, in essence over the observed values of previous MH. As
all the models we compared are additive in previous MH and no transformation
of the outcome was performed, the effect of HAS does not depend on previous MH
and we do not need to average over all values of previous MH. Instead, we
counterfactually set previous MH equal to any arbitrary value, e.g. its mean
value, 75. Given a model that was fit using Markov Chain Monte Carlo, it is
straightforward to obtain a sample of the posterior for the potential outcomes
and for the difference in potential outcomes between the counterfactual states
of exposed and unexposed.444Using built-in functions from the package brms to
obtain a posterior sample of the linear predictor for each potential outcome,
this amounts to the elementary operations subtraction and averaging. As
previous MH drops out of the equation for the differences, this allows
graphing the expected domain-specific effect of HAS by age and calendar-year.
To visualise the expected change in MH by age and calendar-year within each
HAS group, we need to explicitly condition upon previous MH or average over
it. In this case study the difference between the two options is an additive
constant, however, this does not hold for non-linear models.

###  4.2 Results

We compare the performance of the models using 5-fold CV as we did for the
entry and exit models: we calculate the expected log posterior predictive
density out of sample and perform the cross-validation such that approximately
twenty percent of each age-calendar-year and HAS stratum is included in each
fold. We did not stratify on previous MH when allocating the folds.

Table 4.2 summarises the predictive performance. The models can be grouped two
ways. First, we compare the results between complete pooling, partial pooling
and tensor splines. As expected, partial pooling and tensor splines perform
better than complete pooling, with tensor splines providing the best results.
This observation is true within the set of models without HAS, for the models
including a main effect for HAS and also for the models allowing an
interaction between age-period and HAS.

Second, comparing the three models with partial pooling, allowing variation in
the effect of HAS by age and calendar-year performed better than the model
assuming no modification and the model assuming no effect of HAS. The same is
true for tensor splines and the completely pooled estimate also performs worse
than the model that includes a main term for HAS but no effects for age or
calendar-year. For the latter model, domain-specific estimates of expected
change in Mental Health, marginalised over HAS, do vary across the domains as
HAS prevalence varies across the domains. Such marginal estimates can be
considered a _synthetic_ estimator for the effect of age and time on Mental
Health with HAS as an auxiliary variable if the effect of HAS is not of
interest in itself.

Direct comparisons between models that differ both in modelling effects of HAS
and in pooling/smoothing is non-trivial. For instance, complete pooling with a
main effect of HAS does not perform better than tensor splines without a main
effect of HAS. This does not imply that heterogeneity over the domains is more
important than the effect of HAS. Let us presume that the prevalence of HAS
varies across the domains and that HAS has an effect on Mental Health but
there is no direct effect age or calendar-year on Mental Health. Marginalised
over HAS, there will be heterogeneity in the domain-specific expected changes
in Mental Health and this will be ‘detected’ by a tensor spline model that
does not include a term for HAS. The same is true if we assume both an effect
of HAS and a direct effect of the domains. In essence, variation in the
prevalence of HAS over the domains can be absorbed into the heterogeneity of
the Mental Health changes over these domains. Conversely, comparison between
the tensor splines only-model and a model including only HAS, does not provide
a clear answer whether the observed heterogeneity was purely due to
heterogeneity in HAS prevalence.

To directly assess whether HAS has an effect, the posterior estimates for its
main term must be considered. Table 4 shows that results for the main terms
are similar between the completely pooled and the tensor spline model with
effect modification. Both models indicate an average loss of two to three
points in Mental Health each year a person suffers HAS, however, the negative
coefficient for previous Mental Health implies a regression to the mean:
people with lower Mental Health scores improve over time, whereas people with
near-perfect scores tend to score worse the next year. As such, continuous
exposure to HAS results in a shift in equilibrium by -7.24 points: expected
Mental Health is changed by -2.39 in the first year, an extra -2.39
-2.39*-0.33 in the second year … until the negative pressure from HAS equals
the pull towards the mean. This occurs when $\beta_{2}+\beta_{1}\Delta
Y_{it-1}=0$ or $\Delta Y_{it-1}=-\beta_{2}/\beta_{1}$. Using the estimates
from the completely pooled model, this results in a shift of
$-7.24=-(-2.39)/(-0.33)$. Compared to the average and maximum Mental Health
score, 75 respectively 100, this is a clinically important effect.

Table 3:  Comparison of out of sample predictive performance of estimates of age-calendar-year-specific changes in Mental Health, using 5-fold Cross-validation stratified by age-calendar-year and HAS. Table 4: Estimated effect of HAS and regression to the mean.  | posterior mean [95% CI]
---|---
Variable | Complete pooling | Tensor spline with modification
$\hat{\beta_{1}}$ ($Y_{it-1}$) | -0.33 [-0.34; -0.33] | -0.33 [-0.34; -0.33]
$\hat{\beta_{2}}$ (HAS) | -2.39 [-2.70; -2.08] | -2.85 [-3.73; -2.03]

These results indicate that there is an effect of HAS and that there is
heterogeneity of the effect of HAS over age and time. The top panels in
Figures 3.2 and 3.2 show the estimated expected changes in Mental Health for
people with an initial score of 75 for people in HAS and out of HAS. When
tensor splines are used to calculate these counterfactual effects, patterns
are apparent in both counterfactual exposures. Similarly, patterns are
revealed in the bottom panel for the tensor spline-based estimates but not for
partial pooling. These panels describe the pattern in the average effect of
HAS as modified by age and calendar-year. The range of expectation of the
estimates is -3.8 to -1.1, providing a difference in effect size of 2.7
between the least and most affected. This estimated level of heterogeneity is
on par with the size of the average effect. The strongest gradient is in age,
with older people being more affected by HAS than younger people. In addition,
for younger people the effect is decreasing in magnitude since 2009. However,
looking at temporal patterns for younger HAS and younger HAS-free people, it
appears the change in Mental Health is relatively stable in the HAS group but
worsening since 2009 in the HAS-free group.

Figure 5: Predicted change in mental health modelled using Tensor splines,
allowing for a modification of the effect of HAS by age and time. Top left:
expected change in mental health for people the HAS-free group with a previous
MH score of 75. Top right: expected change in mental health for people the HAS
group with a previous MH score of 75. Bottom: average effect of HAS on change
in mental health.

Figure 6: Predicted change in mental health modelled using partial pooling,
allowing for a modification of the effect of HAS by age and time. Top left:
expected change in mental health for people the HAS-free group with a previous
MH score of 75. Top right: expected change in mental health for people the HAS
group with a previous MH score of 75. Bottom: average effect of HAS on change
in mental health.

##  5 Discussion

We have illustrated that SDE methods are useful to visualise age- and time-
specific data and that SDE estimates have better accuracy than direct
estimation and complete pooling. In our examples, tensor splines provided the
most accurate estimates. _By design_ these estimates vary smoothly over age
and calendar time; it should not be a surprise that assuming smoothness
results in smooth estimates. Our preference for these estimates is based both
on a prior belief that such patterns are plausible, the improved out of sample
predictive performance and the way they facilitate hypothesis generation.
Partially pooled estimates also outperformed complete pooling in 5-fold cross-
validation and we prefer them over completely pooled estimates, however,
partial pooling did not reveal patterns and the improvement was substantially
lower than for tensor splines.

We offer an intuitive explanation why smoothing performs well when patterns
exist: a smooth estimate for domain $(a,t)$ ‘borrows strength’ only from those
domains that are quite similar, e.g. $(a,t-1)$, and incur negligible bias if
indeed $\theta_{at}\approx\theta_{at-1}$ and
$\theta_{at}\approx\theta_{a-1t}$. In contrast, partial pooling ‘borrows
strength’ from all other domains. If the heterogeneity between the domains
contains patterns (gradients, clusters, autocorrelations) instead of white
noise, then partially pooled estimates (and completely pooled estimates) are
biased: $E[\hat{\theta}_{at}]\neq\theta_{at}$ but it is simultaneously
possible for the average over the domain-specific estimates to be unbiased:
$E[\sum_{at}\hat{\theta}_{at}]=\sum_{at}\theta_{at}$ as
$|{}E[\sum_{at}\hat{\theta}_{at}-\theta_{at}]|{}=|{}\sum_{at}E[\hat{\theta}_{at}-\theta_{at}]|{}\leq\sum_{at}|{}E[\hat{\theta}_{at}-\theta_{at}]|{}$.

Clearly, the total absolute bias of partially or completely pooled estimates
increases with the strength of the underlying patterns. Therefore, we expect
similar empirical results whereever moderate or strong gradients exist. This
can be examined post-hoc by looking at the amount of variation over the
domain-specific estimates and by comparing models using cross-validation;
comparing within-sample predictive performance is not recommended as this will
favour the most flexible model. Cross-validation has previously been used to
compare the predictive performance of direct estimators, partial pooling and
complete pooling. Wang and Gelman, (2014) observed that partial pooling has
lower prediction error than complete pooling regardless of sample size, is
better than the direct estimator for small samples and performs similar for
large sample sizes. Although they recommend cross-validation over in-sample
training loss, they remark that ‘the improvement of the multilevel model as
given by cross-validation is surprisingly tiny, almost negligible to
unsuspecting eyes’. In our applications, the improvement was clear. Both the
amount of heterogeneity and sample size played a role in this. Although we
have not assessed this directly, it is intuitive that the difference between
partial pooling and tensor splines depends on whether the heterogeneity across
domains is dominated by white noise or by structure. A three-way comparison
between complete pooling, partial pooling and tensor splines is recommended
but further modification could be considered, e.g. if a discontinuity is
suspected around the Global Financial Crisis in 2008, this directly opposes
the assumption that 2008 is equally similar to 2007 as to 2009; the current
approach could be extended by borrowing from interrupted time series
approaches.

The analyses presented in this paper did not adjust for sex, educational
status or region. Incorporating more information through auxiliary data into
the model may improve the reliability of the estimates, and this is a standard
procedure in SDE. For example, if we are not interested in the effect of sex
per se but suspect that sex has a strong influence on the rate of exit, we
could define a _synthetic_ estimator for the age-period domains that uses
information on sex:
$\hat{\theta}_{at}^{s}=w_{atM}\hat{\theta}_{atM}+w_{atF}\hat{\theta}_{atF}$.
Although this estimator is based on smaller domains, it can be efficient,
particularly if the effect of sex is relatively strong, additive on the linear
predictor scale and if the ratio of men to women varied over age-calendar-year
in the sample or in the target population. A similar model-based approach is
_Multilevel Regression and Poststratification_ (MRP), which has recently
become popular as a method to obtain estimates of population groups when the
frequency of subgroups within the sample may differ from the target
population(s). In this technique partial pooling is used to obtain small sub-
domain estimates and subsequently these estimates are weighted to better
represent their contribution to the target population in the domains of
interest (Gelman and Little,, 1997; Downes and Carlin,, 2019). We did not use
post-stratification in this manuscript but the same heuristic arguments apply
to consider smoothing as an alternative to partial pooling for MRP.

APC models

We proposed that the transition probabilities into and out of HAS are more
similar between people of similar ages in the same or adjacent calendar-years,
as opposed to people with many years of age or time between them.
Equivalently, these probabilities are more similar between people of similar
birth-years. Our choice to focus on Age and calendar-year (Period) was
motivated by the structure of the data, which included people of working ages
between 2001 and 2016. Although this covers multiple birth-years (Cohort), we
did not observe these cohorts as part of the working population for equal
periods of time. Nevertheless, Age, Period and Cohort are linked: $A=P-C$. For
the purposes of forecasting and attribution of effects, many models have been
proposed to disentangle APC-effects. Some of these models impose an additive
structure or that only two out of the APC-trio have an effect. Tensor splines
are not additive, however, so we decided to ‘ignore’ the contribution of
Cohort effects; we argue that the use of flexible smoothing splines foregoes
the need to include explicitly all three terms as long as additivity is not
forced upon the estimates. Consider a model that is quadratic and additive in
A, P and C. We can rewrite such a model as one that is quadratic in A and P:

$\displaystyle y$
$\displaystyle=\alpha+\beta_{1}A+\beta_{2}A^{2}+\gamma_{1}P+\gamma_{2}P^{2}+\delta_{1}C+\delta_{2}C^{2}+\epsilon$
$\displaystyle=\alpha+\beta_{1}A+\beta_{2}A^{2}+\gamma_{1}P+\gamma_{2}P^{2}+\delta_{1}\left(P-A\right)+\delta_{2}\left(P^{2}-2AP+A^{2}\right)+\epsilon$
$\displaystyle=\alpha+\left(\beta_{1}-\delta_{1}\right)A+\left(\gamma_{1}+\delta_{1}\right)P+\left(\beta_{2}+\delta_{2}\right)A^{2}+\left(\gamma_{2}+\delta_{2}\right)P^{2}-2\delta_{2}AP+\epsilon$

Hence, removing C induces an ‘interaction’ between A and P. This does not mean
that an interaction between Age and Period proves that a Cohort effect must
exist, unless such interaction can be excluded on other grounds. Similar
results hold for cubic and higher order polynomials. Therefore, one of A, P
and C can safely be ignored when using tensor splines to generate domain-
specific estimates.555Attempts to fit additive models, $s_{1}(A)+s_{2}(P)$,
were abandoned as the Markov Chain Monte Carlo procedure did not seem to
converge. All three need to be considered when interpreting the obtained
patterns.

Forecasting and prediction

Our main motivation for using SDE was to reveal patterns and aid in the
generation of hypotheses. Technically, it is possible to obtain forecasts from
the models using tensor splines. Cross-validating the predictive performance
by leaving out entire calendar-years, demonstrates that tensor splines perform
better than partial pooling and complete pooling. This out of sample result
may seem promising, however, beyond the one-year-ahead forecast the
uncertainty increases rapidly as demonstrated in Figure 12 in appendix.
Effectively, the forecast is similar to one obtained by linearly extrapolating
the temporal trends seen within each age group but placing most weights on the
most recent 20% of the data. Our smoothing (and partial pooling) approach was
not designed for the purpose of forecasting and is unlikely to perform better
than a method explicitly tailored to this task. This is consistent with Booth
and Tickle, (2008): ‘The APC model has been usefully applied in describing the
past, but has been considered less useful in forecasting.’ Nevertheless,
specialised models such as the Lee-Carter model (Lee and Carter,, 1992) can
benefit from adopting smoothing (Currie,, 2013; De Jong and Tickle,, 2006).

Fortunately, SDE is suitable for prediction within-sample. While this is not
relevant for the estimation of entry or exit rates, prediction is an essential
step in the estimation of the expected change in Mental Health. Our estimation
of the domain-specific effect of HAS on Mental Health is a form of semi-
parametric g-computation (Robins,, 1986; Snowden et al.,, 2011): we predicted
the (counterfactual) outcomes for each person using their real age and period
but setting previous Mental Health to 75 and setting HAS first to one and then
to zero. Subtracting the predicted outcomes provides an estimate of the
domain-specific effect of HAS. With this approach we can use SDE methods to
provide accurate domain-specific estimates of _associations_ and to _describe_
the heterogeneity of these estimates. By themselves, such estimates are not
causal estimates. A _causal interpretation_ depends on more assumptions
relating to (unmeasured) confounding, manipulability, absence of spill-over
effects etc.

Confounding, exchangeability and heterogeneity

Since the incidence and escape from HAS depends on age and time, the
prevalence of HAS depends on age and time but other factors could vary too.
For instance, educational levels can vary over time; it is possible that
people today have higher educational levels, on average, than people several
decades ago. It is possible that this is true both within the HAS-free and the
HAS subgroups, and, it is possible that the average educational level has
increased more in the HAS-free group than in the HAS group. If we further
assume that education has an effect on Mental Health, then this by itself is
enough to provide heterogeneity in expected change in Mental Health both over
age, calendar-year and HAS status. No direct effect of HAS is needed, and, the
estimated effects and heterogeneity in estimated effects may be entirely due
to a lack of exchangeability when they are caused by (unmeasured) changes in
educational levels.

Alternative mechanisms can, of course, result in similar patterns to those
described above. For instance, rapid changes in housing costs may have altered
strategies regarding home ownership. Postponing purchase and taking on a
smaller mortgage could be sound economic advise, yet work contrary to prior
expectations and lead to increased stress in those who strive to remain HAS-
free. This mechanism leads to variability in changes in Mental Health but does
not require demographic change or change in observed entry and exit rates but
relies on changes in market prices and behaviour. Indeed, we can imagine the
change in purchase strategy to perfectly counteract the change in pricing, so
that the same people would remain HAS-free regardless of time. These people
would possess different assets with, possibly, different consequences
regarding their satisfaction and Mental Health. Whether we have observed the
purchasing strategies or not, this example mechanism is different from the
education example. It does not suppose change in the distribution of an
individual level covariate but instead supposes a change in market conditions
that affects everyone. Instead of a confounding variable that acts directly on
Mental Health, the market influences Mental Health through the purchase
strategy as a mediator.

For both mechanisms, heterogeneity in Mental Health over age and time is
explained as heterogeneity in covariates (education) or context (house prices)
over age and time. Whenever the ultimate goal of the research is to design
interventions, attributing heterogeneity to observable - and hopefully
manipulable - covariates is ideal. The residual heterogeneity over the domains
is both a nuisance and an opportunity to further explain the observed
patterns. Tables 4.2 and 4 demonstrate that models ignoring the heterogeneity
have lower performance but provide narrower credible intervals for the effect
of HAS. This increased precision is likely spurious; it is well known that
under-estimating the intraclass correlation in multi-level models leads to
overestimates of precision and hence it has been advocated to report estimates
for the effect of interest and estimates of heterogeneity side by side (Merlo
et al.,, 2009, 2018).

Semi-parametric versus non-parametric smoothing

In this paper, we focussed on the technical aspects of obtaining precise
estimates from a small number of observations. We modelled change in Mental
Health as it is plausible that the change in Mental Health is independent of
previous changes in Mental Health, whereas Mental Health itself is highly
auto-correlated and likely to be a Markov Process. However, the Short-form 36
instrument has a limited precision and may be affected by time-invariant
unmeasured characteristics as illustrated in Figure 7. While adjusting for
previous Mental Health is conceptually sound, adjusting the previous SF36
score is not sufficient to remove potential unmeasured confounders $U$
(Glymour et al.,, 2005). It may seem that the ‘fixed effects model’ is more
appropriate as it explicitly aims to account for time-invariant confounders
from the Mental Health measures, however, the standard fixed effects model
assumes that conditional upon exposures, changes in outcomes are white noise;
a ‘zero order’ Markov process instead of a first order process. This encodes
that losses in Mental Health are incurred immediately instead of accumulated
over time, and, that they are reversed (forgotten) immediately when leaving
HAS. The standard fixed effects model does not allow feedback loops. We refer
to Ding and Li, (2019) and Imai and Kim, (2019) for further comparison between
fixed effects (difference in difference) and lagged effects models.

$\mathrm{MentalHealth}_{t}-\mathrm{MentalHealth}_{t-1}$$\mathrm{MentalHealth}_{t}$$\mathrm{SF36}_{t-1}$$\mathrm{HAS}_{t}$$\mathrm{SF36}_{t}-\mathrm{SF36}_{t-1}$$\mathrm{MentalHealth}_{t-1}$$U$$\mathrm{Age-
Period-Cohort}_{t}$$\mathrm{SF36}_{t}$ Figure 7: Direct Acyclic Graph for the
evolution of Mental Health influenced by Housing Affordability Stress and an
Age-Period-Cohort effect wherein SF36 is an imperfect measurement of Mental
Health subject to time-invariant unmeasured personal characteristics.

We assumed a first order Markov Process and our model was additive in previous
Mental Health. This allowed us to set previous Mental Health to 75 when
predicting counterfactual outcomes as it factors out when subtracting the
prediction for a person with HAS from the prediction for the same person
without HAS. We could also have retained the observed previous Mental Health
score and averaged the differences within each domain to provide an estimate
of the marginal effect of HAS. As we’ve restricted the models to be additive
in previous Mental Health, both estimates are identical but more complex
scenarios can be imagined. For example, we could predict the outcomes if the
exposure had been changed in 2001 and use the predicted outcomes as input to
predict the subsequent exposure in 2002 etc. Indeed, the DAG in Figure 4 could
be extended to allow Mental Health as an input into the transition risks of
subsequent HAS. This type of DAG was assumed by Bentley et al., (2018), who
used marginal structural models to examine the effect of social housing versus
private renting or ownership on Mental Health but did not target APC effects.
Another line of research could be to investigate whether HAS is a mediator of
the influence of ‘zeitgeist’ on Mental Health and whether age, cohort and
period operate mainly trough HAS or through other potential mediators.

In our application, smoothing provided a relatively small loss in stated
precision compared to complete pooling and direct estimation, smaller than
what could be expected from dividing the data into non-overlapping aggregates,
e.g. 5-year age groups. As such, the smooth estimates allowed for a
visualisation that revealed patterns and the existence of heterogeneity of
effects. This approach is semi-parametric and can be compared with recent
developments in non-parametric methods such as causal forests (Wager and
Athey,, 2018), which incorporate recursive partitioning into ‘honest’ trees
(Athey and Imbens,, 2016). Honest trees and causal forests are modifications
of classification and regression trees, respectively random forests. These
algorithms (repeatedly) partition the data into nodes that are homogeneous
with respect to the treatment, the outcome or the treatment effect. By design
each node is a convex region and can be seen as a small domain with the
boundaries automatically chosen by the algorithm. In order to obtain unbiased
(‘honest’) estimates of the effect, data must be split into a training set to
grow the tree and a separate estimation dataset to estimate the treatment
effect, which is different from using cross-validation to evaluate the
accuracy of the procedure or to tune parameters to optimise the procedure.

Causal forests have been used to estimate heterogeneity in treatment effects
in the presence of a large number of covariates. In principle, the algorithm
could also be applied to a two-dimensional problem and the resulting effect
estimates could be visualised, potentially revealing trends or clusters
similar to how tensor smoothing may show them. Causal forests do not use a
penalty on wiggle but can be compared to weighted estimators. For each tree
the domain $(a,t)$ must be included in the node that is used to estimate the
treatment effect for this domain and other domains may contribute. The
probability for other domains to be included depends on the structure of the
data but neighbouring domains are more likely to contribute because the nodes
must be convex regions. As a forest is an unweighted ensemble of trees, the
final estimate for the effect in domain $(a,t)$ will be akin to a weighted
estimate with most weight placed on $(a,t)$ and its neighbours; similar to a
kernel that is adapted to both the data and the target domain. If the data
generating process is a smooth function of $a$ and $t$, it is reasonable to
expect that both our tensor splines and the causal forests will reveal
patterns. However, causal forests are fully non-parametric, which foregoes the
potential gains in efficiency by including knowledge or assumptions on
structure such as the existence of conditional heterogeneity.

##  6 Conclusion

We have illustrated that tensor smoothing splines can be used to identify
trends and hot spots in historical Age-Period-Cohort data. As a technique for
small domain estimation it excels when such patterns exist, which is a priori
not known. Cross-validation, stratified by domain, can be used to compare its
performance with other estimators and provide evidence for the existence of
this type of heterogeneity.

Our second case study shows that effect modification by APC on a health effect
of interest can be examined and that the magnitude of effect modification can
be similar to the magnitude of the main effect. This information may help
appreciate differences between studies: researchers often statistically
control for age assuming that the effect of age is an additive confounder
(nuisance) to remove from the estimate of interest. However, our results show
that age may be a modifier of the effect and this may be important when
comparing effect size estimates reported in different studies.

The combination of semi-parametric techniques suitable for small domain
estimation with g-computation enables causal inference in the presence of
strong effect modification and is of interest to develop targetted
interventions. Such applications require support from a suitable Directed
Acyclic Graph; by itself SDE remains useful as an aide to generate hypotheses.

## Code

The source code to generate tables and figures is available on
https://bitbucket.org/Koen_Simons/sde_for_apc/src/master/

## References

  * Athey and Imbens, (2016) Athey, S. and Imbens, G. (2016).  Recursive partitioning for heterogeneous causal effects.  Proceedings of the National Academy of Sciences, 113(27):7353–7360.
  * Bentley et al., (2011) Bentley, R., Baker, E., Mason, K., Subramanian, S., and Kavanagh, A. M. (2011).  Association between housing affordability and mental health: a longitudinal analysis of a nationally representative household survey in Australia.  American journal of epidemiology, 174(7):753–760.
  * Bentley et al., (2018) Bentley, R., Baker, E., Simons, K., Simpson, J. A., and Blakely, T. (2018).  The impact of social housing on mental health: longitudinal analyses using marginal structural models and machine learning-generated weights.  International journal of epidemiology, 47(5):1414–1422.
  * Bentley et al., (2016) Bentley, R. J., Pevalin, D., Baker, E., Mason, K., Reeves, A., and Beer, A. (2016).  Housing affordability, tenure and mental health in Australia and the United Kingdom: a comparative panel analysis.  Housing studies, 31(2):208–222.
  * Besag et al., (1991) Besag, J., York, J., and Mollié, A. (1991).  Bayesian image restoration, with two applications in spatial statistics.  Annals of the institute of statistical mathematics, 43(1):1–20.
  * Booth and Tickle, (2008) Booth, H. and Tickle, L. (2008).  Mortality modelling and forecasting: A review of methods.  Annals of actuarial science, 3(1-2):3–43.
  * Borrowman et al., (2017) Borrowman, L., Kazakevitch, G., and Frost, L. (2017).  How long do households remain in housing affordability stress?  Housing studies, 32(7):869–886.
  * Bürkner, (2017) Bürkner, P.-C. (2017).  brms: An R package for Bayesian multilevel models using Stan.  Journal of Statistical Software, 80(1):1–28.
  * Chambers et al., (2009) Chambers, R., Tzavidis, N., and Salvati, N. (2009).  Borrowing strength over space in small area estimation: Comparing parametric, semi-parametric and non-parametric random effects and M-quantile small area models.
  * Cigdem and Whelan, (2017) Cigdem, M. and Whelan, S. (2017).  Intergenerational transfers and housing tenure - Australian evidence.  International Journal of Housing Policy, 17(2):227–248.
  * Currie, (2013) Currie, I. D. (2013).  Smoothing constrained generalized linear models with an application to the Lee-Carter model.  Statistical Modelling, 13(1):69–93.
  * De Jong and Tickle, (2006) De Jong, P. and Tickle, L. (2006).  Extending Lee-Carter mortality forecasting.  Mathematical Population Studies, 13(1):1–18.
  * Ding and Li, (2019) Ding, P. and Li, F. (2019).  A bracketing relationship between difference-in-differences and lagged-dependent-variable adjustment.  arXiv preprint arXiv:1903.06286.
  * Downes and Carlin, (2019) Downes, M. and Carlin, J. B. (2019).  Multilevel regression and poststratification as a modeling approach for estimating population quantities in large population health studies: A simulation study.  Biometrical Journal.
  * Eilers and Marx, (1996) Eilers, P. H. and Marx, B. D. (1996).  Flexible smoothing with B-splines and penalties.  Statistical science, pages 89–102.
  * Gelman and Little, (1997) Gelman, A. and Little, T. C. (1997).  Poststratification into many categories using hierarchical logistic regression.
  * Glymour et al., (2005) Glymour, M. M., Weuve, J., Berkman, L. F., Kawachi, I., and Robins, J. M. (2005).  When is baseline adjustment useful in analyses of change? An example with education and cognitive change.  American journal of epidemiology, 162(3):267–278.
  * Hernan and Robins, (2010) Hernan, M. A. and Robins, J. M. (2010).  Causal inference.  CRC Boca Raton, FL.
  * Holland, (1986) Holland, P. W. (1986).  Statistics and causal inference.  Journal of the American statistical Association, 81(396):945–960.
  * Imai and Kim, (2019) Imai, K. and Kim, I. S. (2019).  When should we use unit fixed effects regression models for causal inference with longitudinal data?  American Journal of Political Science, 63(2):467–490.
  * Lee and Carter, (1992) Lee, R. D. and Carter, L. R. (1992).  Modeling and forecasting US mortality.  Journal of the American statistical association, 87(419):659–671.
  * Marí-Dell’Olmo et al., (2017) Marí-Dell’Olmo, M., Novoa, A. M., Camprubí, L., Peralta, A., Vásquez-Vera, H., Bosch, J., Amat, J., Díaz, F., Palència, L., Mehdipanah, R., et al. (2017).  Housing policies and health inequalities.  International Journal of Health Services, 47(2):207–232.
  * Mason et al., (2013) Mason, K. E., Baker, E., Blakely, T., and Bentley, R. J. (2013).  Housing affordability and mental health: does the relationship differ for renters and home purchasers?  Social science & medicine, 94:91–97.
  * Merlo et al., (2009) Merlo, J., Ohlsson, H., Lynch, K. F., Chaix, B., and Subramanian, S. (2009).  Individual and collective bodies: using measures of variance and association in contextual epidemiology.  Journal of Epidemiology & Community Health, 63(12):1043–1048.
  * Merlo et al., (2018) Merlo, J., Wagner, P., Austin, P. C., Subramanian, S., and Leckie, G. (2018).  General and specific contextual effects in multilevel regression analyses and their paradoxical relationship: A conceptual tutorial.  SSM-population health, 5:33–37.
  * Nepal et al., (2010) Nepal, B., Tanton, R., and Harding, A. (2010).  Measuring housing stress: how much do definitions matter?  Urban Policy and Research, 28(2):211–224.
  * Opsomer et al., (2008) Opsomer, J. D., Claeskens, G., Ranalli, M. G., Kauermann, G., and Breidt, F. (2008).  Non-parametric small area estimation using penalized spline regression.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(1):265–286.
  * Pfeffermann et al., (2013) Pfeffermann, D. et al. (2013).  New important developments in small area estimation.  Statistical Science, 28(1):40–68.
  * R Core Team, (2018) R Core Team (2018).  R: A Language and Environment for Statistical Computing.  R Foundation for Statistical Computing, Vienna, Austria.
  * Rahman et al., (2013) Rahman, A., Harding, A., Tanton, R., and Liu, S. (2013).  Simulating the characteristics of populations at the small area level: New validation techniques for a spatial microsimulation model in Australia.  Computational Statistics & Data Analysis, 57(1):149–165.
  * Rao and Molina, (2015) Rao, J. N. and Molina, I. (2015).  Small area estimation.  John Wiley & Sons.
  * Robins, (1986) Robins, J. (1986).  A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect.  Mathematical modelling, 7(9-12):1393–1512.
  * Rubin, (1974) Rubin, D. B. (1974).  Estimating causal effects of treatments in randomized and nonrandomized studies.  Journal of educational Psychology, 66(5):688.
  * Singh et al., (2019) Singh, A., Aitken, Z., Baker, E., and Bentley, R. (2019).  Do financial hardship and social support mediate the effect of unaffordable housing on mental health?  Social psychiatry and psychiatric epidemiology, pages 1–9.
  * Snowden et al., (2011) Snowden, J. M., Rose, S., and Mortimer, K. M. (2011).  Implementation of G-computation on a simulated data set: demonstration of a causal inference technique.  American Journal of Epidemiology, 173(7):731–738.
  * Taylor and Tyers, (2017) Taylor, G. and Tyers, R. (2017).  Secular stagnation: Determinants and consequences for Australia.  Economic Record, 93(303):615–650.
  * Trewin, (2002) Trewin, D. (2002).  Measuring Australia’s Progress.  Australian Bureau of Statistics.
  * Wager and Athey, (2018) Wager, S. and Athey, S. (2018).  Estimation and inference of heterogeneous treatment effects using random forests.  Journal of the American Statistical Association, 113(523):1228–1242.
  * Wang and Gelman, (2014) Wang, W. and Gelman, A. (2014).  Difficulty of selecting among multilevel models using predictive accuracy.  Statistics at its Interface, 7(1):1–88.
  * Wood et al., (2013) Wood, S. N., Scheipl, F., and Faraway, J. J. (2013).  Straightforward intermediate rank tensor product smoothing in mixed models.  Statistics and Computing, 23(3):341–360.

## Additional Figures

Figure 8: Sample sizes. Domain-specific estimates using longitudinal data from
the HILDA study must be obtained from relatively small sample sizes. Left:
number of people without Housing Affordability Stress. Right: number of people
in HAS. In both panels the sample size increases in 2012 due to a top-up
sample. Diagonal lines appear due to people aging and remaining in the same
group if they do not drop out of the study and do not enter into HAS or exit
from HAS.

Figure 9: SF36 Mental Health scores. Top left: kernel density plot of the raw
Mental Health scores, which have a skewed distribution. Top right: kernel
density plot of the first differences of the Mental Health scores, which have
a symmetric distribution. Bottom: First differences in Mental Health scores
appear to be more variable for low previous Mental Health scores (conditional
heterogeneity) and their means are negatively correlated with the previous
score (regression to the mean). Figure 10: Posterior estimates for the change
in Mental Health as a function of previous Mental Health for age 50 in 2002,
estimated using tensor spline smoothing and allowing for effect modification
by age and period. Left: HAS-free people, Right: people in HAS. These graphs
illustrate that the model allows both conditional heterogeneity (80% pointwise
CI is widest for lowest previous Mental Health scores), regression to the mean
(negative correlation between change and previous score) and that the effect
of HAS is a shift of the equilibrium equal to -7.24 points (=-2.85/0.33).

Figure 11: Estimated probabilities of entering into Housing Affordability
Stress by age and period. Top: Heatmaps of posterior means for the probability
of exit by APC. Bottom: Distribution of the posterior for selected one-year
age-groups. Left: map obtained from a partial pooling model with iid error.
Right: map obtained using a tensor spline basis. Figure 12: Forecasted
probabilities of exit from Housing Affordability Stress by age. Forecasts
obtained using tensor smoothing splines on age by period with five degrees of
freedom on data spanning 2001 to 2016. Although a forecast can be obtained, it
is similar to a linear extrapolation based on the most recent 20% of the data
and ignores potential cohort effects. Even with these limitations, the
estimated precision decreases rapidly with the forecast horizon.


# Design of a Canted-cosine-theta orbit corrector for the High Luminosity LHC

K. Pepitone, G. Kirby, R. Ruber, A. Ahl, M. Canale, I. Dugic, L. Gentini,
M. Johansson, G. Karlsson, J. Kovacikova, J. Lindström, A. Olsson, M. Olvegård

###### Abstract

The High Luminosity LHC requires dipole orbit correctors grouped in double
aperture magnet assemblies. They provide a field of 3.1 T at 100 A in an
aperture of 70 mm. The current standard design is a classical cosine-theta
layout made with ribbon cable. However, the electric insulation of the ribbon
cable is not radiation-resistant enough to withstand the radiation load
expected in the coming years of LHC operation. A new design, based on a
radiation-resistant cable with polyimide insulator, that can replace the
existing orbit correctors at their end-of-life, is needed. The challenge is to
design a magnet that fits directly into the existing positions and that can
operate with the same busbars, passive quench protection, and power supplies
as existing magnets. We propose a self-protected canted-cosine-theta (CCT)
design. We take the opportunity to explore new concepts for the CCT design to
produce a cost-effective and high-quality design with a more sustainable use
of resources. The new orbit corrector design meets high requirements on the
field quality while keeping within the same mechanical volume and maximum
excitation current.

A collaboration of Swedish universities and Swedish industry has been formed
for the development and production of a prototype magnet following a
concurrent engineering (CE) methodology to reduce the time needed to produce a
functional CCT magnet. The magnet has a 1 m long CCT dipole layout consisting
of two coils. The superconductor is a commercially available 0.33 mm wire with
polyimide insulation in a 6-around-1 cable. The channels in the coil formers,
that determine the CCT layout, allow for $2\times 5$ cable layers. A total of
70 windings makes that the coil current can be kept below 100 A. We will
present the detailed design and preliminary quench simulations.

###### Index Terms:

Canted-cosine-theta, accelerator magnet, quench protection.

##  I Introduction

The existing orbit corrector magnets in the LHC, called MCBC and MCBY [1], are
slated for replacement because of the radiation damage caused by the proton
beams in the accelerator to the ribbon cables [2]. The CCT (Canted-Cosine-
Theta) architecture was chosen for the new designs [3]. Originally developed
in the 1960s, the CCT design is based on the superposition of two solenoids
tilted in opposite directions with respect to the bore axis. The choice of a
radiation resistant polyimide insulated cable will give the correctors a
longer life. CCT magnets offer advantages such as simplified manufacturing
process and require less drawings than standard magnets.

In this paper, we first present, Sec. II, the magnet parameters of the
currently installed magnets, which will impose the conditions for the new
design. Sec. III focuses on the field simulations using COMSOL [4] and the
discharge behavior using ProteCCT [5, 6]. Finally, in Sec. IV, we discuss the
engineering drawings and the mechanical assembly of this magnet.

##  II Magnet design

The characteristics of the original MCBC and MCBY magnets are shown in Table
I. We intend to provide a one-to-one replacement magnet. The essential
parameters to achieve this goal are highlighted in the same table. Note that
the cold inner coil diameter is 70 mm, so the warm diameter is 70.3 mm using
the aluminum 4.0 mm/m contraction.

The power supplies used for the existing magnets can provide a maximum current
of 120 A [7]. Based on the first calculations we would need a current of
approximately 600 A and 10 conductors in each channel to obtain an integrated
field equal to 2.81 Tm for a magnetic length equal to 0.9 m. To comply with
the limitations posed by the power supply we increased the number of
conductors by a factor 7. After discussion with superconducting (SC) cable
manufacturers, we found the possibility to make cables composed of wires
having a 6-around-1 pattern. The 10 cables of 7 wires will be connected in
series and through this the current in each individual wire is reduced from
600 A to around 85 A. However, this new design is very demanding in terms of
cabling since a total of 70 joints must be realized on each coil.

TABLE I: MCBC and MCBY parameters. Parameters | MCBC | MCBY
---|---|---
Coil inner diameter | 56.0 mm | 70.0 mm
Magnetic length | 0.904 m | 0.899 m
Nominal field (at 1.9 K / 4.2 K) | 3.11 T / 2.33 T | 3.00 T / 2.50 T
Integrated field (at 1.9 K / 4.2 K) | 2.81 Tm / 2.10 Tm | 2.70 Tm / 2.25 Tm
Nominal current (at 1.9 K / 4.2 K) | 100 A / 74 A | 88 A / 72 A
Short sample current (at 1.9 K / 4.2 K) | 172 A / 127 A | 162 A / 120 A
Stored energy (at 1.9 K) | 14.2 kJ | 13.6 kJ
Self-inductance | 2.84 H | 5.27 H
DC resistance (RT) | 375 $\mathrm{\SIUnitSymbolOhm}$ | 501 $\mathrm{\SIUnitSymbolOhm}$
Overall length | 1.1 m | 1.1 m

##  III Simulations

The starting point of the simulations is based on the work already realized at
CERN [8]. The initial work was realized using MATLAB and the toolbox Magnetic
field of modulated double helical coils [9]. This toolbox solves, in a few
seconds, the Biot and Savart equation to compute the magnetic field. Based on
the initial results, more detailed simulations of the field and of the
discharge behavior were performed.

###  III-A Field simulations

The exact geometry of the magnet is defined in the COMSOL program including
the number of wires in the cable, dimensions of the cable, total current and
type of material for the formers and the yoke. The formers and the support are
in aluminum; the superconductors are in Niobium-Titanium (NbTi) and the yoke
in iron. All the parameters used for the simulations are given in Table II.

The current design is shown in Fig. 1 and the simulation results are presented
in Fig. 2 and Fig. 3.

Figure 1: 3D view (a) and 2D view (b) of the model defined in COMSOL. Figure 2: 2D view of the magnetic field simulated with COMSOL for a current of ${I}$ = 85 A. Figure 3: 2D view of the mechanical stress simulated with COMSOL for a current of ${I}$ = 85 A. TABLE II: Parameters used in the simulations. Parameters | Values
---|---
Coil inner diameter | 70.0 mm
Overall length | 1.1 m
Integrated field | 2.81 Tm
Magnetic length | 0.857 m
Nominal current (per wire) | 85 A
SC cable configuration | 6-around-1
SC cable diameter | 1.08 mm
SC cable length | 1310 m
SC wire (with / without insulation) | NbTi 0.36 mm / 0.33 mm
Cu to SC ratio | 1.3 $\pm$ 0.1 : 1
Critical current (at 4.2 K) | 137 A
Inner coil inner diameter | 75.7 mm
Inner coil outer diameter | 87.0 mm
Outer coil inner diameter | 94.1 mm
Outer coil outer diameter | 105.4 mm
Pitch | 5.8 mm
Number of turns | 147
Tilt angle | 30∘
Thickness of the formers spars | 2.85 mm
Wall thickness (min) | 0.3 mm
Channel design | 2 x 5 cables
Channel dimensions |  $2.31\times 5.65$ mm2

For a total current of 85 A per wire, a 6-around-1 cable and 10 cables in each
channel, a central field of 3.27 T is obtained. Considering that the magnetic
length is 0.857 m for an overall length below 1.1 m, we obtain an integrated
field of 2.8 Tm. The final integrated field value, which must meet the
specifications, can be reached by adjusting the current during the magnetic
measurements of the first prototype at warm and at cold.

The stress analysis extracted from the simulations show that the mechanical
stress on the formers is very low compared to what aluminum can withstand. We
intend to use aluminum of the type AL6082 with a T6 or T4 heat treatment which
has a stress yield of about 400 MPa at room temperature and 600 MPa when cold.
Our simulations show that the maximum stress should be less than 70 MPa.

###  III-B Quench simulations

The MCBC/MCBY magnets do not have an energy extraction system which means that
our prototype must be self-protected. A solution chosen to passively protect
is to use varistors [10]. Varistors would have to be installed along the
magnet for every 2500 m of cable. If the voltage increases above the varistor
threshold, the diode becomes conductive, and the energy of the magnet is
extracted through the varistor dump resistor and dissipated in the helium
bath. This option for quench protection is illustrated Fig. 4 \- b and details
on the varistors including a comparison with usual resistors can be found here
[10].

An alternative idea of using the central wire of the 6-around-1 cable as a
quench heater or quench detector is also being considered. Tests on the first
prototype will give an idea of the exact current needed to reach the 2.81 Tm.
Based on the results of these tests, we will decide if the central wire of
each cable should remain connected in series with the others or if it can be
used for one of the above mentioned applications.

Even if the magnet has to be self-protected, prototype will be tested with a
standard energy extraction system (see Fig. 4 \- a). Simulations have been
performed to identify the resistor size and to check hot spot temperature.

The simulations were performed using the ProteCCT [5, 6] software. ProteCCT
calculates the eddy currents in the formers and in the support. All heat
transfers are considered by the three-dimensional model, as well as heat
exchanges with the helium bath and with the liquid helium present between the
formers. The different couplings between the superconductors, the formers and
the quench propagation are also considered. A description of the software and
the physics implementation can be found here: [11].

The geometry, in terms of dimensions, number of conductors, excitation current
and all the other parameters cited in Table II were fully defined in the
toolbox. One of the problems of the implementation in ProteCCT is the geometry
of the 6-around-1 cable. The heat transfer between the aluminum formers and
the wires is not equal for the six twisted outer wires and the center wire. We
decided to simulate a cable which is comparable to the 6-around-1 cable. This
means that this cable has the same amount of superconductor, copper and
insulation material as the 6-around-1 cable. The simulation therefore only
involves 10 conductors and not 70 wires. With this approximation, we consider
that the heat transfer is always the same.

The results of the simulation presented in Fig. 5 are post-processed and
correspond to what a single wire would see for a load of 12.2
$\mathrm{\SIUnitSymbolOhm}$. With this discharge resistance, the maximum
voltage to ground is just below 1000 V, which is the limit for the equipment
connected to the magnet (acquisition system and power supply).

ProteCCT calculates the hot spot temperature at the time of discharge and not
at the time of the initial quench. The detection voltage and the validation
time will be set so that the discharge occurs at most 20 ms after the start of
the quench. Fig. 6 shows the hot spot temperature in the conductor as function
of a given current for a given time, extracted from a simple thermal model. If
we assume a constant current of 85 A during 20 ms, we obtain a hot spot
temperature increase of 47 K. If we add this value to the hot spot temperature
calculated by ProteCCT, shown in Fig. 5, we obtain a total temperature of
about 100 K, which is below 200 K, the empirical target hot spot temperature
for this type of magnet.

Figure 4: Standard energy extraction circuit (a), and example passive
extraction circuit using 2 varistors (b). Figure 5: Quench simulation for
${I}$ = 85 A and a dump resistor of 12.2 $\mathrm{\SIUnitSymbolOhm}$. Figure
6: Hot spot temperature in the conductor as a function of the current and the
time.

##  IV CAD model

CAD (Computer-aided design) design is ongoing and include a number of new
challenges compared to existing CCT magnets [8]. These challenges are in three
main areas: (a) the layer jump, where the 70 wires (10 cables) are brought
from the inner former to the outer former (b) joint boxes, where all 70
individual wires are connected in series, and (c) the quench protection as
described in Sec. III-B. Based on the first experience of Scanditronix on
assembling a CCT magnet there is need for improvement of the layer jump, so
that the cables and their location in the channel are properly visible during
assembly. We will use 3D printed parts of the CAD model to develop a good
design for this section. See Fig. 7 \- a for an example of the layer jump.

A multi-layer joint box will be designed and installed on the side of the
magnet. It will consist of three layers of G10 (high-pressure fiberglass
laminate) similar to what is shown in Fig. 7 \- b. Two of these are used for
the 70 splices and the last one for the instrumentation and voltage taps.
These elements will not be impregnated and will be easily accessible in order
to facilitate modifications or fixes, for example a short circuit, on the
splices and connections.

Figure 7: 3D view of the joint box (a) the layer jump (b).

##  V Conclusion

The CCT magnet being designed and manufactured by two Swedish universities and
Swedish industry is now fully simulated. The model will be ready for
manufacturing in the next few weeks and will be tested at FREIA laboratory at
Uppsala University. This magnet is based on the LHC MCBY and MCBC
specifications and has three major improvements compared to other CCT magnets:
improved layer jump, easily accessible joint boxes, and quench protection.

## Acknowledgement

This project has received funding from the European Regional Development Fund
and Region Kronoberg.

The authors would like to express their gratitude to Matthias Mentink, Martin
Novak, Emmanuele Ravaioli, and Mariusz Wozniak for their valuable help on the
quench protection studies and COMSOL simulations.

## References

  * [1] Oliver Sim Brüning, Paul Collier, P Lebrun, Stephen Myers, Ranko Ostojic, John Poole, and Paul Proudlock.  LHC Design Report.  CERN Yellow Reports: Monographs. CERN, Geneva, 2004.
  * [2] Alexandre Louzguiti, Francesco Cerutti, Pierre-Antoine Contat, Daniela Fagundes de Sousa, Glyn Kirby, Benoit Lepoittevin, Mattia Liberale, Jacky Mazet, Emmanuele Ravaioli, Thomas Sahner, Daniel Schoerling, Sandra Sequeira Tavares, Andrea Tsinganis, and Giorgio Vallone.  Design of Radiation Hard Spare Units for the Orbit Corrector Dipoles of LHC.  IEEE Transactions on Applied Superconductivity, 30(4):1–5, 2020\.
  * [3] D.I. Meyer and R. Flasck.  A new configuration for a dipole magnet for use in high energy physics applications.  Nuclear Instruments and Methods, 80(2):339–341, 1970.
  * [4] COMSOL Multiphysics® v. 5.5. www.comsol.com. COMSOL AB, Stockholm, Sweden.
  * [5] STEAM: Simulation of Transient Effects in Accelerator Magnets.  https://espace.cern.ch/steam/.  Accessed: 2021-11-17.
  * [6] Matthias Mentink, Jeroen van Nugteren, Franco Mangiarotti, Michal Duda, and Glyn Kirby.  Quench Behavior of the HL-LHC Twin Aperture Orbit Correctors.  IEEE Transactions on Applied Superconductivity, 28(3):1–6, 2018\.
  * [7] HL-LHC power supplies.  http://sy-dep-epc-lpc.web.cern.ch/converters/lhc120a-10v/general.stm.  Accessed: 2021-11-17.
  * [8] Glyn A. Kirby, Juho Rysti, L. Gentini, Jeroen Van Nugteren, Jaakko Murtomäki, Gijs De Rijk, Ezio Todesco, Karol Scibor, Jacky Mazet, and Juan Carlos Perez.  Hi-Lumi LHC Twin-Aperture Orbit Correctors Magnet System Optimisation.  IEEE Transactions on Applied Superconductivity, 27(4):1–5, 2017\.
  * [9] Loïc Quéval and Raimund Gottkehaskamp.  Analytical Field Calculation of Modulated Double Helical Coils.  IEEE Transactions on Applied Superconductivity, 25(6):1–7, 2015\.
  * [10] T Galvin, G. A. Kirby, D. Coll, F. J. Mangiarotti, W. Wu, X. Ou, Q. Xu, S. Zheng, T. Nakamoto, and K. Suzuki.  Superconducting Magnet Energy Extraction with a Varistor to Reduce Quench Voltages and Hot spots.  In these proceedings, pages TUE–PO1–705–09, 2021.
  * [11] Matthias Mentink.  STEAM-ProteCCT User Manual.  CERN, 7 2019.  https://edms.cern.ch/document/2160020.