20180820 - Excess Mortality in DisMod.tex

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\begin{document}
\title{A Framework for Unifying Excess Morality Inputs into DisMod-AT}
\author{Kyle Foreman \\ \href{mailto:[email protected]}{[email protected]}}
\date{\today}
\maketitle

\section{Overview}

We currently have several ways of incorporating excess mortality ($\chi$) and other-cause mortality ($\omega$) into DisMod:

\begin{enumerate}
  \item Derive $\chi$ and $\omega$ from relative risk of death, prevalence, and all-cause mortality
  \item Use standardized mortality ratios as a proxy for relative risk to derive $\chi$ and $\omega$
  \item Approximate $\chi$ and $\omega$ from cause-specific mortality, prevalence, and all-cause mortality
  \item Use some sort of incidence-based formula combined with the estimate of excess mortality from a global DisMod model to approximate a cause-specific mortality to excess mortality relationship (\emph{Editor's note: I think? Need further clarification.})
\end{enumerate}

This document aims to outline the first three methods and propose a way to standardize them while accounting for biases inherent to each method.

\subsection{The current model}

Start with the basic three-component model:

\begin{figure}[H]
    \label{fig:dismod1}
    \resizebox{150}{!}{%
        \begin{tikzpicture}[%
                >=stealth,
                thick,
                node distance=2cm,
                on grid,
                auto
            ]

            \node[rectangle, draw] (S) {$S$};
            \node[rectangle, draw] (C) [below=of S] {$C$};
            \node[rectangle, draw] (D) [right=of S] {$D$};

            \path[->]
                (S)     edge[bend right=30]     node    {$\iota$}   (C)
                        edge                    node    {$\omega$}  (D)
                (C)     edge[bend right=30]     node    {$\rho$}    (S)
                        edge[right]             node    {$\omega + \chi$}  (D)
                ;
        \end{tikzpicture}
    }
\end{figure}

where prevalence is

\begin{equation}
    p = \frac{C}{S + C}
\end{equation}

and the total population is

\begin{equation}
    N = S + C
\end{equation}

In order to make things easier, we can further decompose the mortality into with-condition ($D_w$) and without-condition ($D_o$) deaths:

\begin{figure}[H]
    \label{fig:dismod2}
    \resizebox{150}{!}{%
        \begin{tikzpicture}[%
                >=stealth,
                thick,
                node distance=2cm,
                on grid,
                auto
            ]

            \node[rectangle, draw] (S) {$S$};
            \node[rectangle, draw] (C) [below=of S] {$C$};
            \node[rectangle, draw] (Do) [right=of S] {$D_o$};
            \node[rectangle, draw] (Dw) [below=of Do] {$D_w$};

            \path[->]
                (S)     edge[bend right=30]     node    {$\iota$}   (C)
                        edge                    node    {$\omega$}  (Do)
                (C)     edge[bend right=30]     node    {$\rho$}    (S)
                        edge[right]             node    {$\omega$}  (Do)
                        edge                    node    {$\chi$}    (Dw)
                ;
        \end{tikzpicture}
    }
\end{figure}

\section{Methods for deriving excess mortality}

\subsection{Relative risk and prevalence}

The most direct method for deriving $\omega$ and $\chi$ is when estimates of relative risk ($RR$), prevalence ($p$), and all-cause mortality ($\mu_{T}$) are available. These are often available for common diseases such as diabetes, heart failure, etc.

Since $RR$ is the probability of dying when you have the disease divided by the probability of dying if you do not have the disease, it is simple to formulate in terms of $\omega$ and $\chi$:

\begin{equation}
    \label{eq:RR}
    \begin{gathered}
    RR = \frac{\textrm{with-condition\ death\ rate}}{\textrm{without-condition\ death\ rate}}
    \\
    \therefore
    \\
    RR = \frac{\omega + \chi}{\omega}
    \end{gathered}
\end{equation}

which we can then solve for $\chi$

\begin{equation}
    \label{eq:chi}
    \chi = \omega \times (RR - 1)
\end{equation}

and $\omega$:

\begin{equation}
    \label{eq:omega}
    \omega = \frac{\chi}{RR - 1}
\end{equation}

We can combine this with information on $\mu_{T}$ and $p$ to then get final estimates of excess mortality.

\begin{equation}
    \begin{gathered}
    \mu_{T} = \textrm{total\ death\ rate}
    \\
    or
    \\
    \mu_{T} = \frac{D_o + D_w}{\textrm{population}}
    \end{gathered}
\end{equation}

so $\mu_{T}$ is the prevalence-adjusted sum of with- and without-condition mortality:

\begin{equation}
    \begin{gathered}
    \mu_{T} = \big[(1 - p) \times \omega\big] + \big[p \times (\omega + \chi)\big]
    \\
    \therefore
    \\
    \mu_{T} = \omega + p \chi
    \end{gathered}
\end{equation}

or

\begin{equation}
    \label{eq:mu_t}
    \omega = \mu_{T} - p \chi
\end{equation}

Combining this with equations \ref{eq:omega} and \ref{eq:chi} above, we get:

\begin{equation}
    \begin{gathered}
    \omega = \mu_{T} \times \frac{1}{1 + p \times (RR-1)} 
    \\
    \&
    \\
    \chi = \mu_{T} \times \frac{RR - 1}{1 + p \times (RR - 1)}
    \end{gathered}
\end{equation}

\subsection{Standardized mortality ratios}

We can incorporate standardized mortality ratios ($SMR$) in much the same way as we utilized $RR$ above. We simply have to adjust for prevalence in a slightly different way.

$SMR$ is very similar to $RR$ (eq \ref{eq:RR}), but the denominator is the death rate in the general population (\emph{i.e.} among both those with and without the condition) instead of solely those that do not have the condition:

\begin{equation}
    \label{eq:SMR}
    SMR = \frac{\textrm{with-condition\ death\ rate}}{\textrm{general\ population\ death\ rate}}
\end{equation}

This metric is often easier to measure for acute or rare conditions which are not amenable to case control or cohort studies. If the prevalence of the condition is low, $SMR$ makes for a good approximation of $RR$, but it gets worse as the prevalence increases. 

\begin{equation}
    SMR = \frac{\omega + \chi}{\big[p \times (\omega + \chi)\big] + \big[(1-p) \times \omega\big]}
\end{equation}

which we can again solve for $\omega$:

\begin{equation}
    \omega = \frac{\chi \times (1 - SMR \times p)}{SMR - 1}
\end{equation}

and $\chi$:

\begin{equation}
    \chi = \frac{\omega \times (SMR - 1)}{1 - SMR \times p}
\end{equation}

Combining those with $\omega = \mu_T - p \chi$ from equation \ref{eq:mu_t} above, we get:

\begin{equation}
    \begin{gathered}
    \omega = \mu_{T} \times \bigg(1 - \frac{p \times (SMR - 1)}{1 - p} \bigg)
    \\
    \&
    \\
    \chi = \mu_{T} \times \frac{SMR - 1}{1 - p}
    \end{gathered}
\end{equation}

\subsection{Cause-specific mortality rates}

\subsubsection{Current method}

For some diseases, we currently treat GBD cause of death estimates as a measure of excess mortality, ignoring the fact that many conditions increase mortality attributable to other causes.

We have discussed in the past using cause-specific mortality to set the lower bound of excess mortality, but in practice that may be difficult to achieve.

Instead, we could estimate the relationship between cause-specific and excess mortality and incorporate that into our DisMod input data.

Currently, the excess mortality $\chi$ is approximated as a function of the 
with-cause mortality $\mu_{c}$ and prevalence $p$:

\begin{equation}
    \chi = \frac{\mu_{c}}{p}
\end{equation}

However, if $\mu_{c}$ is the GBD cause-specific mortality rate for cause $c$

\begin{equation}
    \mu_{c} = \frac{{deaths}_{c}}{N}
\end{equation}

then that is implicitly assuming that ${deaths}_{c}$ captures all of the deaths related to cause $c$.

For most conditions, that is not true. Because of the "one person, one death, one cause" rule (\emph{i.e.} each individual's death is encoded to a single underlying cause from a list of mutually exclusive, collectively exhaustive causes), ${deaths}_{c}$ is constrained to sum to total mortality across causes ${deaths}_T$ while ${D}_{w}$ has no such constraint.

\begin{equation}
    \label{eq:total}
    \begin{gathered}

        \math{deaths}_{T} = \sum_{c}^{C} {deaths}_{c}

        \\

        \math{D}_{T} \neq \sum_{c}^{C} {D}_{w_{c}}

    \end{gathered}
\end{equation}

Rather, the cause-specific mortality ${deaths}_{c}$ captures the fraction of the excess with-condition mortality ${D}_{w}$ that is assigned the underlying cause $c$.

Therefore, what we actually have is an inequality constraint 

\begin{equation}
    D_w \leq {deaths}_c
\end{equation}

or

\begin{equation}
    \chi \leq \frac{\mu_c}{p}
\end{equation}

\subsubsection{Approximating excess mortality from cause-specific mortality}

Since adding an inequality constraint is hard and doesn't actually give us that much information, maybe we can estimate that proportion of excess with-condition mortality that's coded to the condition, which we'll term $\theta$:

\begin{equation}
    \theta = P(excess\ death\ is\ captured\ in\ \mu_c)
\end{equation}

For some conditions (\emph{e.g.} HIV and ischemic heart disease), the ICD rules specify that nearly all people who die of that disease have their death coded to it as the underlying cause of death, so $\theta \approx 1$. 

For other conditions (\emph{e.g.} diabetes or COPD), some of the people who die of that condition are instead coded to something else (e.g. a diabetic who has a heart attack and dies of that - diabetes increased the risk of a heart attack and therefore contributed to excess mortality, but it's not captured as a diabetes death in the GBD cause of death estimates). In these cases, $0 < \theta < 1$.

And finally, some conditions (\emph{e.g.} heart failure or depression) are not considered valid underlying causes of death even though having the condition increases the risk of mortality. In other words, $\theta = 0$. 

Instead of imposing an inequality constraint $\chi p \geq \mu_c$ we could build an estimate of $\theta$ into our model:

\begin{equation}
    \chi = \frac{\mu_{c}}{p \theta}
\end{equation}

or 

\begin{equation}
    p \chi \theta = \mu_{c}
\end{equation}

We therefore would effectively be breaking down the deaths in our model even further, splitting with-condition mortality ${D}_{w}$ into cause-specific mortality ${D}_{c}$ and other excess with-condition mortality that's coded to other underlying causes ${D}_{e}$:

\begin{figure}[H]
    \label{fig:dismod3}
    \resizebox{150}{!}{%
        \begin{tikzpicture}[%
                >=stealth,
                thick,
                node distance=2cm,
                on grid,
                auto
            ]

            \node[rectangle, draw] (S) {$S$};
            \node[rectangle, draw] (C) [below=of S] {$C$};
            \node[rectangle, draw] (Do) [right=of S] {$D_o$};
            \node[rectangle, draw] (Dc) [below=of Do] {$D_c$};
            \node[rectangle, draw] (De) [below=of Dc] {$D_e$};

            \path[->]
                (S)     edge[bend right=30]   node    {$\iota$}          (C)
                        edge                  node    {$\omega$}         (Do)
                (C)     edge[bend right=30]   node    {$\rho$}           (S)
                        edge[right]           node    {$\omega$}         (Do)
                        edge                  node    {$\theta \chi$}    (Dc)
                        edge[left]            node    {$(1-\theta)\chi$} (De)
                ;
        \end{tikzpicture}
    }
\end{figure}

We of course wouldn't need to actually add any additional parameters or components to the model, we would just incorporate an estimate of $\theta$ when inputting GBD cause of death estimates into DisMod, $\chi = \frac{\mu_{c}}{p \theta}$.

In order to make total mortality add up correctly, we'd also update other-cause mortality $\omega$:

\begin{equation}
    \omega = \mu_{T} - \frac{\mu_{c}}{\theta}
\end{equation}

\subsubsection{Estimating $\theta$}

To estimate $\theta$, we could start with simply looking at how often a cause is listed as a contributing versus underlying cause of death

\begin{equation}
    \theta = \frac{{deaths_{underlying}}}{{deaths_{underlying}} + {deaths_{contributing}}}
\end{equation}

which we could derive by age and sex using multiple cause of death (MCD) microdata. We could potentially also include uncertainty on $\theta$ based on sample sizes and differences across data sources.

For causes for which there isn't sufficient data for whatever reason, the fallback would be $\theta = 1$, \emph{i.e.} what we're currently implicitly assuming.

\section{Conclusion}

We could break down our process for incorporating excess mortality data into DisMod into something like this, based on data available for each cause:

\begin{center}
\begin{tabular}{ |c|c|c|c| } 
\hline
\textbf{Data\ type} & $\pmb{\omega}$ & $\pmb{\chi}$ \\
\hline
$RR$
& $\mu_{T} \times \frac{1}{1 + p \times (RR-1)}$ 
& $\mu_{T} \times \frac{RR - 1}{1 + p \times (RR - 1)}$
\\
\hline
$SMR$
& \mu_{T} \times \bigg(1 - \frac{p \times (SMR - 1)}{1 - p} \bigg)
& \mu_{T} \times \frac{SMR - 1}{1 - p}
\\
\hline
$\mu_{c}$
& $\mu_{T} - \frac{\mu_{c}}{\theta}$
& $\frac{\mu_{c}}{p \theta}$
\\
\hline
\end{tabular}
\end{center}

In cases where multiple types of data are available, we could use multiple methods and compare the results. This could, for instance, give us insight into how well MCD data allows us to estimate $\theta$ for something like diabetes where we have all 3 types of data.

\end{document}