kklot / multisurv

title	author	bibliography	output	documentclass	linkcolor	urlcolor	citecolor
Multistate model of marital life course	Van Kinh Nguyen	/Users/knguyen/Documents/zotero_biblatex.bib	pdf_document toc number_sections true true	article	blue	blue	red

Aims

• Estimate four hazard rates of flow below

graph LR;
  A((Virgin))--> B(Sexual debut);
  B--> |*| C{Married};
  C--> D{{Divorce}}
  C-->E[/Widowed/]

Loading

The arrows show which transitions are possible between states in our model.

Methods

Data

We extract age at first sex, age at marriage, and marital statuses variable from DHS. From this, we calculate the time since birth to first sex, from first sex to marriage, from married to divorce or widowed.

We assumed that marriage event occurs after sexual debut; in case the events coincide, we allowed individual to quickly pass through sexual debut and move to married state. Remarried is not considered here as the variable was not collected in the DHS.

Differences in the states transition

• Virgin to sexually debuted and to married: these transitions can be assumed to be exactly observed at the reported AFS and age at married.
• Married to divorce or widowed: we know the age at marriage and the current state, but we don't know when the divorce or the death of spouse occurred. These transitions are thus interval censored.
• Union: we know AFS and current state but don't know the time of union. If we group married and union into one group. This will be treated as interval censor between AFS and current age while the married age is treated as exact observed time.
  • TODO: need to write code to cover this in the likelihood
• Separated: depending on whether age at married is know or not; if known, interval censor between age at married and current age, if not interval censor between AFS and age.
  • TODO: need to write code to cover this in the likelihood

These differences yield different likelihood contributions.

Multistate survival model

During the time before the survey, at a time $t$ the individual is in state $S(t)$. The next state to which the individual moves, and the time of the change, are governed by a set of transition intensities $q_{rs}(t, z(t))$ for each pair of states $r$ and $s$. The intensities may depend on the process time $t$ or individual-specific or time-varying explanatory variables $z(t)$. The intensity represents the instantaneous risk of moving from state $r$ to state $s$

$$q_{ rs }(t, z(t)) = lim_{\delta_t\rightarrow0}\frac{P(S(t+\delta t)=s|S(t)=r)}{\delta t}$$

The intensities form a matrix $Q$ whose rows sum to zero, so that the diagonal entries are defined by $q_{rr} =-\sum_{s\neq r}q_{rs}$.

$$Q(t) = \begin{bmatrix} -q_{VS} & q_{VS} & 0 & 0 & 0 \\ 0 & -q_{SM} & q_{SM} & 0 & 0 \\ 0 & 0 & -(q_{MD} + q_{MW}) & q_{MD} & q_{MW} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}$$

where $V, S, M, D, W$ denotes virgin, sexually debuted, married, divorce, and widowed. To fit a multistate model to data, we estimate matrix Q.

Markov assumption

The Markov assumption is that future evolution only depends on the current state. That is, $q_{rs}(t,z(t),F_t)$ is independent of the observation history $F_t$ of the process up to the time preceding $t$. In a time-homogeneous continuous-time Markov model, a single period of occupancy in state $r$ has an exponential distribution, with rate given by $-q_{rr}$, (or mean $-1/q_{rr}$). The remaining elements of the $r$th row of $Q$ are proportional to the probabilities governing the next state after $r$ to which the individual makes a transition. The probability that the individual’s next move from state $r$ is to state $s$ is $-q_{rs}/q_{rr}$ which altogether forms the transition probability matrix $P(t)$.

$P (t)$ can be calculated by taking the matrix exponential of the transition intensity matrix $Q$. For a time-homogeneous process, the $(r, s)$ entry of $P (t)$, $p_{rs}(t)$, is the probability of being in state $s$ at a time $t + u$ in the future, given the state at time $t$ is $r$. It is difficult to calculate reliably [@molerNineteenDubiousWays2003]. For simpler models, analytic expression for each element of $P (t)$ in terms of $Q$ can be derived. This is faster and avoids the potential numerical instability of calculating the matrix exponential [@van2016multi].

Likelihood

Let $i$ indexes $N$ individuals. The data for individual $i$ consist of a series of times $t_{i1},..., t_{i m_i}$ and corresponding states $S(t_{i1}),..., S(t_{im_i})$, where $m_i$ the number of recorded states which can be be varied between individuals. Given a pair of successive states $S(t_j),S(t_{j+1})$ at times $t_j , t_j +1$. The contribution to the likelihood from this pair of states is

$$L{i,j} = p_{S(t_j) S(t_{j+1})}(t_{j+1} - t_j)$$

which is the entry of the transition matrix $P(t)$ at the $S(t_j)$th row and $S(t_{j+1})$th column, evaluated at $t=t_{j+1} - t_j$. The full likelihood $L(Q)$ is the product of all such terms $L_{i,j}$ over all individuals and all transitions.

$$L(Q) = \prod_{i=1}^{i=N}\prod_{j=1}^{m_i - 1} L_{ij}$$

Depending on how the events and times are defined, $P(t)$ can take different forms.

Exact transition time

Assuming individual responses to DHS is accurate, events "observed" in the model represents exact transition times in between the states, with no transitions occurred between the observation times. For example, response of individual $i$ to the AFS at time $t_{ij}$ (age) marks the exact time of transition from virgin to sexually debuted and that sexually debut events was not occurred in between the time from birth and the AFS. In this case, the likelihood contribution does not require determining the transition probability $P$ but only the intensity matrix $Q$

$$L_{ij} = \exp[q_{S(t_j)S(t_j)}(t_{j+1} - t_j)] \times q_{S(t_j)S(t_{j+1})}$$

since the state is assumed to be $S(t_j)$ throughout the interval between $t_j$ and $t_{j+1}$ with a known transition to state $S(t_{j+1})$ at $t_{j+1}$.

Covariates

Explanatory variables for a particular transition intensity can be modelled a function of these variables. A proportional hazards model where the transition intensity matrix elements $q_{rs}$ of interest can be replaced by [@marshallMultistateModelsDiabetic1995]

$$q (z(t)) = q(0)\exp(\beta^T_{rs} z(t))$$

If the covariates $z(t)$ are time dependent, the contributions to the likelihood of the form $p_{rs}(t - u)$ are replaced by $p_{rs}(t - u,z(u))$ which requires that the value of the covariate is known at every observation time $u$.

Implementation

Model 1 - no random effect - exponential distribution

Using msm package, the likelihood is maximised with crude initial values, which can be set supposing that transitions between states take place only at the observation times. If we observe $n_{rs}$ transitions from state $r$ to state $s$, and a total of $n_r$ transitions from state $r$, then $q_{rs}/q_{rr}$ can be estimated by $n_{rs}/n_r$. Then, given a total of $T_r$ years spent in state $r$, the mean sojourn time $1/q_{rr}$ can be estimated as $T_r/n_r$. Thus, $n_{rs}/T_r$ is a crude estimate of $q_{rs}$.

Preliminary results

Data

In Malawi survey 2015, 17.9% of the data has AFS older than age at married with an average difference of 1.43.

• how to treat this? a separate model? remove?
• Two records with married state but no AFS (removed).

The actual flow in data is

graph LR;
  A((Virgin))--> B(Sexual debut);
  B---> |*| C{Married};
  A-.-> C;
  C--> D{{Divorce}};
  C-->E[/Widowed/];
  C-.-> B;
  B-.->D;
  B-.->E;

Loading

In this data all divorce, separate, and widowed were married.

Fitting model removing all those sexually debuted after married

Those who debuted sexually within a year of marriage age is recoded to the same age at married, the rest is discarded for this analysis.

References

multisurv's Issues

Second model for survival