Diagnosing Markov Assumptions From Earlier History

Why Examine Earlier History?

A Markov model treats the current state as a sufficient summary of the process's past for describing its future. That is a modeling assumption, not something established merely by drawing a state diagram. A practical diagnostic asks whether people who are in the same current state, at comparable times, have different subsequent state changes depending on how they arrived there or what happened earlier.

The comparison must be designed carefully. If one history group is observed earlier in follow-up, has different covariates, or is censored more often, its observed transition frequency may differ for reasons other than history dependence. The aim is therefore to look for a conditional association between earlier history and a future change after aligning the current state and relevant time scale.

Definition: Let $X(t)$ denote the state occupied at time $t$, and let $\mathcal{H}_t$ denote the observed history of the process up to time $t$. The Markov property requires that, for future times $u>t$, the conditional distribution of $X(u)$ given the history through $t$ depends on that history only through the present state $X(t)$. In a time-inhomogeneous model, it may also depend on $t$: $$ P\{X(u)=j\mid X(t)=i,\mathcal{H}_t\} = P\{X(u)=j\mid X(t)=i\}. $$ A diagnostic searches for differences in future state changes by earlier history among individuals with the same current state at comparable times.

For a practical comparison, choose a current state $i$, a time $t$ (or a sufficiently narrow range of comparable times), and a future horizon $h$. Define the outcome as whether a specified change occurs during $(t,t+h]$, or as the state occupied at $t+h$. Then compare that outcome between groups defined by earlier history. The choice of horizon matters: a history association may be evident over a short horizon but not over a longer one, or vice versa.

The earlier material on time origins and the Markov assumption distinguishes calendar time, time since study entry, and time since entering a state. Use the time scale relevant to the model and the question. In particular, a comparison at the same calendar time does not automatically compare people with the same duration in their current state. If duration is part of the earlier history under investigation, it should be examined deliberately rather than inadvertently mixed across groups.

What a History-Based Comparison Tests

At a fixed current state and time, let $H$ identify an earlier-history group, and let $Y$ indicate a specified future state change within $h$ time units. Under the Markov assumption, the conditional distribution of the future given the present state and time should not vary with $H$. For a binary outcome, this implies equal conditional risks across history groups, provided other conditioning factors are held fixed.

P(Y=1\mid X(t)=i,t,H=h_1) = P(Y=1\mid X(t)=i,t,H=h_0).

This equality is the basis of a simple diagnostic. Estimate each group’s future transition risk, calculate a contrast such as a risk difference, and quantify its uncertainty. With several history groups, compare their risks or use a regression model that includes history while controlling for the chosen time scale and measured covariates. A detectable association is evidence against the proposed Markov specification as conditioned; it does not by itself identify why the association occurs.

For a short future interval, the outcome can be defined as a transition to a particular destination. For a longer horizon, it may be more informative to compare the distribution over all possible states, since individuals can change state more than once. The diagnostic should match the scientific question and must account for the actual observation scheme.

Worked Comparisons

Worked Example: Comparing One-Step Transition Risks

Hypothetical data. At the same follow-up time, 120 individuals are observed in state A. All are followed for the next 30 days, and the outcome is transition from A to B during that period. Among 40 individuals who previously visited state C, 12 transition to B. Among 80 individuals without a previous visit to C, 12 transition to B. Assume individuals contribute independent observations, the state and history are recorded correctly, and 30-day outcomes are fully ascertained.

The estimated risks are $12/40=0.300$ and $12/80=0.150$. The estimated risk difference, previous visit minus no previous visit, is $$ \widehat{RD}=0.300-0.150=0.150. $$

Thus, the observed 30-day transition risk is 15 percentage points higher in the group with a previous visit to C. An approximate unpooled standard error is $$ SE(\widehat{RD}) = \sqrt{\frac{0.300(1-0.300)}{40} +\frac{0.150(1-0.150)}{80}} = \sqrt{0.005250+0.001594} = 0.08273. $$

A 95% Wald confidence interval is $$ 0.150\pm1.96(0.08273) = 0.150\pm0.1621 = (-0.0121,\ 0.3121). $$ The interval includes zero and is wide: these data do not provide clear evidence of a difference, but they are also compatible with a substantial positive difference. It is not evidence that the Markov assumption is true.

For a two-sided test of equal risks, the pooled risk under the null is $(12+12)/(40+80)=0.200$. The null standard error is $$ SE_0 = \sqrt{0.200(1-0.200)\left(\frac{1}{40}+\frac{1}{80}\right)} = \sqrt{0.006000} = 0.07746. $$ The test statistic is $z=0.150/0.07746=1.936$, giving a two-sided $p$-value of approximately 0.0528. At $\alpha=0.05$, this test does not reject equality. The conclusion is limited to this state, time, history definition, and 30-day outcome; it is not a general validation of the model.

Worked Example: A Difference Across History Groups

Hypothetical data. At a common assessment time, 100 individuals occupy state A. Each is followed for 14 days for a transition to state B. Among 50 individuals who had a prior transition out of A followed by a return, 15 transition to B during follow-up. Among 50 individuals with no such earlier transition, 8 transition to B. Assume independent individuals, comparable assessment times, correct history classification, and complete 14-day follow-up.

The estimated risks are $15/50=0.300$ and $8/50=0.160$; the risk difference is $0.300-0.160=0.140$, or 14 percentage points. The risk ratio is $0.300/0.160=1.875$. These are descriptive contrasts: the first group has a higher observed 14-day risk, but the comparison alone does not establish a causal effect of the earlier transition.

An approximate standard error for the risk difference is $$ SE = \sqrt{\frac{0.300(0.700)}{50} +\frac{0.160(0.840)}{50}} = \sqrt{0.004200+0.002688} = 0.08299. $$ A 95% Wald interval is $$ 0.140\pm1.96(0.08299) = 0.140\pm0.1627 = (-0.0227,\ 0.3027). $$ The estimated difference is notable, but the interval includes zero and leaves considerable uncertainty. Before interpreting this as evidence of history dependence, check whether the groups are also comparable in time since entering A, relevant measured characteristics, and completeness of follow-up. The calculation is a diagnostic signal, not a substitute for those checks.

Worked Example: How Unequal Times Can Create a Misleading Contrast

Hypothetical data. Suppose the next-interval transition risk is examined at two follow-up times among individuals currently in state A. The history groups have different distributions across those times. At the early time, 1 of 10 individuals with the earlier event and 9 of 90 without it transition. At the late time, 36 of 90 with the earlier event and 4 of 10 without it transition.

Within the early-time group, both risks are $1/10=9/90=0.100$. Within the late-time group, both risks are $36/90=4/10=0.400$. Thus, at each comparable time, the observed risks are equal across history groups.

If time is ignored, the earlier-event group has $(1+36)/(10+90)=37/100=0.370$ transitions, whereas the no-event group has $(9+4)/(90+10)=13/100=0.130$. The crude difference is $0.370-0.130=0.240$, even though there is no difference within either time stratum. The groups simply have different mixes of early and late observations. This example shows why a history comparison must align time or adjust for it before drawing a conclusion about the Markov assumption.

Derivation and Practical Workflow

The diagnostic follows directly from the conditional equality in the definition. Under the Markov assumption, for any two history values $h_1$ and $h_0$, while holding the present state and time fixed, the conditional probability of a specified future event is the same. Subtracting the two equal probabilities gives zero:

\begin{aligned} &P(Y=1\mid X(t)=i,t,H=h_1) - P(Y=1\mid X(t)=i,t,H=h_0)\\ &\qquad = 0. \end{aligned}

Therefore, a nonzero estimated risk difference is a natural diagnostic contrast. It will vary from sample to sample, so its magnitude must be considered together with its uncertainty. Conversely, a confidence interval containing zero does not prove equality: limited sample size or few transitions may make substantial differences hard to detect.

A disciplined analysis proceeds as follows:

Specify the comparison.
Name the current state, the relevant time scale, the earlier-history feature, the future outcome, and the prediction horizon before examining results.

Make groups comparable.
Restrict or stratify by current state and comparable time. Examine measured covariates that could differ between history groups, and decide whether they should be adjusted for.

Estimate and quantify.
Report group-specific risks and a clear contrast with an uncertainty interval. If using a model, state how time, covariates, and history enter it.

Check observation and interpretation.
Review censoring, missing state assessments, history misclassification, sample size, and the range of histories actually represented. Treat the result as a model diagnostic, not a causal conclusion.

Assumptions, Alternatives, and Failure Modes

A simple risk comparison is most interpretable when individuals are independent, state and history are measured consistently, the follow-up horizon is the same, and outcomes are ascertained comparably. If some individuals are censored before the horizon, counting only observed transitions can bias the comparison unless censoring is handled under an appropriate assumption, such as independent censoring conditional on the factors used in the analysis. Missing visits may also conceal transitions. A diagnostic based on recorded data can therefore fail to reveal history dependence when observation is incomplete or differential.

The main alternative to separate stratified comparisons is regression. For a binary future outcome, a regression model can estimate an association with history while adjusting for time and measured covariates; interactions can assess whether the history association varies by time or covariate level. Such a model adds assumptions about functional form and specification. A useful practice is to present interpretable predicted risks or risk contrasts, not only a coefficient, and to check whether conclusions depend on modeling choices.

Neither approach can rule out unmeasured differences between history groups. A history association may reflect omitted covariates, selection into the current state, or differences in observation. Nor does an absence of association for one selected history summary establish the Markov property for every aspect of the full past. The diagnostic is strongest when based on prespecified, scientifically meaningful histories and repeated across relevant states, times, and horizons.

A common mistake is to compare all individuals with and without an earlier event, then treat a crude difference as proof of non-Markov behavior. This reasoning is wrong because the groups may occupy the current state at different times or have different covariates. The correction is to compare future changes conditional on the same current state and comparable times, address relevant measured differences and censoring, and describe any remaining association as evidence against the specified Markov model—not as proof of a causal mechanism.

Key takeaway: A Markov diagnostic asks whether earlier history predicts subsequent state changes after conditioning on the present state and aligning relevant time. A positive finding can expose an inadequate state or time specification, but it must be interpreted alongside uncertainty, observation processes, and group comparability.

Check Your Understanding

Answer each question, then compare with the solutions below.

Why must a history-based comparison hold the current state fixed?
Among 30 people with a specified prior event, 9 change state within 10 days; among 60 without it, 9 change state. Calculate both risks and their risk difference.
What problem is illustrated when history groups have different proportions of early and late assessments?
If a 95% confidence interval for the risk difference includes zero, what can and cannot be concluded?
Name two observation or measurement problems that could weaken a history-based diagnostic.

Solutions. (1) The Markov property conditions on the present state; mixing current states would confound the comparison with different opportunities or destinations for change. (2) The risks are $9/30=0.300$ and $9/60=0.150$; the risk difference is $0.300-0.150=0.150$, or 15 percentage points. (3) A crude contrast may reflect different time composition rather than an association with history at comparable times. (4) The data do not establish a difference at the stated confidence level, but the interval does not prove equal risks or validate the Markov assumption. (5) Examples include differential censoring, missing state assessments, and misclassification of state or earlier history.