Markov Chain Models in Life Course Research (1) Introduction

1. What Are Markov Chain Models For?#

Our lives are a series of transitions between states. Today you might be “employed,” tomorrow “on leave,” and in the future, “retired.” A person might move from “single” to “married,” then to “divorced” or “remarried.” These transitions between social states are central to life course research.

Markov Chain Models are mathematical tools designed to describe how systems evolve between discrete states. They’re widely used beyond social science, such as in:

What unites these applications is the use of state sets and transition rules to represent complex, evolving systems.

According to the definition:

A Markov chain model is a stochastic process model describing transitions between states over time. In Chinese, this translates to: A stochastic process model that describes how states change over time.

So what is a “stochastic process”? And what does “stochastic” mean?

“Stochastic” means that the model involves uncertainty and probability. It does not give a fixed outcome every time. Instead:

You can only describe what is likely to happen, and with what probability.

This is different from a deterministic model, which always gives the same output for the same input.

Here’s a simple example:

Suppose the probability of transitioning from “employed” to “unemployed” is 0.2. This doesn’t mean the person will become unemployed. It means that each time you simulate, there is a 20% chance of unemployment and 80% chance of staying employed or moving elsewhere.

So each time you run the model, the path may vary — this is what we mean by stochasticity.

💡 Term Tip: What is “stochastic”? In this tutorial, we often use the term “stochastic process.” It means the model includes randomness and probabilities, not fixed outcomes. For example, if you start in the “employed” state, there might be a 10% chance you become unemployed next year, and a 90% chance you stay employed. Each simulation is like a lottery draw — the path taken is random. This contrasts with deterministic models, which always produce the same results.

2. What Makes Markov Models Special in Life Course Research?#

In life course research, we analyze how individuals move through key social states over time:

• Education → Employment → Unemployment → Re-employment → Retirement
• Single → Cohabiting → Married → Divorced → Remarried
• Healthy → Ill → Recovered → Disabled → Dead

Compared to other fields, life course models face several unique challenges:

3. Common Markov Models in Life Course Research#

1. Discrete-Time Homogeneous Markov Chains (Basic)#

• Assumption: Transitions happen at fixed intervals, with constant probabilities.
• Use case: Basic state mapping and trajectory prediction.

2. Non-Homogeneous Markov Chains (Age-Dependent)#

• Assumption: Transition probabilities vary by age/time.
• Use case: Comparing youth employment to midlife patterns.

3. Continuous-Time Markov Processes#

• Assumption: State transitions can occur at any time point.
• Use case: Modeling hospital admissions or health events.

4. Multistate Life Tables#

• Assumption: Focus on time spent in each state over a lifetime.
• Use case: Estimating healthy life expectancy, years unemployed, etc.

5. Hidden Markov Models (HMMs)#

• Assumption: True states are latent; we observe outputs.
• Use case: Modeling unobservable mental states or identity trajectories.

4. Transition Probabilities: Where Do They Come From?#

In life course research, transition probabilities are typically derived from longitudinal data, statistical modeling, literature, or expert assumptions. The data structure and research goals determine which method to use.

1️⃣ From Longitudinal Survey Data#

Most common approach. Examples: PSID, SOEP, CFPS, BHPS, HILDA.

Estimated using empirical frequency:

P(A → B) = Count of transitions from A to B / Number currently in A

2️⃣ From Statistical Models#

When data is sparse or we want to control covariates:

• Cox Proportional Hazards (for duration)
• Logistic / Multinomial Logit models
• Conditional Markov Models

These generate probabilities or hazards that can be used to build transition matrices.

3️⃣ From Existing Research or Official Statistics#

Sometimes researchers use external sources:

• Government employment or mortality tables
• WHO/UN health transition tables
• Prior studies reporting full transition matrices

Check time unit compatibility and population match.

4️⃣ From Theory or Expert Simulation#

Used in policy scenarios or theoretical models:

• E.g., simulate impact if “employment → unemployment” drops by 10%
• Useful in system dynamics or agent-based modeling

5. The Simplest Case: Discrete-Time Homogeneous Chains#

Setup:

• State set: S = {Unemployed, Employed, Retired}

• Observe states at fixed intervals (e.g., yearly)

• Assumptions:

  • Markov property: next state only depends on current
  • Homogeneity: transition probabilities do not change over time

Example Transition Matrix:

Retired is an absorbing state — once entered, it cannot be left.

Example: Simulating Xiao Wang’s Career#

import numpy as np

P = np.array([[0.5, 0.4, 0.1],
              [0.1, 0.8, 0.1],
              [0.0, 0.0, 1.0]])
states = ['Unemployed', 'Employed', 'Retired']
current_state = 0
np.random.seed(1)

path = []
for year in range(10):
    path.append(states[current_state])
    current_state = np.random.choice([0,1,2], p=P[current_state])

print("Simulated path for Xiao Wang:", " → ".join(path))

6. Simulation or Explanation: What Is a Markov Model For?#

Markov Models as Simulation Tools#

When you already know the transition matrix, the model is used to simulate potential life paths, generate population forecasts, or run policy experiments.

Markov Models as Explanatory/Statistical Models#

When you estimate the transition matrix from data, the model helps:

• Identify predictors of state transitions
• Test the effects of variables (e.g., gender, education)
• Support causal inference

Comparison to Regression:

When to Use Which?#

Markov models can be both generative and explanatory — they are social science’s version of time-aware regressions and scenario engines.

7. Key Concepts#

Absorbing State#

An absorbing state is one that, once entered, cannot be left. This concept is essential in modeling irreversible life events such as retirement or death.

Definition: A state $s$ is absorbing if:

P(s → s) = 1 and P(s → other) = 0

Everyday Examples:#

In life course studies, absorbing states are:

• Irreversible: once reached, future transitions are no longer possible.
• Structural endpoints: they mark the natural end of certain trajectories (e.g., career or life).
• Useful for simulations: when you need to model how long individuals remain in transient states before absorption.

You can also have multiple absorbing states, such as “retired with pension” vs. “retired without pension,” or “died healthy” vs. “died after chronic illness.”

Parameters in Markov Models#

In statistical modeling, Markov chains are parameterized systems. These parameters define the behavior of transitions, and depending on the model’s complexity, can include several layers.

Where do these parameters come from?#

• They can be manually specified for simulation purposes.
• Or empirically estimated from data using frequency counts or statistical modeling (e.g., MLE, Bayesian).

Homogeneous vs. Non-Homogeneous Markov Chains#

These terms describe whether the transition rules change over time or context:

In life course research, non-homogeneous models are often more realistic because:

• Young adults change jobs more frequently.
• Middle-aged people are more stable in relationships.
• Seniors have higher health transition risks.

However, homogeneous models are still commonly used because they are:

• Easier to estimate.
• Simpler to interpret.
• Adequate for many descriptive or comparative purposes.

We’ll explore the reasons why simpler models are still dominant in a later tutorial.

First-Order vs. Higher-Order Markov Chains#

This dimension describes whether the model remembers past states beyond the immediate previous one:

First-Order: Next state depends only on the current state:

P(Xₜ | Xₜ₋₁)

Higher-Order: Next state depends on multiple past states:

P(Xₜ | Xₜ₋₁, Xₜ₋₂, ..., Xₜ₋ₖ)

Why higher-order might be useful:

• Captures path dependence (e.g., effects of long-term unemployment)
• Allows for duration-dependent transitions (e.g., after 3 years cohabiting, marriage is more likely)

But also why it’s rare:

• Parameter explosion: more possible state combinations
• Harder to estimate: especially with sparse data
• Often approximated using nested states (e.g., “Employed-Year1”, “Employed-Year2”)

Summary Table: Two Key Dimensions#

These are independent dimensions — you can have:

• First-order, homogeneous (simplest)
• First-order, non-homogeneous (common in advanced life course models)
• Higher-order, homogeneous (captures path dependence)
• Higher-order, non-homogeneous (most realistic but rare)

In life course modeling, starting simple (first-order, homogeneous) is typical, and only advancing in complexity as needed by theory, data, or research question.

8. Summary: What You Learned in Part 1#

In this first tutorial, we thoroughly introduced Markov Chain Models in the context of life course research, covering both the theoretical foundations and applied modeling strategies. You now understand:

1. What a Markov chain model is and what it is good for#

• It is a stochastic process model describing transitions between states over time.
• It is ideal for modeling social processes such as education, employment, marriage, and health trajectories.

2. What makes life course modeling unique#

• Social states are often multidimensional, partially observable, and influenced by age and history.
• Compared to fields like engineering or biology, social modeling emphasizes explanation, inequality, and policy relevance.

3. Five common types of Markov models in life course research#

• Discrete-time homogeneous chains (basic)
• Non-homogeneous chains (age-varying)
• Continuous-time processes
• Multistate life table models
• Hidden Markov models (latent state dynamics)

You now know what kinds of questions each model is best suited to address.

4. Where transition probabilities come from#

• Directly from longitudinal data (frequencies)
• Indirectly via statistical modeling (e.g., survival, logit)
• From external sources (e.g., census, research)
• From theory or expert judgment (e.g., simulations)

You understand how different sources match different purposes: empirical analysis, forecasting, or policy modeling.

5. Core concepts: absorbing states, parameters, model orders#

• Absorbing states (e.g., retirement, death) end trajectories.
• Models are parameterized (e.g., probabilities, covariate effects).
• You explored the distinctions between homogeneous/non-homogeneous and first-order/higher-order chains.
• These are independent dimensions: complexity depends on research design.

6. You built and ran a full Python simulation example#

• Created a state space and transition matrix
• Simulated 10-year life trajectory using NumPy
• Connected code to theory — a foundation for later hands-on modeling

This sets the stage for Part 2, where we’ll explore how to:

• Model non-homogeneous transitions
• Incorporate covariates (conditional Markov models)
• Estimate parameters using real longitudinal data

Stay tuned!