Likelihood and Model Fit (Intuition)

Build intuition for likelihood, how models are fit to data, and how likelihood connects estimation to decisions.
Tip

What you’ll build today: a clear, intuitive understanding of likelihood—what it means, how it drives estimation, and how it connects models to real-world decisions.

Learning Objectives

By the end of this lesson, you will be able to:

  • Explain likelihood in plain language
  • Understand how likelihood is used to fit models
  • Connect likelihood to estimation methods (FOCE, SAEM, Bayesian)
  • Recognize what “good fit” actually means

Key Ideas

All model estimation methods are trying to answer the same question:

What parameter values make the observed data most plausible?

This is captured by the likelihood:

\[ L(\theta) = P(\text{data} \mid \theta) \]

  • \(\theta\) = model parameters
  • \(L(\theta)\) = how well those parameters explain the data

Insight: Likelihood is not about whether the model is “true”—it is about how well it explains what we observed.

Warning

A model can have high likelihood and still be wrong in a scientific sense.
Likelihood measures fit to data—not correctness of assumptions.

Why This Lesson Matters

You’ve seen:

  • models (structure)
  • variability (random effects)
  • estimation methods (FOCE, SAEM, Bayesian)

Likelihood is the concept that connects all of them.

Without it:

  • estimation becomes a black box
  • results feel like “software output”

With it:

You understand why the model chose those parameter values.

Worked Example: Two Possible Fits

Suppose two models attempt to explain the same observations.

Compare the models:

  • Model A follows the observations closely
  • Model B misses important features

Model A therefore has higher likelihood.

Interpreting Likelihood

Likelihood answers:

  • How close are predictions to observations?
  • How consistent is the model with the data?

Better fit → higher likelihood
Worse fit → lower likelihood

Visualizing Likelihood

Likelihood compares parameter choices.

flowchart LR

P1["Parameters A"]

-->

L1["Likelihood"]

P2["Parameters B"]

-->

L2["Likelihood"]

L1 -->

BEST["Choose Better Fit"]

L2 -->

BEST

Estimation searches for parameter values that improve likelihood.

Log-Likelihood and Log-Likelihood

In practice, we use:

\[ \log L(\theta)=\log P(\text{data}\mid\theta) \]

The best fit corresponds to the parameter values that maximize this quantity.

We work with log-likelihood because it is:

  • easier to compute
  • numerically stable
  • able to turn products into sums

From Likelihood to Objective Function

Many pharmacometric software packages do not report likelihood directly.

Instead they report an objective function value (OFV):

\[ OFV = -2\log L \]

where:

  • \(L\) is the likelihood
  • lower OFV indicates better agreement with the data

This transformation makes optimization easier and creates a common scale for comparing models.

You do not need to calculate OFV manually.

NoteKey Takeaway

Higher likelihood ↔︎ Lower OFV

These contain the same information expressed on different scales.

Expanding the Idea: Noise Matters

Real data are noisy.

So models are not expected to pass exactly through every observation.

Instead, likelihood evaluates:

  • how close predictions are to observations
  • whether those differences are consistent with expected variability

This is why residual error models matter.

Imagine repeating the same experiment.

Even with identical parameters:

  • observations would change slightly
  • the likelihood would change accordingly

A good model therefore does not explain every point perfectly.

It explains the data within expected variability.

Insight

A “good fit” is not a perfect curve—it is a model that explains the data within expected variability.

Note

Overfitting can increase likelihood locally, but harm generalizability.

Connecting to Estimation Methods

Different methods work with the likelihood in different ways:

  • FO / FOCE → linear approximations
  • SAEM → simulation-based approximation
  • Bayesian → full posterior (likelihood × prior)

But:

They are all trying to maximize (or explore) the same likelihood.

Why Likelihood Matters for Decisions

Likelihood affects:

  • parameter estimates
  • uncertainty
  • predictions

If likelihood is misleading:

  • parameters may be biased
  • predictions may be wrong
  • decisions may fail

Strategies

  • Evaluate fit visually and statistically
  • Consider variability, not just mean fit
  • Compare models using likelihood-based criteria such as changes in objective function value (ΔOFV), AIC, and BIC
  • Use likelihood as a guide, not a guarantee

Common Mistakes

  • Treating likelihood as proof of correctness
  • Ignoring model assumptions
  • Overfitting to maximize likelihood
  • Assuming higher likelihood always means better model for decisions

Practice Problems

  1. What does likelihood represent?
  2. Why isn’t perfect fit required?
  3. How do estimation methods differ in handling likelihood?
  1. The probability of observing the data given parameter values.
  2. Because data contain variability and noise.
  3. They use different approximations or frameworks to evaluate likelihood.

Summary

Likelihood:

  • measures how plausible the observed data are under a model
  • drives parameter estimation
  • connects models to observations

But it does not:

  • guarantee correctness
  • replace scientific reasoning
  • Likelihood = “how plausible is this model given the data?”
  • Higher likelihood = better fit (not necessarily better science)
  • Always consider variability
  • Use likelihood to compare models—not to blindly trust them