.. _bda-example-serial-dilution-assay:

========================================================================
Example: serial dilution assay
========================================================================

**Part 5 · Stage 15 · 🌊 Basis Functions & Gaussian Processes**  ·  Lesson 124 of 144  ·  *advanced*

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.. important::

   **✨ AI-generated content.** This page was written with the assistance of an
   AI language model and is provided as a learning aid. Despite careful
   review, it may still contain mistakes, omissions, or out-of-date
   information. Whether you are new to the topic, a team lead, or a senior
   practitioner, treat it as a starting point rather than an authoritative
   reference: read it critically and independently verify anything you act on
   (code, commands, figures, and factual claims) against official
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Nonlinear calibration, done properly
--------------------------------------

A serial dilution assay measures the concentration of a compound — an allergen, an antibody, a
pollutant — by reading an instrument at several **dilutions** of a sample and comparing against a
calibration curve. Gelman, Chew and Shnaidman used it to open the nonlinear-models stage because it
exposes, in one applied problem, why linear-normal thinking fails and what replaces it.

Why the standard approach wastes data
---------------------------------------

The relation between concentration and instrument reading is **nonlinear and heteroscedastic**: it
saturates at high concentrations and its measurement noise grows with the signal. The common practice
copes badly — it **discards** every reading that falls above or below the instrument's detection limits,
throwing away a large fraction of the data, and weights the survivors equally despite their unequal
precision. Estimates are noisy, and a sample whose readings all lie below the limit yields **no estimate
at all**.

The Bayesian model
--------------------

Model the full nonlinear calibration curve and the unknown concentrations **jointly**, with a variance
that grows with the mean. A standard form maps concentration :math:`x` to expected reading through a
four-parameter curve:

.. math::

   \mathrm{E}[y \mid x] = \beta_1 + \frac{\beta_2}{1 + (x/\beta_3)^{-\beta_4}}, \qquad
   \mathrm{sd}(y \mid x) \propto \bigl(\mathrm{E}[y \mid x]\bigr)^{\alpha},

the second equation encoding heteroscedasticity — noise scaling with signal — so measurements are
weighted by their actual precision rather than equally.

.. code-block:: python

   import pymc as pm
   with pm.Model():
       b = pm.Normal("b", 0, 5, shape=4)                    # calibration-curve parameters
       alpha = pm.HalfNormal("alpha", 1)                    # variance-scaling exponent
       conc = pm.HalfNormal("conc", 10, shape=n_unknown)    # unknown concentrations (parameters)
       mu = b[0] + b[1] / (1 + (x / b[2]) ** (-b[3]))       # nonlinear curve
       sigma = pm.Deterministic("sigma", (mu ** alpha))     # signal-dependent noise
       pm.Normal("y", mu, sigma, observed=readings)         # ALL readings, none discarded
       idata = pm.sample()

Why it wins
-------------

Using **all** the data — including the below-detection readings, which are informative even when
imprecise — the joint model achieves **much lower standard errors** than the discard-and-average
standard, and returns an estimate even when every measurement is off-scale, where the classical method
simply fails. Three general lessons open the stage. Real relationships are often **nonlinear**, and
forcing linearity discards structure. Measurement noise is often **not constant**, and modelling its
dependence on the signal is what makes the weighting correct. And a **joint** model — curve and
concentrations together, with uncertainty flowing between them — beats a two-step fit that treats the
calibration as fixed. Part V builds on all three, moving from this parametric nonlinearity toward
splines, Gaussian processes and models with no fixed functional form at all.

.. hint::

   **Related lessons:** :doc:`Splines and weighted sums of basis functions <126-splines-and-weighted-sums-of-basis-functions>`  ·  :doc:`Example: population toxicokinetics <125-example-population-toxicokinetics>`  ·  :doc:`Bayesian analysis of classical regression <092-bayesian-analysis-of-classical-regression>`  ·  :doc:`Unequal variances and correlations <097-unequal-variances-and-correlations>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/12/09/example-serial-dilution-assay/ <https://insightful-data-lab.com/2025/12/09/example-serial-dilution-assay/>`__ (insightful-data-lab.com).

.. tags:: purpose: reference, topic: data analysis, domain: bayesian, level: advanced