If you are still fitting ELISA data with linear or log-linear regression, you are leaving accuracy on the table — especially at the top and bottom of your standard curve. Immunoassays do not produce straight lines. They produce sigmoidal dose-response curves, and the right way to model them is with a four-parameter logistic (4PL) or five-parameter logistic (5PL) fit.

This guide explains why, shows the actual equations, and tells you when to upgrade from 4PL to 5PL.

Why ELISA standard curves are not linear

An ELISA measures optical density (OD) as a function of analyte concentration. At very low concentrations the signal sits near a background plateau. At very high concentrations the signal saturates near a maximum plateau. In between, the signal rises (or falls) steeply over a narrow concentration range.

That shape is sigmoidal. A straight line can only approximate a small slice of the middle. Outside that narrow window, linear regression systematically over- or under-predicts concentration.

Common consequences of forcing ELISA data into a linear model:

Low-end bias: concentrations near the LLOQ are pulled away from the true value.
High-end saturation ignored: samples near the ULOQ are overestimated.
Poor back-calculation: standards re-interpolated through a linear fit often fall outside the 80–120% recovery window.

Log-transforming the concentration axis helps visually, but it does not fix the underlying curvature. The asymptotes remain, and the middle section is still only locally straight.

The 4PL model: the default for ELISA

The four-parameter logistic curve describes the sigmoidal relationship between concentration and OD:

OD = D + (A - D) / (1 + (C / EC50)^B)

Parameter	Meaning
A	Lower asymptote (background plateau)
D	Upper asymptote (saturation plateau)
B	Hill slope (steepness of the transition)
EC50	Inflection point; concentration giving halfway response

4PL is the standard choice for ELISA, MSD, Luminex, and other sandwich or competitive immunoassays because it captures both plateaus and the steep transition between them.

Key advantages over linear fits:

Valid across the full range: both asymptotes are fitted parameters, not ignored.
Invertible: you can solve for concentration from any measured OD.
Weighted fitting: you can weight by 1/Y or 1/Y² to handle heteroscedasticity common in colorimetric readouts.

When to use 5PL instead

5PL adds a fifth parameter, the asymmetry factor E:

OD = D + (A - D) / (1 + (C / EC50)^B)^E

Use 5PL when the curve is visibly asymmetric — one asymptote is approached more gradually than the other. This often happens when:

the upper plateau is shallower than the lower plateau,
the hook effect or steric hindrance distorts the high end,
residual plots from a 4PL fit show a systematic S-shape rather than random scatter.

5PL costs one extra degree of freedom, so it is only justified when it clearly reduces residual scatter or improves standard recovery. If the 4PL residuals look random, stay with 4PL.

How to judge the fit: do not trust R² alone

A bad 4PL can still return R² > 0.99. Look at these instead:

Back-calculated standard recovery: each standard should recover 80–120% of its nominal concentration; 85–115% in the middle of the range.
Residual plot: should scatter randomly around zero. A U-shape means the model is wrong.
EC50 position: should sit inside the standard range, not far outside it.
Assay window: the OD difference between the top and bottom standards should be large enough. A ΔOD of ~0.3 suggests the assay barely responded.
Replicate %CV: under 15% for samples, under 10% for standards.

Practical decision tree

Situation	Recommended model
Standard curve has clear top and bottom plateaus	4PL
Curve is visibly asymmetric	5PL
Only a narrow working range is used	Linear or quadratic (rarely)
Residuals from 4PL show systematic curvature	5PL
Standards recover poorly at one end	Check blank handling first, then try 5PL

Doing it without a spreadsheet

Labreador's ELISA module fits 4PL and 5PL standard curves directly in your browser. Paste or upload the raw plate grid, mark blanks and standards, and the tool returns:

fitted parameters with confidence intervals,
back-calculated standard recovery and %bias,
sample interpolation with LLOQ/ULOQ flags,
replicate %CV and a Method Health panel.

No upload, no account, no data leaves your machine.

If you use Labreador in your work, please cite:

Labreador - Bioassay Analysis Platform (Version 1.6.0) [Computer software]. Zenodo. https://doi.org/10.5281/zenodo.21676846

4PL vs 5PL for ELISA Standard Curves: Why Log-Linear Regression Fails