# Values below the detection limit: why LOD/2 is a shortcut, and how to analyse non-detects

> A value below the limit of detection is censored, known only to be lower than the limit. Count them, then use a censored-data model, or a rank test when all share one limit, instead of LOD/2. Python and R code.

Canonical: https://www.licklider.ai/guides/values-below-detection-limit/
Section: Guides › Standard curves, detection limits and omics
Author: Licklider · Published 2026-11-11 · Last reviewed 2026-11-11 (UTC)

> **Quick answer.** A sample reported as "below the limit of detection" (LOD) has a concentration somewhere between zero and the LOD: the value is **censored**, not zero and not exactly LOD/2. First **report how many values are below the limit in each group**. Then analyse with a method that uses that information, chosen in advance: a **censored-data model** on log concentrations, which estimates the ratio of geometric means between groups and assumes a similar log-SD in each group (with about 12 animals per group, read its confidence interval rather than a p-value near 0.05), or, when every sample shares one detection limit, a **Mann–Whitney test** with the non-detects tied at the bottom. If a group is almost all non-detects, compare the proportion detected as well, and treat any ratio as rough. These methods assume independent animals; for paired or repeated samples, use a paired or mixed model. Substituting LOD/2 is a shortcut that changes the spread of the data and is discouraged by the methods literature; if you use it, say so and check that the conclusion does not depend on it.

**Terms used below.** *Limit of detection (LOD)*: the lowest concentration the assay can distinguish from zero. *Lower limit of quantification (LLOQ)*: the lowest concentration measured with acceptable accuracy and precision; values between the LOD and the LLOQ are detected but not reliably quantified. *Non-detect*: a sample below the LOD. *Censored value*: a value known only to lie below (left-censored) or above (right-censored) a limit. *Substitution*: replacing non-detects with a fixed number such as LOD/2 or LOD/√2. *Censored-data model*: a regression that uses non-detects as "below this limit" (a Tobit model, or a survival-type model fitted with left censoring). *Geometric mean*: the mean on the log scale, converted back. *Ratio of geometric means*: how many times higher the concentrations in one group are than in the other; the usual effect for log-normal data. *Log-SD*: the standard deviation on the log scale. *Maximum likelihood*: fitting a model by finding the parameter values that make the data most probable. *Power*: the chance of detecting an effect that is really there. *Log-normal*: skewed values whose logarithms follow a normal distribution, as concentrations often do. *Tobit model*: another name for a censored-data regression. *Reporting limit*: the limit below which a laboratory reports "not detected", here the LOD. *Welch's t-test*: compares two means without assuming equal spread. *Fisher's exact test*: compares two proportions in small samples. *Tied at the bottom*: all non-detects given the same lowest rank. *95% confidence interval (CI)*: the range of effects compatible with the data. *Independent*: each value from a different animal, with no pairing. *Sensitivity check*: repeating the analysis another way (here with substitution) to show that the conclusion does not change.

## Which analysis for your data

| Your data | Analysis | Report |
| --- | --- | --- |
| A few non-detects (under about 10%) | A censored-data model; substitution only as a stated sensitivity check | Number below the limit per group; ratio of geometric means with 95% CI |
| More non-detects, one common limit | Censored-data model on log concentrations; or Mann–Whitney with non-detects tied | As above; for the rank test, say which group tended to be higher (a median cannot be estimated for a group with more than half non-detects) |
| Several detection limits (different plates or dilutions) | Censored-data model, which takes each sample's own limit | As above |
| Most values below the limit in a group | Compare the proportion detected (Fisher's exact test) as well, and treat any ratio as rough | Both |

Helsel puts the case against substitution directly: "any substitution of a constant fraction of reporting limits will distort estimates of the standard deviation, and therefore all (parametric) hypothesis tests using that statistic", and "substituting values for nondetects should be used rarely" ([Helsel 2006](https://doi.org/10.1016/j.chemosphere.2006.04.051)). In an epidemiological comparison, Lubin and colleagues found that substitution "can be biased unless the percentage of measurements below detection limits is small (5-10%)" ([Lubin et al. 2004](https://doi.org/10.1289/ehp.7199)). For a rank test, Helsel notes that with one reporting limit, "censored observations are represented as a tied rank lower than the rank for the lowest detected observation" ([Helsel 2012](https://doi.org/10.1002/9781118162729)). The LOD/√2 rule often attributed to Hornung and Reed was proposed for estimating an average: "when the data are not highly skewed, replacement of nondetectable values by L/√2 should produce very good estimates", while LOD/2 "appears warranted only when the data are highly skewed" ([Hornung & Reed 1990](https://doi.org/10.1080/1047322X.1990.10389587)); neither rule was proposed for comparing groups.

## Worked example: a cytokine with 3 non-detects in the control group

Constructed data: a cytokine in 12 control and 12 treated mice, LOD 5 pg/mL. By construction concentrations are log-normal and twice as high with treatment; 3 control values and no treated values were below the LOD. The random seed (3) was the lowest, counting up from 1, that gave three or four control non-detects and at most one treated non-detect, the typical pattern for this model. Stored code and data: [cytokine.csv](/guides/values-below-detection-limit/cytokine.csv), [model.py](/guides/values-below-detection-limit/model.py) (the concentration model), [make_data.py](/guides/values-below-detection-limit/make_data.py), [analysis.py](/guides/values-below-detection-limit/analysis.py), [analysis.R](/guides/values-below-detection-limit/analysis.R) and [the analysis output](/guides/values-below-detection-limit/analysis_output.txt); analysis.py and analysis.R print the same output.

| Analysis | Result |
| --- | --- |
| Values below the LOD | Control 3 of 12; treated 0 of 12 |
| LOD/2 substitution, Welch's t-test on log values | Ratio 2.05; control log-SD 1.145; p = 0.070 |
| Mann–Whitney test (non-detects tied at the bottom) | p = 0.012 |
| **Censored-data (left-censored log-normal) model** | **Ratio 1.97 (95% CI 0.98 to 3.98)**, log-SD 0.860, p = 0.058 |

The substitution and censored-model analyses estimate a similar effect, about twofold (the rank test gives a p-value, not a ratio), but substitution places three values at an invented 2.5 pg/mL, which changes the control group's spread (log-SD 1.145 after substitution, against 1.07 for the detected control values alone; one high value, 114 pg/mL, contributes much to either). The censored model gives the ratio with a confidence interval that uses what is known about the non-detects; it assumes the same log-SD in both groups (0.860 here), so check the spread of the detected log values in each group: here 1.07 in controls and 0.59 with treatment, a difference driven partly by the one high value and within chance for 9 and 12 values (the true log-SDs are equal by construction). If the spreads clearly differ in your data (for example more than twofold), allow a separate log-SD per group (in R, add `+ strata(group)`; here that gives a ratio of 2.10, 95% CI 0.98 to 4.51) and say so. With 12 mice per group, this example cannot separate the groups: the interval is compatible with no effect and with a fourfold effect. The Mann–Whitney p-value (0.012) is smaller because it tests a different question (whether one group tends to have higher values, not the ratio); choose one primary analysis in advance and report the other as supporting. In 10,000 simulated experiments per scenario with 12 mice per group ([simulation.py](/guides/values-below-detection-limit/simulation.py), [the simulation output](/guides/values-below-detection-limit/simulation_output.txt)):

| Scenario (control values below the LOD) | LOD/2 substitution | Mann–Whitney | Censored-data model |
| --- | --- | --- | --- |
| No effect, 28% (false positives) | 5.1% | 4.7% | 7.0% |
| Twofold effect, 28% (power) | 51.7% | 48.8% | **58.7%** |
| No effect, 50% (false positives) | 4.7% | 4.3% | 5.7% |
| Twofold effect, 50% (power) | 50.2% | 47.2% | **54.6%** |

The censored model estimated the ratio without bias (median 2.00 in both effect scenarios) and had the highest power, partly because its false-positive rate was 7.0% rather than 5% with 12 mice per group and 28% of control values censored; so read its confidence interval rather than a p-value near 0.05. In these two-group scenarios substitution's ratio estimate stayed close to the truth, but it distorted the spread: with half the control values censored, the median control log-SD after substitution was 0.73 against a true 0.80. That distortion matters most for descriptive statistics, correlations and regressions, where Helsel and Lubin found the largest problems.

```python
# Condensed: analysis.py also runs the substitution and Mann-Whitney comparisons
import numpy as np, pandas as pd
from scipy import stats, optimize

d = pd.read_csv("cytokine.csv")   # mouse, group, conc_pg_ml, detected: for a non-detect enter the LOD in conc_pg_ml AND detected = 0
# change "treated" below to your treatment label; every other label is treated as the reference group
y, ev = np.log(d.conc_pg_ml.to_numpy()), d.detected.to_numpy()
x = (d.group == "treated").to_numpy().astype(float)
def nll(q):                                          # left-censored log-normal: a non-detect contributes P(X < LOD)
    z = (y - q[0] - q[1] * x) / np.exp(q[2])
    return -(ev * (stats.norm.logpdf(z) - q[2]) + (1 - ev) * stats.norm.logcdf(z)).sum()
fit = optimize.minimize(nll, [np.log(8), 0, np.log(0.8)], method="BFGS")
assert fit.success, "model did not converge: check that each group has some detected values"
assert d.group.nunique() == 2, "this snippet compares two groups; use a factor-coded model for more"
E = np.eye(3) * 1e-4; p0 = fit.x                     # numerical Hessian -> standard error
H = np.array([[(nll(p0 + a + b) - nll(p0 + a - b) - nll(p0 - a + b) + nll(p0 - a - b)) / 4e-8 for b in E] for a in E])
se = np.sqrt(np.linalg.inv(H)[1, 1])
assert np.isfinite(se), "standard error not available: too few detected values for this model"
print(np.exp(p0[1]), np.exp(p0[1] + np.array([-1.96, 1.96]) * se), 2 * stats.norm.sf(abs(p0[1] / se)))   # ratio, 95% CI, p
```

```r
library(survival)                                    # used only as a fitting engine for the censored likelihood
d <- read.csv("cytokine.csv"); d$group <- factor(d$group, levels = c("control", "treated"))
fit <- survreg(Surv(conc_pg_ml, detected, type = "left") ~ group, data = d, dist = "lognormal")
exp(cbind(coef(fit), confint(fit)))["grouptreated", ]   # ratio of geometric means with 95% CI (Wald)
summary(fit)$table["grouptreated", "p"]               # p-value
```

## Before you analyse

1. **Record non-detects as "below the limit", with the limit**, not as 0, blank or LOD/2. Enter the limit as the value and a 0/1 detection indicator, as in the example file (columns mouse, group, conc_pg_ml, detected). Take the LOD from the kit insert or from your own blank replicates, and say which you used; in analysis.py and analysis.R the constant `LOD` is used only for the LOD/2 comparison, so change it to your limit. Average technical replicate wells first; n is the number of animals.
2. **Know which limit applies.** Values between the LOD and the LLOQ are detected but not reliably quantified; ICH M10 defines quantification limits for bioanalysis ([ICH M10](https://database.ich.org/sites/default/files/M10_Guideline_Step4_2022_0524.pdf)). Decide in advance whether you treat values below the LLOQ as censored at the LLOQ.
3. **Count non-detects per group** before choosing the analysis, and report the counts.
4. **Diluted samples have their own limit** (the LOD times the dilution factor); put each non-detect's own limit in conc_pg_ml, and the censored-data model uses it.
5. **Check the log-normal assumption and the spread** on the detected values (a plot on the log scale and the log-SD in each group, printed by analysis.py); with heavy censoring, the model leans on these assumptions for the unseen values.
6. **Paired or repeated samples** (the same animal before and after, several time points) need a paired or mixed censored model, which this page does not cover; ask a statistician or use a package for censored mixed models.
7. **If you substitute anyway**, say so, give the fraction substituted, and show that LOD/√2 or the censored model gives the same conclusion.

## Check what an AI assistant ran

1. **Non-detects set to zero**, which makes logs impossible and pulls means down. Ask for the counts and a censored analysis.
2. **Non-detects deleted**, which removes the lowest values and biases the group upward. Ask for them back as censored values.
3. **LOD/2 substituted without saying so.** Ask how many values were substituted and for a censored model as a check.
4. **A t-test on raw concentrations** of skewed data with many identical LOD values. Ask for log concentrations and a censored model, or a rank test.

## Report it

Methods:

> Cytokine concentrations below the limit of detection ([5] pg/mL) were treated as left-censored. Log concentrations were compared between groups with a censored-data (Tobit) log-normal regression fitted by maximum likelihood, assuming a common log-SD (survival package version 3.5-8, R 4.3.3; replace with your versions), giving the ratio of geometric means.

Results (bracketed values are from the constructed example):

> [3] of [12] control and [0] of [12] treated values were below the limit of detection. The ratio of geometric means (treated/control) was [1.97] (95% CI [0.98] to [3.98]; p = [0.058]), so the data are compatible with no effect and with a fourfold increase (with 12 animals per group, give more weight to the interval than to the p-value).

Replace the bracketed values with your own.

## References

- Helsel DR (2006). Fabricating data: how substituting values for nondetects can ruin results, and what can be done about it. *Chemosphere* 65:2434–2439. [doi:10.1016/j.chemosphere.2006.04.051](https://doi.org/10.1016/j.chemosphere.2006.04.051)
- Lubin JH, Colt JS, Camann D, et al. (2004). Epidemiologic evaluation of measurement data in the presence of detection limits. *Environmental Health Perspectives* 112:1691–1696. [doi:10.1289/ehp.7199](https://doi.org/10.1289/ehp.7199)
- Helsel DR (2012). *Statistics for Censored Environmental Data Using Minitab and R*, 2nd edition. Wiley. [doi:10.1002/9781118162729](https://doi.org/10.1002/9781118162729)
- Hornung RW, Reed LD (1990). Estimation of average concentration in the presence of nondetectable values. *Applied Occupational and Environmental Hygiene* 5:46–51. [doi:10.1080/1047322X.1990.10389587](https://doi.org/10.1080/1047322X.1990.10389587)
- ICH (2022). M10: Bioanalytical method validation and study sample analysis. [database.ich.org](https://database.ich.org/sites/default/files/M10_Guideline_Step4_2022_0524.pdf) (checked 11 November 2026)

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## How this page was made

- Written and published by Licklider. Recommendations cite peer-reviewed methods papers, the ICH M10 guideline and software documentation listed under References; cited passages were checked against the source text.
- The worked example uses constructed data. Its numbers were produced by stored code (Python 3.13 with SciPy 1.18.1; R 4.3.3 with survival 3.5-8), and the two languages print identical output; the simulation was run in Python.
- Separate review passes checked the statistics, the citations and readability before publication.