What it does

From quantal (yes/no: died/survived, effect seen/not seen) dose–response data it computes the dose that affects 50 % and 90 % of the subjects (LD50, LD90) together with the 95 % confidence interval. The same mathematics gives the ED50 when the response is a "desired effect" rather than "death"; the ratio of the two is the therapeutic index.

Typical uses in pharmaceutical technology

• Comparing the acute toxicity of a new formulation (e.g. nanoparticles, liposomes, solid dispersion) with that of the free active substance
• Determining the intrinsic toxicity of the carrier/excipient system (vehicle)
• Finding the ED50 in an efficacy study to choose the dose range; safety margin as LD50/ED50
• Bioassays: insecticidal/antiparasitic efficacy, larval or microbial mortality and other results of the "x out of n" kind
• Teaching: a table-free, step-by-step counterpart of Finney's probit analysis

Data entry

• Each row is one dose group: Dose (positive, in a single unit such as mg/kg), Response (number affected/dead in that group), Number of Animals (total subjects in the group). The mortality percentage is computed from these; do not enter percentages.
• At least 3 doses are required; for a reliable slope 4–5 doses and a partial response (between 0 % and 100 %) in at least two groups are recommended. A slope cannot be estimated from 0 % and 100 % groups alone.
• Choosing the doses geometrically (e.g. 10, 15, 22, 33, 50) makes the log-dose axis equally spaced; probit/logit work on that axis.
• Rows with dose ≤ 0 or zero animals are not analysed. The control group (dose 0) is therefore not read; see Limitations.

Which method?

• Probit / Logit (classical) — the reference method for LD50. A generalised linear model on the log-dose axis, solved by IRLS with binomial weighting according to the number of animals at each dose (the modern counterpart of Finney's tables). The 95 % confidence interval is given by Fieller's method. Probit assumes that tolerance is log-normally distributed; logit is a distribution with slightly heavier tails. The two usually come out very close; the traditional report in toxicology is the probit.
• Logistic regression (4PL) — the four-parameter Hill curve. Designed for continuous responses (IC50, EC50, cell viability); it does not use the number of animals and its asymptotes are not constrained to 0–100 %. With quantal data keep it for visual comparison only.
• Linear regression — describes the mortality percentage as a straight line against dose. It does not represent the sigmoid relationship; it reads LD50 from the intercept of the line and may give negative or out-of-range values. For teaching/comparison only; not to be reported.
• Descriptive statistics — a summary of the entered columns, not an LD calculation.

Reading the results

• LD50 and 95 % CI: the value to report. The narrower the interval, the more reliable the estimate; the LD90 interval is naturally wider because it lies in the tail of the curve.
• "CI could not be computed": the slope does not differ significantly from zero (Finney's g ≥ 1 case). It usually means too few doses, very small groups or no partial response at all; even if an LD50 figure is shown, it is unreliable.
• Slope (β₁): if large, the population is homogeneous and the dose–response is steep; if small, tolerance varies widely between individuals. Its standard error (±) is the uncertainty of the coefficient.
• R² is not a selection criterion here: with quantal data R² measures the difference in proportions between groups, and in 4PL it is artificially high because of the four parameters. Judge the methods by CI width and the convergence warning.
• If the probit and logit LD50 diverge markedly, the data cannot determine the shape of the curve; an additional dose group is needed.
• No result carrying a "did not converge" warning should be reported.

Prerequisites

• Subjects must be independent and assigned to the dose groups at random; repeated observations of the same animal are not quantal data.
• The response must be binary and defined within a single observation window (e.g. 24 h, 14 days).
• Group sizes are preferably equal; if not, probit/logit take this into account by weighting, 4PL does not.
• Species, sex, age and route of administration must be uniform within one analysis; mixed groups distort the slope.
• There should be no deaths in the control group. If there are, Abbott's correction is required; this application does not apply that correction, you must correct the data beforehand.

Limitations

• Do not extrapolate. An LD90 beyond the highest dose tested is an extension of the curve; the confidence interval reflects this only partly.
• Small groups give wide intervals. With groups of 5 animals the 95 % CI often spans a two-fold range; report the estimate with its interval, not as a single number.
• Only a single endpoint is modelled. Time to death, dose–time interaction or sub-lethal effects do not enter this analysis.
• There is no comparison test. Whether the LD50 values of two formulations differ can be judged from the overlap of the confidence intervals, but this does not replace a formal test (e.g. parallelism test, likelihood ratio).
• Units are the user's responsibility. The result is labelled "mg/kg"; data entered in a different unit must be read in that unit.