First off, what is LD50? LD50 is the median lethal dose of a toxic substance, i.e.,
that dose of a chemical which kills half the members of a tested population. Basically,
what we have is a predictor that is the dose of a chemical and a binary response
variable that indicates whether the individual dies or not.

Although, this is stated in terms
of lethality and deaths, the concept is a bit more general, i.e., what is the value of the
continuous predictor that results in a probability of .5 of being in the 1 category. An
example is, what is the test score needed for an individual to have a 50/50 chance of
being admitted to a special program?

Let’s say that you have a logit model that looks like this,

Logit(Y) = constant + coef*X

Or a probit model that looks like this,

Probit(Y) = constant + coef*X

In either case, the computation of LD50 is the same. Here is the formula,

LD50 = -constant/coef

Example LD50 for logit model.

Logit(Y) = -8.30019 + .13257*XLD50 = 8.30019/.13257 = 62.61

Example LD50 for probit model.

Probit(Y) = -4.95764 + .07925*XLD50 = 4.95764/.07925 = 62.56

Please note that although the LD50 formula is the same for both logit and probit models, the LD50 values
are not the same since they come from different probability distributions.

What if instead of a LD50, I want to compute an LD60 or LD75? This requires a small change
to the formulas. However, the formula for a logit model is no longer the same as for a
probit model.

Logit:   LDp = (log(p/(1-p))-constant)/coefProbit:   LDp = (invnormal(p)-constant)/coef

Example LD75 for logit model.

Logit(Y) = -8.30019 + .13257*XLD75 = (log(.75/(1-.75)) – -8.30019)/.13257 = (1.09861 – -8.30019)/.13257
= 70.9

Example LD75 for probit model.

Probit(Y) = -4.95764 + .07925*XLD75 = (invnormal(.75) – -4.95764)/.07925 = (.67449 – -4.95764)/.07925 = 71.1