Mathematical Overview and Summary

The logistic function has its roots in the early 19th century, introduced by Pierre François Verhulst, a Belgian mathematician, in 1838. Verhulst developed the logistic function as a model for population growth, addressing the limitations of exponential growth models by incorporating a carrying capacity that populations cannot exceed. This model, known as the logistic growth model, can be expressed as:

where $P(t)$ is the population at time $t$, $K$ is the carrying capacity, $P_0$ is the initial population size, and $r$ is the growth rate.

The logistic function gained further prominence in the 20th century through its application in various fields, including biology and bio-assay. Joseph Berkson's 1944 paper, "Application of the Logistic Function to Bio-Assay," that we have reviewed was pivotal in popularizing its use for estimating drug potency. Berkson argued that the logistic function was a more statistically manageable and theoretically sound alternative to the probit function, which had been traditionally used based on the assumption that individual susceptibilities to a drug follow a normal distribution.

Traditional Method: Probit Function

The probit function is based on the cumulative distribution function (CDF) of the normal distribution. If we assume that individual susceptibilities to a drug are normally distributed, the proportion $P$ of individuals affected by dose $X$ can be expressed as:

where $\Phi$ is the CDF of the standard normal distribution. The probit model relates the probability $P$ to the dose $X$ by:

The term "probit" is derived from "probability unit.”

Logistic Function

The logistic function is given by:

where $a$ and $b$ are parameters to be estimated from the data. This function models the probability $P$ of an event (e.g., an individual being affected by a drug) given the dose $X$.

Logit Transformation

The logistic function can be linearized using the logit transformation. The logit of a probability $P$ is defined as:

This transformation converts the S-shaped curve of the logistic function into a straight line, which simplifies the analysis and estimation of parameters $a$ and $b$ . The term "logit" combines "logistic" and "unit," referring to the logarithm of the odds.

Why it helps with linearization?

To see how this aligns with the logistic function, let's start with the logistic function itself:

Now, taking the logit of both sides:

Substitute $P(X)$ from the logistic function:

Simplifying the Expression

First, simplify the denominator $1 - P(X)$:

Combine the terms over a common denominator:

Now, the logit function becomes:

Further Simplification

The fractions have a common denominator $1 + e^{-(a + bX)}$, which can be canceled out:

Since $\frac{1}{e^{-(a + bX)}} = e^{a + bX}$, we get:

The natural logarithm $\ln$ and the exponential function $$ e $$ are inverses of each other, so we get:

Conclusion

By applying the logit transformation to the logistic function, we linearize the relationship between the dose $X$ and the probability $P$:

This linearization is particularly useful for statistical analysis, as it allows us to use linear regression techniques to estimate the parameters $a$ and $b$.

Comparison: Logistic vs. Probit Functions

Similarity:

• Both the logistic and probit functions produce S-shaped (sigmoidal) curves.
• They are both used to model the probability of a binary outcome (e.g., success/failure, affected/not affected).

Differences:

• The logistic function uses the logit transformation, while the probit function uses the inverse of the cumulative normal distribution.
• The logistic function is computationally simpler, as it does not require the use of the error function or numerical integration involved in the normal CDF.

Mathematical Properties

1. Logistic Function:
• CDF: $P(X) = \frac{1}{1 + e^{-(a + bX)}}$
• Logit Transformation: $\text{logit}(P) = \ln\left(\frac{P}{1 - P}\right) = a + bX$
2. Probit Function:
• CDF: $P(X) = \Phi\left(\frac{X - \mu}{\sigma}\right)$
• Inverse Probit Transformation: $\text{probit}(P) = \Phi^{-1}(P) = \frac{X - \mu}{\sigma}$

Fitting the Models

Maximum Likelihood Estimation (MLE):

• Both logistic and probit models can be fitted using MLE.
• For the logistic model, we maximize the likelihood of observing the given data under the logistic function.
• For the probit model, we maximize the likelihood under the normal distribution.

Least Squares:

• The logistic function can be linearized using the logit transformation, allowing for least squares fitting.
• The probit function does not linearize as easily, making least squares fitting less straightforward.

Conclusion

Berkson's paper argues that the logistic function, due to its mathematical properties and ease of computation, is a viable and often superior alternative to the probit function for bio-assay purposes. The logistic function's ability to be easily linearized via the logit transformation simplifies parameter estimation and makes it an attractive choice for analyzing dose-response data.