Why probability should be converted to logarithm of odds (logit) in logistic regression analysis?

In logistic regression analysis, probability is converted to odds, p/(1-p), and odds is converted to logarithm. Binomial distribution, either event of interest happens or doesn’t happen, is analysed by multiple regression analysis.

Probability is between 0 and 1. Logarithm of odds (logit) diverges from minus infinity to plus infinity. See charts to understand.

Probability is real between 0 and 1. Take probability horizontal axis and odds vertical axis, respectively. Vertical axis ranges from 0 to infinity, as following chart.

Fig1. probability and logit
Fig1. probability and odds

Next, take odds horizontal axis and logarithm of odds vertical axis, respectively. Vertical axis ranges from minus infinity to plus infinity as following chart.

Fig2. logit and logarithm of logit
Fig2. odds and logarithm of odds (logit)

At last, take probability horizontal axis and logit vertical axis, respectively. Although probability ranges only from 0 to 1, logit diverges all real number as following chart.

Fig3. probability and logarithm of logit
Fig3. probability and logit

How to calculate Fisher’s exact test with logarithm?

Chi-square test is known to compare between ratios with two-by-two table. But you couldn’t use chi-square test if total number was smaller than 20 or expected value was smaller than 5.

Even if you couldn’t use chi-square test, you could use Fisher’s exact test and calculate accurate p-value. Although the test has reliability, it requires huge amount of calculation with factorial function and software may overflow. You would easily calculate it with conversion to the logarithm first. Next, you could add or subtract the logarithm. At last, you could convert the result to the power of e, the base of natural logarithm.

  TRUE FALSE Marginal total
POSITIVE a b a + b
NEGATIVE c d c + d
Marginal total a + c b + d N 
\displaystyle \begin{array} {rcl} P &=& \frac{(a+b)!(c+d)!(a+c)!(b+d)!}{N!a!b!c!d!}\vspace{0.2in}\\&=& \exp \left[ LN \left( \frac{(a+b)!(c+d)!(a+c)!(b+d)!}{N!a!b!c!d!} \right) \right]\vspace{0.2in}\\ &=& \exp [ LN((a+b)!) + LN((c+d)!) + LN((a+c)!) + LN((b+d)!)\vspace{0.2in}\\& & - LN(N!) - LN(a!) - LN(b!) - LN(c!) - LN(d!) ]\end{array}