 # Estimated Logistic Regression

Qualitative (categorical) dependent variables are the dummy variables used as dependent variables rather than independent variables. Recall that a dummy variable is a variable that takes on the value 0 or 1.The logistic transformation is the most widely used. It takes the probability that an event happens “p” divided by the probability that the event will not happen “1-p”.. A logit is the natural logarithm of the odds of an event happening. The logistic transformation tends to linearize the relationship between the independent and dependent variables.

For example, if the probability of a company going bankrupt is 0.6, 0.6/ (1- 0.6)=1.5, the odds of a company becoming bankrupt are 1.5 times more than the probability of the company not becoming bankrupt. Hence, we should use the logit model or discriminant analysis when estimating the probability of bankruptcy. The event probability can be calculated as follows:

If p is binary, we use the maximum likelihood method to estimate logistic regression coefficients instead of using least squares. The maximum likelihood method maximizes the likelihood function for the data. The Bernoulli distribution is chosen as the probability distribution because p is binary. The maximum likelihood method is iterative. Each iteration will result in a higher log-likelihood until the difference in the log-likelihood of two successive iterations is the same. At this point, the iterating process will stop.

## Interpreting Logistic Regression Models

In a logit model, the slope coefficient is the change in the logit that the event happens per unit change in the independent variable. The exponent of the slope coefficient is the ratio of odds that the event will happen with a unit increase in the independent variable. For a logit regression, the test of the hypothesis regression coefficient is significantly different from zero is the same as that of an ordinary linear regression. The overall performance of a logit regression can be evaluated by examining the likelihood chi-square test statistic. Since logistic regression cannot be fitted using at least-square approach, logistic regression has no equivalent measure for the coefficient of determination. Researchers have proposed $$Pseudo-R^{2}$$ to compare different specifications of the same model but are not suitable when comparing models with different datasets.

## Question

Which of the following measures is least likely used to interpret a logistic regression model:

1. $$R^{2}$$
2. $$Pseudo-R^{2}$$
3. P-value

### Solution

The Correct answer is A. $$R^{2}$$ is not used because logistic regression cannot be
fitted using a least-square approach.

B is incorrect. The $$Pseudo-R^{2}$$ is used in logistic regression to
compare different specifications of the same model and is an alternative for $$R^{2}$$ .

C is incorrect. The p-value is used to evaluate the overall statistical significance of
a model.

LOS 2(n) Interpret an estimated logistic regression.

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