You use regression when you want to fit a model to predict some value or outcome. Multinomial regression is where you have more than one input to predict one output.
Here is an example of regression:
Suppose you had the following data:
Height Weight Age for 100 people.
You might want to see if you can predict someone's age if you are given their height and weight. You call age the dependent variable because it depends on height and weight, and you can height and weight the independent variables.
When you fit your model, you get something like this:
Age = Intercept + A * Height * B * Weight
The model gives you the values of A and B and Intercept, so you can predict the Age of someone if you know their Height and Weight.
This is multinomial regression where the dependent variable is continuous, that is, it can be any value within reason and it varies.
Logistic regression is used when you want a binary outcome, for example 1 meaning success and 0 failure.
Using the same example, you might want to predict someone's sex based on their height, weight and age. In this model, you would have three independent variables making it multinomial and your outcome would be 1 = male, 0 = not male, that is female.
The logistic regression model gives you the probability that, given someone's age, weight and height, that they are male (in this example, the higher the probability, the greater the chance that they person is male).
Again, you get a number of coefficients, but this time you use them to calculate a LOGIT:
LOGIT = Intercept + A * Height + B * Weight * C * Age
The LOGIT is also written as:
LOGIT(P) = Log [( P )/ ( 1 - P )]
So if you fit your model and get your values for A, B, C and the INTERCEPT, you can calculate the LOGIT value from some data and with this, calculate the probability P, that they are male. This probability P is given by:
P = 1 / ( 1 + e**-(LOGIT))
where ** is to the power of.
This is a much better way to model categories or classications, which are often used for understanding risk in banks and marketing, where you want to know the likelihood that someone will buy a product or default on a loan payment.
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