How to Understand a Multinomial Model
The Basics

Multinomial models look all big and scary and everything, but the idea is really pretty simple.  They refer to situations in which there can be multiple causes for a single event and allow you to estimate in the idenpendent contribution of each of those causes.

Consider the following cheesy example.  Say Todd asks Betty to go to the homecoming dance with him.  What's the probability that Betty will say yes? To come up with an answer to that question, you might want to consider some of the reasons Betty might have for saying yes:

Each of these different reasons for saying "yes" could be represented in a tree diagram:

If any of those three things happens she'll say yes.  The probability of her saying "yes" then depends on the independent probability of each of those things happening.  So there's some probability that Betty really likes Todd.  Let's call that probability "L".  Similarly let's call the probability that she's desperate "D" and let's call the probability that she's have mercy on Todd "M".

For each of those cases you can also define the probability of that thing not happening.  The probability that Betty doesn't like Todd is one minus the probability of liking Todd (1-L).  Why?  Liking Todd and not liking Todd represent all the possibilities.  She either likes him or she doesn't.  So if there's an 80% chance she likes Todd, there must be a 20% (1-.80) chance she doesn't like Todd. So we can also define these probabilities: That allows us to redraw the original diagram, but this time include variables representing the different probabilities.

Notice, that in this theory of Betty's behavior, her being desperate only comes into play if she doesn't like Todd in the first place.  That makes, sense, if she likes Todd, she will go out with him even if she's not desperate.  So we can say there are three ways Betty could say "Yes":

Mathematically these can be represented as probabilities in the following way
 
L Probability of liking Todd
(1-L)D Probability that she both doesn't like Todd and is desperate
(1-L)(1-D)M Probability that she doesn't like Todd, is not desperate, and nonetheless takes mercy on him
 
Notice that anytime you follow two or more branches, you multiply their probabilities together.  So (1-L) times D is the probability that the events Betty doesn't like Todd and Betty is desperate both occur at the same time.  The total probability of Betty saying "yes" to Todd is determined by adding up all the ways Betty could say "yes".
P(Betty says "yes") = L + (1-L)D + (1-L)(1-D)M
Translation: Betty could say yes because she likes Todd [i.e. L], because she doesn't like Todd but she's desperate [i.e. (1-L)D], or because she doesn't like Todd, isn't desperate, but has mercy on him [i.e. (1-L)(1-D)M].

Deriving Parameter Estimates

Once you've understood the above section (and ask questions if you don't) the next thing we need to do is figure out how multinomial models allow you to derive estimates of the parameters.  The symbols above (i.e. L, D, M) are called parameters.  The real goal of multinomial modeling is to derive estimates of the parameters.  The estimates will tell you the probability that Betty likes Todd, the probability that she's desperate etc.  In other words, all we know is that Betty said "yes", how can we figure out why she said "yes". (Or in reality, why a bunch of Bettys would say yes to a bunch of Todds).

This is done by setting up situations where according to your theory the tree diagram would not be quite the same any more.  For instance, imagine Todd asked Betty out, but before that happened, you let Betty know that there were other girls Todd could go out with who would say "yes".  That would eliminate the need for Betty to show mercy on him, and the only reason to say "yes" now is either because she likes him or because she's desperate.

The equation representing the probability of saying "yes" now becomes:

P(Betty says "yes") = L + (1-L)D
You could also set up a situation in which you let Betty think there are other guys who will ask her out even if she says "no" to Todd.  This would rule out desperation as a motive.  Now the diagram would look like this:
P(Betty says "yes") = L + (1-L)M
Having set up these three different situations you can now run the experiment (on multiple Betty's and Todd's) using the three different situations.  Let's say you ran the experiment on 100 Betty-Todd combos in each of the three situations.  You might find that So you end up with three equations representing these results: To gain estimates of the three parameters, a model fitting program is used.  The program basically does the following things: Once that's done, the program provides you with the best estimate of the parameters you were trying to estimate (i.e. L, D, M).  Obviously, this is kind of a silly example, but multinomial modeling in principle can be used to model all sorts of interesting psychological phenomenon.

Summary: Understanding the "language" of multinomial models

From what we've just seen you can use to following basic rules when interpretting a multinomial model.

L
This is the probability that Betty likes Todd.
This is the probability that Betty does not like Todd
P(Betty says "yes") = L + (1-L)M
This means Betty will say "yes" if she likes Todd or
if she doesn't like Todd but has mercy on him.
Final Advice on Understanding Multinomial Models

First, multinomial models aren't just math for math sake. They represent interesting psychological processes.  So before you proceed with a manuscript, always make sure you understand what those processes are.  Be prepared to take notes as you read, and for every symbol (i.e. parameter) write down what the symbol stands for in your notes.

If a tree diagram isn't presented, draw your own.  Any time you see a letter by itself, that represents the probability of that process happening.  Any time you see one minus a letter, that represents the probability of that process not happening (there are more complicated cases too, but we won't get into them here).

Once you've drawn the tree, make sure that you understand the difference between the trees for the different situations that they set up.  There will always be more than one equation (or tree) representing different situations.  Try to understand how those situations differ (e.g. if she has other prospects there's no need for her to be desperate).  That's a big part of understanding the model.

Once you've got the basics down, reading about these models will be a snap. They look intimidating sometimes, but they really aren't so difficult to understand once you get some practice with them.