What method should I use to identify which variables differentiate between objects of two different classes?

To illustrate the problem posed by the question: Consider the problem of differentiating between consumers who belong to two different segments. I could use a naive or a sophisticated approach as outlined below.

The question is: what is the benefit to using the sophisticated approach?

Naive Approach

Compute the summary statistics (e.g., means, frequencies) of all the available variables (e.g., income, age etc) and then see how the groups differ on these variables by inspecting the table of summary statistics.

Sophisticated Approach

Use logistic regression to identify which variables (e.g. income, age, etc) predict class membership.

Why use the sophisticated approach over the naive one?

• Could you explain a little more what you mean by 'differentiating' between them? Are you looking for a predictive model for which segment a consumer most likely belongs to based on these variables? Or just to characterize each group? – Scortchi - Reinstate Monica Apr 19 '13 at 13:50
• I am looking ultimately for a predictive model but it seems to me that characterizing (or describing) a group is one way to predict membership (which is what the naive approach uses). – stats_student Apr 19 '13 at 15:17
• Your first approach isn't really well enough defined to be able to assess how it would perform. But one thing it seems to ignore entirely is association between the predictor variables, which can completely reverse an apparent relation. – Scortchi - Reinstate Monica Apr 19 '13 at 16:09
• Sure, it ignores association but we are not interested in causal relationships but simply in prediction. So, I do not see why or how ignoring association is problematic. To take an extreme example- if one variable correlates perfectly with another then it does not matter which one we pick to make predictions as either one would suffice. – stats_student Apr 19 '13 at 16:32
• You've missed my point, which is that the distribution of a response conditional on several predictors jointly can't be inferred from the distribution of that response conditional on each of those predictors marginally. – Scortchi - Reinstate Monica Apr 20 '13 at 1:31

1 Answer

Take a simple case: two categorical predictors, $x_1$ & $x_2$, of segment, $y$.

For $x_1=0$ :

$\begin{array}{ccc} & y=0 & y=1 \\ x_2=0 & 1 & 2 \\ x_2=1 & 4 & 2 \end{array}$

For $x_1=1$ :

$\begin{array}{ccc} & y=0 & y=1 \\ x_2=0 & 2 & 4 \\ x_2=1 & 2 & 1 \end{array}$

The first approach you describe ignores all the data in the body of the three-way contingency table & looks only at the marginal tables:

$\begin{array}{ccc} & y=0 & y=1 \\ x_1=0 & 5 & 4 \\ x_1=1 & 4 & 5 \end{array}$

&

$\begin{array}{ccc} & y=0 & y=1 \\ x_2=0 & 3 & 6 \\ x_2=1 & 6 & 3 \end{array}$

I don't know what predictions you would make from these, say for an individual with $x_1=1$, $x_2=1$ - because you haven't said - but it's clear that any information about association or interaction between the predictors has been lost. For continuous variables you would, I gather, discard still more information by reducing the whole marginal distribution to its mean.

Logistic regression will of course fit $\log\frac{\Pr(Y=1)}{1-\Pr(Y=1)}=\log 2 + x_2\log\frac{1}{4}$, using all information in the body of the table to either estimate parameters or assess model fit; any predictive modelling approach worth its salt will do something similar.

Not that it isn't worthwhile to describe the characteristics of each segment, either as an end in itself or as a prelude to a predictive model.