Accounting for the presence of a variable in regression?

So, as an example, let's say that I have data on a small auction that I conducted.
Let's say I am selling a car and there are 5 bidders, and I have data on each of their bids, who won, who participated, etc. Now, let's say that I have 5 different cars, so the auction occurs 5 separate times. For example, in one auction, all 5 buyers participate while in another only 3 buyers participate. I am wondering how to model out this situation for expected revenue (continuous variable -> linear model), while accounting for which buyers were present. I am modeling the data for all 5 of the auction together, but want to account for the presence or absence of a buyer in an auction. Is this simply a matter of including a dummy variable for each buyer, where 0 is absent and 1 is present? Or is there some alternative solution like tree models or other advanced tools that can be utilized to examine this problem.

EDIT 1: I have more than 5 "auctions", but I just specified that number to keep things simple.

EDIT 2: I'd be interested in hearing about non dummy-variable solutions to this problem.

I'm afraid your example as it is put is fundamentally undoable. If you have only five cars to sell, you simply don't have enough data to work out the separate effects of the presence or absence of each of your five buyers, regardless of what analytical technique you use.

If you had more data (lots of auctions with different combinations of the five buyers), you could model it as you say by setting up a dummy variable for each buyer, where 0 is absent and 1 is present. I don't think any more sophisticated approach would be necessary. The one complication would be that the price of the car is unlikely to be normally distributed, so you need to make sure that any inference you do takes that into account.

• Thanks. Yeah, in real life, there are actually more than 5 data points. Jan 19 '12 at 18:19

Rather than having a variable for each buyer, why don't you just make a variable that is the total number of times a buyer participated? If you look at the p-value of that variable's co-efficient, you'll know whether the number of times someone participated in the auction is significant for predicting revenue or not. If its not significant, than it probably doesn't matter how often they participate (maybe it matters more some how much money they choose to shell out if they make one big purchase over other smaller ones).

The next thing you can do is interpret the coefficients. I personally find this a little trickier for a model with a continuous response compared to a logit or cox model (so please look into this further). The co-efficient should tell you: for each unit increase in a variable, the response will increase by the B1 times (where B1 is the coefficient). I would interpret the B1 value as:

• If the coefficient is significant and positive than the more times someone participates the more revenue you'll get (and the beta might indicate by how much). The further from 0 the larger the impact.
• The opposite is true if your beta is negative (the more times you participate the less you contribute to revenue; perhaps some other factor is more predictive).
• By "participate" do you mean put in a bid in the auction? So if a buyer puts in 6 bids, their value is 6 for that auction? So you would still need a variable for each buyer, it just will have a bit more information than a dummy. Jan 19 '12 at 20:05
• My interpretation of "participate" was whether someone showed up, not the number of bids that were placed. Jan 19 '12 at 20:14