# "Pick 3 out of 12 statements" - is linear regression possible in this case?

Suppose I'm trying to figure out what makes people shop at a particular store. I conduct a survey where they are asked:

1. how often they shop there
2. to pick 3 out of a list of 12 characteristics that they most associate with the store, e.g. "well-known brand", "good quality", "good loyalty programs", etc.

I was thinking of doing a linear regression of Q1 on Q2. in order to figure out which of these factors are significant when making a decision. Denote by $Y$ the frequency and by $X_i$ a binary variable indicating whether the respondent associated factor $i$ with the store:

$$Y \sim 1 + X_1 + ... + X_{12}$$

The problem with this seems to be perfect multicollinearity: $\sum_i X_i = 3$. Curiously, when I ran the regression in R, I checked the model with car::vif and didn't get alarmingly large values (mostly around 2 for each coefficient).

Apart from that, it's difficult to interpret the results... the values of the coefficients don't really correspond to anything, they only give a rough idea of the factors' relative importance. For example, if the average value of $Y$ is 4.5, I get an intercept of 4.0 and the coefficients all approximately 0.1-0.2, so that picking three of them would roughly correspond to the average. In a sense, the regression compares the respondents to a hypothetical person who "didn't select anything", even though that's impossible.

• Is this model even appropriate at all, given the collinearity problem? If not, what might be a good alternative?
• I was wondering if forcibly removing the intercept from the model might be a good idea here to try and salvage it?
• You didn't need an intercept in the first place, because it's automatically included in the $X_i,$ as you observed. Thus, "forcibly removing" it doesn't change the model--it only causes R to use a different model matrix. If indeed the variables are collinear (that is, everybody has provided three answers), then R has already dropped one of your variables already--most likely $X_{12}.$
– whuber
Commented Apr 5, 2018 at 14:47
• @whuber So do I have multicollinearity here or not? The answer says I don't, but I can't see how the definition isn't satisfied (for all observations $-3 + \sum X_i = 0$)... on the other hand R didn't remove any variables (I checked), and the vif doesn't indicate multicollinearity.. what gives? Commented Apr 6, 2018 at 8:51
• If R removed no variables, then the situation is not as you described it!
– whuber
Commented Apr 6, 2018 at 13:24
• @whuber I suppose there might've been some people who failed to actually select three options. You're saying that if this weren't the case, R would detect it? Commented Apr 6, 2018 at 14:43
• You can easily check: tabulate the values of $\sum X_i.$ If you only get the value $3$, then you have collinearity.
– whuber
Commented Apr 6, 2018 at 14:54

• I would agree with your interpretation...though I would say the sum of the 3 chosen coefficients gives the average deviation for such respondents from the mean frequency. As for multicollinearity, you are correct...but you don't have that here. For example, there is no way to have $X1 = c_2·X_2 + c_3·X_3 + ··· + c_{12}X_{12}$ with the data as you've described it. Commented Apr 5, 2018 at 15:14