# How can adding a 2nd IV make the 1st IV significant?

I have what is probably a simple question, but it is baffling me right now, so I am hoping you can help me out.

I have a least squares regression model, with one independent variable and one dependent variable. The relationship is not significant. Now I add a second independent variable. Now the relationship between the first independent variable and the dependent variable becomes significant.

How does this work? This is probably demonstrating some issue with my understanding, but to me, but I do not see how adding this second independent variable can make the first significant.

• This is a very widely discussed topic on this site. This is probably due to collinearity. Do a search for "collinearity" and you will find dozens of relevant threads. I suggest reading some of the answers to stats.stackexchange.com/questions/14500/… – Macro May 14 '12 at 18:16
• possible duplicate of significant predictors become non-significant in multiple logistic regression. There are many threads this is effectively a duplicate of -- that was the closest one I could find in under two minutes – Macro May 14 '12 at 18:20
• This is sort of the opposite problem of the one in the thread @macro just found, but the reasons are very similar. – Peter Flom - Reinstate Monica May 14 '12 at 18:28
• @Macro, I think you're right that this may be a duplicate, but I think that the issue here is slightly different from the 2 questions above. The OP doesn't refer to the significance of the model-as-a-whole, nor to variables becoming non-significant w/ additional IV's. I suspect this is not about multicollinearity, but about power or possibly suppression. – gung - Reinstate Monica May 14 '12 at 18:29
• also, @gung, suppression in a linear models only occurs when there is collinearity - the difference is about interpretation, so "this is not about multicollinearity but about possibly suppression" sets up a misleading dichotomy – Macro May 14 '12 at 18:44

Although collinearity (of predictor variables) is a possible explanation, I would like to suggest it is not an illuminating explanation because we know collinearity is related to "common information" among the predictors, so there is nothing mysterious or counter-intuitive about the side effect of introducing a second correlated predictor into the model.

Let us then consider the case of two predictors that are truly orthogonal: there is absolutely no collinearity among them. A remarkable change in significance can still happen.

Designate the predictor variables $X_1$ and $X_2$ and let $Y$ name the predictand. The regression of $Y$ against $X_1$ will fail to be significant when the variation in $Y$ around its mean is not appreciably reduced when $X_1$ is used as the independent variable. When that variation is strongly associated with a second variable $X_2$, however, the situation changes. Recall that multiple regression of $Y$ against $X_1$ and $X_2$ is equivalent to

1. Separately regress $Y$ and $X_1$ against $X_2$.

2. Regress the $Y$ residuals against the $X_1$ residuals.

The residuals from the first step have removed the effect of $X_2$. When $X_2$ is closely correlated with $Y$, this can expose a relatively small amount of variation that had previously been masked. If this variation is associated with $X_1$, we obtain a significant result.

All this might perhaps be clarified with a concrete example. To begin, let's use R to generate two orthogonal independent variables along with some independent random error $\varepsilon$:

n <- 32
set.seed(182)
u <-matrix(rnorm(2*n), ncol=2)
u0 <- cbind(u[,1] - mean(u[,1]), u[,2] - mean(u[,2]))