# Why does the intercept column in model.matrix replace the first factor?

I'm trying to convert my factor column to dummy variables:

str(cards$pointsBin) # Factor w/ 5 levels ".lte100",".lte150",..: 3 2 3 1 4 4 2 2 4 4 ... labels <- model.matrix(~ pointsBin, data=cards) head(labels) # (Intercept) pointsBin.lte150 pointsBin.lte200 pointsBin.lte250 pointsBin.lte300 # 741 1 0 0 0 0 # 407 1 1 0 0 0 # 676 1 0 0 1 0 # 697 1 1 0 0 0 # 422 1 0 1 0 0 # 300 1 0 1 0 0  There is no column for the first value of my factor (".lte100"), which is what the first row should be categorized as. How do I get this data back? And what does the Intercept column that seems to be all 1's mean? • When you have "K" dummy variables then your resulting model will have a.) the intercept term (which is a column of ones) and b.) "K-1" additional columns. The reason is because otherwise the columns of the resulting matrix would not be linearly independent (and, as a result, you wouldn't be able to do OLS). Oct 1, 2015 at 5:34 • Why 'not meaningful'? It's the same model with the same goodness of fit, just parameterized in a different way. Oct 1, 2015 at 5:44 • @digitgopher: When you run a regression and end up with a model like this:$\hat{y} = \beta_{0} + \beta_{1}*x_{1}$, you're technically ending up with a model like this:$\hat{y} = \beta_{0}*x_{0} + \beta_{1}*x_{1}$, where this new term$x_{0}\$ is always equal to "1" (hence the column of ones). If you were to eliminate this column of ones when running a regular regression, you'd end up with a biased model since you'd, in effect, be forcing every single model through the origin. Oct 1, 2015 at 7:33
• @SteveS: In fact R's so friendly that if you try remove the intercept - 1 when you have a single categorical predictor represented as a factor (as in this question), it'll assume you don't really mean that & switch to using sum-to-zero coding; which is of course just a different parametrization. Too friendly, if you ask me. Oct 1, 2015 at 8:56
• @SteveS: Thanks. I should have checked: it switches to cell-means coding. It doesn't do what you might expect, which is fit the forced-through-the-origin model you quite rightly warn against (it will do that though, when the column's of numeric type). Oct 1, 2015 at 11:33

Consider the following:

require(mlbench)



The first two commands are identical. The last two commands specifies not to produce the intercept and keeps the two dummy variables produced.

• Thank you - this explains the code I wanted. But it doesn't explain why or help me understand. Oct 2, 2015 at 3:16
• @digitgopher: See ats.ucla.edu/stat/r/library/contrast_coding.htm for an explanation of reference-level coding & other coding schemes. Oct 2, 2015 at 9:45

In statistics, when we have a factor variable with $$k$$ levels, we need to convert it to $$k - 1$$ indicator variables. We choose one level as the baseline, and then have an indicator variable for each of the remaining levels.

First let me explain why this isn't throwing away any information. Say there are levels A, B, C, and we have $$I_B$$ and $$I_C$$, the indicators for being in level B and C. An individual is in level A if and only if $$I_B = 0$$ (not in B) and $$I_C = 0$$ (not in C). So we still have kept track of the individuals in level A. This works for any number of levels.

Now as you note we could code the same information with three indicator variables $$I_A$$, $$I_B$$ and $$I_C$$. The reason why we don't is known as multicollinearity. In short, the columns of the regression matrix won't be linearly independent. This means the matrix $$X^T X$$ is not invertible, so we can't perform linear regression, as we need to calculate this inverse to calculate the regression estimates $$\hat{\beta} = \left(X^T X\right)^{-1}X^T y$$.

To take a really simple example, say we have two individuals, one in each of two levels A and B, and an intercept term in our regression.

The regression matrix $$X$$ will be $$\begin{equation} X = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix} \end{equation}$$ Then \begin{align} X^T X & = \begin{bmatrix} 1 & 1\\ 1 & 0\\ 0 & 1\\ \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix}\\ & = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\\ \end{bmatrix} \end{align}

You can then see that adding the second and third columns of $$X^T X$$ gives the first column, so the inverse can't be computed.

This applies in general the Wikipedia page on multicollinearity gives more of an explanation.

If there is an exact linear relationship (perfect multicollinearity) among the independent variables, at least one of the columns of $$X$$ is a linear combination of the others, and so the rank of $$X$$ (and therefore of $$X^T X$$) is less than $$k+1$$, and the matrix $$X^T X$$ will not be invertible.

It is ok to include an indicator for all levels of the factor if you don't include the intercept term, and this encodes the exact same information. (The intercept term minus all other indicators gives the indicator for the baseline factor). But if you have more than one factor in the model this runs into the multicollinearity problem, so you can only include all levels if there is only one factor variable and no intercept term in the model.