R语言lm函数如何处理因子水平？C_Cdqrls采用何种算法？

Great question! Let’s unpack this step by step to clear up your confusion.

How lm() Handles Factor Levels & the Role of C_Cdqrls #

First, let’s confirm what you already know: when you fit a model with lm() that includes factors, the first step is converting those factors into a dummy variable matrix via model.matrix() (which under the hood calls C_modelmatrix). That part checks out.

Now, the critical piece: lm() is always doing linear regression—full stop—even when your response variable is a factor. The C_Cdqrls function it calls is purely implementing a QR-decomposition-based least squares algorithm to solve the linear system (X\beta = y) for the coefficients that minimize squared error.

When your response y is a factor, lm() first converts it to numeric values (e.g., a 2-level factor becomes 0/1, a multi-level factor becomes integer codes for each level) before passing it to C_Cdqrls. The C function has no idea the original y was a factor—it just sees a numeric vector and solves for the least squares fit on the design matrix X.

Linear Regression vs. Discriminant Analysis: Not Mathematically Equivalent #

Your hunch about ISLR Chapter 4.4’s discriminant analysis is a common point of confusion, but the two methods are fundamentally different:

• Discriminant analysis is a classification method rooted in Bayesian probability. It assumes each class’s features follow a normal distribution, then calculates posterior probabilities to assign observations to classes. Its goal is to maximize separation between groups.
• lm() with a factor response is just linear regression applied to a numeric encoding of the factor. For binary factors, this is a linear probability model (estimating (E[Y|X] = X\beta)), which can produce predicted values outside the 0-1 range. For multi-level factors, it’s fitting a separate linear model for each level’s integer code.

The core mathematical objectives are totally distinct: least squares minimizes prediction error, while discriminant analysis maximizes class likelihood.

Model Matrix Differences: Numeric vs. Factor y #

Let’s use your example to make the model matrix difference concrete. Here’s a reproducible snippet:

set.seed(123)
x <- rnorm(10)
y_num <- rnorm(10)
y_fac <- factor(rep(c("A", "B"), each=5))
z <- rnorm(10)

# Model matrix when y is numeric
mm_num <- model.matrix(z ~ x*y_num)
colnames(mm_num)
# Output: "(Intercept)" "x"           "y_num"       "x:y_num"

# Model matrix when y is a factor
mm_fac <- model.matrix(z ~ x*y_fac)
colnames(mm_fac)
# Output: "(Intercept)" "x"           "y_facB"      "x:y_facB"

When y is numeric, the model matrix includes the raw y values and their interaction with x. When y is a factor, it gets converted to dummy variables (here, y_facB represents the "B" level vs. the reference "A" level), and the interaction term is between x and that dummy variable.

In both cases, though, lm() passes this matrix to C_Cdqrls to run least squares regression—no discriminant analysis is involved at any step.

Why the Confusion? #

It’s easy to mix these up because binary linear regression can sometimes produce coefficient signs similar to logistic regression or linear discriminant analysis. But the underlying math and assumptions are worlds apart: lm() makes no distributional assumptions about the response (beyond the usual linear regression error assumptions), while discriminant analysis relies heavily on normality assumptions for each class.

内容的提问来源于stack exchange，提问作者Christoph