R，线性回归，lm，约

Question

I would like use the linear regression to estimate the Concentration from the count using linear regression this is a sample of my dataset:

Concentration   count#0
Ctcf                    3153
Err                 2228
Nkx3-2              4
Isl/                    6
Engrailed               10
Dr                  14
Usf                 461
Dach1/Dac               4185
POS_C(8)    139664      1143
POS_A(128)  2234624     8897
POS_F(0.125)    2182            20
POS_D(2)    34916           220
POS_B(32)   558656      3359
POS_E(0.5)  8729            21

I am wondering if is better to use lm and then predict or to use approx . and approxfun ? I am not an expert in Statistics and I didn't find any explanation on Internet. Thanks!

Answer 1

lm is what you use if you want to fit an ordinary linear regression (LR). If you believe that your response can be well described by a linear combination of your predictors then LR might be appropriate. You don't need the data to be normal for an LR to work but you do need (approximate) normality if you're going to compute test statistics for the parameters and such. Also if you're interested in inference and coefficient interpretation don't forget to check the usual diagnostics (residuals have mean 0, common variance and no trends, outliers, multicollinearity, normality, etc).

The actual model for an LR is Y = X %*% beta + e where Y , beta , and e are vectors, X is a matrix, and %*% denotes matrix multiplication. This notation assumes that the first column of X is all 1's. By default lm uses a QR decomposition which allows it to avoid computing the inverse of t(X) %*% X and even t(X) %*% X , which is a big time saver if X is large.

lm finds [but not by direct computation] solve(t(X) %*% X) %*% t(X) %*% Y which gives us the unique (provided X is full rank) estimate of beta .

You definitely do not want to use anything else if a plain LR is all you want.