predict.lm() in a loop. warning: prediction from a rank-deficient fit may be misleading

Question

This R code throws a warning

# Fit regression model to each cluster y <- list()  length(y) <- k vars <- list()  length(vars) <- k f <- list() length(f) <- k  for (i in 1:k) {   vars[[i]] <- names(corc[[i]][corc[[i]]!= "1"])   f[[i]]  <- as.formula(paste("Death ~", paste(vars[[i]], collapse= "+")))   y[[i]]  <- lm(f[[i]], data=C1[[i]]) #training set   C1[[i]] <- cbind(C1[[i]], fitted(y[[i]]))   C2[[i]] <- cbind(C2[[i]], predict(y[[i]], C2[[i]])) #test set } 

I have a training data set (C1) and a test data set (C2). Each one has 129 variables. I did k means cluster analysis on the C1 and then split my data set based on cluster membership and created a list of different clusters (C1[[1]], C1[[2]], ..., C1[[k]]). I also assigned a cluster membership to each case in C2 and created C2[[1]],..., C2[[k]]. Then I fit a linear regression to each cluster in C1. My dependant variable is "Death". My predictors are different in each cluster and vars[[i]] (i=1,...,k) shows a list of predictors' name. I want to predict Death for each case in test data set (C2[[1]],..., C2[[k]). When I run the following code, for some of the clusters.

I got this warning:

In predict.lm(y[[i]], C2[[i]]) : prediction from a rank-deficient fit may be misleading 

I read a lot about this warning but I couldn't figure out what the issue is.

Answer 1

You can inspect the predict function with body(predict.lm). There you will see this line:

if (p < ncol(X) && !(missing(newdata) || is.null(newdata)))      warning("prediction from a rank-deficient fit may be misleading") 

This warning checks if the rank of your data matrix is at least equal to the number of parameters you want to fit. One way to invoke it is having some collinear covariates:

data <- data.frame(y=c(1,2,3,4), x1=c(1,1,2,3), x2=c(3,4,5,2), x3=c(4,2,6,0), x4=c(2,1,3,0)) data2 <- data.frame(x1=c(3,2,1,3), x2=c(3,2,1,4), x3=c(3,4,5,1), x4=c(0,0,2,3)) fit <- lm(y ~ ., data=data)  predict(fit, data2)        1        2        3        4  4.076087 2.826087 1.576087 4.065217  Warning message: In predict.lm(fit, data2) :   prediction from a rank-deficient fit may be misleading 

Notice that x3 and x4 have the same direction in data. One is the multiple of the other. This can be checked with length(fit$coefficients) > fit$rank

Another way is having more parameters than available variables:

fit2 <- lm(y ~ x1*x2*x3*x4, data=data) predict(fit2, data2) Warning message: In predict.lm(fit2, data2) :   prediction from a rank-deficient fit may be misleading 

Answer 2

This warning:

In predict.lm(model, test) :   prediction from a rank-deficient fit may be misleading 

Gets thrown from R's predict.lm. See: http://stat.ethz.ch/R-manual/R-devel/library/stats/html/predict.lm.html

Understand rank deficiency: Ask R to tell you the rank of a matrix:

train <- data.frame(y=c(1234, 325, 152, 403),                     x1=c(3538, 324, 382, 335),                     x2=c(2985, 323, 223, 288),                     x3=c(8750, 322, 123, 935)) test <- data.frame(x1=c(3538, 324, 382, 335),                     x2=c(2985, 323, 223, 288),                     x3=c(8750, 322, 123, 935)) library(Matrix) cat(rankMatrix(train), "\n")   #prints 4 cat(rankMatrix(test), "\n")    #prints 3 

A matrix that does not have "full rank" is said to be "rank deficient". A matrix is said to have full rank if its rank is either equal to its number of columns or to its number of rows (or to both).

The problem is that predict.lm will throw this warning even if your matrices are full rank (not rank deficient) because predict.lm pulls a fast one under the hood, by throwing out what it considers useless features, modifying your full rank input to be rank-deficient. It then complains about it through a warning.

Also this warning seems to be a catch-all for other situations like for example you have too many input features and your data density is too sparse and it's offering up it's opinion that predictions are brittle.

Example of passing full rank matrices, yet predict.lm still complains of rank deficiency

train <- data.frame(y=c(1,2,3,4),                         x1=c(1,1,2,3),                         x2=c(3,4,5,2),                         x3=c(4,2,6,0),                         x4=c(2,1,3,0)                    ) test <- data.frame(x1=c(1, 2,  3,  9),                    x2=c(3, 5,  1, 15),                    x3=c(5, 9,  5, 22),                    x4=c(9, 13, 2, 99)) library(Matrix) cat(rankMatrix(train), "\n")    #prints 4, is full rank, good to go cat(rankMatrix(test), "\n")     #prints 4, is full rank, good to go myformula = as.formula("y ~ x1+x2+x3+x4") model <- lm(myformula, train) predict(model, test)      #Warning: prediction from a rank-deficient fit may be misleading 

workaround:

Assuming predict is returning good predictions, you can ignore the warning. predict.lm offers up it's opinion given insufficient perspective and here you are.

So disable warnings on the predict step like this:

options(warn=-1)      #turn off warnings predict(model, test) options(warn=1)      #turn warnings back on