NA produced in linear regression model

Question

I have read similar posts to this but my problem is not resolved by the answers given. I want to do a v simple linear regression to see if bite incidence is related to district, zone (vacc or control) and year. As you can see in the output one of the districts RORYA is given NA coefficients, and I get the message "Coefficients: (1 not defined because of singularities)". I have read up on this and it seems its to do with co-linearity of factors. One solution given is to add -1 to the call, which removes the intercept but does not solve my problem as RORYA district still has NAs in the summary output.

Another solution I have tried is changing the order of the explanatory variables in the call. This does change things...Rorya district suddenly has coefficients but the Zone variable becomes NA'd. Neither of which is good as I would like a coefficent for all the explanatory variables.

I was wondering whether anyone might know why this is happening and whether there is a solution to this problem so that all the variables can have coefficients?

Thanks in advance.

A Reproducible example:

df <- structure(list(DISTRICT = structure(c(1L, 6L, 5L, 3L, 2L, 4L, 
1L, 6L, 5L, 3L, 2L, 4L, 1L, 6L, 5L, 3L, 2L, 4L, 1L, 6L, 5L, 3L, 
2L, 4L), .Label = c("BUNDA", "MASWA", "MUSOMA", "RORYA", "SERENGETI", 
"TARIME"), class = "factor"), zone = structure(c(2L, 2L, 2L, 
1L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 
2L, 2L, 1L, 1L, 1L), .Label = c("c", "v"), class = "factor"), 
year = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 
2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L), .Label = c("2010", 
"2011", "2012", "2013"), class = "factor"), bites = c(7.461327937, 
NA, NA, NA, 35.16164185, 26.39109338, 57.89990479, 1.47191729, 
3.608371422, 51.36718605, NA, 16.21167165, 46.85713945, 15.89670673, 
5.212092054, 259.8137381, 30.80276062, 20.73585909, 10.44585911, 
9.420270656, 7.617673001, 307.4586643, 27.31565565, 30.16124958
), deaths = c(0, NA, NA, NA, 0, 1.508062479, 0.298453117, 
0, 0, 0, NA, 2.262093719, 0.298453117, 0.294383458, 0, 2.233355915, 
0.581184163, 1.131046859, 0.298453117, 0.588766916, 1.202790474, 
2.977807887, 0, 1.885078099)), .Names = c("DISTRICT", "zone", 
"year", "bites", "deaths"), row.names = c(NA, -24L), class = "data.frame")

Code:

summary(df )
names(df)
attach(df)
is.numeric(year)
df$year  <- as.factor(as.character(df$year))
is.factor(df$year)

model1 <- lm(bites ~   zone + DISTRICT-1 +year, data = df)
summary(model1)

> sessionInfo()
R version 3.1.0 (2014-04-10)
Platform: x86_64-apple-darwin13.1.0 (64-bit)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] grid      stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ggplot2_1.0.0

loaded via a namespace (and not attached):
[1] colorspace_1.2-4 digest_0.6.4     gtable_0.1.2     MASS_7.3-34      munsell_0.4.2   plyr_1.8.1       proto_0.3-10     Rcpp_0.11.2     
[9] reshape2_1.4     scales_0.2.4     stringr_0.6.2    tools_3.1.0  

Answer 1

It seems that some of the regressors are a linear combination of other regressors, since the rank of the fitted model is lower than the rank of the matrix of regressors:

model1$rank
# [1] 9
ncol(model.matrix(model1))
# [1] 10

The matrix X'X is not invertible (the determinant is zero), as required to obtain the ordinary least squares estimates:

det(crossprod(model.matrix(model1)))
# [1] -4.069976e-11

By default, this issue is not reported as an error by lm, but it can be caught with the option singular.ok = FALSE:

lm(bites ~  zone + DISTRICT-1 +year, data = df, singular.ok = FALSE)
# Error in lm.fit(x, y, offset = offset, singular.ok = singular.ok, ...) : 
#   singular fit encountered

The function lm does not include all the variables when they are defined as factors in order to avoid multicollinearity among them, but in this case it seems that the problem still remains.

You should probably need to remove the regressor zone. You may get further clues in this post.