I am trying to tweak contrast coding on a linear model where I want to know if each level of a factor is significantly different from the grand mean.
Let’s say the factor has levels "A", "B" and "C". The most common control-treatment contrasts obviously set "A" as the reference level, and compare "B" and "C" to that. This is not what I want, because level "A" does not show up in model summary.
Deviation coding also doesn’t seem to give me what I want, since it sets the contrast matrix for level "C" to [-1,-1,-1], and now this level does not show up in model summary.
set.seed(1)
y <- rnorm(6, 0, 1)
x <- factor(rep(LETTERS[1:3], each = 2))
fit <- lm(y ~ x, contrasts = list(x = contr.sum))
summary(fit)
In addition, the reported level names have changed from "A", "B" to "1" and "2".
Call:
lm(formula = y ~ x, contrasts = list(x = contr.sum))
Residuals:
1 2 3 4 5 6
-0.405 0.405 -1.215 1.215 0.575 -0.575
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.02902 0.46809 -0.062 0.954
x1 -0.19239 0.66198 -0.291 0.790
x2 0.40885 0.66198 0.618 0.581
Residual standard error: 1.147 on 3 degrees of freedom
Multiple R-squared: 0.1129, Adjusted R-squared: -0.4785
F-statistic: 0.1909 on 2 and 3 DF, p-value: 0.8355
Am I missing something? Should I add a dummy variable that is equal to the grand mean, so that I can use this as the reference level?
I saw a similar question (but maybe more demanding) asked last year, but without solution (yet): models with 'differences from mean' for all coefficients on categorical variables; get 'contrast coding' to do it?.
The accepted answer here works, but the author has not provided an explanation. I have asked about it on the stats SE: https://stats.stackexchange.com/questions/600798/understanding-the-process-of-tweaking-contrasts-in-linear-model-fitting-to-show
This answer shows you how to obtain the following coefficient table:
# Estimate Std. Error t value Pr(>|t|)
#(Intercept) -0.02901982 0.4680937 -0.06199574 0.9544655
#A -0.19238543 0.6619845 -0.29061922 0.7902750
#B 0.40884591 0.6619845 0.61760645 0.5805485
#C -0.21646049 0.6619845 -0.32698723 0.7651640
Amazing, isn't it? It mimics what you see from summary(fit), i.e.,
# Estimate Std. Error t value Pr(>|t|)
#(Intercept) -0.02901982 0.4680937 -0.06199574 0.9544655
#x1 -0.19238543 0.6619845 -0.29061922 0.7902750
#x2 0.40884591 0.6619845 0.61760645 0.5805485
But now we have all factor levels displayed.
lm summary does not display all factor levels?In 2016, I answered this Stack Overflow question: `lm` summary not display all factor levels and since then, it has become the target for marking duplicated questions on similar topics.
To recap, the basic idea is that in order to have a full-rank design matrix for least squares fitting, we must apply contrasts to a factor variable. Let's say that the factor has N levels, then no matter what type of contrasts we choose (see ?contrasts for a list), it reduces the raw N dummy variables to a new set of N - 1 variables. Therefore, only N - 1 coefficients are associated with an N-level factor.
However, we can transform the N - 1 coefficients back to the original N coefficients using the contrasts matrix. The transformation enables us to obtain a coefficient table for all factor levels. I will now demonstrate how to do this, based on OP's reproducible example:
set.seed(1)
y <- rnorm(6, 0, 1)
x <- factor(rep(LETTERS[1:3], each = 2))
fit <- lm(y ~ x, contrasts = list(x = contr.sum))
In this example, the sum-to-zero contrast is applied to factor x. To know more on how to control contrasts for model fitting, see my answer at How to set contrasts for my variable in regression analysis with R?.
For a factor variable of N levels subject to sum-to-zero contrasts, we can use the following function to get the N x (N - 1) transformation matrix that maps the (N - 1) coefficients estimated by lm back to the N coefficients for all levels.
ContrSumMat <- function (fctr, sparse = FALSE) {
if (!is.factor(fctr)) stop("'fctr' is not a factor variable!")
N <- nlevels(fctr)
Cmat <- contr.sum(N, sparse = sparse)
dimnames(Cmat) <- list(levels(fctr), seq_len(N - 1))
Cmat
}
For the example 3-level factor x, this matrix is:
Cmat <- ContrSumMat(x)
# 1 2
#A 1 0
#B 0 1
#C -1 -1
The fitted model fit reports 3 - 1 = 2 coefficients for this factor. We can extract them as:
## coefficients After Contrasts
coef_ac <- coef(fit)[2:3]
# x1 x2
#-0.1923854 0.4088459
Therefore, the level-specific coefficients are:
## coefficients Before Contrasts
coef_bc <- (Cmat %*% coef_ac)[, 1]
# A B C
#-0.1923854 0.4088459 -0.2164605
## note that they sum to zero as expected
sum(coef_bc)
#[1] 0
Similarly, we can get the covariance matrix after contrasts:
var_ac <- vcov(fit)[2:3, 2:3]
# x1 x2
#x1 0.4382235 -0.2191118
#x2 -0.2191118 0.4382235
and transform it to the one before contrasts:
var_bc <- Cmat %*% var_ac %*% t(Cmat)
# A B C
#A 0.4382235 -0.2191118 -0.2191118
#B -0.2191118 0.4382235 -0.2191118
#C -0.2191118 -0.2191118 0.4382235
Interpretation:
The model intercept coef(fit)[1] is the grand mean.
The computed coef_bc is the deviation of each level's mean from the grand mean.
The diagonal entries of var_bc gives the estimated variance of these deviations.
We can then proceed to compute t-statistics and p-values for these coefficients, as follows.
## standard error of point estimate `coef_bc`
std.error_bc <- sqrt(diag(var_bc))
# A B C
#0.6619845 0.6619845 0.6619845
## t-statistics (Null Hypothesis: coef_bc = 0)
t.stats_bc <- coef_bc / std.error_bc
# A B C
#-0.2906192 0.6176065 -0.3269872
## p-values of the t-statistics
p.value_bc <- 2 * pt(abs(t.stats_bc), df = fit$df.residual, lower.tail = FALSE)
# A B C
#0.7902750 0.5805485 0.7651640
## construct a coefficient table that mimics `coef(summary(fit))`
stats.tab_bc <- cbind("Estimate" = coef_bc,
"Std. Error" = std.error_bc,
"t value" = t.stats_bc,
"Pr(>|t|)" = p.value_bc)
# Estimate Std. Error t value Pr(>|t|)
#A -0.1923854 0.6619845 -0.2906192 0.7902750
#B 0.4088459 0.6619845 0.6176065 0.5805485
#C -0.2164605 0.6619845 -0.3269872 0.7651640
We can also augment it by including the result for the grand mean (i.e., the model intercept).
## extract statistics of the intercept
intercept.stats <- coef(summary(fit))[1, , drop = FALSE]
# Estimate Std. Error t value Pr(>|t|)
#(Intercept) -0.02901982 0.4680937 -0.06199574 0.9544655
## augment the coefficient table
stats.tab <- rbind(intercept.stats, stats.tab_bc)
# Estimate Std. Error t value Pr(>|t|)
#(Intercept) -0.02901982 0.4680937 -0.06199574 0.9544655
#A -0.19238543 0.6619845 -0.29061922 0.7902750
#B 0.40884591 0.6619845 0.61760645 0.5805485
#C -0.21646049 0.6619845 -0.32698723 0.7651640
We can also print this table with significance stars.
printCoefmat(stats.tab)
# Estimate Std. Error t value Pr(>|t|)
#(Intercept) -0.02902 0.46809 -0.0620 0.9545
#A -0.19239 0.66199 -0.2906 0.7903
#B 0.40885 0.66199 0.6176 0.5805
#C -0.21646 0.66199 -0.3270 0.7652
Emm? Why are there no stars? Well, in this example all p-values are very large. The stars will show up if p-values are small. Here is a convincing demo:
fake.tab <- stats.tab
fake.tab[, 4] <- fake.tab[, 4] / 100
printCoefmat(fake.tab)
# Estimate Std. Error t value Pr(>|t|)
#(Intercept) -0.02902 0.46809 -0.0620 0.009545 **
#A -0.19239 0.66199 -0.2906 0.007903 **
#B 0.40885 0.66199 0.6176 0.005805 **
#C -0.21646 0.66199 -0.3270 0.007652 **
#---
#Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Oh, this is so beautiful. For the meaning of these stars, see my answer at: Interpeting R significance codes for ANOVA table?
It should be possible to write a function (or even an R package) to perform such table transformation. However, it might take great effort to make such function flexible enough, to handle:
all type of contrasts (this is easy to do);
complicated model terms, like interaction between a factor and other numeric/factor variables (this seems really involving!!).
So, I will stop here for the moment.
Are the model scores that I get from the lm's summary still accurate, even though it isn't displaying all levels of the factor?
Yes, they are. lm conducts accurate least squares fitting.
In addition, the transformation of coefficient table does not affect R-squares, degree of freedom, residuals, fitted values, F-statistics, ANOVA table, etc.