Explanations supplement

Contents

1. Sub-models
2. Comparison arrows
3. Confounded effects in joint tests

Sub-models

Estimated marginal means (EMMs) and other statistics computed by the emmeans package are model-based: they depend on the model that has been fitted to the data. In this section we discuss a provision whereby a different underlying model may be considered. The submodel option in update() can project EMMs and other statistics to an alternative universe where a simpler version of the model has been fitted to the data. Another way of looking at this is that it constrains certain external effects to be zero – as opposed to averaging over them as is otherwise done for marginal means.

Two things to know before getting into details:

1. The submodel option uses information from the fixed-effects portion of the model matrix
2. Not all model classes are supported for the submodel option.

Now some details. Suppose that we have a fixed-effects model matrix \(X\), and let \(X_1\) denote a sub-matrix of \(X\) whose columns correspond to a specified sub-model. (Note: if there are weights, use \(X = W^{1/2}X^*\), where \(X^*\) is the model matrix without the weights incorporated.) The trick we use is what is called the alias matrix: \(A = (X_1'X_1)^-X_1'X\) where \(Z^-\) denotes a generalized inverse of \(Z\). It can be shown that \((X_1'X_1)^-X_1' = A(X'X)^-X'\); thus, in an ordinary fixed-effects regression model, \(b_1 = Ab\) where \(b_1\) and \(b\) denote the regression coefficients for the sub-model and full model, respectively. Thus, given a matrix \(L\) such that \(Lb\) provides estimates of interest for the full model, the corresponding estimates for the sub-model are \(L_1b_1\), where \(L_1\) is the sub-matrix of \(L\) consisting of the columns corresponding to the columns of \(X_1\). Moreover, \(L_1b_1 = L_1(Ab) = (L_1A)b\); that is, we can replace \(L\) by \(L_1A\) to obtain estimates from the sub-model. That’s all that update(..., submodel = ...) does.

Here are some intuitive observations:

1. Consider the excluded effects, \(X_2\), consisting of the columns of \(X\) other than \(X_1\). The corresponding columns of the alias matrix are regression coefficients treating \(X_2\) as the response and \(X_1\) as the predictors.
2. Thus, when we obtain predictions via these aliases, we are predicting the effects of \(X_2\) based on \(X_1\).
3. The columns of the new linear predictor \(\tilde L = L_1A\) depend only on the columns of \(L_1\), and hence not on other columns of \(L\).

These three points provide three ways of saying nearly the same thing, namely that we are excluding the effects in \(X_2\). Note that in a rank-deficient situation, there are different possible generalized inverses, and so in (1), \(A\) is not unique. However, the predictions in (2) are unique. In ordinary regression models, (1), (2), and (3) all apply and will be the same as predictions from re-fitting the model with model matrix \(X_1\); however, in generalized linear models, mixed models, etc., re-fitting will likely produce somewhat different results. That is because fitting such models involves iterative weighting, and the re-fitted models will probably not have the same weights. However, point (3) will still hold: the predictions obtained with a submodel will involve only the columns of \(L_1\) and hence constrain all effects outside of the sub-model to be zero. Therefore, when it really matters to get the correct estimates from the stated sub-model, the user should actually fit that sub-model unless the full model is an ordinary linear regression.

A technicality: Most writers define the alias matrix as \((X_1'X_1)^-X_1'X_2\), where \(X_2\) denotes that part of \(X\) that excludes the columns of \(X_1\). We are including all columns of \(X\) here just because it makes the notation very simple; the \(X_1\) portion of \(X\) just reduces to the identity (at least in the case where \(X_1\) is full-rank).

A word on computation: Like many matrix expressions, we do not compute \(A\) directly as shown. Instead, we use the QR decomposition of \(X_1\), obtainable via the R call Z <- qr(X1). Then the alias matrix is computed via A <- qr.coef(Z, X). In fact, nothing changes if we use just the \(R\) portion of \(X = QR\), saving us both memory and computational effort. The exported function .cmpMM() extracts this \(R\) matrix, taking care of any pivoting that might have occurred. And in an lm object, the QR decomposition of \(X\) is already saved as a slot. The qr.coef() function works just fine in both the full-rank and rank-deficient cases, but in the latter situation, some elements of A will be NA; those correspond to “excluded” predictors, but that is another way of saying that we are constraining their regression coefficients to be zero. Thus, we can easily clean that up via A[is.na(A)] <- 0.

If we specify submodel = "minimal", the software figures out the sub-model by extracting terms involving only factors that have not already been averaged over. If the user specifies submodel = "type2", an additional step is performed: Let \(X_1\) have only the highest-order effect in the minimal model, and \(X_0\) denote the matrix of all columns of \(X\) whose columns do not contain the effect in \(X_1\). We then replace \(Z\) by the QR decomposition of \([I - X_0(X_0'X_0)^-X_0']X_1^*\). This projects \(X_1^*\) onto the null space of \(X_0\). The net result is that we obtain estimates of just the \(X_1^*\) effects, after adjusting for all effects that don’t contain it (including the intercept if present). Such estimates have very limited use in data description, but provide a kind of “Type II” analysis when used in conjunction with joint_tests(). The "type2" calculations parallel those documented by SAS for obtaining type II estimable functions in SAS PROC GLM. However, we (as well as car::Anova()) define “contained” effects differently from SAS, treating covariates no differently than factors.

Comparison arrows

The plot() method for emmGrid objects offers the option comparisons = TRUE. If used, the software attempts to construct “comparison arrows” whereby two estimated marginal means (EMMs) differ significantly if, and only if, their respective comparison arrows do not overlap. In this section, we explain how these arrows are obtained.

First, please understand these comparison arrows are decidedly not the same as confidence intervals. Confidence intervals for EMMs are based on the statistical properties of the individual EMMs, whereas comparison arrows are based on the statistical properties of differences of EMMs.

Let the EMMs be denoted \(m_1, m_2, ..., m_k\). For simplicity, let us assume that these are ordered: \(m_1 \le m_2 \le \cdots \le m_k\). Let \(d_{ij} = m_j - m_i\) denote the difference between the \(i\)th and \(j\)th EMM. Then the \((1 - \alpha)\) confidence interval for the true difference \(\delta_{ij} = \mu_j - \mu_i\) is \[ d_{ij} - e_{ij}\quad\mbox{to}\quad d_{ij} + e_{ij} \] where \(e_{ij}\) is the “margin of error” for the difference; i.e., \(e_{ij} = t\cdot SE(d_{ij})\) for some critical value \(t\) (equal to \(t_{\alpha/2}\) when no multiplicity adjustment is used). Note that \(d_{ij}\) is statistically significant if, and only if, \(d_{ij} > e_{ij}\).

Now, how to get the comparison arrows? These arrows are plotted with origins at the \(m_i\); we have an arrow of length \(L_i\) pointing to the left, and an arrow of length \(R_i\) pointing to the right. To compare EMMs \(m_i\) and \(m_j\) (and remembering that we are supposing that \(m_i \le m_j\)), we propose to look to see if the arrows extending right from \(m_i\) and left from \(m_j\) overlap or not. So, ideally, if we want overlap to be identified with statistical non-significance, we want \[ R_i + L_j = e_{ij} \quad\mbox{for all } i < j \]

If we can do that, then the two arrows will overlap if, and only if, \(d_{ij} < e_{ij}\).

This is easy to accomplish if all the \(e_{ij}\) are equal: just set all \(L_i = R_j = \frac12e_{12}\). But with differing \(e_{ij}\) values, it may or may not even be possible to obtain suitable arrow lengths.

The code in emmeans uses an ad hoc weighted regression method to solve the above equations. We give greater weights to cases where \(d_{ij}\) is close to \(e_{ij}\), because those are the cases where it is more critical that we get the lengths of the arrows right. Once the regression equations are solved, we test to make sure that \(R_i + L_j < d_{ij}\) when the difference is significant, and \(\ge d_{ij}\) when it is not. If one or more of those checks fails, a warning is issued.

That’s the essence of the algorithm. Note, however, that there are a few complications that need to be handled:

• For the lowest EMM \(m_1\), \(L_1\) is completely arbitrary because there are no right-pointing arrows with which to compare it; in fact, we don’t even need to display that arrow. The same is true of \(R_k\) for the largest EMM \(m_k\). Moreover, there could be additional unneeded arrows when other \(m_i\) are equal to \(m_1\) or \(m_k\).
• Depending on the number \(k\) of EMMs and the number of tied minima and maxima, the system of equations could be under-determined, over-determined, or just right.
• It is possible that the solution could result in some \(L_i\) or \(R_j\) being negative. That would result in an error.

In summary, the algorithm does not always work (in fact it is possible to construct cases where no solution is possible). But we try to do the best we can. The main reason for trying to do this is to encourage people to not ever use confidence intervals for the \(m_i\) as a means of testing the comparisons \(d_{ij}\). That is almost always incorrect.

What is better yet is to simply avoid using comparison arrows altogether and use pwpp() or pwpm() to display the P values directly.

m = c(6.1, 4.5, 5.4,    6.3, 5.5, 6.7)
se2 = c(.3, .4, .37,  .41, .23, .48)^2
lev = list(A = c("a1","a2","a3"), B = c("b1", "b2"))
foo = emmobj(m, diag(se2), levels = lev, linfct = diag(6))
plot(foo, CIs = FALSE, comparisons = TRUE)

mkmat <- function(V, rho = 0, indexes = list(1:3, 4:6)) {
    sd = sqrt(diag(V))
    for (i in indexes)
        V[i,i] = (1 - rho)*diag(sd[i]^2) + rho*outer(sd[i], sd[i])
    V
}
# Intraclass correlation = 0.3
foo3 = foo
foo3@V <- mkmat(foo3@V, 0.3)
plot(foo3, CIs = FALSE, comparisons = TRUE)

foo6 = foo
foo6@V <- mkmat(foo6@V, 0.6)
plot(foo6, CIs = FALSE, comparisons = TRUE)

## Warning: Comparison discrepancy in group "1", a1 b1 - a2 b2:
##     Target overlap = 0.443, overlap on graph = -0.2131

foo8 = foo
foo8@V <- mkmat(foo8@V, 0.8)
plot(foo8, CIs = FALSE, comparisons = TRUE)

## Error: Aborted -- Some comparison arrows have negative length!
## (in group "1")

plot(foo8, CIs = FALSE, comparisons = TRUE, by = "B")

pwpp(foo6, sort = FALSE)

pwpm(foo6)

##       a1 b1  a2 b1  a3 b1  a1 b2  a2 b2  a3 b2
## a1 b1 [6.1] <.0001 0.1993 0.9988 0.6070 0.8972
## a2 b1   1.6  [4.5] 0.0958 0.0208 0.2532 0.0057
## a3 b1   0.7   -0.9  [5.4] 0.5788 0.9999 0.2641
## a1 b2  -0.2   -1.8   -0.9  [6.3] 0.1439 0.9204
## a2 b2   0.6   -1.0   -0.1    0.8  [5.5] 0.0245
## a3 b2  -0.6   -2.2   -1.3   -0.4   -1.2  [6.7]
## 
## Row and column labels: A:B
## Upper triangle: P values   adjust = "tukey"
## Diagonal: [Estimates] (estimate) 
## Lower triangle: Comparisons (estimate)   earlier vs. later

Confounded effects in joint tests

When a design has empty cells, this creates estimability issues; and in the context of joint_tests(), this means that certain main- and interaction effects cannot be estimated, leading to reduced degrees of freedom in the joint tests. Instead, the estimable parts of these lost effects are pooled together into a set of contrasts labeled (confounded) in the output (as long as showconf = TRUE). Here is a simple illustration based on the warpbreaks dataset with the last cell removed.

options(contrasts = c("contr.treatment", "contr.poly"))   ## ensure system default
w <- warpbreaks[1:40, ]   ### no data for (wool = B, tension = H)
w.lm <- lm(breaks ~ wool * tension, data = w)
w.rg <- ref_grid(w.lm)
(jt <- joint_tests(w.rg))

##  model term   df1 df2 F.ratio p.value note
##  tension        1  35   6.064  0.0189    e
##  wool:tension   1  35   3.740  0.0613    e
##  (confounded)   2  35   2.266  0.1187     
## 
## e: df1 reduced due to non-estimability

With no empty cells, we would have had 1 d.f. for wool, 2 d.f. for tension, and 2 d.f. for wool:tension, for a total of 5 d.f. (6 cells, 6 - 1 = 5 d.f. for contrasts among them). With the missing cell, we can estimate only 5 - 1 = 4 d.f. worth of effects, and the (confounding) part accounts for 2 of them that are not purely main effects nor interaction effects.

How do we obtain these extra 2 d.f.? Well, first of all, it is easy to obtain a joint test of all comparisons among the cell means

(test.all <- test(contrast(w.rg, "consec"), joint = TRUE))

##  df1 df2 F.ratio p.value note
##    4  35   4.332  0.0060    e
## 
## e: df1 reduced due to non-estimability

What we do is simply pool together all the contrasts that were tested in jt, and compare them with the above. To do this, we obtain the estimable functions associated with jt:

(ef <- attr(jt, "est.fcns"))

## $tension
##      (Intercept) woolB  tensionM tensionH woolB:tensionM woolB:tensionH
## [1,]           0     0 0.8944272        0      0.4472136              0
## 
## $`wool:tension`
##      (Intercept) woolB tensionM tensionH woolB:tensionM woolB:tensionH
## [1,]           0     0        0        0              1              0

Then we can get a joint test of all these together via some trickery:

tmp <- w.rg
tmp@linfct <- do.call(rbind, ef)      # combine EFs into a matrix
tmp@grid <- tmp@grid[1:2, ]           # replace the @linfct and @grid slots
(test.ef <- test(tmp, joint = TRUE))  #  -- that's enough to get the test

##  df1 df2 F.ratio p.value
##    2  35   6.398  0.0043

(Note: In case this isn’t obvious, we couldn’t have just pooled together the F ratios for those effects, because they are type-III tests which are not necessarily independent.) So now we have test.all with the 4 d.f. worth of all effects comparing the cells, and test.ef with just the effects in the top part of the jt table. We can get the unexplained part of test.all by d.f.-weighted subtraction:

(test.all$df1 * test.all$F.ratio  -  test.ef$df1 * test.ef$F.ratio) /
    (test.all$df1 - test.ef$df1)

## [1] 2.266

This gives us the F.ratio shown in the (confounded) row of the jt results above. That’s all there is to it!

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