# How should I explain the lack of ERROR in a repeated measures ANOVA table using REML?

I have conducted a repeated measures ANOVA using a Linear mixed-effects model fit with REML in R. When I report the results I include the ANOVA table output by anova(model_object)

anova(O2.rmtest)
numDF denDF   F-value p-value
(Intercept)       1   131 168.08616  <.0001
mmolO2.L          1   131   8.35259  0.0045
Lake              2   131  13.83037  <.0001
mmolO2.L:Lake     2   131   1.00465  0.3690


The editor of the journal wants to know why this table lacks an Error term.

What is the proper explanation for the lack of an error term in this output and is there any other information that I should be including so that readers can properly evaluate these results?

• The path of least resistance: take out the ANOVA table. Report F-ratios for the tests that you care about, not the ones that fill out an ANOVA table. – Jake Westfall Oct 4 '13 at 2:22
• "The path of least resistance" is ok if you are sure where it takes you and you don't get disoriented. Else, keep on the pavement and follow the signs... – FairMiles Oct 4 '13 at 13:55

You solve a "classical ANOVA" (linear model) by means of least squares, this is, partitioning the sums of squares (among explicative factors) and minimizing the residual sum of squares (unexplained variation). An ANOVA table resumes that partition into all components in the model.

In a linear mixed model you are solving the problem through maximum likelihood (or REML eventually), very loosely speaking by finding the parameters that maximize the probability of observing the data (its likelihood) if several assumptions true.

The "ANOVA" table that you are getting is, then, not that partition of the sums of squares, but a list of parameters in the model (for the fixed factors) followed by Wald tests of the null hypotheses that they are equal to zero. See ?anova.lme (if you used lme function of the nlme package as I suspect) for some details of what it informs you if applied to a single model [BTW, it will inform you something different if applied to two or more models, see same help page].

If you summary(model) you get more information, including the estimated random/residual variations (both ~Errors in your model), which may do make sense to report when describing your fitted model.

[BTW2: be sure to understand what the parameters in the model (and lines in that table) represent compared to what factors represent in a classic ANOVA table; e.g., dummy or treatment coding is default in R]

• Only people with Miles in their username may answer this question. :) – Jeremy Miles Oct 3 '13 at 22:59

ANOVA is a strange word, because it means many different things. When people fit a general linear model with categorical predictors, they often call it ANOVA, and they get sums of squares (including error sums of squares).

From ?anova

When given a single argument it produces a table which tests whether the model terms are
significant.


So the editor is expecting the table sums of squares such as you get from anova, something like:

> x <- runif(100)
> y <- runif(100)
> anova(lm(y ~ x))
Analysis of Variance Table

Response: y
Df Sum Sq  Mean Sq F value Pr(>F)
x          1 0.0023 0.002314  0.0303 0.8623
Residuals 98 7.4958 0.076487


And these sums of squares should sum to the total sums of squares:

> var(y) * (length(y)-1)
 7.498077


You don't have this, because you didn't do a general linear model (or what the editor is thinking of as anova) and so you don't have sums of squares.

You could try explaining this. But I like to take the path of least resistance when it comes to dealing with statistical issues with editors and I would just rename the table. You could call it Type III tests of fixed effects (I think that's what SAS and SPSS call it), or something like 'significance tests of each predictor'. I'd also remove the intercept from it (unless you're really interested in that) and I'd be tempted to remove mmolO2.L as well, if (as I'd assume) you have that in the parameter estimates already.