# Working with messy data

#### emmeans package, Version 2.0.0

## Contents

1. Issues with observational data
2. Mediating covariates
3. Mediating factors and weights
4. Nuisance factors
5. Counterfactuals and G-Computation
6. Sub-models
7. Nested fixed effects
   1. Avoiding mis-identified nesting

Index of all vignette topics

## Issues with observational data

In experiments, we control the conditions under which observations are made. Ideally, this leads to balanced datasets and clear inferences about the effects of those experimental conditions. In observational data, factor levels are observed rather than controlled, and in the analysis we control for those factors and covariates. It is possible that some factors and covariates lie in the causal path for other predictors. Observational studies can be designed in ways to mitigate some of these issues; but often we are left with a mess. Using EMMs does not solve the inherent problems in messy, undesigned studies; but they do give us ways to compensate for imbalance in the data, and allow us to estimate meaningful effects after carefully considering the ways in which they can be confounded.

As an illustration, consider the `nutrition` dataset provided with the package. These data are used as an example in Milliken and Johnson (1992), Analysis of Messy Data, and contain the results of an observational study on nutrition education. Low-income mothers are classified by race, age category, and whether or not they received food stamps (the `group` factor); and the response variable is a gain score (post minus pre scores) after completing a nutrition training program. First, let’s fit a model than includes all main effects and 2-way interactions, and obtain its “type II” ANOVA:

    nutr.lm <- lm(gain ~ (age + group + race)^2, data = nutrition)
    car::Anova(nutr.lm)

    ## Note: model has aliased coefficients
    ##       sums of squares computed by model comparison

    ## Anova Table (Type II tests)
    ##
    ## Response: gain
    ##             Sum Sq Df F value    Pr(>F)
    ## age          82.37  3  0.9614    0.4145
    ## group       658.13  1 23.0441 6.105e-06
    ## race         11.17  2  0.1956    0.8227
    ## age:group    91.58  3  1.0688    0.3663
    ## age:race     87.30  3  1.0189    0.3880
    ## group:race  113.70  2  1.9906    0.1424
    ## Residuals  2627.47 92

There is definitely a `group` effect and a hint of and interaction with `race`. Here are the EMMs for those two factors, along with their counts:

    emmeans(nutr.lm, ~ group * race, calc = c(n = ".wgt."))

    ##  group      race     emmean   SE df  n lower.CL upper.CL
    ##  FoodStamps Black      4.71 2.37 92  7  0.00497     9.41
    ##  NoAid      Black     -2.19 2.49 92 14 -7.13690     2.76
    ##  FoodStamps Hispanic nonEst   NA NA  1       NA       NA
    ##  NoAid      Hispanic nonEst   NA NA  2       NA       NA
    ##  FoodStamps White      3.61 1.16 92 52  1.31252     5.90
    ##  NoAid      White      2.26 2.39 92 31 -2.48897     7.00
    ##
    ## Results are averaged over the levels of: age
    ## Confidence level used: 0.95

Hmmmm. The EMMs when `race` is “Hispanic” are not given; instead they are flagged as non-estimable. What does that mean? Well, when using a model to make predictions, it is impossible to do that beyond the linear space of the data used to fit the model. And we have no data for three of the age groups in the Hispanic population:

    with(nutrition, table(race, age))

    ##           age
    ## race        1  2  3  4
    ##   Black     2  7 10  2
    ##   Hispanic  0  0  3  0
    ##   White     5 16 51 11

We can’t make predictions for all the cases we are averaging over in the above EMMs, and that is why some of them are non-estimable. The bottom line is that we simply cannot include Hispanics in the mix when comparing factor effects. That’s a limitation of this study that cannot be overcome without collecting additional data. Our choices for further analysis are to focus only on Black and White populations; or to focus only on age group 3. For example (the latter):

    emmeans(nutr.lm, pairwise ~ group | race, at = list(age = "3")) |>
        summary(by = NULL)

    ## Note: adjust = "tukey" was changed to "sidak"
    ## because "tukey" is only appropriate for one set of pairwise comparisons

    ## $emmeans
    ##  group      race     emmean   SE df lower.CL upper.CL
    ##  FoodStamps Black      7.50 2.67 92     2.19   12.807
    ##  NoAid      Black     -3.67 2.18 92    -8.00    0.666
    ##  FoodStamps Hispanic   0.00 5.34 92   -10.61   10.614
    ##  NoAid      Hispanic   2.50 3.78 92    -5.01   10.005
    ##  FoodStamps White      5.42 0.96 92     3.51    7.326
    ##  NoAid      White     -0.20 1.19 92    -2.57    2.173
    ##
    ## Confidence level used: 0.95
    ##
    ## $contrasts
    ##  contrast           race     estimate   SE df t.ratio p.value
    ##  FoodStamps - NoAid Black       11.17 3.45 92   3.237  0.0050
    ##  FoodStamps - NoAid Hispanic    -2.50 6.55 92  -0.382  0.9739
    ##  FoodStamps - NoAid White        5.62 1.53 92   3.666  0.0012
    ##
    ## P value adjustment: sidak method for 3 tests

(We used trickery with providing a `by` variable, and then taking it away, to make the output more compact.) Evidently, the training program has been beneficial to the Black and White groups in that age category. There is no conclusion for the Hispanic group – for which we have very little data.

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## Mediating covariates

The `framing` data in the mediation package has the results of an experiment conducted by Brader et al. (2008) where subjects were given the opportunity to send a message to Congress regarding immigration. However, before being offered this, some subjects (`treat = 1`) were first shown a news story that portrays Latinos in a negative way. Besides the binary response (whether or not they elected to send a message), the experimenters also measured `emo`, the subjects’ emotional state after the treatment was applied. There are various demographic variables as well. Let’s a logistic regression model, after changing the labels for `educ` to shorter strings.

    framing <- mediation::framing
    levels(framing$educ) <- c("NA","Ref","< HS", "HS", "> HS","Coll +")
    framing.glm <- glm(cong_mesg ~ age + income + educ + emo + gender * factor(treat),
        family = binomial, data = framing)

The conventional way to handle covariates like `emo` is to set them at their means and use those means for purposes of predictions and EMMs. These adjusted means are shown in the following plot.

    emmip(framing.glm, treat ~ educ | gender, type = "response")

This plot gives the impression that the effect of `treat` is reversed between male and female subjects; and also that the effect of education is not monotone. Both of these are counter-intuitive.

######

However, note that the covariate `emo` is measured post-treatment. That suggests that in fact `treat` (and perhaps other factors) could affect the value of `emo`; and if that is true (as is in fact established by mediation analysis techniques), we should not pretend that `emo` can be set independently of `treat` as was done to obtain the EMMs shown above. Instead, let `emo` depend on `treat` and the other predictors – easily done using `cov.reduce` – and we obtain an entirely different impression:

    emmip(framing.glm, treat ~ educ | gender, type = "response",
        cov.reduce = emo ~ treat*gender + age + educ + income)

The reference grid underlying this plot has different `emo` values for each factor combination. The plot suggests that, after taking emotional response into account, male (but not female) subjects exposed to the negative news story are more likely to send the message than are females or those not seeing the negative news story. Also, the effect of `educ` is now nearly monotone.

######

By the way, the results in this plot are the same is what you would obtain by refitting the model with an adjusted covariate

    emo.adj <- resid(lm(emo ~ treat*gender + age + educ + income, data = framing))

… and then using ordinary covariate-adjusted means at the means of `emo.adj`. This is a technique that is often recommended.

If there is more than one mediating covariate, their settings may be defined in sequence; for example, if `x1`, `x2`, and `x3` are all mediating covariates, we might use

    emmeans(..., cov.reduce = list(x1 ~ trt, x2 ~ trt + x1, x3 ~ trt + x1 + x2))

(or possibly with some interactions included as well).

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## Mediating factors and weights

A mediating covariate is one that is in the causal path; likewise, it is possible to have a mediating factor. For mediating factors, the moral equivalent of the `cov.reduce` technique described above is to use weighted averages in lieu of equally-weighted ones in computing EMMs. The weights used in these averages should depend on the frequencies of mediating factor(s). Usually, the `"cells"` weighting scheme described later in this section is the right approach. In complex situations, it may be necessary to compute EMMs in stages.

As described in the “basics” vignette, EMMs are usually defined as equally-weighted means of reference-grid predictions. However, there are several built-in alternative weighting schemes that are available by specifying a character value for `weights` in a call to `emmeans()` or related function. The options are `"equal"` (the default), `"proportional"`, `"outer"`, `"cells"`, and `"flat"`.

The `"proportional"` (or `"prop"` for short) method weights proportionally to the frequencies (or model weights) of each factor combination that is averaged over. The `"outer"` method uses the outer product of the marginal frequencies of each factor that is being averaged over. To explain the distinction, suppose the EMMs for `A` involve averaging over two factors `B` and `C`. With `"prop"`, we use the frequencies for each combination of `B` and `C`; whereas for `"outer"`, first obtain the marginal frequencies for `B` and for `C` and weight proportionally to the product of these for each combination of `B` and `C`. The latter weights are like the “expected” counts used in a chi-square test for independence. Put another way, outer weighting is the same as proportional weighting applied one factor at a time; the following two would yield the same results:

    emmeans(model, "A", weights = "outer")
    emmeans(model, c("A", "B"), weights = "prop") |>  emmeans(weights = "prop")

Using `"cells"` weights gives each prediction the same weight as occurs in the model; applied to a reference grid for a model with all interactions, `"cells"`-weighted EMMs are the same as the ordinary marginal means of the data. With `"flat"` weights, equal weights are used, except zero weight is applied to any factor combination having no data. Usually, `"cells"` or `"flat"` weighting will not produce non-estimable results, because we exclude empty cells. (That said, if covariates are linearly dependent with factors, we may still encounter non-estimable cases.)

Here is a comparison of predictions for `nutr.lm` defined above, using different weighting schemes:

    sapply(c("equal", "prop", "outer", "cells", "flat"), \(w)
        emmeans(nutr.lm, ~ race, weights = w) |> predict())

    ##         equal     prop    outer     cells      flat
    ## [1,] 1.258929 1.926554 2.546674 0.3809524 0.6865079
    ## [2,]       NA       NA       NA 1.6666667 1.2500000
    ## [3,] 2.932008 2.522821 3.142940 2.7951807 1.6103407

In the other hand, if we do `group * race` EMMs, only one factor (`age`) is averaged over; thus, the results for `"prop"` and `"outer"` weights will be identical in that case.

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## Nuisance factors

Consider a situation where we have a model with 15 factors, each at 5 levels. Regardless of how simple or complex the model is, the reference grid consists of all combinations of these factors – and there are \(5^{15}\) of these, or over 30 billion. If there are, say, 100 regression coefficients in the model, then just the `linfct` slot in the reference grid requires \(100\times5^{15}\times8\) bytes of storage, or almost 23,000 gigabytes. Suppose in addition the model has a multivariate response with 5 levels. That multiplies both the rows and columns in `linfct`, increasing the storage requirements by a factor of 25. Either way, your computer can’t store that much – so this definitely qualifies as a messy situation!

The `ref_grid()` function now provides some relief, in the way of specifying some of the factors as “nuisance” factors. The reference grid is then constructed with those factors already averaged-out. So, for example with the same scenario, if only three of those 15 factors are of primary interest, and we specify the other 12 as nuisance factors to be averaged, that leaves us with only \(3^5=125\) rows in the reference grid, and hence \(125\times100\times8=10,000\) bytes of storage required for `linfct`. If there is a 5-level multivariate response, we’ll have 625 rows in the reference grid and \(25\times1000=250,000\) bytes in `linfct`. Suddenly a horribly unmanageable situation becomes quite manageable!

But of course, there is a restriction: nuisance factors must not interact with any other factors – not even other nuisance factors. And a multivariate response (or an implied multivariate response, e.g., in an ordinal model) can never be a nuisance factor. Under that condition, the average effects of a nuisance factor are the same regardless of the levels of other factors, making it possible to pre-average them by considering just one case.

We specify nuisance factors by listing their names in a `nuisance` argument to `ref_grid()` (in `emmeans()`, this argument is passed to `ref_grid)`). Often, it is much more convenient to give the factors that are not nuisance factors, via a `non.nuisance` argument. If you do specify a nuisance factor that does interact with others, or doesn’t exist, it is quietly excluded from the nuisance list.

######

Time for an example. Consider the `mtcars` dataset standard in R, and the model

    mtcars.lm <- lm(mpg ~ factor(cyl)*am + disp + hp + drat + log(wt) + vs +
                      factor(gear) + factor(carb), data = mtcars)

And let’s construct two different reference grids:

    rg.usual <- ref_grid(mtcars.lm)
    rg.usual

    ## 'emmGrid' object with variables:
    ##     cyl = 4, 6, 8
    ##     am = 0, 1
    ##     disp = 230.72
    ##     hp = 146.69
    ##     drat = 3.5966
    ##     wt = 3.2172
    ##     vs = 0, 1
    ##     gear = 3, 4, 5
    ##     carb = 1, 2, 3, 4, 6, 8

    nrow(linfct(rg.usual))

    ## [1] 216

    rg.nuis = ref_grid(mtcars.lm, non.nuisance = "cyl")
    rg.nuis

    ## 'emmGrid' object with variables:
    ##     cyl = 4, 6, 8
    ##     am = 0, 1
    ## Nuisance factors that have been collapsed by averaging:
    ##     disp(1), hp(1), drat(1), wt(1), vs(2), gear(3), carb(6)

    nrow(linfct(rg.nuis))

    ## [1] 6

Notice that we left `am` out of `non.nuisance` and hence included it in `nuisance`. However, it interacts with `cyl`, so it was not allowed as a nuisance factor. But `rg.nuis` requires 1/36 as much storage. There’s really nothing else to show, other than to demonstrate that we get the same EMMs either way, with slightly different annotations:

    emmeans(rg.usual, ~ cyl * am)

    ##  cyl am emmean   SE df lower.CL upper.CL
    ##    4  0   19.0 4.29 14    9.823     28.2
    ##    6  0   19.7 3.32 14   12.556     26.8
    ##    8  0   29.0 5.98 14   16.130     41.8
    ##    4  1   15.4 4.29 14    6.206     24.6
    ##    6  1   27.3 4.90 14   16.741     37.8
    ##    8  1   11.2 5.56 14   -0.718     23.1
    ##
    ## Results are averaged over the levels of: vs, gear, carb
    ## Confidence level used: 0.95

    emmeans(rg.nuis, ~ cyl * am)

    ##  cyl am emmean   SE df lower.CL upper.CL
    ##    4  0   19.0 4.29 14    9.823     28.2
    ##    6  0   19.7 3.32 14   12.556     26.8
    ##    8  0   29.0 5.98 14   16.130     41.8
    ##    4  1   15.4 4.29 14    6.206     24.6
    ##    6  1   27.3 4.90 14   16.741     37.8
    ##    8  1   11.2 5.56 14   -0.718     23.1
    ##
    ## Results are averaged over the levels of: 7 nuisance factors
    ## Confidence level used: 0.95

By default, the pre-averaging is done with equal weights. If we specify `wt.nuis` as anything other than `"equal"`, they are averaged proportionally. As described above, this really amounts to `"outer"` weights since they are averaged separately. Let’s try it to see how the estimates differ:

    predict(emmeans(mtcars.lm, ~ cyl * am, non.nuis = c("cyl", "am"),
                    wt.nuis = "prop"))

    ## [1] 16.51254 17.17869 26.45709 12.90600 24.75053  8.70546

    predict(emmeans(mtcars.lm, ~ cyl * am, weights = "outer"))

    ## [1] 16.51254 17.17869 26.45709 12.90600 24.75053  8.70546

These are the same as each other, but different from the equally-weighted EMMs we obtained before. By the way, to help make things consistent, if `weights` is character, `emmeans()` passes `wt.nuis = weights` to `ref_grid` (if it is called), unless `wt.nuis` is also specified.

There is a trick to get `emmeans` to use the smallest possible reference grid: Pass the `specs` argument to `ref_grid()` as `non.nuisance`. But we have to quote it to delay evaluation, and also use `all.vars()` if (and only if) `specs` is a formula:

    emmeans(mtcars.lm, ~ gear | am, non.nuis = quote(all.vars(specs)))

    ## am = 0:
    ##  gear emmean   SE df lower.CL upper.CL
    ##     3   15.2 2.65 14     9.56     20.9
    ##     4   22.4 3.36 14    15.20     29.6
    ##     5   30.0 4.54 14    20.28     39.7
    ##
    ## am = 1:
    ##  gear emmean   SE df lower.CL upper.CL
    ##     3   10.6 4.49 14     1.02     20.3
    ##     4   17.8 2.62 14    12.19     23.4
    ##     5   25.4 4.05 14    16.72     34.1
    ##
    ## Results are averaged over the levels of: 6 nuisance factors, cyl
    ## Confidence level used: 0.95

Observe that `cyl` was passed over as a nuisance factor because it interacts with another factor.

### Limiting the size of the reference grid

We have just seen how easily the size of a reference grid can get out of hand. The `rg.limit` option (set via `emm_options()` or as an optional argument in `ref_grid()` or `emmeans()`) serves to guard against excessive memory demands. It specifies the number of allowed rows in the reference grid. But because of the way `ref_grid()` works, this check is made before any multivariate-response levels are taken into account. If the limit is exceeded, an error is thrown:

    ref_grid(mtcars.lm, rg.limit = 200)

    ## Error: The rows of your requested reference grid would be 216, which exceeds
    ## the limit of 200 (not including any multivariate responses).
    ## Your options are:
    ##   1. Specify some (or more) nuisance factors using the 'nuisance' argument
    ##      (see ?ref_grid). These must be factors that do not interact with others.
    ##   2. Add the argument 'rg.limit = <new limit>' to the call. Be careful,
    ##      because this could cause excessive memory use and performance issues.
    ##      Or, change the default via 'emm_options(rg.limit = <new limit>)'.

The default `rg.limit` is 10,000. With this limit, and if we have 1,000 columns in the model matrix, then the size of `linfct` is limited to about 80MB. If in addition, there is a 5-level multivariate response, the limit is 2GB – darn big, but perhaps manageable. Even so, I suspect that the 10000-row default may be to loose to guard against some users getting into a tight situation.

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## Counterfactuals and G-Computation

G-computation is a method for model-based causal inference originated by JM Robins (Mathematical Modelling, 1986), and we want to remove confounding of treatment effects due to time-varying covariates and such. The idea is that, under certain assumptions, we can use the model to predict every subject’s response to each treatment – not just the treatment they received. To do this, we make several copies of the whole dataset, substituting the actual treatment(s) with each of the possible treatment levels; these provide us with counterfactual predictions. We then average those predictions over each copy of the dataset. Typically, this averaging is done on the response scale; that is the interesting case because on the link scale, everything is linear and we can obtain basically the same results using ordinary `emmeans()` computations with proportional weights.

An additional consideration is that when we average each of the counterfactual datasets, we are trying to represent the entire covariate distribution, rather than conditioning on the cases in the dataset. So it is a good idea to broaden the covariance estimate using, say, a sandwich estimate.

This kind of computation has just a little bit in common with nuisance variables, in that the net result is that we can sweep several predictors out of the reference grid just by averaging them away. For this to make sense, the predictors averaged-away will have been observed before treatment so that their effects are separate from the treatment effects.

The implementation of this in emmeans is via the `counterfactuals` argument in `ref_grid()` (but usually passed from `emmeans()`). We simply specify the factor(s) we want to keep. This creates an index variable `.obs.no.` to keep track of the observations in the dataset, and then the reference grid (before averaging) consists of every observation of the dataset in combination with the `counterfactuals` combinations.

#####

As an example, consider the `neuralgia` data, where we have a binary response, pain, a treatment of interest (two active treatments and placebo), and pre-treatment predictors of sex, age, and duration of the condition. We will include the `vcovHC()` covariance estimate in the sandwich package.

    neuralgia.glm <- glm(Pain ~ Sex + Age + Duration + Treatment,
                         data = neuralgia, family = binomial)
    emmeans(neuralgia.glm, "Treatment", counterfactuals = "Treatment",
            vcov. = sandwich::vcovHC)

    ##  Treatment  prob     SE  df asymp.LCL asymp.UCL
    ##  A         0.283 0.0792 Inf    0.1280     0.439
    ##  B         0.221 0.0669 Inf    0.0894     0.352
    ##  P         0.754 0.1040 Inf    0.5500     0.958
    ##
    ## Results are averaged over the levels of: actual_Treatment
    ## Confidence level used: 0.95

Note that the results are already on the response (probability) scale, which is the default. Let’s compare this with what we get without using counterfactuals (i.e., predicting at each covariate average):

    emmeans(neuralgia.glm, "Treatment", weights = "prop", type = "response")

    ##  Treatment  prob     SE  df asymp.LCL asymp.UCL
    ##  A         0.196 0.1050 Inf    0.0617     0.475
    ##  B         0.126 0.0822 Inf    0.0323     0.384
    ##  P         0.855 0.0852 Inf    0.6053     0.958
    ##
    ## Results are averaged over the levels of: Sex
    ## Confidence level used: 0.95
    ## Intervals are back-transformed from the logit scale

These results are markedly different; the counterfactual method produces smaller differences between each of the active treatments and placebo.

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## Sub-models

We have just seen that we can assign different weights to the levels of containing factors. Another option is to constrain the effects of those containing factors to zero. In essence, that means fitting a different model without those containing effects; however, for certain models (not all), an `emmGrid` may be updated with a `submodel` specification so as to impose such a constraint. For illustration, return again to the nutrition example, and consider the analysis of `group` and `race` as before, after removing interactions involving `age`:

    emmeans(nutr.lm, pairwise ~ group | race, submodel = ~ age + group*race) |>
            summary(by = NULL)

    ## Note: adjust = "tukey" was changed to "sidak"
    ## because "tukey" is only appropriate for one set of pairwise comparisons

    ## $emmeans
    ##  group      race     emmean    SE df lower.CL upper.CL
    ##  FoodStamps Black      4.91 2.060 92    0.817    9.003
    ##  NoAid      Black     -3.01 1.580 92   -6.148    0.133
    ##  FoodStamps Hispanic  -1.18 5.410 92  -11.935    9.567
    ##  NoAid      Hispanic   1.32 3.880 92   -6.382    9.014
    ##  FoodStamps White      4.10 0.901 92    2.308    5.886
    ##  NoAid      White     -1.44 1.110 92   -3.654    0.771
    ##
    ## Results are averaged over the levels of: age
    ## submodel: ~ age + group + race + group:race
    ## Confidence level used: 0.95
    ##
    ## $contrasts
    ##  contrast           race     estimate   SE df t.ratio p.value
    ##  FoodStamps - NoAid Black        7.92 2.62 92   3.021  0.0098
    ##  FoodStamps - NoAid Hispanic    -2.50 6.55 92  -0.382  0.9739
    ##  FoodStamps - NoAid White        5.54 1.27 92   4.364  0.0001
    ##
    ## Results are averaged over the levels of: age
    ## P value adjustment: sidak method for 3 tests

If you like, you may confirm that we would obtain exactly the same estimates if we had fitted that sub-model to the data, except we continue to use the residual variance from the full model in tests and confidence intervals. Without the interactions with `age`, all of the marginal means become estimable. The results are somewhat different from those obtained earlier where we narrowed the scope to just age 3. These new estimates include all ages, averaging over them equally, but with constraints that the interaction effects involving `age` are all zero.

######

There are two special character values that may be used with `submodel`. Specifying `"minimal"` creates a submodel with only the active factors:

    emmeans(nutr.lm, ~ group * race, submodel = "minimal")

    ##  group      race     emmean    SE df lower.CL upper.CL
    ##  FoodStamps Black     5.000 2.020 92    0.988    9.012
    ##  NoAid      Black    -1.929 1.430 92   -4.765    0.908
    ##  FoodStamps Hispanic  0.000 5.340 92  -10.614   10.614
    ##  NoAid      Hispanic  2.500 3.780 92   -5.005   10.005
    ##  FoodStamps White     4.769 0.741 92    3.297    6.241
    ##  NoAid      White    -0.516 0.960 92   -2.422    1.390
    ##
    ## Results are averaged over the levels of: age
    ## submodel: ~ group + race + group:race
    ## Confidence level used: 0.95

This submodel constrains all effects involving `age` to be zero. Another interesting option is `"type2"`, whereby we in essence analyze the residuals of the model with all contained or overlapping effects, then constrain the containing effects to be zero. So what is left if only the interaction effects of the factors involved. This is most useful with `joint_tests()`:

    joint_tests(nutr.lm, submodel = "type2")

    ##  model term df1 df2 F.ratio p.value note
    ##  age          3  92   0.961  0.4145
    ##  group        1  92  23.044  <.0001
    ##  race         2  92   0.196  0.8227
    ##  age:group    3  92   1.069  0.3663
    ##  age:race     3  92   1.019  0.3880  d e
    ##  group:race   2  92   1.991  0.1424
    ##
    ## d: df1 reduced due to linear dependence
    ## e: df1 reduced due to non-estimability

These results are identical to the type II anova obtained at the beginning of this example.

More details on how `submodel` works may be found in `vignette("xplanations")`

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## Nested fixed effects

A factor `A` is nested in another factor `B` if the levels of `A` have a different meaning in one level of `B` than in another. Often, nested factors are random effects—for example, subjects in an experiment may be randomly assigned to treatments, in which case subjects are nested in treatments—and if we model them as random effects, these random nested effects are not among the fixed effects and are not an issue to `emmeans`. But sometimes we have fixed nested factors.

######

Here is an example of a fictional study of five fictional treatments for some disease in cows. Two of the treatments are administered by injection, and the other three are administered orally. There are varying numbers of observations for each drug. The data and model follow:

    cows <- data.frame (
        route = factor(rep(c("injection", "oral"), c(5, 9))),
        drug = factor(rep(c("Bovineumab", "Charloisazepam",
                  "Angustatin", "Herefordmycin", "Mollycoddle"), c(3,2,  4,2,3))),
        resp = c(34, 35, 34,   44, 43,      36, 33, 36, 32,   26, 25,   25, 24, 24)
    )
    cows.lm <- lm(resp ~ route + drug, data = cows)

The `ref_grid` function finds a nested structure in this model:

    cows.rg <- ref_grid(cows.lm)
    cows.rg

    ## 'emmGrid' object with variables:
    ##     route = injection, oral
    ##     drug = Angustatin, Bovineumab, Charloisazepam, Herefordmycin, Mollycoddle
    ## Nesting structure:  drug %in% route
    ## Some things are non-estimable (null space dim = 1)

When there is nesting, `emmeans` computes averages separately in each group

    route.emm <- emmeans(cows.rg, "route")
    route.emm

    ##  route     emmean    SE df lower.CL upper.CL
    ##  injection   38.9 0.591  9     37.6     40.3
    ##  oral        28.0 0.449  9     27.0     29.0
    ##
    ## Results are averaged over the levels of: drug
    ## Confidence level used: 0.95

… and insists on carrying along any grouping factors that a factor is nested in:

    drug.emm <- emmeans(cows.rg, "drug")
    drug.emm

    ##  drug           route     emmean    SE df lower.CL upper.CL
    ##  Bovineumab     injection   34.3 0.747  9     32.6     36.0
    ##  Charloisazepam injection   43.5 0.915  9     41.4     45.6
    ##  Angustatin     oral        34.2 0.647  9     32.8     35.7
    ##  Herefordmycin  oral        25.5 0.915  9     23.4     27.6
    ##  Mollycoddle    oral        24.3 0.747  9     22.6     26.0
    ##
    ## Confidence level used: 0.95

Here are the associated pairwise comparisons:

    pairs(route.emm, reverse = TRUE)

    ##  contrast         estimate    SE df t.ratio p.value
    ##  oral - injection    -10.9 0.742  9 -14.671  <.0001
    ##
    ## Results are averaged over the levels of: drug

    pairs(drug.emm, by = "route", reverse = TRUE)

    ## route = injection:
    ##  contrast                    estimate    SE df t.ratio p.value
    ##  Charloisazepam - Bovineumab     9.17 1.180  9   7.757  <.0001
    ##
    ## route = oral:
    ##  contrast                    estimate    SE df t.ratio p.value
    ##  Herefordmycin - Angustatin     -8.75 1.120  9  -7.805  0.0001
    ##  Mollycoddle - Angustatin       -9.92 0.989  9 -10.030  <.0001
    ##  Mollycoddle - Herefordmycin    -1.17 1.180  9  -0.987  0.6026
    ##
    ## P value adjustment: tukey method for comparing a family of 3 estimates

In the latter result, the contrast itself becomes a nested factor in the returned `emmGrid` object. That would not be the case if there had been no `by` variable.

#### Graphs with nesting

It can be very helpful to take advantage of special features of ggplot2 when graphing results with nested factors. For example, the default plot for the `cows` example is not ideal:

    emmip(cows.rg, ~ drug | route)

We can instead remove `route` from the call and instead handle it with ggplot2 code to use separate x scales:

    require(ggplot2)
    emmip(cows.rg, ~ drug) + facet_wrap(~ route, scales = "free_x")

Similarly with `plot.emmGrid()`:

    plot(drug.emm, PIs = TRUE) +
        facet_wrap(~ route, nrow = 2, scales = "free_y")

### Auto-identification of nested factors – avoid being trapped!

`ref_grid()` and `emmeans()` tries to discover and accommodate nested structures in the fixed effects. It does this in two ways: first, by identifying factors whose levels appear in combination with only one level of another factor; and second, by examining the `terms` attribute of the fixed effects. In the latter approach, if an interaction `A:B` appears in the model but `A` is not present as a main effect, then `A` is deemed to be nested in `B`. Note that this can create a trap: some users take shortcuts by omitting some fixed effects, knowing that this won’t affect the fitted values. But such shortcuts do affect the interpretation of model parameters, ANOVA tables, etc., and I advise against ever taking such shortcuts. Here are some ways you may notice mistakenly-identified nesting:

- A message is displayed when nesting is detected
- A `str()` listing of the `emmGrid` object shows a nesting component
- An `emmeans()` summary unexpectedly includes one or more factors that you didn’t specify
- EMMs obtained using `by` factors don’t seem to behave right, or give the same results with different specifications

To override the auto-detection of nested effects, use the `nesting` argument in `ref_grid()` or `emmeans()`. Specifying `nesting = NULL` will ignore all nesting. Incorrectly-discovered nesting can be overcome by specifying something akin to `nesting = "A %in% B, C %in% (A * B)"` or, equivalently, `nesting = list(A = "B", C = c("A", "B"))`.

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