`demean()`

computes group- and de-meaned versions of a variable that can be
used in regression analysis to model the between- and within-subject effect.
`degroup()`

is more generic in terms of the centering-operation. While
`demean()`

always uses mean-centering, `degroup()`

can also use the mode or
median for centering.

## Usage

```
demean(
x,
select,
by,
suffix_demean = "_within",
suffix_groupmean = "_between",
add_attributes = TRUE,
verbose = TRUE,
group = NULL
)
degroup(
x,
select,
by,
center = "mean",
suffix_demean = "_within",
suffix_groupmean = "_between",
add_attributes = TRUE,
verbose = TRUE,
group = NULL
)
detrend(
x,
select,
by,
center = "mean",
suffix_demean = "_within",
suffix_groupmean = "_between",
add_attributes = TRUE,
verbose = TRUE,
group = NULL
)
```

## Arguments

- x
A data frame.

- select
Character vector (or formula) with names of variables to select that should be group- and de-meaned.

- by
Character vector (or formula) with the name of the variable(s) that indicates the group- or cluster-ID. For cross-classified designs,

`by`

can also identify two or more variables as group- or cluster-IDs. See also section*De-meaning for cross-classified designs*below.- suffix_demean, suffix_groupmean
String value, will be appended to the names of the group-meaned and de-meaned variables of

`x`

. By default, de-meaned variables will be suffixed with`"_within"`

and grouped-meaned variables with`"_between"`

.- add_attributes
Logical, if

`TRUE`

, the returned variables gain attributes to indicate the within- and between-effects. This is only relevant when printing`model_parameters()`

- in such cases, the within- and between-effects are printed in separated blocks.- verbose
Toggle warnings and messages.

- group
Deprecated. Use

`by`

instead.- center
Method for centering.

`demean()`

always performs mean-centering, while`degroup()`

can use`center = "median"`

or`center = "mode"`

for median- or mode-centering, and also`"min"`

or`"max"`

.

## Value

A data frame with the group-/de-meaned variables, which get the suffix
`"_between"`

(for the group-meaned variable) and `"_within"`

(for the
de-meaned variable) by default.

## Heterogeneity Bias

Mixed models include different levels of sources of variability, i.e.
error terms at each level. When macro-indicators (or level-2 predictors,
or higher-level units, or more general: *group-level predictors that
vary within and across groups*) are included as fixed effects (i.e.
treated as covariate at level-1), the variance that is left unaccounted for
this covariate will be absorbed into the error terms of level-1 and level-2
(

*Bafumi and Gelman 2006; Gelman and Hill 2007, Chapter 12.6.*): "Such covariates contain two parts: one that is specific to the higher-level entity that does not vary between occasions, and one that represents the difference between occasions, within higher-level entities" (

*Bell et al. 2015*). Hence, the error terms will be correlated with the covariate, which violates one of the assumptions of mixed models (iid, independent and identically distributed error terms). This bias is also called the

*heterogeneity bias*(

*Bell et al. 2015*). To resolve this problem, level-2 predictors used as (level-1) covariates should be separated into their "within" and "between" effects by "de-meaning" and "group-meaning": After demeaning time-varying predictors, "at the higher level, the mean term is no longer constrained by Level 1 effects, so it is free to account for all the higher-level variance associated with that variable" (

*Bell et al. 2015*).

## Panel data and correlating fixed and group effects

`demean()`

is intended to create group- and de-meaned variables for panel
regression models (fixed effects models), or for complex
random-effect-within-between models (see *Bell et al. 2015, 2018*), where
group-effects (random effects) and fixed effects correlate (see
*Bafumi and Gelman 2006*). This can happen, for instance, when analyzing
panel data, which can lead to *Heterogeneity Bias*. To control for correlating
predictors and group effects, it is recommended to include the group-meaned
and de-meaned version of *time-varying covariates* (and group-meaned version
of *time-invariant covariates* that are on a higher level, e.g. level-2
predictors) in the model. By this, one can fit complex multilevel models for
panel data, including time-varying predictors, time-invariant predictors and
random effects.

## Why mixed models are preferred over fixed effects models

A mixed models approach can model the causes of endogeneity explicitly
by including the (separated) within- and between-effects of time-varying
fixed effects and including time-constant fixed effects. Furthermore,
mixed models also include random effects, thus a mixed models approach
is superior to classic fixed-effects models, which lack information of
variation in the group-effects or between-subject effects. Furthermore,
fixed effects regression cannot include random slopes, which means that
fixed effects regressions are neglecting "cross-cluster differences in the
effects of lower-level controls (which) reduces the precision of estimated
context effects, resulting in unnecessarily wide confidence intervals and
low statistical power" (*Heisig et al. 2017*).

## Terminology

The group-meaned variable is simply the mean of an independent variable
within each group (or id-level or cluster) represented by `by`

. It represents
the cluster-mean of an independent variable. The regression coefficient of a
group-meaned variable is the *between-subject-effect*. The de-meaned variable
is then the centered version of the group-meaned variable. De-meaning is
sometimes also called person-mean centering or centering within clusters.
The regression coefficient of a de-meaned variable represents the
*within-subject-effect*.

## De-meaning with continuous predictors

For continuous time-varying predictors, the recommendation is to include both their de-meaned and group-meaned versions as fixed effects, but not the raw (untransformed) time-varying predictors themselves. The de-meaned predictor should also be included as random effect (random slope). In regression models, the coefficient of the de-meaned predictors indicates the within-subject effect, while the coefficient of the group-meaned predictor indicates the between-subject effect.

## De-meaning with binary predictors

For binary time-varying predictors, there are two recommendations. First
is to include the raw (untransformed) binary predictor as fixed effect
only and the *de-meaned* variable as random effect (random slope).
The alternative would be to add the de-meaned version(s) of binary
time-varying covariates as additional fixed effect as well (instead of
adding it as random slope). Centering time-varying binary variables to
obtain within-effects (level 1) isn't necessary. They have a sensible
interpretation when left in the typical 0/1 format (*Hoffmann 2015,
chapter 8-2.I*). `demean()`

will thus coerce categorical time-varying
predictors to numeric to compute the de- and group-meaned versions for
these variables, where the raw (untransformed) binary predictor and the
de-meaned version should be added to the model.

## De-meaning of factors with more than 2 levels

Factors with more than two levels are demeaned in two ways: first, these
are also converted to numeric and de-meaned; second, dummy variables
are created (binary, with 0/1 coding for each level) and these binary
dummy-variables are de-meaned in the same way (as described above).
Packages like **panelr** internally convert factors to dummies before
demeaning, so this behaviour can be mimicked here.

## De-meaning interaction terms

There are multiple ways to deal with interaction terms of within- and between-effects.

A classical approach is to simply use the product term of the de-meaned variables (i.e. introducing the de-meaned variables as interaction term in the model formula, e.g.

`y ~ x_within * time_within`

). This approach, however, might be subject to bias (see*Giesselmann & Schmidt-Catran 2020*).Another option is to first calculate the product term and then apply the de-meaning to it. This approach produces an estimator "that reflects unit-level differences of interacted variables whose moderators vary within units", which is desirable if

*no*within interaction of two time-dependent variables is required. This is what`demean()`

does internally when`select`

contains interaction terms.A third option, when the interaction should result in a genuine within estimator, is to "double de-mean" the interaction terms (

*Giesselmann & Schmidt-Catran 2018*), however, this is currently not supported by`demean()`

. If this is required, the`wmb()`

function from the**panelr**package should be used.

To de-mean interaction terms for within-between models, simply specify
the term as interaction for the `select`

-argument, e.g. `select = "a*b"`

(see 'Examples').

## De-meaning for cross-classified designs

`demean()`

can also handle cross-classified designs, where the data has two
or more groups at the higher (i.e. second) level. In such cases, the
`by`

-argument can identify two or more variables that represent the
cross-classified group- or cluster-IDs. The de-meaned variables for
cross-classified designs are simply subtracting all group means from each
individual value, i.e. *fully cluster-mean-centering* (see *Guo et al. 2024*
for details). Note that de-meaning for cross-classified designs is *not*
equivalent to de-meaning of nested data structures from models with three or
more levels, i.e. de-meaning is supposed to work for models like
`y ~ x + (1|group1) + (1|group2)`

, but *not* for models like
`y ~ x + (1|group1/group2)`

.

## Analysing panel data with mixed models using lme4

A description of how to translate the formulas described in *Bell et al. 2018*
into R using `lmer()`

from **lme4** can be found in
this vignette.

## References

Bafumi J, Gelman A. 2006. Fitting Multilevel Models When Predictors and Group Effects Correlate. In. Philadelphia, PA: Annual meeting of the American Political Science Association.

Bell A, Fairbrother M, Jones K. 2019. Fixed and Random Effects Models: Making an Informed Choice. Quality & Quantity (53); 1051-1074

Bell A, Jones K. 2015. Explaining Fixed Effects: Random Effects Modeling of Time-Series Cross-Sectional and Panel Data. Political Science Research and Methods, 3(1), 133–153.

Gelman A, Hill J. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. Analytical Methods for Social Research. Cambridge, New York: Cambridge University Press

Giesselmann M, Schmidt-Catran, AW. 2020. Interactions in fixed effects regression models. Sociological Methods & Research, 1–28. https://doi.org/10.1177/0049124120914934

Guo Y, Dhaliwal J, Rights JD. 2024. Disaggregating level-specific effects in cross-classified multilevel models. Behavior Research Methods, 56(4), 3023–3057.

Heisig JP, Schaeffer M, Giesecke J. 2017. The Costs of Simplicity: Why Multilevel Models May Benefit from Accounting for Cross-Cluster Differences in the Effects of Controls. American Sociological Review 82 (4): 796–827.

Hoffman L. 2015. Longitudinal analysis: modeling within-person fluctuation and change. New York: Routledge

## See also

If grand-mean centering (instead of centering within-clusters)
is required, see `center()`

. See `performance::check_heterogeneity_bias()`

to check for heterogeneity bias.

## Examples

```
data(iris)
iris$ID <- sample(1:4, nrow(iris), replace = TRUE) # fake-ID
iris$binary <- as.factor(rbinom(150, 1, .35)) # binary variable
x <- demean(iris, select = c("Sepal.Length", "Petal.Length"), by = "ID")
head(x)
#> Sepal.Length_between Petal.Length_between Sepal.Length_within
#> 1 5.691429 3.285714 -0.5914286
#> 2 5.941667 4.272222 -1.0416667
#> 3 5.941667 4.272222 -1.2416667
#> 4 5.920513 3.964103 -1.3205128
#> 5 5.691429 3.285714 -0.6914286
#> 6 5.812500 3.507500 -0.4125000
#> Petal.Length_within
#> 1 -1.885714
#> 2 -2.872222
#> 3 -2.972222
#> 4 -2.464103
#> 5 -1.885714
#> 6 -1.807500
x <- demean(iris, select = c("Sepal.Length", "binary", "Species"), by = "ID")
#> Categorical predictors (binary, Species) have been coerced to numeric
#> values to compute de- and group-meaned variables.
head(x)
#> Sepal.Length_between binary_between Species_between Species_setosa_between
#> 1 5.691429 0.2571429 0.7714286 0.4571429
#> 2 5.941667 0.3333333 1.3333333 0.1944444
#> 3 5.941667 0.3333333 1.3333333 0.1944444
#> 4 5.920513 0.2307692 1.0000000 0.2564103
#> 5 5.691429 0.2571429 0.7714286 0.4571429
#> 6 5.812500 0.4250000 0.9000000 0.4250000
#> Species_versicolor_between Species_virginica_between Sepal.Length_within
#> 1 0.3142857 0.2285714 -0.5914286
#> 2 0.2777778 0.5277778 -1.0416667
#> 3 0.2777778 0.5277778 -1.2416667
#> 4 0.4871795 0.2564103 -1.3205128
#> 5 0.3142857 0.2285714 -0.6914286
#> 6 0.2500000 0.3250000 -0.4125000
#> binary_within Species_within Species_setosa_within Species_versicolor_within
#> 1 -0.2571429 -0.7714286 0.5428571 -0.3142857
#> 2 -0.3333333 -1.3333333 0.8055556 -0.2777778
#> 3 -0.3333333 -1.3333333 0.8055556 -0.2777778
#> 4 0.7692308 -1.0000000 0.7435897 -0.4871795
#> 5 -0.2571429 -0.7714286 0.5428571 -0.3142857
#> 6 -0.4250000 -0.9000000 0.5750000 -0.2500000
#> Species_virginica_within
#> 1 -0.2285714
#> 2 -0.5277778
#> 3 -0.5277778
#> 4 -0.2564103
#> 5 -0.2285714
#> 6 -0.3250000
# demean interaction term x*y
dat <- data.frame(
a = c(1, 2, 3, 4, 1, 2, 3, 4),
x = c(4, 3, 3, 4, 1, 2, 1, 2),
y = c(1, 2, 1, 2, 4, 3, 2, 1),
ID = c(1, 2, 3, 1, 2, 3, 1, 2)
)
demean(dat, select = c("a", "x*y"), by = "ID")
#> a_between x_y_between a_within x_y_within
#> 1 2.666667 4.666667 -1.6666667 -0.6666667
#> 2 2.333333 4.000000 -0.3333333 2.0000000
#> 3 2.500000 4.500000 0.5000000 -1.5000000
#> 4 2.666667 4.666667 1.3333333 3.3333333
#> 5 2.333333 4.000000 -1.3333333 0.0000000
#> 6 2.500000 4.500000 -0.5000000 1.5000000
#> 7 2.666667 4.666667 0.3333333 -2.6666667
#> 8 2.333333 4.000000 1.6666667 -2.0000000
# or in formula-notation
demean(dat, select = ~ a + x * y, by = ~ID)
#> a_between x_y_between a_within x_y_within
#> 1 2.666667 4.666667 -1.6666667 -0.6666667
#> 2 2.333333 4.000000 -0.3333333 2.0000000
#> 3 2.500000 4.500000 0.5000000 -1.5000000
#> 4 2.666667 4.666667 1.3333333 3.3333333
#> 5 2.333333 4.000000 -1.3333333 0.0000000
#> 6 2.500000 4.500000 -0.5000000 1.5000000
#> 7 2.666667 4.666667 0.3333333 -2.6666667
#> 8 2.333333 4.000000 1.6666667 -2.0000000
```