Fit "within-between" and several other regression variants for panel data in a multilevel modeling framework.

wbm(formula, data, id = NULL, wave = NULL, model = "w-b",
detrend = FALSE, use.wave = FALSE, wave.factor = FALSE,
min.waves = 2, family = gaussian, balance.correction = FALSE,
dt.random = TRUE, dt.order = 1, pR2 = TRUE, pvals = TRUE,
t.df = "Satterthwaite", weights = NULL, offset = NULL,
interaction.style = c("double-demean", "demean", "raw"),
scale = FALSE, scale.response = FALSE, n.sd = 1,
dt_random = dt.random, dt_order = dt.order,
balance_correction = balance.correction, ...)

## Arguments

formula Model formula. See details for crucial info on panelr's formula syntax. The data, either a panel_data object or data.frame. If data is not a panel_data object, then the name of the individual id column as a string. Otherwise, leave as NULL, the default. If data is not a panel_data object, then the name of the panel wave column as a string. Otherwise, leave as NULL, the default. One of "w-b", "within", "between", "contextual". See details for more on these options. Adjust within-subject effects for trends in the predictors? Default is FALSE, but some research suggests this is a better idea (see Curran and Bauer (2011) reference). Should the wave be included as a predictor? Default is FALSE. Should the wave variable be treated as an unordered factor instead of continuous? Default is FALSE. What is the minimum number of waves an individual must have participated in to be included in the analysis? Default is 2 and any valid number is accepted. "all" is also acceptable if you want to include only complete panelists. Use this to specify GLM link families. Default is gaussian, the linear model. Correct between-subject effects for unbalanced panels following the procedure in Curran and Bauer (2011)? Default is FALSE. Should the detrending procedure be performed with a random slope for each entity? Default is TRUE but for short panels FALSE may be better, fitting a trend for all entities. If detrending using detrend, what order polynomial would you like to specify for the relationship between time and the predictors? Default is 1, a linear model. Calculate a pseudo R-squared? Default is TRUE, but in some cases may cause errors or add computation time. Calculate p values? Default is TRUE but for some complex linear models, this may take a long time to compute using the pbkrtest package. For linear models only. User may choose the method for calculating the degrees of freedom in t-tests. Default is "Satterthwaite", but you may also choose "Kenward-Roger". Kenward-Roger standard errors/degrees of freedom requires the pbkrtest package. If using weights, either the name of the column in the data that contains the weights or a vector of the weights. this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if more than one is specified their sum is used. See model.offset. The best way to calculate interactions in within models is in some dispute. The conventional way ("demean") is to first calculate the product of the variables involved in the interaction before those variables have their means subtracted and then subtract the mean of the product from the product term (see Schunk and Perales (2017)). Giesselmann and Schmidt-Catran (2018) show this method carries between-entity differences that within models are designed to model out. They suggest an alternate method ("double-demean") in which the product term is first calculated using the de-meaned lower-order variables and then the subject means are subtracted from this product term. Another option is to simply use the product term of the de-meaned variables ("raw"), but Giesselmann and Schmidt-Catran (2018) show this method biases the results towards zero effect. The default is "double-demean" but if emulating other software is the goal, "demean" might be preferred. If TRUE, reports standardized regression coefficients. Default is FALSE. Should the response variable also be rescaled? Default is FALSE. How many standard deviations should you divide by for standardization? Default is 1, though some prefer 2. Deprecated. Equivalent to dt.random. Deprecated. Equivalent to dt.order. Deprecated. Equivalent to balance.correction. Additional arguments provided to lme4::lmer(), lme4::glmer(), or lme4::glmer.nb().

## Value

A wbm object, which inherits from merMod.

## Details

Formula syntax

The within-between models, and multilevel panel models more generally, distinguish between time-varying and time-invariant predictors. These are, as they sound, variables that are either measured repeatedly (in every wave) in the case of time-varying predictors or only once in the case of time-invariant predictors. You need to specify these separately in the formula to tell the model which variables you expect to change over time and which will not. The primary way of doing so is via the | operator.

As an example, we can look at the WageData included in this package. We will create a model that predicts the logarithm of the individual's wages (lwage) with their union status (union), which can change over time, and their race (blk; dichotomized as black or non-black), which does not change throughout the period of study. Our formula will look like this:

lwage ~ union | blk

Put time-varying variables before the first | and time-invariant variables afterwards. You can specify lags like lag(union) for time-varying variables; for more than 1 lag, include the number: lag(union, 2).

After the first | go the time-invariant variables. Note that if you put a time-varying variable here, what you get is the observed value rather than one adjusted to isolate within-entity effects. You may also take a time-varying variable --- let's say weeks worked (wks) --- and use imean(wks) to include the individual's mean across all waves as a predictor while omitting the per-wave measures.

There is also a place for a second |. Here you can specify cross-level interactions (within-level interactions can be specified here as well). If I wanted the interaction term for union and blk --- to see whether the effect of union status depended on one's race --- I would specify the formula this way:

lwage ~ union | blk | union * blk

Another use for the post-second | section of the formula is for changing the random effects specification. By default, only a random intercept is specified in the call to lme4::lmer()/lme4::glmer(). If you would like to specify other random slopes, include them here using the typical lme4 syntax:

lwage ~ union | blk | (union | id)

You can also include the wave variable in a random effects term to specify a latent growth curve model:

lwage ~ union | blk + t | (t | id)

One last thing to know: If you want to use the second | but not the first, put a 1 or 0 after the first, like this:

lwage ~ union | 1 | (union | id)

Of course, with no time-invariant variables, you need no | operators at all.

Models

As a convenience, wbm does the heavy lifting for specifying the within-between model correctly. As a side effect it only takes a few easy tweaks to specify the model slightly differently. You can change this behavior with the model argument.

By default, the argument is "w-b" (equivalently, "within-between"). This means, for each time-varying predictor, you have two types of variables in the model. The "between" effect is represented by the individual-level mean for each entity (e.g., each respondent to a panel survey). The "within" effect is represented by each wave's measure with the individual-level mean subtracted. Some refer to this as "de-meaning." Thinking in a Hausman test framework --- with the within-between model as described here --- you should expect the within and between coefficients to be the same if a random effects model were appropriate.

The contextual model is very similar (use argument "contextual"). In some situations, this will be more intuitive to interpret. Empirically, the only difference compared to the within-between specification is that the contextual model does not subtract the individual-level means from the wave-level measures. This also changes the interpretation of the between-subject coefficients: In the contextual model, they are the difference between the within and between effects. If there's no difference between within and between effects, then, the coefficients will be 0.

To fit a random effects model, use either "between" or "random". This involves no de-meaning and no individual-level means whatsoever.

To fit a fixed effects model, use either "within" or "fixed". Any between-subjects terms in the formula will be ignored. The time-varying variables will be de-meaned, but the individual-level mean is not included in the model.

## References

Allison, P. (2009). Fixed effects regression models. Thousand Oaks, CA: SAGE Publications. https://doi.org/10.4135/9781412993869.d33

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, 133–153. https://doi.org/10.1017/psrm.2014.7

Curran, P. J., & Bauer, D. J. (2011). The disaggregation of within-person and between-person effects in longitudinal models of change. Annual Review of Psychology, 62, 583–619. https://doi.org/10.1146/annurev.psych.093008.100356

Giesselmann, M., & Schmidt-Catran, A. (2018). Interactions in fixed effects regression models (Discussion Papers of DIW Berlin No. 1748). DIW Berlin, German Institute for Economic Research. Retrieved from https://ideas.repec.org/p/diw/diwwpp/dp1748.html

Schunck, R., & Perales, F. (2017). Within- and between-cluster effects in generalized linear mixed models: A discussion of approaches and the xthybrid command. The Stata Journal, 17, 89–115. https://doi.org/10.1177/1536867X1701700106

wbm_stan() for a Bayesian estimation option.

## Examples

data("WageData")
wages <- panel_data(WageData, id = id, wave = t)
model <- wbm(lwage ~ lag(union) + wks | blk + fem | blk * lag(union),
data = wages)
summary(model)#> MODEL INFO:
#> Entities: 595
#> Time periods: 2-7
#> Dependent variable: lwage
#> Model type: Linear mixed effects
#> Specification: within-between
#>
#> MODEL FIT:
#> AIC = 1386.31, BIC = 1448.11
#> Pseudo-R² (fixed effects) = 0.13
#> Pseudo-R² (total) = 0.74
#> Entity ICC = 0.7
#>
#> WITHIN EFFECTS:
#> ---------------------------------------------------------
#>                     Est.   S.E.   t val.      d.f.      p
#> ---------------- ------- ------ -------- --------- ------
#> lag(union)          0.06   0.03     2.28   2972.01   0.02
#> wks                -0.00   0.00    -1.51   2994.31   0.13
#> ---------------------------------------------------------
#>
#> BETWEEN EFFECTS:
#> ---------------------------------------------------------------
#>                            Est.   S.E.   t val.     d.f.      p
#> ----------------------- ------- ------ -------- -------- ------
#> (Intercept)                6.60   0.23    28.53   589.99   0.00
#> imean(lag(union))         -0.03   0.03    -0.80   589.98   0.42
#> imean(wks)                 0.00   0.00     0.91   589.99   0.36
#> blk                       -0.23   0.06    -3.85   589.98   0.00
#> fem                       -0.44   0.05    -8.89   589.98   0.00
#> ---------------------------------------------------------------
#>
#> CROSS-LEVEL INTERACTIONS:
#> -------------------------------------------------------------
#>                         Est.   S.E.   t val.      d.f.      p
#> -------------------- ------- ------ -------- --------- ------
#> lag(union):blk         -0.13   0.12    -1.03   2971.99   0.31
#> -------------------------------------------------------------
#>
#> p values calculated using Satterthwaite d.f.
#>
#> RANDOM EFFECTS:
#> ------------------------------------
#>   Group      Parameter    Std. Dev.
#> ---------- ------------- -----------
#>     id      (Intercept)     0.354
#>  Residual                  0.2326
#> ------------------------------------