Fit a within-between (hybrid) model — fit_within

Puts the within-person and the between-person component of a predictor into one model, so the two effects can be compared directly and the gap between them tested. Person-centering alone cannot do this: it estimates the within effect and discards the between effect entirely.

Usage

fit_within_between(
  data,
  y,
  x,
  id,
  time = NULL,
  model = c("within_between", "within", "between", "contextual"),
  estimator = c("ols", "reml", "ml"),
  scope = "pooled",
  subgroup = NULL,
  random = NULL,
  cluster = NULL,
  conf_level = 0.95,
  test_prop = 0.2,
  min_train = 10L,
  min_test = 1L
)

Arguments

data: Data frame.
y: Outcome column name.
x: Predictors: names, numeric positions, a:b range, formula, or data frame.
id: Person/unit ID column.
time: Optional ordering column.
model: Parameterization: "within_between", "within", "between", or "contextual". See Details.
estimator: "ols" (cluster-robust least squares), or "reml" / "ml" for a mixed model via lme4.
scope: "pooled", "subgroup", or "all". A between-person term is constant inside a person, so individual scope is available only for model = "within".
subgroup: Optional subgroup mapping: an find_subgroups() result, a grouping column in data, or a named vector of labels per person.
random: Extra grouping columns to give random intercepts, e.g. random = "course" for a cross-classified design. Mixed estimators only.
cluster: Column(s) to cluster standard errors on, defaulting to id. Up to two, for two-way clustering. OLS only.
conf_level: Confidence level for the reported intervals.
test_prop: Proportion of each person's ordered rows held out.
min_train: Minimum complete training rows per person.
min_test: Minimum complete held-out rows per person.

Value

An idiographic_wb object, which is also an idiographic_fit. coefs() gains component and variable columns; contextual() returns the within/between comparison.

Details

Each predictor x is split into a person mean (the between component) and the deviation from it (the within component).

Model parameterizations

within_between: Both components as separate terms (the default). The contextual effect is their difference, tested as a contrast.
within: The within component only – the fixed-effects estimator. The only parameterization that also works at individual scope.
between: The between component only.
contextual: The raw predictor plus the person mean. Algebraically equivalent to within_between, but the coefficient on the person mean is the contextual effect directly, rather than a contrast of two.

Estimators

"ols" (the default, base R) fits by least squares with cluster-robust standard errors; rows within a person are not independent, and treating them as independent gives roughly 77% coverage where 95% is claimed. Pass cluster to cluster on something other than – or in addition to – the person; with two clustering variables the Cameron-Gelbach-Miller two-way estimator is used, which is what a design with people crossed by courses needs.

"reml" and "ml" fit a mixed model with lme4 instead, adding a random intercept for id and for anything named in random. This is what makes a cross-classified design – (1 | person) + (1 | course) – expressible, and it makes the model's variance components available through variance_components(). The fixed-effect estimates agree closely with the OLS ones; what differs is the standard errors and the variance decomposition.

Person means are computed on the training rows and applied to the held-out rows, so the reported metrics stay honest.

Examples

fit <- fit_within_between(srl, y = "effort", x = c("efficacy", "planning"),
                          id = "name", time = "day")
coefs(fit)
#>    scope          model estimator subject subgroup             term component
#> 1 pooled within_between       ols    .all     .all      (Intercept)     .none
#> 2 pooled within_between       ols    .all     .all  efficacy_within    within
#> 3 pooled within_between       ols    .all     .all  planning_within    within
#> 4 pooled within_between       ols    .all     .all efficacy_between   between
#> 5 pooled within_between       ols    .all     .all planning_between   between
#>   variable   estimate  std_error statistic      p_value   conf_low  conf_high
#> 1    .none 21.1716609 7.12357772  2.972054 5.321378e-03  6.7100293 35.6332925
#> 2 efficacy  0.2680674 0.04661939  5.750126 1.648335e-06  0.1734250  0.3627098
#> 3 planning  0.3319056 0.04783083  6.939157 4.578176e-08  0.2348039  0.4290074
#> 4 efficacy  0.4601588 0.13552830  3.395297 1.719875e-03  0.1850217  0.7352959
#> 5 planning  0.2257383 0.14281198  1.580668 1.229501e-01 -0.0641854  0.5156621
contextual(fit)
#>   variable   within   between   contextual     S.E.            95% CI       p
#> -----------------------------------------------------------------------------
#>   efficacy   0.2681    0.4602       0.1921   0.1361   [-0.084, 0.468]   0.167
#>   planning   0.3319    0.2257      -0.1062   0.1496   [-0.410, 0.197]   0.483
#> 
#> scope = pooled,  model = within_between,  estimator = ols,  subject = .all,  subgroup = .all

# The contextual parameterization reports the same gap as one coefficient.
fit_within_between(srl, y = "effort", x = "efficacy", id = "name",
                   model = "contextual")
#> MODEL INFO
#>   Outcome         effort
#>   Predictors      efficacy
#>   Person ID       name (36 people)
#>   Specification   contextual
#>   Estimator       OLS, cluster-robust on name
#>   Scope           pooled
#> 
#> MODEL FIT (held out)
#>   Rows        1,150
#>   RMSE        21.9825
#>   R-squared   0.3304
#> 
#> WITHIN EFFECTS
#>                Est.     S.E.   t val.        p
#> ----------------------------------------------
#>   efficacy   0.4164   0.0552     7.55   <1e-04
#> 
#> CONTEXTUAL EFFECTS (between - within)
#>                Est.     S.E.           95% CI        p
#> ------------------------------------------------------
#>   efficacy   0.2102   0.1021   [0.003, 0.417]   0.0469
#>   An interval excluding zero means the two processes differ.