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Tilted (self-censoring-adjusted) within-person VAR(1) at a fixed value of the sensitivity parameter

Source: R/fit_tilt.R
fit_tilt.Rd

Sensitivity analysis for motif M2 (self-censoring). The selection model \(P(R_t = 1 \mid X_t = x) = \Phi_N(\alpha + \delta' x)\) is held at a fixed sensitivity vector \(\delta\) that is never estimated from the fit. The response intercept is calibrated so that the model-implied response rate given the observed history matches the observed one, and complete pairs are reweighted by \(w_t = P(R_t = 1 \mid x_{t-1}) / P(R_t = 1 \mid x_t)\) (inverse-probability weighting with a stabilized numerator). Weighted within-person regression and intercept calibration are iterated to a fixed point; the population parameters are a fixed point when \(\delta\) is the true selection vector.

Usage

fit_tilt(
  data,
  vars,
  delta,
  id = "id",
  time = "time",
  day = NULL,
  R = "R",
  propensity = c("common", "person"),
  pairs = NULL,
  probe = NULL,
  max_iter = 200L,
  tol = 1e-05,
  start = NULL,
  damping = 0.5,
  min_pairs = 5L
)

Arguments

data

Long data frame with one row per scheduled prompt.

vars

Names of the state columns.

delta

The sensitivity parameter: a numeric vector (length = number of variables) of selection coefficients in probit units per unit of the state; a scalar is applied to the first variable. Negative values encode "high states are skipped".

id, time

Names of the person and prompt-index columns.

day

Optional name of a day column; pairs are formed only within a day (the overnight gap is not a lag-1 transition).

R

Name of the response-indicator column (0/1, no NA). If the default name is not among the columns of data, a prompt counts as answered when all vars are non-missing; any other name must exist.

propensity

"common" (one intercept) or "person" (one intercept per person; use when a person-propensity motif M4 is declared together with M2).

pairs

Optional precomputed result of make_pairs.

probe

Optional name of a randomized-probe indicator column; probe prompts are answered by design, receive weight one and are excluded from the intercept calibration.

max_iter, tol

Fixed-point iteration control.

start

Optional pairs_fit (typically the fit at a neighboring sensitivity value) used as the starting point of the iteration.

damping

Step size of the damped fixed-point update (1 = undamped).

min_pairs

Passed to fit_pairs.

Value

An object of class c("tilt_fit", "pairs_fit"): a

fit_pairs object with additional elements delta,

alpha (calibrated intercept(s)), iterations,

converged, weights, max_weight, propensity, and ess, the effective number of complete pairs \((\sum w)^2 / \sum w^2\) (Kish). The weights are not truncated: profiles in which ess falls far below the number of pairs, or in which a few pairs carry very large weights, rest on few observations and should be read with caution. The cluster-robust covariance treats the weights as fixed (the uncertainty of the calibrated intercept and of the parameters entering the weights is not propagated).

See also

tilt_profile fits a grid of sensitivity values; calibrate_delta chooses one from a design feature.

Examples

sim <- simulate_ema(N = 40, n_prompts = 30, motifs = "M2", delta = -1, seed = 1)
f0 <- fit_pairs(sim$data, sim$vars)      # treats skips as ignorable
f <- fit_tilt(sim$data, sim$vars, delta = -1)
f
#> Within-person VAR(1) from 709 complete adjacent pairs, 40 persons (half-panel jackknife) 
#> Lagged coefficients Phi (rows = outcome at t, columns = predictor at t-1):
#>           NegA   PosA Stress Fatigue
#> NegA     0.300 -0.022  0.271  -0.058
#> PosA    -0.084  0.433 -0.050   0.113
#> Stress   0.106 -0.049  0.474   0.133
#> Fatigue  0.030 -0.133 -0.017   0.307
#> Contemporaneous partial correlations (innovations):
#>           NegA   PosA Stress Fatigue
#> NegA     1.000 -0.423  0.311   0.023
#> PosA    -0.423  1.000  0.095   0.002
#> Stress   0.311  0.095  1.000   0.186
#> Fatigue  0.023  0.002  0.186   1.000
#> Between-person means (dynamics-recovered): 2.369 3.833 2.849 2.991 
f$ess / f$n_pairs                          # effective share of pairs
#> [1] 0.9331293
c(untilted = f0$Phi[1, 1], tilted = f$Phi[1, 1], truth = default_params()$Phi[1, 1])
#> untilted   tilted    truth 
#> 0.272030 0.299884 0.400000 
# person-specific response intercepts (motif M4 declared with M2)
fp <- fit_tilt(sim$data, sim$vars, delta = -1, propensity = "person")
summary(fp$alpha)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   2.524   3.271   3.563   3.573   3.924   4.399