recoverability(dm_graph("M1"))
#> Recoverability report for motifs: M1
#>
#> * transition kernel (Phi, Psi, contemporaneous network)
#> [recoverable] complete adjacent pairs, within-person (person intercepts)
#> condition: R_t _||_ X_t | {X_{t-1},eta} and R_{t-1} _||_ X_t | {X_{t-1},eta, R_t}
#> * person mean via observed within-person mean
#> [NOT recoverable] biased
#> condition: R_t _||_ X_t | {eta}
#> * person mean via recovered dynamics
#> [recoverable] mu_i = (I - Phi)^{-1} c_i from the recovered kernel
#> condition: transition kernel recoverable
#> * between-person law (mu, Sigma_mu), person-weighted
#> [recoverable] one recovered mean per person (dynamics-recovered means), persons weighted equally; positivity assumed
#> condition: a person-mean estimator is available and P(R_t = 1 | eta) > 0
#> * between-person law, prompt-weighted (pooling answered prompts)
#> [NOT recoverable] biased: response rate depends on the person's states
#> condition: R_t _||_ X_t | {empty set}
#> * silence test (coefficient of R_t in X_{t+1} ~ X_{t-1} + R_t, within person)
#> null expected (test valid as a test of the recoverable class)
#> condition: R_t _||_ X_{t+1} | {X_{t-1},eta,R_{t-1},R_{t+1}}
A table over the taxonomy
The loop below reproduces the recoverability table of the
accompanying article for all single motifs and the combinations that
matter in practice.
decls <- list("M0", "M1", "M2", "M3", "M4", "M5", "M6", c("M1", "M3"), c("M1", "M4"),
c("M3", "M4"), c("M2", "M4"), c("M2", "M3"))
tab <- do.call(rbind, lapply(decls, function(m) {
r <- recoverability(dm_graph(m))
data.frame(motifs = paste(m, collapse = "+"),
kernel = r$recoverable[1], mean_obs = r$recoverable[2], mean_dyn = r$recoverable[3],
law_person = r$recoverable[4], law_prompt = r$recoverable[5],
silence_null = attr(r, "silence_null"))
}))
#> M5 declared without 'context': a latent context is assumed
tab
#> motifs kernel mean_obs mean_dyn law_person law_prompt silence_null
#> 1 M0 TRUE TRUE TRUE TRUE TRUE TRUE
#> 2 M1 TRUE FALSE TRUE TRUE FALSE TRUE
#> 3 M2 FALSE FALSE FALSE FALSE FALSE FALSE
#> 4 M3 TRUE TRUE TRUE TRUE TRUE TRUE
#> 5 M4 TRUE TRUE TRUE TRUE FALSE TRUE
#> 6 M5 FALSE FALSE FALSE FALSE FALSE FALSE
#> 7 M6 TRUE TRUE FALSE TRUE TRUE FALSE
#> 8 M1+M3 TRUE FALSE TRUE TRUE FALSE FALSE
#> 9 M1+M4 TRUE FALSE TRUE TRUE FALSE TRUE
#> 10 M3+M4 TRUE TRUE TRUE TRUE FALSE TRUE
#> 11 M2+M4 FALSE FALSE FALSE FALSE FALSE FALSE
#> 12 M2+M3 FALSE FALSE FALSE FALSE FALSE FALSE
Three patterns organize the table.
- M0, M1, M3, M4 and their combinations are
recoverable. Skipping may depend on the previous state, on the
previous response, or on a person’s stable propensity, and the answered
adjacent pairs with person intercepts still recover the dynamics. Two
casualties remain: under M1 (alone or combined) the observed
within-person mean is biased, because skipping depends on the previous
state, so the person mean must be recovered from the dynamics; and the
prompt-weighted between-person law is biased under M1 and M4, whenever
the response rate depends on a person’s states or on
zeta.
- M2 and latent M5 are not recoverable.
Self-censoring and an unobserved context make
R_t depend on
X_t itself, and no conditioning set of observed variables
blocks the path. Structural non-recoverability does not say how large
the bias of a given estimator is: in the additive simulations of Yu
(2026) the lagged coefficients are barely affected by a serially
independent latent context (the shift lands in the intercepts) but are
biased under a persistent one, while the means are biased in both cases,
and under self-censoring everything is biased. This is where the
sensitivity analysis of vignette("sensitivity-analysis")
comes in.
- M6 is asymmetric. Reactivity leaves the kernel
recoverable (complete pairs estimate the assessment-conditioned kernel),
but the intercept of the answered pairs absorbs the reactivity shift, so
the person mean must be read from the answered prompts rather than from
the dynamics.
The silence_null column shows where the silence test is
a valid test of the recoverable class: everywhere among M0, M1, M3, M4
and their combinations except M1 + M3 (under M6 the kernel is
recoverable, but reactivity makes the test reject by construction),
where conditioning on an answered prompt at t + 1 opens a
collider at R_{t+1}: its parents are R_t
(burden) and X_t (lagged-state dependence), and
X_t drives X_{t+1}. The report says so in
words:
r13 <- recoverability(dm_graph(c("M1", "M3")))
r13$estimator[grepl("silence", r13$estimand)]
#> [1] "null violated although the kernel is recoverable (collider at R_{t+1}); use the sensor-gap test or check for fatigue first"
Observed contexts, sensors and probes
An observed context turns latent M5 into a recoverable mechanism: the
context enters the conditioning set (covariate adjustment,
fit_pairs(covariates = )), or the pairs are reweighted so
that the context-marginal kernel is recovered
(fit_ipw()).
recoverability(dm_graph("M5", context = "observed"))[1, c("recoverable", "estimator")]
#> recoverable
#> 1 TRUE
#> estimator
#> 1 complete pairs, context-conditional kernel (covariate adjustment); context-marginal kernel by stabilized IPW on C
A sensor adds the sensor-gap row, which is valid also under M1 + M3
because nothing is conditioned on at t + 1:
rs <- recoverability(dm_graph(c("M1", "M3"), sensor = TRUE))
rs$estimator[grepl("sensor", rs$estimand)]
#> [1] "null expected (test valid as a test of the recoverable class)"
A probe adds a row for the kernel estimated from probe prompts only.
Because Z_t = 1 forces R_t = 1, complete pairs
restricted to probe prompts are not selected on X_t even
under self-censoring; this is what
calibrate_delta(method = "probe") exploits.
rp <- recoverability(dm_graph("M2", probe = TRUE))
rp[rp$estimand == "transition kernel from probe prompts", c("recoverable", "estimator")]
#> recoverable
#> 2 TRUE
#> estimator
#> 2 complete pairs restricted to probe prompts (Z_t = 1); delta calibrated by the probe contrast
