---
title: "Was silence informative? The silence test, the sensor-gap test and the fatigue check"
author: "Hsiu-Ting Yu"
output:
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{Was silence informative? The silence test, the sensor-gap test and the fatigue check}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 4.2)
library(silentema)
```

## The question

A declared dm-graph says which mechanism is assumed (see `vignette("dm-graphs")`). The tests in this vignette ask the data whether that assumption is tenable. They exploit one fact: under every recoverable mechanism, the response indicator at prompt *t* carries no information about the states beyond what the observed history already carries. If skipped prompts turn out to predict what is observed afterwards, silence was informative.

Two tests and one descriptive check are available.

| Function | Regression (within person) | Null holds under | Needs |
|---|---|---|---|
| `silence_test()` | `X_{t+1} ~ X_{t-1} + R_t` over triples with *t-1* and *t+1* answered | M0, M1, M3, M4 and combinations without M1 + M3 | nothing beyond the states |
| `sensor_gap_test()` | `S_t ~ X_{t-1} + R_t` over prompts with an answered predecessor | all recoverable motifs, including M1 + M3 | an always-observed sensor |
| `fatigue_check()` | response rate after an answered vs a skipped prompt; by study quarter | (descriptive) | the response indicators |

The two tests use person fixed effects (within-person demeaning), so stable between-person differences in level or in compliance cannot produce a spurious result. The fatigue check reports both a pooled contrast, which between-person differences in compliance inflate, and the mean of the person-specific contrasts, which they do not.

## The silence test

Take every triple of consecutive scheduled prompts (*t-1*, *t*, *t+1*) in which the two outer prompts were answered. The middle prompt may or may not have been answered. Regress the state at *t+1* on the state at *t-1* and the response indicator at *t*, within person. If the middle skip is uninformative, its coefficient is zero.

```{r silence}
sim2 <- simulate_ema(N = 60, n_prompts = 40, motifs = "M2", compliance = 0.7, delta = -1, seed = 1)
st <- silence_test(sim2$data, sim2$vars)
st
```

The data were generated with self-censoring on negative affect (`delta = -1`: the probit propensity to respond falls by one unit per unit of negative affect). The coefficient of `R_t` for `NegA` is negative: after a *skipped* prompt (`R_t = 0`), negative affect at *t+1* is higher than after an answered one, given the state at *t-1*. The states hidden by skips were higher than the reported ones, which is exactly what self-censoring produces. The joint Wald test over all four variables combines the evidence.

The result object is a data frame with attributes:

```{r silence-attr}
attr(st, "joint")
c(triples = attr(st, "n_triples"), skipped_middle = attr(st, "n_skipped"))
```

Under a recoverable mechanism the coefficients are near zero and the joint test does not reject:

```{r silence-null}
sim1 <- simulate_ema(N = 60, n_prompts = 40, motifs = "M1", compliance = 0.7, seed = 1)
silence_test(sim1$data, sim1$vars)
```

### Reading the sign

The sign of the coefficient of `R_t` on the self-censoring variable is the sign of the *tilt*: a negative coefficient means that the hidden states were higher (people skip when the state is high), a positive one that they were lower. This sign is what `tilt_profile()` and `calibrate_delta()` later quantify. Coefficients on the other variables reflect the same tilt propagated through the contemporaneous and lagged associations.

### Inference

With at least 10 persons the default is a cluster-robust covariance by person with *t*(*G* - 1) and *F*(*q*, *G* - 1) reference distributions (Cameron & Miller, 2015). For a single person (or very few), `se = "dayblock"` resamples person-day blocks with replacement; it needs a `day` column.

```{r single}
one <- simulate_ema(N = 1, n_prompts = 200, motifs = "M2", compliance = 0.7, delta = -1, days = 8, seed = 5)
silence_test(one$data, one$vars, day = "day", B = 100, seed = 1)
```

With one person and a few dozen triples the test has little power; the coefficient for `NegA` has the expected sign but its bootstrap interval covers zero. Single-case analyses should lean on the sensitivity analysis rather than on the test.

### Curvature

Under M1 the selection on `X_t` through `R_{t+1}` makes the conditional mean of `X_{t+1}` given `X_{t-1}` slightly nonlinear; `poly = 2` adds squared lagged states. In the simulation studies of Yu (2026) the linear test kept its size within Monte Carlo error under M1, so `poly = 1` is the default.

```{r poly}
silence_test(sim1$data, sim1$vars, poly = 2)$p
```

### The M1 + M3 collider

When lagged-state dependence and burden are both present, 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}`. Given that the prompt at *t+1* was answered despite a skip at *t*, the state at *t* must have favored responding, so the state at *t+1* is shifted, and the silence test rejects in large samples although the transition kernel is recoverable. `recoverability()` warns about it. The coefficient has the sign *opposite* to the self-censoring signature (positive when high states lower the propensity to respond), which is one reason to run `fatigue_check()` before interpreting a rejection. Under M2 + M3 the analogous collider (with `X_{t+1}` as a parent of `R_{t+1}`) works *against* the self-censoring signal, so the test loses power. In both cases the sensor-gap test is the better instrument.

```{r collider}
sim13 <- simulate_ema(N = 300, n_prompts = 56, motifs = c("M1", "M3"), compliance = 0.7, kappa_R = 1, seed = 2)
silence_test(sim13$data, sim13$vars)
recoverability(dm_graph(c("M1", "M3")))$estimator[6]
```

## The sensor-gap test

An always-observed channel that loads on the states (an accelerometer, heart rate, phone-usage features) turns the question around: instead of asking what happens *after* a skip, ask what the sensor recorded *at* the skipped prompt. `sensor_gap_test()` regresses the sensor at *t* on the state at *t-1* and `R_t`, within person, over all prompts whose predecessor was answered. Nothing at *t+1* is conditioned on, so the test is valid under M1 + M3 as well.

```{r sensor}
sim2s <- simulate_ema(N = 60, n_prompts = 40, motifs = "M2", compliance = 0.7, delta = -1,
                      sensor_cor = 0.6, seed = 3)
sg <- sensor_gap_test(sim2s$data, sim2s$vars, sensor = "S")
sg
attr(sg, "loading")
```

The sensor gap is negative: at skipped prompts the sensor recorded higher values of what it measures. The `loading` attribute gives the within-person regression of the sensor on the states at answered prompts; the calibration in `vignette("sensitivity-analysis")` uses it to translate the gap into a sensitivity value.

Under a recoverable mechanism the gap is null:

```{r sensor-null}
sim1s <- simulate_ema(N = 60, n_prompts = 40, motifs = c("M1", "M3"), compliance = 0.7, sensor_cor = 0.6, seed = 3)
sensor_gap_test(sim1s$data, sim1s$vars, sensor = "S")
```

## The fatigue check

`fatigue_check()` is descriptive. It reports the response rate after an answered and after a skipped prompt, pooled and as the mean of the person-specific differences, and the response rate by study quarter. A large persistence contrast (the `kappa_R` term of the simulator) and, when compliance drifts over the study (a negative `burden` term), a declining rate over quarters are what burden (M3) produces; the example below uses `kappa_R` only, so the quarterly rates stay flat. Some serial dependence in `R` also arises under M1, M2 and M4 through the autocorrelated states and the person propensities, so the check does not discriminate M3 from the other motifs by itself.

```{r fatigue}
sim3 <- simulate_ema(N = 60, n_prompts = 40, motifs = "M3", compliance = 0.7, kappa_R = 1, seed = 4)
fatigue_check(sim3$data)
fatigue_check(sim2$data)
```

Under burden (first output) the response rate after a skipped prompt is far below the rate after an answered one, pooled and within person. Under self-censoring alone (second output) the pooled contrast is smaller and the person-mean contrast smaller still: skips cluster in time because the states that cause them are autocorrelated, not because a skip itself lowers the next propensity.

Where burden is declared together with self-censoring, the observed persistence is the quantity that `calibrate_delta(method = "postskip", burden = "fit")` matches.

## How often do the tests reject? A small Monte Carlo

The loop below repeats the silence test on new data sets under three mechanisms. It is deliberately small (the accompanying article uses 500 replications per cell and larger samples); the point is to show the pattern: size near the nominal level under M0 and M1, power under M2.

```{r mc}
rej <- function(motifs, reps = 60, ...) {
  mean(vapply(seq_len(reps), function(r) {
    s <- simulate_ema(N = 60, n_prompts = 40, motifs = motifs, compliance = 0.7, seed = 1000 + r, ...)
    attr(silence_test(s$data, s$vars), "joint")["p"] < 0.05
  }, logical(1)))
}
c(M0 = rej("M0"), M1 = rej("M1"), M2 = rej("M2", delta = -1))
```

With 60 replications the Monte Carlo standard error of a rejection rate near .05 is about .03, so the first two numbers are compatible with the nominal level; the third shows the power the test has at this sample size and tilt. In the simulation studies of Yu (2026) (500 replications per cell, *N* = 100, *T* = 56) the rejection rate under the recoverable motifs stayed between 4% and 7% with two mild excesses (8.8% under burden at 70% compliance and 7.4% under lagged-state dependence at 85%), compatible with Monte Carlo error and the known anticonservatism of cluster-robust standard errors.

## What to do with a rejection

A rejection says that the declared recoverable class is not tenable (or, under M1 + M3, that the collider is at work). It does not say which non-recoverable mechanism is present. The recommended continuation is

1. check for burden with `fatigue_check()`; if it is substantial and M1 is plausible, prefer the sensor-gap test;
2. if the sensor-gap test also rejects, or no sensor exists and burden is small, treat the skips as self-censoring and run the sensitivity analysis of `vignette("sensitivity-analysis")`, using the sign of the silence test as the sign of the tilt;
3. record both tests in the `missingness_declaration()`.

A non-rejection is weaker evidence: at typical sample sizes the tests have limited power against small tilts, and the identified sets of the sensitivity analysis remain the honest summary when self-censoring is substantively plausible.

## References

Cameron, A. C., & Miller, D. L. (2015). A practitioner's guide to cluster-robust inference. *Journal of Human Resources, 50*, 317-372. https://doi.org/10.3368/jhr.50.2.317

Yu, H.-T. (2026). What skipped prompts hide: Detecting, diagnosing, and correcting informative nonresponse in ecological momentary assessment. Manuscript under review.