Censoring Types and Masked Causes

Two Independent Sources of Information Loss

Masked-cause series system data involves two orthogonal forms of incomplete observation:

1. Censoring of the failure time — we may not observe \(T_i\) exactly. The system may be observed only as “still working at \(t\)” (right-censored), “already failed by \(t\)” (left-censored), or “failed somewhere in \((t, t_{\text{upper}})\)” (interval-censored).
2. Masking of the failure cause — even when \(T_i\) is observed, the component that failed is often only known up to a candidate set \(C_i\) that satisfies conditions C1, C2, and C3.

maskedhaz handles all four censoring types uniformly via numerical integration when needed. This vignette walks through each observation type with a worked example, then cross-validates the numerical machinery against maskedcauses’ closed-form exponential series likelihood.

The omega Column

Data frames passed to loglik(), score(), hess_loglik(), and fit() must include an omega column indicating the observation type per row:

Candidate sets are Boolean columns x1, x2, …, xm. Right-censored rows are allowed to have all-FALSE candidate columns because they carry no cause-of-failure information.

Contribution Formulas

For a row of type \(\omega\), the log-likelihood contribution is:

• Exact: \(\log\left(\sum_{j \in C_i} h_j(t_i; \theta_j)\right) - H_{\text{sys}}(t_i; \theta)\)
• Right: \(-H_{\text{sys}}(t_i; \theta)\)
• Left: \(\log \int_0^{t_i} \left[\sum_{j \in C_i} h_j(u; \theta_j)\right] S_{\text{sys}}(u; \theta) \, du\)
• Interval: \(\log \int_{t_i}^{t_{\text{upper},i}} \left[\sum_{j \in C_i} h_j(u; \theta_j)\right] S_{\text{sys}}(u; \theta) \, du\)

The exact and right contributions are closed-form regardless of the component family. Left and interval contributions require integration of the candidate-weighted instantaneous hazard over the observation interval, which maskedhaz does via stats::integrate().

Worked Examples

library(maskedhaz)
#> Registered S3 method overwritten by 'dist.structure':
#>   method          from        
#>   mean.exp_series maskedcauses

model <- dfr_series_md(components = list(
  dfr_weibull(shape = 2, scale = 6),
  dfr_exponential(0.08),
  dfr_exponential(0.12)
))

set.seed(100)
df <- rdata(model)(theta = c(2, 6, 0.08, 0.12), n = 400, tau = 5, p = 0.4)
table(df$omega)
#> 
#> exact right 
#>   317    83

result <- fit(model)(df, par = c(1.5, 7, 0.05, 0.1))
rbind(truth = c(2, 6, 0.08, 0.12), estimate = round(coef(result), 3))
#>           [,1] [,2] [,3]  [,4]
#> truth    2.000    6 0.08 0.120
#> estimate 2.487    6 0.08 0.108

df_left <- data.frame(
  t     = c(2, 3, 4, 3.5, 5),
  omega = "left",
  x1    = c(TRUE,  TRUE,  FALSE, TRUE,  TRUE),
  x2    = c(FALSE, TRUE,  TRUE,  FALSE, TRUE),
  x3    = c(TRUE,  FALSE, TRUE,  TRUE,  FALSE),
  stringsAsFactors = FALSE
)
df_left
#>     t omega    x1    x2    x3
#> 1 2.0  left  TRUE FALSE  TRUE
#> 2 3.0  left  TRUE  TRUE FALSE
#> 3 4.0  left FALSE  TRUE  TRUE
#> 4 3.5  left  TRUE FALSE  TRUE
#> 5 5.0  left  TRUE  TRUE FALSE

ll_fn <- loglik(model)
ll_fn(df_left, par = c(2, 6, 0.08, 0.12))
#> [1] -4.62688

df_interval <- data.frame(
  t       = c(1.0, 2.0, 3.5),
  t_upper = c(2.5, 4.0, 5.0),
  omega   = "interval",
  x1      = c(TRUE,  FALSE, TRUE),
  x2      = c(TRUE,  TRUE,  FALSE),
  x3      = c(FALSE, TRUE,  TRUE),
  stringsAsFactors = FALSE
)
df_interval
#>     t t_upper    omega    x1    x2    x3
#> 1 1.0     2.5 interval  TRUE  TRUE FALSE
#> 2 2.0     4.0 interval FALSE  TRUE  TRUE
#> 3 3.5     5.0 interval  TRUE FALSE  TRUE
ll_fn(df_interval, par = c(2, 6, 0.08, 0.12))
#> [1] -5.506764

df_mixed <- data.frame(
  t       = c(3,    5,    2,    1.5,  4.5),
  t_upper = c(NA,   NA,   NA,   3.0,  NA),
  omega   = c("exact", "right", "left", "interval", "exact"),
  x1      = c(TRUE,  FALSE, TRUE,  TRUE,  FALSE),
  x2      = c(FALSE, FALSE, TRUE,  FALSE, TRUE),
  x3      = c(TRUE,  FALSE, FALSE, TRUE,  TRUE),
  stringsAsFactors = FALSE
)

ll_fn(df_mixed, par = c(2, 6, 0.08, 0.12))
#> [1] -10.0332

Cross-Validation Against a Reference Implementation

Both maskedhaz and maskedcauses implement the series_md protocol — the same likelihood, the same data schema, the same C1–C2–C3 conditions. They differ only in how the likelihood is computed. maskedcauses ships a closed-form reference implementation for exponential-series models via exp_series_md_c1_c2_c3(). maskedhaz computes the same likelihood numerically (even for exponential components, the left/interval paths go through integrate()). Comparing the two evaluates the numerical machinery against a trusted analytical baseline.

library(maskedcauses)
#> 
#> Attaching package: 'maskedcauses'
#> The following objects are masked from 'package:maskedhaz':
#> 
#>     component_hazard, ncomponents

# Same three-component exponential series in both packages
rates <- c(0.1, 0.2, 0.3)
exp_model_haz <- dfr_series_md(
  components = list(dfr_exponential(), dfr_exponential(), dfr_exponential()),
  n_par = c(1L, 1L, 1L)
)
exp_model_mc <- exp_series_md_c1_c2_c3()

# A single-row data frame of each observation type
df_cv <- data.frame(
  t       = c(3.0, 5.0, 2.0, 1.5),
  t_upper = c(NA,  NA,  NA,  3.0),
  omega   = c("exact", "right", "left", "interval"),
  x1      = c(TRUE,  FALSE, TRUE,  TRUE),
  x2      = c(FALSE, FALSE, TRUE,  FALSE),
  x3      = c(TRUE,  FALSE, FALSE, TRUE),
  stringsAsFactors = FALSE
)

# Evaluate both log-likelihoods at the same parameters
ll_haz <- loglik(exp_model_haz)(df_cv, par = rates)
ll_mc  <- likelihood.model::loglik(exp_model_mc)(df_cv, par = rates)

c(maskedhaz = ll_haz, maskedcauses = ll_mc,
  difference = abs(ll_haz - ll_mc))
#>    maskedhaz maskedcauses   difference 
#>    -8.595121    -8.595121     0.000000

The two log-likelihoods agree to machine precision on this example (integrate()’s default relative tolerance of \(10^{-8}\) gave us all the digits). This is a direct test of the numerical machinery: maskedcauses uses closed-form expressions for each contribution (which exist for exponential components), while maskedhaz uses integrate() for the left and interval contributions. The fact that they match end-to-end means the integration path is correct.

Because both packages return the same fisher_mle class on fit(), the cross-validation also works at the estimation level — you can fit the same data with both, compare coef(), vcov(), and logLik() side by side, and any agreement (or disagreement) is directly interpretable because the return types are identical.

Identifiability Under Masking

Identifiability of individual component rates in an exponential series system depends on the masking structure, and it’s worth being precise about the limits.

# A realistic-sized dataset with partial masking
set.seed(2024)
df_id <- likelihood.model::rdata(exp_model_mc)(
  theta = rates, n = 300, tau = 5, p = 0.5
)
result_id <- suppressWarnings(
  fit(exp_model_haz)(df_id, par = c(0.5, 0.5, 0.5))
)

cat("individual MLE:", round(coef(result_id), 3),
    "  (truth:", rates, ")\n")
#> individual MLE: 0.109 0.219 0.282   (truth: 0.1 0.2 0.3 )
cat("sum of MLE:    ", round(sum(coef(result_id)), 3),
    "  (truth:", sum(rates), ")\n")
#> sum of MLE:     0.611   (truth: 0.6 )

With p = 0.5 (each non-failed component in the candidate set half the time), the individual rates are recovered well — candidate sets of varying composition carry real information about which component is which. We can verify this isn’t seed luck by checking the log-likelihood at very different rate vectors with the same sum:

ll_fn <- loglik(exp_model_haz)
c(truth    = ll_fn(df_id, par = c(0.10, 0.20, 0.30)),
  balanced = ll_fn(df_id, par = c(0.20, 0.20, 0.20)),
  skewed   = ll_fn(df_id, par = c(0.05, 0.05, 0.50)))
#>     truth  balanced    skewed 
#> -556.6886 -563.5536 -601.0104

The three parameter vectors all have sum \(0.6\), but the log-likelihoods differ substantially. The candidate-set structure is informative about the ratios of rates, not just the sum.

That said, individual identifiability degrades as masking grows. In the limit p = 1 (full masking, every candidate set is \(\{1, 2, 3\}\)), only the sum \(\sum_j \lambda_j\) remains identifiable — the candidate sets tell you nothing about which component failed, and the likelihood reduces to a function of the total rate alone. And under pure right-censoring (no exact failures), no candidate set information is ever carried, so only the sum is identified regardless of p. The series_md protocol doesn’t preclude fitting under these conditions; it just returns a flat likelihood surface and noisy individual estimates.

Mixing distribution families (as in vignette("custom-components")) sidesteps this fragility entirely — the qualitatively different hazard shapes pin down each component’s parameters through temporal signal, even when masking is heavy.

Summary

• The omega column is the single switch between contribution formulas
• Exact and right contributions are closed-form; left and interval use integrate()
• Candidate sets encode whatever diagnostic information narrows the cause
• Numerical likelihood agrees with maskedcauses closed-form to integrator tolerance
• Exponential-only systems have identifiability limitations — mixed families don’t