Transform and validate a scale-derived response variable

Takes a discrete (or continuous) response on the scale \(0, 1, \ldots, K\) (where \(K =\) ncuts) and converts it to a pair of interval endpoints on the open unit interval \((0, 1)\). Each observation is classified into one of four censoring types following the complete likelihood used in this package:

\(\delta = 0\): Uncensored (exact): the observation is a continuous value already in \((0, 1)\). The likelihood contribution is the density \(f(y_i | \theta)\). Endpoints: \(l_i = u_i = y_i\) (or \(y_i / K\) when on the scale).
\(\delta = 1\): Left-censored: the latent value is below some upper bound \(u_i\). The contribution is \(F(u_i | \theta)\). When the observation is at the scale minimum (\(y = 0\)), the upper bound is \(u_i = \mathrm{lim} / K\). When the user forces \(\delta = 1\) on a non-boundary observation (\(y \neq 0\)), the upper bound is \(u_i = (y + \mathrm{lim}) / K\), preserving observation- specific variation. In both cases \(l_i = \epsilon\).
\(\delta = 2\): Right-censored: the latent value is above some lower bound \(l_i\). The contribution is \(1 - F(l_i | \theta)\). When the observation is at the scale maximum (\(y = K\)), the lower bound is \(l_i = (K - \mathrm{lim}) / K\). When the user forces \(\delta = 2\) on a non-boundary observation (\(y \neq K\)), the lower bound is \(l_i = (y - \mathrm{lim}) / K\), preserving observation- specific variation. In both cases \(u_i = 1 - \epsilon\).
\(\delta = 3\): Interval-censored: the standard case for scale data. The contribution is \(F(u_i | \theta) - F(l_i | \theta)\) with midpoint interval endpoints \([(y - \mathrm{lim})/K,\; (y + \mathrm{lim})/K]\).

Usage

brs_check(y, ncuts = 100L, lim = 0.5, delta = NULL)

Arguments

y

Numeric vector: the raw response. Can be either integer scores on the scale \(\{0, 1, \ldots, K\}\) or continuous values already in \((0, 1)\).

ncuts

Integer: number of scale categories \(K\) (default 100). Must be \(\geq \max(y)\).

lim

Numeric: half-width \(h\) of the uncertainty region (default 0.5). Controls the width of the interval around each scale point.

delta

Integer vector or NULL. If NULL (default), censoring types are inferred automatically from the boundary rules described above.

If provided, must have the same length as y, with every element in {0, 1, 2, 3}. The supplied values override the automatic classification on a per-observation basis, and the endpoint formulas adapt to non-boundary observations as described in the table above.

This parameter is used internally by the simulation functions when the analyst forces a specific censoring type (e.g., brs_sim(..., delta = 2)).

Value

A numeric matrix with \(n\) rows and 5 columns:

left: Lower endpoint \(l_i\) on \((0, 1)\), clamped to \([\epsilon, 1 - \epsilon]\).
right: Upper endpoint \(u_i\) on \((0, 1)\), clamped to \([\epsilon, 1 - \epsilon]\).
yt: Midpoint approximation \(y_t\) for starting-value computation (does not enter the likelihood).
y: Original response value (preserved unchanged).
delta: Censoring indicator: 0 = exact (density), 1 = left-censored \(F(u)\), 2 = right-censored \(1 - F(l)\), 3 = interval-censored \(F(u) - F(l)\).

Details

Automatic classification (delta = NULL):

If the entire input vector is already in \((0, 1)\) (i.e., all values satisfy \(0 < y < 1\)), all observations are treated as uncensored (\(\delta = 0\)).

Otherwise, for scale (integer) data:

\(y = 0\): left-censored (\(\delta = 1\)).
\(y = K\): right-censored (\(\delta = 2\)).
\(0 < y < K\): interval-censored (\(\delta = 3\)).

User-supplied delta (delta vector):

When the delta argument is provided, the user-supplied censoring indicators override the automatic boundary-based rules on a per-observation basis. This is the mechanism used by brs_sim when the analyst forces a specific censoring type in Monte Carlo studies.

The endpoint formulas for each delta value are:

\(\delta\)	Condition	\(l_i\) (left)	\(u_i\) (right)
0	\(y \in (0, 1)\)	\(y\)	\(y\)
0	\(y\) on scale	\(y / K\)	\(y / K\)
1	\(y = 0\) (boundary)	\(\epsilon\)	\(\mathrm{lim} / K\)
1	\(y \neq 0\) (forced)	\(\epsilon\)	\((y + \mathrm{lim}) / K\)
2	\(y = K\) (boundary)	\((K - \mathrm{lim}) / K\)	\(1 - \epsilon\)
2	\(y \neq K\) (forced)	\((y - \mathrm{lim}) / K\)	\(1 - \epsilon\)
3	midpoint interval	\((y - \mathrm{lim}) / K\)	\((y + \mathrm{lim}) / K\)

All endpoints are clamped to \([\epsilon, 1 - \epsilon]\) with \(\epsilon = 10^{-5}\) to avoid boundary issues in the beta likelihood.

The midpoint approximation yt is computed as:

\(y_t = y\) when \(y \in (0, 1)\) (continuous data).
\(y_t = y / K\) when \(y\) is on the integer scale.

This value is used exclusively as an initialization aid for starting-value computation and does not enter the likelihood.

Interaction with the fitting pipeline:

This function is called internally by .extract_response() when the data does not carry the "is_prepared" attribute. If data has already been processed by brs_prep or by simulation with forced delta (brs_sim with delta != NULL), the pre-computed columns are used directly and brs_check() is skipped.

References

Lopes, J. E. (2023). Modelos de regressao beta para dados de escala. Master's dissertation, Universidade Federal do Parana, Curitiba. URI: https://hdl.handle.net/1884/86624.

Hawker, G. A., Mian, S., Kendzerska, T., and French, M. (2011). Measures of adult pain: Visual Analog Scale for Pain (VAS Pain), Numeric Rating Scale for Pain (NRS Pain), McGill Pain Questionnaire (MPQ), Short-Form McGill Pain Questionnaire (SF-MPQ), Chronic Pain Grade Scale (CPGS), Short Form-36 Bodily Pain Scale (SF-36 BPS), and Measure of Intermittent and Constant Osteoarthritis Pain (ICOAP). Arthritis Care and Research, 63(S11), S240-S252. doi:10.1002/acr.20543

Hjermstad, M. J., Fayers, P. M., Haugen, D. F., et al. (2011). Studies comparing Numerical Rating Scales, Verbal Rating Scales, and Visual Analogue Scales for assessment of pain intensity in adults: a systematic literature review. Journal of Pain and Symptom Management, 41(6), 1073-1093. doi:10.1016/j.jpainsymman.2010.08.016

Examples

# Scale data with boundary observations
y <- c(0, 3, 5, 7, 9, 10)
brs_check(y, ncuts = 10)
#>         left   right      yt  y delta
#> [1,] 0.00001 0.05000 0.00001  0     1
#> [2,] 0.25000 0.35000 0.30000  3     3
#> [3,] 0.45000 0.55000 0.50000  5     3
#> [4,] 0.65000 0.75000 0.70000  7     3
#> [5,] 0.85000 0.95000 0.90000  9     3
#> [6,] 0.95000 0.99999 0.99999 10     2

# Force all observations to be exact (delta = 0)
brs_check(y, ncuts = 10, delta = rep(0L, length(y)))
#>         left   right      yt  y delta
#> [1,] 0.00001 0.00001 0.00001  0     0
#> [2,] 0.30000 0.30000 0.30000  3     0
#> [3,] 0.50000 0.50000 0.50000  5     0
#> [4,] 0.70000 0.70000 0.70000  7     0
#> [5,] 0.90000 0.90000 0.90000  9     0
#> [6,] 0.99999 0.99999 0.99999 10     0

# Force delta = 1 on non-boundary observations:
# endpoints use actual y values, preserving variation
y2 <- c(30, 60)
brs_check(y2, ncuts = 100, delta = c(1L, 1L))
#>       left right  yt  y delta
#> [1,] 1e-05 0.305 0.3 30     1
#> [2,] 1e-05 0.605 0.6 60     1
#  left = (eps, eps), right = (30.5/100, 60.5/100)