Fit the linearized Mazur hyperbola (Hinds et al., 2026)

Linearizes D = 1/(1 + k t) to ln(1/D - 1) - ln(t) = ln(k) + e so that ln(k) has a closed-form per-subject estimator (the geometric mean of (1/D - 1)/t), a one-way random-effects model on the transformed scale has closed-form MLEs, and condition means can be compared with an exact F-test (anova.beezdiscounting_linear()); the test is exact when the random-effects variance estimate g-hat > 0, and when g-hat = 0 the statistic is still reported, with a warning. No numerical optimization is involved.

Usage

fit_dd_linear(
  data,
  y_var = "y",
  x_var = "x",
  id_var = "id",
  factors = NULL,
  response_scale = c("proportion", "percent", "amount"),
  ll = NULL,
  boundary = c("clamp", "drop", "error"),
  eps = NULL,
  conf_level = 0.95
)

Arguments

data: Long data frame with subject id, delay, and indifference point columns.
y_var, x_var, id_var: Column names for indifference point, delay, subject.
factors: Optional name of ONE categorical column giving the experimental condition. Must be between-subject: every id appears under exactly one level; frames with an id under several levels are rejected because the exact F-test assumes independent units per condition.
response_scale, ll: As in fit_dd_tmb().
boundary: How to treat y in {0, 1}: "clamp" (default; exact 0 becomes eps, exact 1 becomes 1 - eps, nothing else moves), "drop" (per-subject estimates only unless the design stays balanced), "error".
eps: Where exact 0 / 1 are placed under boundary = "clamp" (eps and 1 - eps); default 1/(2 * ll) if ll is given, else 0.005.
conf_level: Confidence level for per-subject ln(k) intervals (a single number strictly between 0 and 1); also the default level of confint(parm = "subject").

Value

An object of class beezdiscounting_linear: a list with subjects (per-unit tibble of ln k estimates, intervals and log-likelihoods), re (closed-form random-effects MLEs mu, sigma2, g, or NULL if the design is unbalanced), design (factor name and levels), transform (boundary bookkeeping; the counts refer to the retained units), data (the long frame: id, x, y, the factor, and the derived columns condition, unit, y_lin, d_used, log_jac; factors may not name one of the derived columns, and other user columns are not carried), conf_level, and call.

Details

Indifference points at exactly 0 or 1 are undefined under the transform; by default they are moved to eps and 1 - eps respectively (boundary = "clamp"). Only exact 0 and 1 are touched — interior points, however close to a bound, are used as observed. This is a package decision; the paper does not address boundary values. Use boundary = "drop" or "error" when you would rather not impute them.

References

Hinds, D., Tegge, A. N., Stein, J. S., LaConte, S. M., McClure, S. M., & Ferreira, M. A. R. (2026). To linearize or not to linearize: That is the Mazur delay discounting question. Journal of Mathematical Psychology, 130, 103006. doi:10.1016/j.jmp.2026.103006

Examples

fit <- fit_dd_linear(dd_ip)
#> Warning: Clamped out-of-range y to [0, 1]: 4 value(s) > 1 set to 1, 0 value(s) < 0 set to 0.
fit
#> Linearized Mazur discounting fit (Hinds et al., 2026)
#>   100 units, 6 delays each
#>   boundary = "clamp" (eps = 0.005): 129 point(s) at 0/1, 129 clamped, 0 dropped
#>   Population ln k (mu):
#>  (all) 
#> -1.077 
#>   sigma2 = 1.090, g = 4.901, logLik(raw) = 932.52