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tptest 1.1.0 (2026-10-01)

This release corrects several computations. Results from version 1.0.3 for the inverse and log-quadratic forms, for Fieller intervals, for generalized linear models and for the cubic form should be recomputed.

Corrected computations

• Fieller interval for the inverse form (y = b1*x + b2/x): the covariance term in the boundary points of the set for b2/b1 had the wrong sign. The boundary points are now (b1*b2 - s12*T^2 +/- T*sqrt(D)) / (b1^2 - s11*T^2), which matches a brute-force inversion of the defining inequality (in the audit example the package gave [1.9391, 2.0923] where the correct interval is [1.9142, 2.0693]; grid inversion gives [1.91425, 2.06926]).
• Log-quadratic form: the regressor named in vars[1] is ln(x), so the automatic data range is the minimum and maximum of that column. Version 1.0.3 took the logarithm of that column a second time and evaluated the slopes at log(log(x)). User-supplied min and max are now on the ln(x) scale by default; the new argument bounds_scale = "levels" accepts them in levels of x. The turning point is reported in levels, exp(-b1/(2*b2)).
• Fieller critical value: t(df) with the residual degrees of freedom of the model is used for lm-type models (normal for glm and for the coefs path), consistent with the Sasabuchi p-value and the delta-method interval in the same object. fieller_ci() gains a df argument. confint() at a level other than the one used in the call now uses the same distribution instead of always using the normal.
• Fieller edge cases: when the denominator coefficient is not significant at the chosen level but the discriminant is positive, the confidence set is the union of two rays and is reported as type = "two_rays" with both boundary points (version 1.0.3 reported the whole real line). For the inverse form, when the set for b2/b1 contains zero the set for x* = sqrt(b2/b1) is reported as (0, hi] (lo = 0) instead of NA. The types "ray" and "empty" are added; "entire_real_line" is replaced by "unbounded", which is now used only when the set is the whole (positive) line.
• Inverse form, delta-method standard error: the gradient of x* = sqrt(b2/b1) was written with sqrt(b1) and sqrt(b2) separately and returned NaN for an inverse U shape (b1 < 0, b2 < 0). It is now (-0.5*sqrt(b2/b1)/b1, 0.5/(b1*sqrt(b2/b1))).
• Degrees of freedom: t(df.residual) is used for lm and plm models. For glm objects and for the coefs path the normal distribution is used; this is the package's choice, Lind and Mehlum (2010, Section 2) only state that the test is asymptotically valid for generalized linear models. The new argument df overrides this choice (Inf forces the normal distribution).
• Cubic form: the two-endpoint test requires the slope to be monotone on the interval (Lind and Mehlum 2010, footnote 3), which does not hold for a cubic across its inflection point. The interval is now split at the inflection point (the estimate, treated as fixed, so the sub-interval tests do not have exact size) and the test is applied on each monotone sub-interval; the results are in sasabuchi$segments and are labelled as a package extension. t_overall and p_overall are then NA. When the inflection point lies outside the interval the single segment is the test of equation (9) of the paper with H = 3 and is reported as the overall test.
• Inverse form: min must be positive (the slope -1/x^2 is not defined at 0); an error is raised otherwise.
• Automatic data range: the minimum and maximum are taken over the estimation sample (the rows used by the model, from its model frame), not over all rows of data. data is no longer required for the range when the regressor can be recovered from the model.

Changed

• Log-quadratic form: the delta-method confidence interval for the turning point is computed on the ln(x) scale and exponentiated (package choice; version 1.0.3 applied the delta method to exp(-b1/(2*b2)) directly). tp_log, tp_log_se and bounds_levels are returned in addition, and the Fieller set is now available for this form (the exponentiated quadratic set).

Output changes

• When the fitted extremum lies outside the interval the Sasabuchi statistic min(-t_l, t_h) (or min(t_l, -t_h)) and its p-value are reported (both are negative and above 0.5, respectively) together with the message that H0 cannot be rejected, instead of NA and the message "trivial failure".
• shape describes the fitted curve on the interval ("U shape", "Inverse U shape", or a monotone label when the extremum is outside); the new element alternative gives the alternative hypothesis tested.
• p_min and p_max are the one-sided p-values of the two component tests under the tested alternative and are labelled "P (one-sided)".
• sasabuchi$outside and df are returned; the print method shows the distribution used.
• plot() draws the log-quadratic form on the ln(x) scale.

Tests

• A testthat suite pins the endpoint t-statistics, the Sasabuchi statistic, the delta-method standard errors and the Fieller sets for the quadratic, inverse and log-quadratic forms to hand calculations from lm() coefficients and covariance matrices and to brute-force inversion of the Fieller inequality, and compares the coefs-path results approximately with Table 1 of Lind and Mehlum (2010) (the covariance of the coefficients is backed out from the rounded standard errors printed there).

tptest 1.0.0

Initial Release

This is the first release of the tptest package, a port of the Stata tptest command for R.

Features

• tptest(): Main function for turning point and inflection point tests

  • Sasabuchi (1980) / Lind-Mehlum (2010) test for U-shape detection
  • Support for quadratic, cubic, inverse, and log-quadratic forms
  • Delta-method standard errors and confidence intervals
  • Works with lm, glm, and other standard model objects

• fieller_ci(): Fieller (1954) confidence intervals for turning points

  • More robust when denominator coefficient has high uncertainty
  • Handles bounded, unbounded, and degenerate cases

• twolines_test(): Simonsohn (2018) two-lines test

  • Alternative validation for U-shaped relationships
  • Splits data at turning point and tests slope signs

• Parametric bootstrap confidence intervals

• S3 methods: print(), summary(), plot(), coef(), confint()

Data

• ekc: Environmental Kuznets Curve example dataset
  • Simulated panel data for 50 countries over 10 years
  • Demonstrates inverse U-shaped relationship

Documentation

• Complete roxygen2 documentation with examples
• References to key papers with DOIs
• README with usage examples