Universal Turning Point and Inflection Point Tests.
tptest: Universal Turning Point and Inflection Point Tests
Overview
The tptest package implements tests for U-shaped and inverse U-shaped relationships in regression analysis. It provides a comprehensive framework for detecting turning points and inflection points in time series and panel data.
Key Features
- Sasabuchi (1980) / Lind-Mehlum (2010) Test: Rigorous test for U-shape or inverse U-shape
- Multiple Functional Forms: Quadratic, cubic, inverse, and log-quadratic specifications
- Delta-Method Standard Errors: Proper inference for turning point estimates
- Fieller Confidence Intervals: Robust intervals when denominator is uncertain
- Simonsohn (2018) Two-Lines Test: Alternative U-shape validation
- Parametric Bootstrap: Bootstrap confidence intervals
Installation
# Install from CRAN (when available)
install.packages("tptest")
# Install development version from GitHub
devtools::install_github("muhammedalkhalaf/tptest")
Usage
library(tptest)
# Simulate data with U-shaped relationship
set.seed(42)
n <- 200
x <- runif(n, 1, 10)
y <- 50 - 8*x + 0.5*x^2 + rnorm(n, sd = 5)
dat <- data.frame(y = y, x = x, x_sq = x^2)
# Fit quadratic model
fit <- lm(y ~ x + x_sq, data = dat)
# Test for U-shape
result <- tptest(fit, vars = c("x", "x_sq"), data = dat)
print(result)
Output
==========================================
Turning Point Test (Lind & Mehlum 2010)
==========================================
Model form: Quadratic: y = b1*x + b2*x^2
Data interval: [1.023, 9.987]
Detected shape: U shape
Turning point (x*): 8.234
Delta-method SE: 0.5123
95% CI: [7.230, 9.238]
------------------------------------------
Sasabuchi (1980) Test
------------------------------------------
Lower bound Upper bound
Interval 1.0230 9.9870
Slope -6.2456 1.7532
t-value -12.4532 3.4521
P>|t| 0.0000 0.0003
Overall test: t = 3.4521, p = 0.000312 ***
-> Strong evidence of U shape (p < 0.01)
------------------------------------------
*** p<0.01, ** p<0.05, * p<0.10
Environmental Kuznets Curve Example
# Load example data
data(ekc)
# Fit model
fit <- lm(emissions ~ gdp + gdp_sq, data = ekc)
# Test for inverse U-shape
result <- tptest(fit, vars = c("gdp", "gdp_sq"),
fieller = TRUE, data = ekc)
summary(result)
plot(result)
References
Lind, J. T., & Mehlum, H. (2010). With or without U? The appropriate test for a U-shaped relationship. Oxford Bulletin of Economics and Statistics, 72(1), 109-118. https://doi.org/10.1111/j.1468-0084.2009.00569.x
Sasabuchi, S. (1980). A test of a multivariate normal mean with composite hypotheses determined by linear inequalities. Biometrika, 67(2), 429-439.
Fieller, E. C. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society: Series B, 16(2), 175-185. https://doi.org/10.1111/j.2517-6161.1954.tb00159.x
Simonsohn, U. (2018). Two lines: A valid alternative to the invalid testing of U-shaped relationships with quadratic regressions. Advances in Methods and Practices in Psychological Science, 1(4), 538-555.
License
GPL-3