Performs turning point and inflection point tests for
U-shaped and inverse U-shaped relationships in regression models.
Implements the Sasabuchi (1980) test as extended by Lind and Mehlum (2010)
with support for quadratic, cubic, log-quadratic, and inverse functional
forms. Features include delta-method standard errors, Fieller confidence
intervals, Simonsohn (2018) two-lines test, and parametric bootstrap.
Designed for post-estimation analysis of linear models, panel models,
and quantile regression.
References: Lind and Mehlum (2010)
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.
# Install from CRAN (when available)
install.packages("tptest")
# Install development version from GitHub
devtools::install_github("muhammedalkhalaf/tptest")
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)
==========================================
Turning Point Test (Lind and Mehlum 2010)
==========================================
Model form: Quadratic: y = b1*x + b2*x^2
Data interval: [1.002, 9.9]
Distribution: t(197)
Fitted shape on the interval: U shape
Turning point (x*): 8.0909
Delta-method SE: 0.336815
95% CI: [7.42667, 8.75513]
------------------------------------------
Sasabuchi (1980) Test
------------------------------------------
Lower bound Upper bound
Interval 1.0022 9.9000
Slope -6.8191 1.7403
t-value -12.9977 3.3549
P (one-sided) 0.0000 0.0005
Tested alternative: U shape
Overall test: t = 3.3549, p = 0.000476 ***
-> Strong evidence of U shape (p < 0.01)
------------------------------------------
*** p<0.01, ** p<0.05, * p<0.10
The interval is taken from the estimation sample of the model unless min
and max are supplied. The t distribution with the residual degrees of
freedom is used for lm-type models; the normal distribution is used for
glm objects and when coefficients are passed through coefs.
form |
Model | Turning point | Notes |
|---|---|---|---|
quadratic |
y = b1*x + b2*x^2 |
-b1/(2*b2) |
vars = c("x", "x_sq") |
inverse |
y = b1*x + b2/x |
sqrt(b2/b1) |
vars = c("x", "x_inv"); U shape when b1 > 0, inverse U when b1 < 0 |
logquadratic |
y = b1*ln(x) + b2*ln(x)^2 |
exp(-b1/(2*b2)) |
vars names the ln(x) and ln(x)^2 regressors; bounds are on the ln(x) scale unless bounds_scale = "levels"; the turning point and its intervals are reported in levels of x |
cubic |
y = b1*x + b2*x^2 + b3*x^3 |
roots of the slope | the slope is not monotone across the inflection point, so the two-endpoint test is applied on each monotone sub-interval (a package extension, not part of Lind and Mehlum 2010) |
# 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)
Lind, J. T. and 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.
GPL-3