mrfDepth (version 1.0.17)

cmltest: Test for linearity of the conditional median in simple regression

Description

A test based on regression depth for linearity of the conditional median z given the simple regression dataset x. The computation of the regression depth of z with respect to x is done by the function rdepth. The test is only valid when x contains no duplicates.

Usage

cmltest(x, z)

Value

pval

The \(p\)-value of the hypothesis test.

Arguments

x

An \(n\) by 2 regression data matrix.
The first column is the explanatory variable, the second column corresponds to the response variable.

z

A matrix with one row containing an intercept and slope.

Author

P. Segaert

Details

The following hypothesis test is performed:
\(H_0\): The data come from a model with: \(med(x_2|x_1) = z_1 + z_2 * x_1\)
The test statistic being used is the regression depth of z with respect to x.

References

Van Aelst S., Rousseeuw P.J., Hubert M., Struyf A. (2002). The deepest regression method. Journal of Multivariate Analysis, 81, 138--166.

Rousseeuw P.J., Struyf A. (2002). A depth test for symmetry. In: Goodness-of-Fit Tests and Model Validity, Birkhäuser Boston, pages 401--412.

See Also

rdepth, rdepthmedian

Examples

data(stars)

# Compute the least squares fit. Due to outliers
# this fit will be bad and thus H0 should be rejected. 
temp <- lsfit(x = stars[,1], y = stars[,2])$coefficients
intercept <- temp[1]
slope <- temp[2]
z <- matrix(c(intercept, slope), nrow = 1)
pvalue1 <- cmltest(x = stars[!duplicated(stars), ], z = z)
pvalue1

# Let's now test the deepest regression line. 
result <- rdepthmedian(x = stars)
pvalue2 <- cmltest(x = stars[!duplicated(stars), ], z = matrix(result$deepest, nrow = 1))
pvalue2

plot(stars)
abline(a = intercept, b = slope)
abline(result$deepest, col = "red")
text(x = 3.8, y = 5.3, labels = paste("p-value", round(pvalue1, digits = 3)))
text(x = 4.45, y = 4.8, labels = paste("p-value", round(pvalue2, digits = 3)),
     col = "red")