Imagine a data set of only 7 pairs of observations¹ for two methods X and Y. Imagine that the target is to reject the null Hypothesis of slope = 1 and intercept = 0.

Method	Sample 1	Sample 2	Sample 3	Sample 4	Sample 5	Sample 6	Sample 7
X	38.0	39.8	38.0	26.9	37.5	33.2	36.9
Y	30.8	33.7	26.1	21.5	33.9	26.9	29.7

With Bayesian Deming regression paired with a MD test this is possible. The simulation is run with df = 1 to provide maximal robustness. The rstan sampling was performed with set.seed(20240225) on Debian Trixie with amd64 architecture.

The classical CI approach has no hope, see the purple HDI-CI box. The data set is too small, even for the Bayesian Deming regression. But the result of a Bayesian Deming regression can be tested with the Mahalanobis distance MD method. The power of the MD testing method is so much higher that it is still possible to reject the null hypothesis, even with such a reduced data set. The probability of the MD test (the Chi-sq. p-value with df=2 is printed in the figure above) is extremely low and highly significant.

Here below the regression plot drawn with the {rstanbdp} package

For matter of comparison here the results of Deming and PBequi analytical regressions obtained with the package {mcr}. With both methods a MD approach via bootstrap is impossible because of the small size of the sample.

Here below the table for the classical (frequentist) Deming regression results

	EST	SE	LCI	UCI
Intercept	-7.393668	10.8756262	-35.350355	20.563020
Slope	1.016203	0.3022048	0.239361	1.793046

The CIs are slightly wider than with the Bayesian Deming method and also a little shifted compared to the HDI intervals. This is not surprising since slope and intercept are not normally distributed. HDI for the CI can mitigate the right side excess previously reported.

Here below the table for the PBequi regression results.

	EST	SE	LCI	UCI
Intercept	-7.306061	111.345365	-293.528433	278.916312
Slope	1.030303	3.017419	-6.726219	8.786825

Apparently PBequi is not able to provide meaningful CIs for such a small data set.