Dent and Blackie regression test: DBK

The regression of the observed versus the reference data gives a result of slope = 1 and intercept = 0 in case of perfect agreement. We want to test the hypothesis that the estimated slope and intercept are an expression of this result with the paired sets of reference and actual data. Standard regression significance tests are not correct since the slope and intercept need to be tested simultaneously. The DBK test follows Dent and Blackie (1979) to compute the simultaneous probability that these two parameters in the linear regression have the values indicated. This is done as follows.

The linear regression model is

$$Y(t) = a + bX(t)$$ (5)

where X are the reference (expected) data and Y the actual (observed) data. The test of the hypothesis that $a = 0$ and $b = 1$ requires estimation of the simultaneous probability of obtaining these values under the null hypothesis of data coming from the same population. This is accomplished by computing the statistics

$$F = \frac{(n - 2) \left( na^2 - 2n \overline{X} a(b-1) + \sum X^2_i (b-1)^2 \right)}{2nS^2}$$ (6)

where $a$ and $b$ are the estimated parameters of the regression model, $n$ is the number of <x,y> pairs, $\overline{X}$ is the mean of the reference data, and $S$ is the residual mean squared error of the regression model, computed as

$$S = \frac{\sum (Y_i - \hat{Y}_i)^2}{n - 2}$$ (7)

with

$$\hat{Y}_i = \overline{Y} + b(X_i - \overline{X})$$ (8)

This test needs a point-by-point matching of the two time series; if the lengths of reference and actual data do not match, a warning is issued, and only a number of pairs corresponding to the length of the shortest one is used, assuming they start at the same point in time (or other ordering variable).