1. Suppose a contingency-table to have been formed for two characters A and B which have been assigned in some way into classes A
1
...A
r
, B
1
...B
s
, the table having
r
columns,
s
rows, and giving the frequencies of occurrence of all combinations like A
m
B
n
. Let (A
m
) (B
n
) denote the frequencies of A
m
and B
n
(A
m
B
n
) the frequency of their combination. Extract from the general contingency-table any four adjacent frequencies, say— (A
m
B
n
), (A
m
+ 1
B
n
) (A
m
B
n
+1
), (A
m
+ 1
B
n
+ 1
).
The backbone modeling in ground-motion characterization (GMC) is a useful methodology to describe the epistemic uncertainty in median ground-motion predictions. The approach uses a backbone ground-motion model (GMM) and populates the GMC logic tree with the scaled and/or adjusted versions of the backbone GMM to capture the epistemic uncertainty in median ground motions. The scaling and/or adjustment should represent the specific features and uncertainties involved in source, path, and site effects at the target site. The identification of the backbone model requires different considerations specific to the nature of the ground-motion hazard problem. In this article, we present a scaled backbone modeling approach that considers the magnitude- and distance-scaling predictors as well as their correlation to address the epistemic uncertainty in median ground-motion predictions. This approach results in a trivariate normal distribution to fully define a range of epistemic uncertainty in a model sample space. The simultaneous consideration of magnitude and distance scaling while defining the epistemic uncertainty and the methodology followed for the simplified representation of trivariate normal distribution in ground-motion logic tree are the two important features in our procedure. We first present the proposed approach that is followed by a case study for Central and Eastern North America (CENA) stable continental region. The case study discusses the underlying assumptions and limitations of the proposed approach.
On a property which holds good for all groupings of a normal distribution of frequency for two variables, with applications to the study of contingency-tables for the inheritance of unmeasured qualities
