estimate_ICA_OrdCont {Surrogate} | R Documentation |
Estimate ICA in Ordinal-Continuous Setting
Description
estimate_ICA_OrdCont()
estimates the individual causal association (ICA)
for a sample of individual causal treatment effects with a continuous
surrogate and an ordinal true endpoint. The ICA in this setting is defined as
follows,
R^2_H = \frac{I(\Delta S; \Delta T)}{H(\Delta T)}
where
I(\Delta S; \Delta T)
is the mutual information and H(\Delta T)
the entropy.
Usage
estimate_ICA_OrdCont(delta_S, delta_T)
Arguments
delta_S |
(numeric) Vector of individual causal treatment effects on the surrogate. |
delta_T |
(integer) Vector of individual causal treatment effects on the true endpoint. |
Value
(numeric) Estimated ICA
Individual Causal Association
Many association measures can operationalize the ICA. For each setting, we consider one default definition for the ICA which follows from the mutual information.
Continuous-Continuous
The ICA is defined as the squared informational coefficient of correlation
(SICC or R^2_H
), which is a transformation of the mutual information
to the unit interval:
R^2_h = 1 - e^{-2 \cdot I(\Delta S; \Delta T)}
where 0 indicates independence, and 1 a functional relationship between
\Delta S
and \Delta T
. If (\Delta S, \Delta T)'
is bivariate
normal, the ICA equals the Pearson correlation between \Delta S
and
\Delta T
.
Ordinal-Continuous
The ICA is defined as the following transformation of the mutual information:
R^2_H = \frac{I(\Delta S; \Delta T)}{H(\Delta T)},
where I(\Delta S; \Delta T)
is the mutual information and H(\Delta T)
the entropy.
Ordinal-Ordinal
The ICA is defined as the following transformation of the mutual information:
R^2_H = \frac{I(\Delta S; \Delta T)}{\min \{H(\Delta S), H(\Delta T) \}},
where I(\Delta S; \Delta T)
is the mutual information, and H(\Delta S)
and H(\Delta T)
the entropy of \Delta S
and \Delta T
,
respectively.