estimate_ICA_OrdCont: 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)

Value

(numeric) Estimated ICA

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.

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.