# Assess surrogacy in the causal-inference single-trial setting in the binary-binary case

### Description

The function `ICA.BinBin`

quantifies surrogacy in the single-trial causal-inference framework (individual causal association and causal concordance) when both the surrogate and the true endpoints are binary outcomes. See **Details** below.

### Usage

1 2 3 | ```
ICA.BinBin(pi1_1_, pi1_0_, pi_1_1, pi_1_0, pi0_1_, pi_0_1,
Monotonicity=c("General"), Sum_Pi_f = seq(from=0.01, to=0.99, by=.01),
M=10000, Volume.Perc=0, Seed=sample(1:100000, size=1))
``` |

### Arguments

`pi1_1_` |
A scalar or vector that contains values for |

`pi1_0_` |
A scalar or vector that contains values for |

`pi_1_1` |
A scalar or vector that contains values for |

`pi_1_0` |
A scalar or vector that contains values for |

`pi0_1_` |
A scalar or vector that contains values for |

`pi_0_1` |
A scalar or vector that contains values for |

`Monotonicity` |
Specifies which assumptions regarding monotonicity should be made: |

`Sum_Pi_f` |
A scalar or vector that specifies the grid of values |

`M` |
The number of runs that are conducted for a given value of |

`Volume.Perc` |
Note that the marginals that are observable in the data set a number of restrictions on the unidentified correlations. For example, under montonicity for |

`Seed` |
The seed to be used to generate |

### Details

In the continuous normal setting, surroagacy can be assessed by studying the association between the individual causal effects on *S* and *T* (see `ICA.ContCont`

). In that setting, the Pearson correlation is the obvious measure of association.

When *S* and *T* are binary endpoints, multiple alternatives exist. Alonso et al. (2014) proposed the individual causal association (ICA; *R_{H}^{2}*), which captures the association between the individual causal effects of the treatment on *S* (*Δ_S*) and *T* (*Δ_T*) using information-theoretic principles.

The function `ICA.BinBin`

computes *R_{H}^{2}* based on plausible values of the potential outcomes. Denote by *\bold{Y}'=(T_0,T_1,S_0,S_1)* the vector of potential outcomes. The vector *\bold{Y}* can take 16 values and the set of parameters *π_{ijpq}=P(T_0=i,T_1=j,S_0=p,S_1=q)* (with *i,j,p,q=0/1*) fully characterizes its distribution.

However, the parameters in *π_{ijpq}* are not all functionally independent, e.g., *1=π_{\cdot\cdot\cdot\cdot}*. When no assumptions regarding monotonicity are made, the data impose a total of *7* restrictions, and thus only *9* proabilities in *π_{ijpq}* are allowed to vary freely (for details, see Alonso et al., 2014). Based on the data and assuming SUTVA, the marginal probabilites *π_{1 \cdot 1 \cdot}*, *π_{1 \cdot 0 \cdot}*, *π_{\cdot 1 \cdot 1}*, *π_{\cdot 1 \cdot 0}*, *π_{0 \cdot 1 \cdot}*, and *π_{\cdot 0 \cdot 1}* can be computed (by hand or using the function `MarginalProbs`

). Define the vector

*\bold{b}'=(1, π_{1 \cdot 1 \cdot}, π_{1 \cdot 0 \cdot}, π_{\cdot 1 \cdot 1}, π_{\cdot 1 \cdot 0}, π_{0 \cdot 1 \cdot}, π_{\cdot 0 \cdot 1})*

and *\bold{A}* is a contrast matrix such that the identified restrictions can be written as a system of linear equation

*\bold{A π} = \bold{b}.*

The matrix *\bold{A}* has rank *7* and can be partitioned as *\bold{A=(A_r | A_f)}*, and similarly the vector *\bold{π}* can be partitioned as *\bold{π^{'}=(π_r^{'} | π_f^{'})}* (where *f* refers to the submatrix/vector given by the *9* last columns/components of *\bold{A/π}*). Using these partitions the previous system of linear equations can be rewritten as

*\bold{A_r π_r + A_f π_f = b}.*

The following algorithm is used to generate plausible distributions for *\bold{Y}*. First, select a value of the specified grid of values (specified using `Sum_Pi_f`

in the function call). For *k=1* to *M* (specified using `M`

in the function call), generate a vector *π_f* that contains *9* components that are uniformly sampled from hyperplane subject to the restriction that the sum of the generated components equals `Sum_Pi_f`

(the function `RandVec`

, which uses the `randfixedsum`

algorithm written by Roger Stafford, is used to obtain these components). Next, *\bold{π_r=A_r^{-1}(b - A_f π_f)}* is computed and the *π_r* vectors where all components are in the *[0;\:1]* range are retained. This procedure is repeated for each of the `Sum_Pi_f`

values. Based on these results, *R_H^2* is estimated. The obtained values can be used to conduct a sensitivity analysis during the validation exercise.

The previous developments hold when no monotonicity is assumed. When monotonicity for *S*, *T*, or for *S* and *T* is assumed, some of the probabilities of *π* are zero. For example, when montonicity is assumed for *T*, then *P(T_0 <= T_1)=1*, or equivantly, *π_{1000}=π_{1010}=π_{1001}=π_{1011}=0*. When monotonicity is assumed, the procedure described above is modified accordingly (for details, see Alonso et al., 2014). When a general analysis is requested (using `Monotonicity=c("General")`

in the function call), all settings are considered (no monotonicity, monotonicity for *S* alone, for *T* alone, and for both for *S* and *T*.)

To account for the uncertainty in the estimation of the marginal probabilities, a vector of values can be specified from which a random draw is made in each run (see **Examples** below).

### Value

An object of class `ICA.BinBin`

with components,

`Pi.Vectors` |
An object of class |

`R2_H` |
The vector of the |

`Theta_T` |
The vector of odds ratios for |

`Theta_S` |
The vector of odds ratios for |

`H_Delta_T` |
The vector of the entropies of |

`Monotonicity` |
The assumption regarding monotonicity that was made. |

`Volume.No` |
The 'volume' of the parameter space when monotonicity is not assumed. Is only provided when the argument |

`Volume.T` |
The 'volume' of the parameter space when monotonicity for |

`Volume.S` |
The 'volume' of the parameter space when monotonicity for |

`Volume.ST` |
The 'volume' of the parameter space when monotonicity for |

### Author(s)

Wim Van der Elst, Paul Meyvisch, Ariel Alonso & Geert Molenberghs

### References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2015). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.

### See Also

`ICA.ContCont`

, `MICA.ContCont`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
## Not run: # Time consuming code part
# Compute R2_H given the marginals specified as the pi's, making no
# assumptions regarding monotonicity (general case)
ICA <- ICA.BinBin(pi1_1_=0.2619048, pi1_0_=0.2857143, pi_1_1=0.6372549,
pi_1_0=0.07843137, pi0_1_=0.1349206, pi_0_1=0.127451, Seed=1,
Monotonicity=c("General"), Sum_Pi_f = seq(from=0.01, to=.99, by=.01), M=10000)
# obtain plot of the results
plot(ICA, R2_H=TRUE)
# Example 2 where the uncertainty in the estimation
# of the marginals is taken into account
ICA_BINBIN2 <- ICA.BinBin(pi1_1_=runif(10000, 0.2573, 0.4252),
pi1_0_=runif(10000, 0.1769, 0.3310),
pi_1_1=runif(10000, 0.5947, 0.7779),
pi_1_0=runif(10000, 0.0322, 0.1442),
pi0_1_=runif(10000, 0.0617, 0.1764),
pi_0_1=runif(10000, 0.0254, 0.1315),
Monotonicity=c("General"),
Sum_Pi_f = seq(from=0.01, to=0.99, by=.01),
M=50000, Seed=1)
# Plot results
plot(ICA_BINBIN2)
## End(Not run)
``` |

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