R: Assess surrogacy in the causal-inference single-trial setting...

ICA.BinBin {Surrogate}

R Documentation

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

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 `P(T=1,S=1\|Z=0)`, i.e., the probability that `S=T=1` when under treatment `Z=0`. A vector is specified to account for uncertainty, i.e., rather than keeping `P(T=1,S=1\|Z=0)` fixed at one estimated value, a distribution can be specified (see examples below) from which a value is drawn in each run.
`pi1_0_`	A scalar or vector that contains values for `P(T=1,S=0\|Z=0)`.
`pi_1_1`	A scalar or vector that contains values for `P(T=1,S=1\|Z=1)`.
`pi_1_0`	A scalar or vector that contains values for `P(T=1,S=0\|Z=1)`.
`pi0_1_`	A scalar or vector that contains values for `P(T=0,S=1\|Z=0)`.
`pi_0_1`	A scalar or vector that contains values for `P(T=0,S=1\|Z=1)`.
`Monotonicity`	Specifies which assumptions regarding monotonicity should be made: `Monotonicity=c("General")`, `Monotonicity=c("No")`, `Monotonicity=c("True.Endp")`, `Monotonicity=c("Surr.Endp")`, or `Monotonicity=c("Surr.True.Endp")`. See Details below. Default `Monotonicity=c("General")`.
`Sum_Pi_f`	A scalar or vector that specifies the grid of values `G={g_{1},\: g_{2},\:...,\: g_{k}}` to be considered when the sensitivity analysis is conducted. See Details below. Default `Sum_Pi_f = seq(from=0.01, to=0.99, by=.01)`.
`M`	The number of runs that are conducted for a given value of `Sum_Pi_f`. This argument is not used when `Volume.Perc=0`. Default `M=10000`.
`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 `S` and `T`, it holds that `\pi_{0111}<=min(\pi_{0\cdot1\cdot}, \pi_{\cdot1\cdot1})` and `\pi_{1100}<=min(\pi_{1\cdot0\cdot}, \pi_{\cdot1\cdot0})`. For example, when `min(\pi_{0\cdot1\cdot}, \pi_{\cdot1\cdot1})=0.10` and `min(\pi_{1\cdot0\cdot}, \pi_{\cdot1\cdot0})=0.08`, then all valid `\pi_{0111}<=0.10` and all valid `\pi_{1100}<=0.08`. The argument `Volume.Perc` specifies the fraction of the 'volume' of the paramater space that is explored. This volume is computed based on the grids G={0, 0.01, ..., maximum possible value for the counterfactual probability at hand}. E.g., in the previous example, the 'volume' of the parameter space would be `11*9=99`, and when e.g., the argument `Volume.Perc=1` is used a total of `99` runs will be conducted for each given value of `Sum_Pi_f`. Notice that when monotonicity is not assumed, relatively high values of `Volume.Perc` will lead to a large number of runs and consequently a long analysis time.
`Seed`	The seed to be used to generate `\pi_r`. Default `Seed=sample(1:100000, size=1)`.

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 (\Delta_S) and T (\Delta_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 \pi_{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 \pi_{ijpq} are not all functionally independent, e.g., 1=\pi_{\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 \pi_{ijpq} are allowed to vary freely (for details, see Alonso et al., 2014). Based on the data and assuming SUTVA, the marginal probabilites \pi_{1 \cdot 1 \cdot}, \pi_{1 \cdot 0 \cdot}, \pi_{\cdot 1 \cdot 1}, \pi_{\cdot 1 \cdot 0}, \pi_{0 \cdot 1 \cdot}, and \pi_{\cdot 0 \cdot 1} can be computed (by hand or using the function MarginalProbs). Define the vector

\bold{b}'=(1, \pi_{1 \cdot 1 \cdot}, \pi_{1 \cdot 0 \cdot}, \pi_{\cdot 1 \cdot 1}, \pi_{\cdot 1 \cdot 0}, \pi_{0 \cdot 1 \cdot}, \pi_{\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 \pi} = \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{\pi} can be partitioned as \bold{\pi^{'}=(\pi_r^{'} | \pi_f^{'})} (where f refers to the submatrix/vector given by the 9 last columns/components of \bold{A/\pi}). Using these partitions the previous system of linear equations can be rewritten as

\bold{A_r \pi_r + A_f \pi_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 \pi_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{\pi_r=A_r^{-1}(b - A_f \pi_f)} is computed and the \pi_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 \pi are zero. For example, when montonicity is assumed for T, then P(T_0 <= T_1)=1, or equivantly, \pi_{1000}=\pi_{1010}=\pi_{1001}=\pi_{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 `data.frame` that contains the valid `\pi` vectors.
`R2_H`	The vector of the `R_H^2` values.
`Theta_T`	The vector of odds ratios for `T`.
`Theta_S`	The vector of odds ratios for `S`.
`H_Delta_T`	The vector of the entropies of `\Delta_T`.
`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.Perc` is used (i.e., when it is not equal to `0`.
`Volume.T`	The 'volume' of the parameter space when monotonicity for `T` is assumed. Is only provided when the argument `Volume.Perc` is used.
`Volume.S`	The 'volume' of the parameter space when monotonicity for `S` is assumed. Is only provided when the argument `Volume.Perc` is used.
`Volume.ST`	The 'volume' of the parameter space when monotonicity for `S` and `T` is assumed. Is only provided when the argument `Volume.Perc` is used.

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.

Examples

## 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)

[Package Surrogate version 3.3.0 Index]