# SobolIndicesAll-class: Estimating All Sobol Indices of Certain Order Using... In SobolSensitivity: Computing the Sobol Sensitivity Indices

## Description

Sobol [1] proposed a definition called Sobol Indices for estimating the importance of single variable or multiple variales' interaction. We have derived the formulas for main effect Sobol indices by using sensitivity analysis under GLM of three link functions in `SobolIndicesAll` class, and compute the sobol indices of all possible variables interactions of given order by using this algorithm.

## Usage

 ```1 2``` ```SobolIndicesAll(xdata, orderinput=1, beta=0, link=c("identity","log","logit")) summary(object) ```

## Arguments

 `xdata` A data set of class 'matrix' or 'data.frame' which only includes the variables or features. `orderinput` A integer; the order of the interaction of the variables which are of interest for computing all possible interactions' sobol indices main effect. `beta` A vector; the coefficients of the variables estimated by the regression model. `link` A character; the link function used under the GLM model. `object` An object of the `SobolIndicesAll` class.

## Details

The proposed algorithm for computing the Sobol Indices is to use a simple strategy under the GLM model with independent or multivariate normal inputs:

g(E(Y|X))=β_0 + X β_1

where X is the data matrix of the varibles or features, g(.) is the link function under GLM, and β=(β_0, β_1) is the vector of intercept and coefficients estimates in GLM. Note that β_0=0 if there is no intercept in the setting of fitting GLM.

We derive the conditional expectations of the response with respect to the input subsets, and then estimate the main effect Sobol' sensitivity indices directly as follows by using closed formulas or (approximate) numerically using empirical variance estimates for a large number of GLMs:

S_P=Var(E(Y|X_P))/Var(Y)

where P is the index set for the subset of variables of interest.

The results can enable us to perform ANOVA-type variance decomposition analysis on data with multicollinearity issue, not only under Gaussian regression but also under other types of GLMs such as Poisson and logistic regression. The resulting main effect Sobol indices for all the variables interaction main effect (of order `orderinput`) of interest (all possible index sets of length `orderinput`) are stored in the `sobol.indices.all` slot.

## Value

The `SobolIndicesAll` function computes all the main effect Sobol indices for variables interactions of order `orderinput`, constructs and returns an object of the `SobolIndicesAll` class.

## Objects from the Class

Objects should be created using the `SobolIndicesAll` constructor.

## Slots

`xdata`:

A data set of class 'matrix' or 'data.frame' which only includes the variables or features.

`orderinput`

A integer which is the order of the interaction of the variables of interest for computing all possible k-order interactions' main effect Sobol indices.

`beta`:

A vector which are the coefficients of the variables in a GLM model.

`link`:

A character which is the link function used under the GLM model.

`sobol.indices.all`:

A list or a numeric object which stores sobol indices of variables interactions of order `orderinput`.

## Methods

summary

`(object = "SobolIndicesAll")`: ...

## Author(s)

Min Wang <wang.1807@mbi.osu.edu>

## References

[1] Sobol, I. M. (1990). On sensitivity estimation for nonlinear mathematical models, Matematicheskoe Modelirovanie, 2, 112-118.

[2] Lu, R., Wang D., Wang, M. and Rempala, G. (2016). Identifying Gene-gene Interactions Using Sobol Sensitivity Indices, submitted.

`identitySIfunction`, `logSIfunction` and `logitSIfunction` to get a complete list of the functions under different link functions to compute the sobol indices.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```showClass("SobolIndicesAll") # simulate xdata and beta xdata <- matrix(rnorm(20*5, 1), ncol=5) beta <- runif(6, min=-1, max=1) # all paired variables interactions are of interest orderinput <- 2 # link function is logit link (binomial, etc.) link <- "logit" # apply the proposed method siall <- SobolIndicesAll(xdata, orderinput=orderinput, beta, link="logit") # Review the results summary(siall) ```