# ARsens.power: Power of the Anderson-Rubin (1949) Test with Sensitivity... In ivmodel: Statistical Inference and Sensitivity Analysis for Instrumental Variables Model

## Description

`ARsens.power` computes the power of sensitivity analysis, which is based on an extension of Anderson-Rubin (1949) test and allows IV be possibly invalid within a certain range.

## Usage

 ```1 2``` ```ARsens.power(n, k, beta, gamma, Zadj_sq, sigmau, sigmav, rho, alpha = 0.05, deltarange = deltarange, delta = NULL) ```

## Arguments

 `n` Sample size. `k` Number of exogenous variables. `beta` True causal effect minus null hypothesis causal effect. `gamma` Regression coefficient for effect of instruments on treatment. `Zadj_sq` Variance of instruments after regressed on the observed variables. `sigmau` Standard deviation of potential outcome under control (structural error for y). `sigmav` Standard deviation of error from regressing treatment on instruments. `rho` Correlation between u (potential outcome under control) and v (error from regressing treatment on instrument). `alpha` Significance level. `deltarange` Range of sensitivity allowance. A numeric vector of length 2. `delta` True value of sensitivity parameter when calculating the power. Usually take delta = 0 for the favorable situation or delta = NULL for unknown delta.

## Value

Power of sensitivity analysis for the proposed study, which extends the Anderson-Rubin (1949) test with possibly invalid IV. The power formula is derived in Jiang, Small and Zhang (2015).

## Author(s)

Yang Jiang, Hyunseung Kang, and Dylan Small

## References

Anderson, T.W. and Rubin, H. (1949), Estimation of the parameters of a single equation in a complete system of stochastic equations, Annals of Mathematical Statistics, 20, 46-63.
Wang, X., Jiang, Y., Small, D. and Zhang, N (2017), Sensitivity analysis and power for instrumental variable studies, (under review of Biometrics).

See also `ivmodel` for details on the instrumental variables model.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21``` ```# Assume we calculate the power of sensitivity analysis in a study with # one IV (l=1) and the only exogenous variable is the intercept (k=1). # Suppose the difference between the null hypothesis and true causal # effect is 1 (beta=1). # The sample size is 250 (n=250), the IV variance is .25 (Zadj_sq =.25). # The standard deviation of potential outcome is 1(sigmau= 1). # The coefficient of regressing IV upon exposure is .5 (gamma= .5). # The correlation between u and v is assumed to be .5 (rho=.5). # The standard deviation of first stage error is .4 (sigmav=.4). # The significance level for the study is alpha = .05. # power of sensitivity analysis under the favorable situation, # assuming the range of sensitivity allowance is (-0.1, 0.1) ARsens.power(n=250, k=1, beta=1, gamma=.5, Zadj_sq=.25, sigmau=1, sigmav=.4, rho=.5, alpha = 0.05, deltarange=c(-0.1, 0.1), delta=0) # power of sensitivity analysis with unknown delta, # assuming the range of sensitivity allowance is (-0.1, 0.1) ARsens.power(n=250, k=1, beta=1, gamma=.5, Zadj_sq=.25, sigmau=1, sigmav=.4, rho=.5, alpha = 0.05, deltarange=c(-0.1, 0.1)) ```