ivmodel (version 1.9.1)

ARsens.size: Sample Size Calculator for the Power of the Anderson-Rubin (1949) Test with Sensitivity Analysis

Description

ARsens.size computes the minimum sample size required for achieving certain 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

ARsens.size(power, k, beta, gamma, Zadj_sq, sigmau, sigmav, rho, 
            alpha = 0.05, deltarange = deltarange, delta = NULL)

Value

Minimum sample size required for achieving certain 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).

Arguments

power

The desired power over a constant.

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 covariates.

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 instruments).

alpha

Significance level.

deltarange

Range of sensitivity allowance. A numeric vector of length 2.

delta

True value of sensitivity parameter when calculating power. Usually take delta = 0 for the favorable situation or delta = NULL for unknown delta.

Author

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

See also ivmodel for details on the instrumental variables model.

Examples

# Assume we performed a sensitivity analysis in a study with one 
# IV (l=1) and the only exogenous variable is the intercept (k=1). 
# We want to calculate the minimum sample size needed for this 
# sensitivity analysis to have an at least 0.8 power.

# Suppose the difference between the null hypothesis and true causal 
# effect is 1 (beta=1).
# 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.

# minimum sample size for sensitivity analysis under the favorable 
# situation, assuming the range of sensitivity allowance is (-0.1, 0.1)
ARsens.size(power=0.8, 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)

# minimum sample size for sensitivity analysis with unknown delta, 
# assuming the range of sensitivity allowance is (-0.1, 0.1)
ARsens.size(power=0.8, k=1, beta=1, gamma=.5, Zadj_sq=.25, sigmau=1, 
    sigmav=.4, rho=.5, alpha = 0.05, deltarange=c(-0.1, 0.1))