Minimum detectable slope for mediator in cox regression based on Vittinghoff, Sen and McCulloch's (2009) method

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Description

Calculate minimal detectable slope for mediator given sample size and power in cox regression based on Vittinghoff, Sen and McCulloch's (2009) method.

Usage

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minEffect.VSMc.cox(n, 
                   power, 
                   sigma.m, 
                   psi, 
                   corr.xm, 
                   alpha = 0.05, 
                   verbose = TRUE)

Arguments

n

sample size.

power

power for testing b_2=0 for the cox regression \log(λ)=\log(λ_0)+b1 x_i + b2 m_i, where λ is the hazard function and λ_0 is the baseline hazard function.

sigma.m

standard deviation of the mediator.

psi

the probability that an observation is uncensored, so that the number of event d= n * psi, where n is the sample size.

corr.xm

correlation between the predictor x and the mediator m.

alpha

type I error rate.

verbose

logical. TRUE means printing minimum absolute detectable effect; FALSE means not printing minimum absolute detectable effect.

Details

The test is for testing the null hypothesis b_2=0 versus the alternative hypothesis b_2\neq 0 for the cox regressions:

\log(λ)=\log(λ_0)+b_1 x_i + b_2 m_i

Vittinghoff et al. (2009) showed that for the above cox regression, testing the mediation effect is equivalent to testing the null hypothesis H_0: b_2=0 versus the alternative hypothesis H_a: b_2\neq 0, if the correlation corr.xm between the primary predictor and mediator is non-zero.

The full model is

\log(λ)=\log(λ_0)+b_1 x_i + b_2 m_i

The reduced model is

\log(λ)=\log(λ_0)+b_1 x_i

Vittinghoff et al. (2009) mentioned that if confounders need to be included in both the full and reduced models, the sample size/power calculation formula could be accommodated by redefining corr.xm as the multiple correlation of the mediator with the confounders as well as the predictor.

Value

b2

minimum absolute detectable effect.

res.uniroot

results of optimization to find the optimal sample size.

Note

The test is a two-sided test. Code for one-sided tests will be added later.

Author(s)

Weiliang Qiu stwxq@channing.harvard.edu

References

Vittinghoff, E. and Sen, S. and McCulloch, C.E.. Sample size calculations for evaluating mediation. Statistics In Medicine. 2009;28:541-557.

See Also

powerMediation.VSMc.cox, ssMediation.VSMc.cox

Examples

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  # example in section 6 (page 547) of Vittinghoff et al. (2009).
  # minimum effect is = log(1.5) = 0.4054651

  minEffect.VSMc.cox(n = 1399, power = 0.7999916, 
    sigma.m = sqrt(0.25 * (1 - 0.25)), psi = 0.2, corr.xm = 0.3, 
    alpha = 0.05, verbose = TRUE)

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