Package 'powerMediation'

Minimum detectable slope

Description

Calculate minimal detectable slope given sample size and power for simple linear regression.

Usage

minEffect.SLR(n, 
              power, 
              sigma.x, 
              sigma.y, 
              alpha = 0.05, 
              verbose = TRUE)

Arguments

n	sample size.
power	power for testing if $\lambda=0$ for the simple linear regression $y_i=\gamma+\lambda x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma_{e}^2).$
sigma.x	standard deviation of the predictor $sd(x)=\sigma_x$.
sigma.y	marginal standard deviation of the outcome $sd(y)=\sigma_y$. (not the conditional standard deviation $sd(y|x)$)
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 $\lambda=0$ versus the alternative hypothesis $\lambda\neq 0$ for the simple linear regressions:

$y_i=\gamma+\lambda x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

Value

lambda.a	minimum absolute detectable effect.
res.uniroot	results of optimization to find the optimal minimum absolute detectable effect.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Dupont, W.D. and Plummer, W.D.. Power and Sample Size Calculations for Studies Involving Linear Regression. Controlled Clinical Trials. 1998;19:589-601.

See Also

power.SLR, power.SLR.rho, ss.SLR, ss.SLR.rho.

Examples

minEffect.SLR(n = 100, power = 0.8, sigma.x = 0.2, sigma.y = 0.5, 
    alpha = 0.05, verbose = TRUE)

---

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

Description

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

Usage

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

Arguments

n	sample size.
power	power for testing $b_2=0$ for the linear regression $y_i=b0+b1 x_i + b2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma_e^2)$.
sigma.m	standard deviation of the mediator.
sigma.e	standard deviation of the random error term in the linear regression $y_i=b0+b1 x_i + b2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma_e^2)$.
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 linear regressions:

$y_i=b_0+b_1 x_i + b_2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

Vittinghoff et al. (2009) showed that for the above linear 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

$y_i=b_0+b_1 x_i + b_2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

The reduced model is

$y_i=b_0+b_1 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

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. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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, ssMediation.VSMc

Examples

# example in section 3 (page 544) of Vittinghoff et al. (2009).
  # minimum effect is =0.1
  minEffect.VSMc(n = 863, power = 0.8, sigma.m = 1, 
    sigma.e = 1, corr.xm = 0.3, alpha = 0.05, verbose = TRUE)

---

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

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

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 $\log(\lambda)=\log(\lambda_0)+b1 x_i + b2 m_i$, where $\lambda$ is the hazard function and $\lambda_0$ 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:

$\log(\lambda)=\log(\lambda_0)+b_1 x_i + b_2 m_i$

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

$\log(\lambda)=\log(\lambda_0)+b_1 x_i + b_2 m_i$

The reduced model is

$\log(\lambda)=\log(\lambda_0)+b_1 x_i$

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. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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

# 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)

---

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

Description

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

Usage

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

Arguments

n	sample size.
power	power for testing $b_2=0$ for the logistic regression $\log(p_i/(1-p_i))=b0+b1 x_i + b2 m_i$.
sigma.m	standard deviation of the mediator.
p	the marginal prevalence of the outcome.
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 logistic regressions:

$\log(p_i/(1-p_i))=b_0+b_1 x_i + b_2 m_i$

Vittinghoff et al. (2009) showed that for the above logistic 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

$\log(p_i/(1-p_i))=b_0+b_1 x_i + b_2 m_i$

The reduced model is

$\log(p_i/(1-p_i))=b_0+b_1 x_i$

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. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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.logistic, ssMediation.VSMc.logistic

Examples

# example in section 4 (page 545) of Vittinghoff et al. (2009).
  # minimum effect is log(1.5)= 0.4054651 

  minEffect.VSMc.logistic(n = 255, power = 0.8, sigma.m = 1, 
    p = 0.5, corr.xm = 0.5, alpha = 0.05, verbose = TRUE)

---

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

Description

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

Usage

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

Arguments

n	sample size.
power	power for testing $b_2=0$ for the poisson regression $\log(E(Y_i))=b0+b1 x_i + b2 m_i$.
sigma.m	standard deviation of the mediator.
EY	the marginal mean of the outcome
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 poisson regressions:

$\log(E(Y_i))=b_0+b_1 x_i + b_2 m_i$

Vittinghoff et al. (2009) showed that for the above poisson 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

$\log(E(Y_i))=b_0+b_1 x_i + b_2 m_i$

The reduced model is

$\log(E(Y_i))=b_0+b_1 x_i$

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. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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.poisson, ssMediation.VSMc.poisson

Examples

# example in section 5 (page 546) of Vittinghoff et al. (2009).
  # minimum effect is = log(1.35) = 0.3001046
  minEffect.VSMc.poisson(n = 1239, power = 0.7998578, 
    sigma.m = sqrt(0.25 * (1 - 0.25)), 
    EY = 0.5, corr.xm = 0.5, 
    alpha = 0.05, verbose = TRUE)

---

Power for testing slope for simple linear regression

Description

Calculate power for testing slope for simple linear regression.

Usage

power.SLR(n, 
          lambda.a, 
          sigma.x, 
          sigma.y, 
          alpha = 0.05, 
          verbose = TRUE)

Arguments

n	sample size.
lambda.a	regression coefficient in the simple linear regression $y_i=\gamma+\lambda x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma_{e}^2).$
sigma.x	standard deviation of the predictor $sd(x)$.
sigma.y	marginal standard deviation of the outcome $sd(y)$. (not the marginal standard deviation $sd(y|x)$)
alpha	type I error rate.
verbose	logical. TRUE means printing power; FALSE means not printing power.

Details

The power is for testing the null hypothesis $\lambda=0$ versus the alternative hypothesis $\lambda\neq 0$ for the simple linear regressions:

$y_i=\gamma+\lambda x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

Value

power	power for testing if $b_2=0$.
delta	$\lambda\sigma_x\sqrt{n}/\sqrt{\sigma_y^2-(\lambda\sigma_x)^2}$.
s	$\sqrt{\sigma_y^2-(\lambda\sigma_x)^2}$.
t.cr	$\Phi^{-1}(1-\alpha/2)$, where $\Phi$ is the cumulative distribution function of the standard normal distribution.
rho	correlation between the predictor $x$ and outcome $y$ $=\lambda\sigma_x/\sigma_y$.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Dupont, W.D. and Plummer, W.D.. Power and Sample Size Calculations for Studies Involving Linear Regression. Controlled Clinical Trials. 1998;19:589-601.

See Also

minEffect.SLR, power.SLR.rho, ss.SLR.rho, ss.SLR.

Examples

power.SLR(n=100, lambda.a=0.8, sigma.x=0.2, sigma.y=0.5, 
    alpha = 0.05, verbose = TRUE)

---

Power for testing slope for simple linear regression

Description

Calculate power for testing slope for simple linear regression.

Usage

power.SLR.rho(n, 
              rho2, 
              alpha = 0.05, 
              verbose = TRUE)

Arguments

n	sample size.
rho2	square of the correlation between the outcome and the predictor.
alpha	type I error rate.
verbose	logical. TRUE means printing power; FALSE means not printing power.

Details

The power is for testing the null hypothesis $\lambda=0$ versus the alternative hypothesis $\lambda\neq 0$ for the simple linear regressions:

$y_i=\gamma+\lambda x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

Value

power	power for testing if $b_2=0$.
delta	$\sqrt{n}/\sqrt{1/\rho^2-1}$.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Dupont, W.D. and Plummer, W.D.. Power and Sample Size Calculations for Studies Involving Linear Regression. Controlled Clinical Trials. 1998;19:589-601.

See Also

minEffect.SLR, power.SLR, ss.SLR.rho, ss.SLR.

Examples

power.SLR.rho(n=100, rho2=0.6, alpha = 0.05, verbose = TRUE)

---

Power Calculation for Interaction Effect in 2x2 Two-Way ANOVA Given Effect Sizes

Description

Power calculation for interaction effect in 2x2 two-way ANOVA given effect sizes.

Usage

powerInteract2by2(n, tauBetaSigma, alpha = 0.05, nTests = 1, verbose = FALSE)

Arguments

n	integer. Number of subjects per group.
tauBetaSigma	Effect sizes $\left(\tau\beta\right)_{ij}/\sigma, i=1, \ldots, a, j=1,\ldots, b$, where $a=b=2$ and $\sigma$ is the standard deviation of random error. Rows are for factor 1 and columns are for factor 2. Note that $\sum_{i=1}^a \left(\tau\beta\right)_{ij} = \sum_{j=1}^b \left(\tau\beta\right)_{ij}=0$. We can get $\left(\tau\beta\right)_{11}=\theta$, $\left(\tau\beta\right)_{12}=-\theta$, $\left(\tau\beta\right)_{21}=-\theta$, $\left(\tau\beta\right)_{22}=\theta$. So tauBetaSigma=$\theta/\sigma$
alpha	family-wise type I error rate.
nTests	integer. For high-throughput omics study, we perform two-way ANOVA for each of 'nTests' probes. We use Bonferroni correction to control for family-wise type I error rate. That is, for each probe, type I error rate would be alpha/nTests.
verbose	logical. Indicating if intermediate results should be printed out.

Details

We assume the following model:

$y_{ijk}=\mu+\tau_i + \beta_j + \left(\tau\beta\right)_{ij} + \epsilon_{ijk},$

where $i=1,\ldots, a, j=1,\ldots, b, k=1, \ldots, n$, $\sum_{i=1}^{a}\tau_i = 0$, $\sum_{j=1}^{b}\beta_j = 0$, $\sum_{i=1}^{a} \left(\tau\beta\right)_{ij} = 0$, $\sum_{j=1}^{b} \left(\tau\beta\right)_{ij} = 0$, and $\epsilon_{ijk}\stackrel{i.i.d}{\sim} N\left(0, \sigma^2\right)$.

The group means are

$\mu_{ij} = \mu+\tau_i + \beta_j + \left(\tau\beta\right)_{ij}, i=1 \ldots, a, j=1,\ldots, b.$

Note that $\mu = \sum_{i=1}^{a}\sum_{j=1}^b \mu_{ij} / (ab)$, $\tau_i = \sum_{j=1}^b \mu_{ij}/b - \mu$, and $\beta_j = \sum_{i=1}^a \mu_{ij}/a - \mu$.

The null hypothesis $H_0$: all $\left(\tau\beta\right)_{ij}, i=1, \ldots, a, j=1,\ldots, b$ are equal to zero. The alternative hypothesis $H_a$: at least one $\left(\tau\beta\right)_{ij}$ is different from zero.

The F test statistic is

$F=MS_{AB}/MS_{E}\stackrel{H_a}{\sim} F_{(a-1)(b-1), ab(n - 1), ncp},$

where ncp is the non-centrality parameter of the F test statistic:

$ncp=n\sum_{i=1}^{a}\sum_{j=1}^{b}\left[\frac{\left(\tau\beta\right)_{ij}}{\sigma}\right]^2.$

For the scenario $a=b=2$, we have $\left(\tau\beta\right)_{11}=\theta$, $\left(\tau\beta\right)_{12}=-\theta$, $\left(\tau\beta\right)_{21}=-\theta$, $\left(\tau\beta\right)_{22}=\theta$. Hence, the non-centrality parameter can be simplified to

$ncp=4n\left(\frac{\theta}{\sigma}\right)^2.$

The power for testing the null hypothesis $H_0$ versus the alternative hypothesis $H_a$ is

$power=Pr\left(F > F_0 | H_a\right),$

where the rejection region boundary $F_0$ satisfies:

$Pr\left(F > F_0 | H_0\right) = \alpha/nTests.$

Value

A list with 5 elements:

power	the power of the two-way ANOVA test
df1	the first degree of freedom of the F test statistic (df1=(a-1)(b-1))
df2	the second degree of freedom of the F test statistic (df1=a*b(n-1))
F0	the rejection region boundary
ncp	the non-centrality parameter

Author(s)

Weiliang Qiu [email protected]

References

Chow SC, Shao J, and Wang H. Sample size calculations in clinical research. 2nd edition. Chapman & Hall/CRC. 2008

Montgomery DC. Design and Analysis of Experiments. 8th edition. John Wiley & Sons. Inc.

Examples

n = 25
tauBetaSigma = 0.3

# power = 0.8437275
res2 = powerInteract2by2(n = n, tauBetaSigma = tauBetaSigma, 
    alpha = 0.05, nTests = 1, verbose = TRUE)

---

Calculating power for simple logistic regression with binary predictor

Description

Calculating power for simple logistic regression with binary predictor.

Usage

powerLogisticBin(n, 
                 p1, 
                 p2, 
                 B, 
                 alpha = 0.05)

Arguments

n	total number of sample size.
p1	$pr(diseased|X=0)$, i.e. the event rate at $X=0$ in logistic regression $logit(p) = a + b X$, where $X$ is the binary predictor.
p2	$pr(diseased|X=1)$, the event rate at $X=1$ in logistic regression $logit(p) = a + b X$, where $X$ is the binary predictor.
B	$pr(X=1)$, i.e. proportion of the sample with $X=1$
alpha	Type I error rate.

Details

The logistic regression mode is

$\log(p/(1-p)) = \beta_0 + \beta_1 X$

where $p=prob(Y=1)$, $X$ is the binary predictor, $p_1=pr(diseased | X=0)$, $p_2=pr(diseased| X = 1)$, $B=pr(X=1)$, and $p = (1 - B) p_1+B p_2$. The sample size formula we used for testing if $\beta_1=0$, is Formula (2) in Hsieh et al. (1998):

$n=(Z_{1-\alpha/2}[p(1-p)/B]^{1/2} + Z_{power}[p_1(1-p_1)+p_2(1-p_2)(1-B)/B]^{1/2})^2/[ (p_1-p_2)^2 (1-B) ]$

where $n$ is the required total sample size and $Z_u$ is the $u$-th percentile of the standard normal distribution.

Value

Estimated power.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Hsieh, FY, Bloch, DA, and Larsen, MD. A SIMPLE METHOD OF SAMPLE SIZE CALCULATION FOR LINEAR AND LOGISTIC REGRESSION. Statistics in Medicine. 1998; 17:1623-1634.

See Also

powerLogisticBin

Examples

## Example in Table I Design (Balanced design with high event rates) 
    ## of Hsieh et al. (1998 )
    ## the power = 0.95
    powerLogisticBin(n = 1281, p1 = 0.4, p2 = 0.5, B = 0.5, alpha = 0.05)

---

Calculating power for simple logistic regression with continuous predictor

Description

Calculating power for simple logistic regression with continuous predictor.

Usage

powerLogisticCon(n, 
                 p1, 
                 OR, 
                 alpha = 0.05)

Arguments

n	total sample size.
p1	the event rate at the mean of the continuous predictor $X$ in logistic regression $logit(p) = a + b X$.
OR	Expected odds ratio. $\log(OR)$ is the change in log odds for the difference between at the mean of $X$ and at one SD above the mean.
alpha	Type I error rate.

Details

The logistic regression mode is

$\log(p/(1-p)) = \beta_0 + \beta_1 X$

where $p=prob(Y=1)$, $X$ is the continuous predictor, and $\log(OR)$ is the the change in log odds for the difference between at the mean of $X$ and at one SD above the mean. The sample size formula we used for testing if $\beta_1=0$ or equivalently $OR=1$, is Formula (1) in Hsieh et al. (1998):

$n=(Z_{1-\alpha/2} + Z_{power})^2/[ p_1 (1-p_1) [log(OR)]^2 ]$

where $n$ is the required total sample size, $OR$ is the odds ratio to be tested, $p_1$ is the event rate at the mean of the predictor $X$, and $Z_u$ is the $u$-th percentile of the standard normal distribution.

Value

Estimated power.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Hsieh, FY, Bloch, DA, and Larsen, MD. A SIMPLE METHOD OF SAMPLE SIZE CALCULATION FOR LINEAR AND LOGISTIC REGRESSION. Statistics in Medicine. 1998; 17:1623-1634.

See Also

SSizeLogisticCon

Examples

## Example in Table II Design (Balanced design (1)) of Hsieh et al. (1998 )
    ## the power is 0.95
    powerLogisticCon(n=317, p1=0.5, OR=exp(0.405), alpha=0.05)

---

Power calculation for longitudinal study with 2 time point

Description

Power calculation for testing if mean changes for 2 groups are the same or not for longitudinal study with 2 time point.

Usage

powerLong(es, 
          n, 
          rho = 0.5, 
          alpha = 0.05)

Arguments

es	effect size of the difference of mean change.
n	sample size per group.
rho	correlation coefficient between baseline and follow-up values within a treatment group.
alpha	Type I error rate.

Details

The power formula is based on Equation 8.31 on page 336 of Rosner (2006).

$power=\Phi\left(-Z_{1-\alpha/2}+\frac{\delta\sqrt{n}}{\sigma_d \sqrt{2}}\right)$

where $\sigma_d = \sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2$, $\delta=|\mu_1 - \mu_2|$, $\mu_1$ is the mean change over time $t$ in group 1, $\mu_2$ is the mean change over time $t$ in group 2, $\sigma_1^2$ is the variance of baseline values within a treatment group, $\sigma_2^2$ is the variance of follow-up values within a treatment group, $\rho$ is the correlation coefficient between baseline and follow-up values within a treatment group, and $Z_u$ is the u-th percentile of the standard normal distribution.

We wish to test $\mu_1 = \mu_2$.

When $\sigma_1=\sigma_2=\sigma$, then formula reduces to

$power=\Phi\left(-Z_{1-\alpha/2} + \frac{|d|\sqrt{n}}{2\sqrt{1-\rho}}\right)$

where $d=\delta/\sigma$.

Value

power for testing for difference of mean changes.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Rosner, B. Fundamentals of Biostatistics. Sixth edition. Thomson Brooks/Cole. 2006.

See Also

ssLong, ssLongFull, powerLongFull.

Examples

# Example 8.34 on page 336 of Rosner (2006)
    # power=0.75
    powerLong(es=5/15, n=75, rho=0.7, alpha=0.05)

---

Power calculation for testing if mean changes for 2 groups are the same or not for longitudinal study with more than 2 time points

Description

Power calculation for testing if mean changes for 2 groups are the same or not for longitudinal study with more than 2 time points.

Usage

powerLong.multiTime(es, m, nn, sx2, rho = 0.5, alpha = 0.05)

Arguments

es	effect size
m	number of subjects
nn	number of observations per subject
sx2	within subject variance
rho	within subject correlation
alpha	type I error rate

Details

We are interested in comparing the slopes of the 2 groups $A$ and $B$:

$\beta_{1A} = \beta_{1B}$

where

$Y_{ijA}=\beta_{0A}+\beta_{1A} x_{jA} + \epsilon_{ijA}, j=1, \ldots, nn; i=1, \ldots, m$

and

$Y_{ijB}=\beta_{0B}+\beta_{1B} x_{jB} + \epsilon_{ijB}, j=1, \ldots, nn; i=1, \ldots, m$

The power calculation formula is (Equation on page 30 of Diggle et al. (1994)):

$power=\Phi\left[ -z_{1-\alpha} + \sqrt{\frac{m nn s_x^2 es^2}{2(1-\rho)}} \right]$

where $es=d/\sigma$, $d$ is the meaninful differnce of interest, $sigma^2$ is the variance of the random error, $\rho$ is the within-subject correlation, and $s_x^2$ is the within-subject variance.

Value

power

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Diggle PJ, Liang KY, and Zeger SL (1994). Analysis of Longitundinal Data. page 30. Clarendon Press, Oxford

See Also

ssLong.multiTime

Examples

# power=0.8
  powerLong.multiTime(es=0.5/10, m=196, nn=3, sx2=4.22, rho = 0.5, alpha = 0.05)

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Power calculation for longitudinal study with 2 time point

Description

Power calculation for testing if mean changes for 2 groups are the same or not for longitudinal study with 2 time point.

Usage

powerLongFull(delta, 
              sigma1, 
              sigma2, 
              n, 
              rho = 0.5, 
              alpha = 0.05)

Arguments

delta	absolute difference of the mean changes between the two groups: $\delta=|\mu_1 - \mu_2|$ where $\mu_1$ is the mean change over time $t$ in group 1, $\mu_2$ is the mean change over time $t$ in group 2.
sigma1	the standard deviation of baseline values within a treatment group
sigma2	the standard deviation of follow-up values within a treatment group
n	sample size per group
rho	correlation coefficient between baseline and follow-up values within a treatment group.
alpha	Type I error rate.

Details

The power formula is based on Equation 8.31 on page 336 of Rosner (2006).

$power=\Phi\left(-Z_{1-\alpha/2}+\frac{\delta\sqrt{n}}{\sigma_d \sqrt{2}}\right)$

where $\sigma_d = \sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2$, $\delta=|\mu_1 - \mu_2|$, $\mu_1$ is the mean change over time $t$ in group 1, $\mu_2$ is the mean change over time $t$ in group 2, $\sigma_1^2$ is the variance of baseline values within a treatment group, $\sigma_2^2$ is the variance of follow-up values within a treatment group, $\rho$ is the correlation coefficient between baseline and follow-up values within a treatment group, and $Z_u$ is the u-th percentile of the standard normal distribution.

We wish to test $\mu_1 = \mu_2$.

Value

power for testing for difference of mean changes.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Rosner, B. Fundamentals of Biostatistics. Sixth edition. Thomson Brooks/Cole. 2006.

See Also

ssLong, ssLongFull, powerLong.

Examples

# Example 8.33 on page 336 of Rosner (2006)
    # power=0.80
    powerLongFull(delta=5, sigma1=15, sigma2=15, n=85, rho=0.7, alpha=0.05)

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Power for testing mediation effect (Sobel's test)

Description

Calculate power for testing mediation effect based on Sobel's test.

Usage

powerMediation.Sobel(n, 
                     theta.1a, 
                     lambda.a, 
                     sigma.x, 
                     sigma.m,
                     sigma.epsilon, 
                     alpha = 0.05, 
                     verbose = TRUE)

Arguments

n	sample size.
theta.1a	regression coefficient for the predictor in the linear regression linking the predictor $x$ to the mediator $m$ ($m_i=\theta_0+\theta_{1a} x_i + e_i, e_i\sim N(0, \sigma^2_e)$).
lambda.a	regression coefficient for the mediator in the linear regression linking the predictor $x$ and the mediator $m$ to the outcome $y$ ($y_i=\gamma+\lambda_{a} m_i+ \lambda_2 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{\epsilon})$).
sigma.x	standard deviation of the predictor.
sigma.m	standard deviation of the mediator.
sigma.epsilon	standard deviation of the random error term in the linear regression linking the predictor $x$ and the mediator $m$ to the outcome $y$ ($y_i=\gamma+\lambda_a m_i+ \lambda_2 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{\epsilon})$).
alpha	type I error.
verbose	logical. TRUE means printing power; FALSE means not printing power.

Details

The power is for testing the null hypothesis $\theta_1\lambda=0$ versus the alternative hypothesis $\theta_{1a}\lambda_a\neq 0$ for the linear regressions:

$m_i=\theta_0+\theta_{1a} x_i + e_i, e_i\sim N(0, \sigma^2_e)$

$y_i=\gamma+\lambda_a m_i+ \lambda_2 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{\epsilon})$

Test statistic is based on Sobel's (1982) test:

$Z=\frac{\hat{\theta}_{1a}\hat{\lambda_a}}{\hat{\sigma}_{\theta_{1a}\lambda_a}}$

where $\hat{\sigma}_{\theta_{1a}\lambda_a}$ is the estimated standard deviation of the estimate $\hat{\theta}_{1a}\hat{\lambda_a}$ using multivariate delta method:

$\sigma_{\theta_{1a}\lambda_a}=\sqrt{\theta_{1a}^2\sigma_{\lambda_a}^2+\lambda_a^2\sigma_{\theta_{1a}}^2}$

and $\sigma_{\theta_{1a}}^2=\sigma_e^2/(n\sigma_x^2)$ is the variance of the estimate $\hat{\theta}_{1a}$, and $\sigma_{\lambda_a}^2=\sigma_{\epsilon}^2/(n\sigma_m^2(1-\rho_{mx}^2))$ is the variance of the estimate $\hat{\lambda_a}$, $\sigma_m^2$ is the variance of the mediator $m_i$.

From the linear regression $m_i=\theta_0+\theta_{1a} x_i+e_i$, we have the relationship $\sigma_e^2=\sigma_m^2(1-\rho^2_{mx})$. Hence, we can simply the variance $\sigma_{\theta_{1a}, \lambda_a}$ to

$\sigma_{\theta_{1a}\lambda_a}=\sqrt{\theta_{1a}^2\frac{\sigma_{\epsilon}^2}{n\sigma_m^2(1-\rho_{mx}^2)}+\lambda_a^2\frac{\sigma_{m}^2(1-\rho_{mx}^2)}{n\sigma_x^2}}$

Value

power	power of the test for the parameter $\theta_{1a}\lambda_a$
delta	$\theta_1\lambda/(sd(\hat{\theta}_{1a})sd(\hat{\lambda}_a))$

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

References

Sobel, M. E. Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology. 1982;13:290-312.

See Also

ssMediation.Sobel, testMediation.Sobel

Examples

powerMediation.Sobel(n=248, theta.1a=0.1701, lambda.a=0.1998, 
   sigma.x=0.57, sigma.m=0.61, sigma.epsilon=0.2, 
   alpha = 0.05, verbose = TRUE)

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Power for testing mediation effect in linear regression based on Vittinghoff, Sen and McCulloch's (2009) method

Description

Calculate Power for testing mediation effect in linear regression based on Vittinghoff, Sen and McCulloch's (2009) method.

Usage

powerMediation.VSMc(n, 
                    b2, 
                    sigma.m, 
                    sigma.e, 
                    corr.xm, 
                    alpha = 0.05, 
                    verbose = TRUE)

Arguments

n	sample size.
b2	regression coefficient for the mediator $m$ in the linear regression $y_i=b0+b1 x_i + b2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma_e^2)$.
sigma.m	standard deviation of the mediator.
sigma.e	standard deviation of the random error term in the linear regression $y_i=b0+b1 x_i + b2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma_e^2)$.
corr.xm	correlation between the predictor $x$ and the mediator $m$.
alpha	type I error rate.
verbose	logical. TRUE means printing power; FALSE means not printing power.

Details

The power is for testing the null hypothesis $b_2=0$ versus the alternative hypothesis $b_2\neq 0$ for the linear regressions:

$y_i=b_0+b_1 x_i + b_2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

Vittinghoff et al. (2009) showed that for the above linear regression, testing the mediation effect is equivalent to testing the null hypothesis $H_0: b_2=0$ versus the alternative hypothesis $H_a: b_2\neq 0$.

The full model is

$y_i=b_0+b_1 x_i + b_2 m_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

The reduced model is

$y_i=b_0+b_1 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{e})$

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

power	power for testing if $b_2=0$.
delta	$b_2\sigma_m\sqrt{1-\rho_{xm}^2}/\sigma_e$, where $\sigma_m$ is the standard deviation of the mediator $m$, $\rho_{xm}$ is the correlation between the predictor $x$ and the mediator $m$, and $\sigma_e$ is the standard deviation of the random error term in the linear regression.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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

minEffect.VSMc, ssMediation.VSMc

Examples

# example in section 3 (page 544) of Vittinghoff et al. (2009).
  # power=0.8
  powerMediation.VSMc(n = 863, b2 = 0.1, sigma.m = 1, sigma.e = 1, 
    corr.xm = 0.3, alpha = 0.05, verbose = TRUE)

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Power for testing mediation effect in cox regression based on Vittinghoff, Sen and McCulloch's (2009) method

Description

Calculate Power for testing mediation effect in cox regression based on Vittinghoff, Sen and McCulloch's (2009) method.

Usage

powerMediation.VSMc.cox(n, 
                        b2, 
                        sigma.m, 
                        psi, 
                        corr.xm, 
                        alpha = 0.05, 
                        verbose = TRUE)

Arguments

n	sample size.
b2	regression coefficient for the mediator $m$ in the cox regression $\log(\lambda)=\log(\lambda_0)+b1 x_i + b2 m_i$, where $\lambda$ is the hazard function and $\lambda_0$ 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 power; FALSE means not printing power.

Details

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

$\log(\lambda)=\log(\lambda_0)+b_1 x_i + b_2 m_i$

where $\lambda$ is the hazard function and $\lambda_0$ is the baseline hazard function.

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 $H_a: b_2\neq 0$.

The full model is

$\log(\lambda)=\log(\lambda_0)+b_1 x_i + b_2 m_i$

The reduced model is

$\log(\lambda)=\log(\lambda_0)+b_1 x_i$

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

power	power for testing if $b_2=0$.
delta	$b_2\sigma_m\sqrt{(1-\rho_{xm}^2) psi}$

, where $\sigma_m$ is the standard deviation of the mediator $m$, $\rho_{xm}$ is the correlation between the predictor $x$ and the mediator $m$, and $psi$ is the probability that an observation is uncensored, so that the number of event $d= n * psi$, where $n$ is the sample size.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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

minEffect.VSMc.cox, ssMediation.VSMc.cox

Examples

# example in section 6 (page 547) of Vittinghoff et al. (2009).
  # power = 0.7999916
  powerMediation.VSMc.cox(n = 1399, b2 = log(1.5), 
    sigma.m = sqrt(0.25 * (1 - 0.25)), psi = 0.2, corr.xm = 0.3,
    alpha = 0.05, verbose = TRUE)

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Power for testing mediation effect in logistic regression based on Vittinghoff, Sen and McCulloch's (2009) method

Description

Calculate Power for testing mediation effect in logistic regression based on Vittinghoff, Sen and McCulloch's (2009) method.

Usage

powerMediation.VSMc.logistic(n, 
                             b2, 
                             sigma.m, 
                             p, 
                             corr.xm, 
                             alpha = 0.05, 
                             verbose = TRUE)

Arguments

n	sample size.
b2	regression coefficient for the mediator $m$ in the logistic regression $\log(p_i/(1-p_i))=b0+b1 x_i + b2 m_i$.
sigma.m	standard deviation of the mediator.
p	the marginal prevalence of the outcome.
corr.xm	correlation between the predictor $x$ and the mediator $m$.
alpha	type I error rate.
verbose	logical. TRUE means printing power; FALSE means not printing power.

Details

The power is for testing the null hypothesis $b_2=0$ versus the alternative hypothesis $b_2\neq 0$ for the logistic regressions:

$\log(p_i/(1-p_i))=b0+b1 x_i + b2 m_i$

Vittinghoff et al. (2009) showed that for the above logistic regression, testing the mediation effect is equivalent to testing the null hypothesis $H_0: b_2=0$ versus the alternative hypothesis $H_a: b_2\neq 0$.

The full model is

$\log(p_i/(1-p_i))=b_0+b_1 x_i + b_2 m_i$

The reduced model is

$\log(p_i/(1-p_i))=b_0+b_1 x_i$

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

power	power for testing if $b_2=0$.
delta	$b_2\sigma_m\sqrt{(1-\rho_{xm}^2) p (1-p)}$

, where $\sigma_m$ is the standard deviation of the mediator $m$, $\rho_{xm}$ is the correlation between the predictor $x$ and the mediator $m$, and $p$ is the marginal prevalence of the outcome.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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

minEffect.VSMc.logistic, ssMediation.VSMc.logistic

Examples

# example in section 4 (page 545) of Vittinghoff et al. (2009).
  # power = 0.8005793
  powerMediation.VSMc.logistic(n = 255, b2 = log(1.5), sigma.m = 1, 
    p = 0.5, corr.xm = 0.5, alpha = 0.05, verbose = TRUE)

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Power for testing mediation effect in poisson regression based on Vittinghoff, Sen and McCulloch's (2009) method

Description

Calculate Power for testing mediation effect in poisson regression based on Vittinghoff, Sen and McCulloch's (2009) method.

Usage

powerMediation.VSMc.poisson(n, 
                            b2, 
                            sigma.m, 
                            EY, 
                            corr.xm, 
                            alpha = 0.05, 
                            verbose = TRUE)

Arguments

n	sample size.
b2	regression coefficient for the mediator $m$ in the poisson regression $\log(E(Y_i))=b0+b1 x_i + b2 m_i$.
sigma.m	standard deviation of the mediator.
EY	the marginal mean of the outcome.
corr.xm	correlation between the predictor $x$ and the mediator $m$.
alpha	type I error rate.
verbose	logical. TRUE means printing power; FALSE means not printing power.

Details

The power is for testing the null hypothesis $b_2=0$ versus the alternative hypothesis $b_2\neq 0$ for the poisson regressions:

$\log(E(Y_i))=b0+b1 x_i + b2 m_i$

Vittinghoff et al. (2009) showed that for the above poisson regression, testing the mediation effect is equivalent to testing the null hypothesis $H_0: b_2=0$ versus the alternative hypothesis $H_a: b_2\neq 0$.

The full model is

$\log(E(Y_i))=b_0+b_1 x_i + b_2 m_i$

The reduced model is

$\log(E(Y_i))=b_0+b_1 x_i$

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

power	power for testing if $b_2=0$.
delta	$b_2\sigma_m\sqrt{(1-\rho_{xm}^2) EY}$

, where $\sigma_m$ is the standard deviation of the mediator $m$, $\rho_{xm}$ is the correlation between the predictor $x$ and the mediator $m$, and $EY$ is the marginal mean of the outcome.

Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set alpha=0.10 to obtain one-sided test at 5% significance level.

Author(s)

Weiliang Qiu [email protected]

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

minEffect.VSMc.poisson, ssMediation.VSMc.poisson

Examples

# example in section 5 (page 546) of Vittinghoff et al. (2009).
  # power = 0.7998578
  powerMediation.VSMc.poisson(n = 1239, b2 = log(1.35), 
    sigma.m = sqrt(0.25 * (1 - 0.25)), EY = 0.5, corr.xm = 0.5,
    alpha = 0.05, verbose = TRUE)

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