Ophthalmology Data

Description

The Ophthalmology data set is described in Example 14.41 on page 807 in Rosner (2006).

Usage

data(Oph)

Format

A data frame with 354 observations on the following 3 variables.

times

a numeric vector recording the survival/censoring time for each event/censoring.

status

a numeric vector recording if a observed time is event time (status=1) or censoring time (status=0).

group

a factor with levels C (indicating control group) and E (indicating experimental group).

Details

This data set was from a clinical trial (Berson et al., 1993) conducted to test the efficacy of different vitamin supplements in preventing visual loss in patients with retinitis pigmentosa. Rosner (2006) used the data from this clinical trial to illustrate the analysis of survival data (Sections 14.9-14.12 of Rosner (2006)).

The data set consists of two groups of participants: (1) the experimental group (i.e., group E in which participants receiving 15,000 IU of vitamin A per day) and (2) the control group (i.e., group C in which participants receiving 75 IU of vitamin A per day).

The participants were enrolled over a 2-year period (1984-1987) and followed for a maximum of 6 years. The follow-up was terminated in September 1991. Some participants dropped out of the study before September 1991 and had not failed. Dropouts were due to death, other diseases, or side effects possibly due to the study medications, or unwillingness to comply (take study medications). There are 6 time points (at 1st year, 2nd year, 3rd year, 4th year, 5-th year, and 6-th year) in this data set.

Rosner (2006, page 786) defined the participants who do not reach a disease endpoint during their period of follow-up as censored observations. A participant has been censored at time t if the participant has been followed up to time t and has not failed. Noninformative censoring is assumed. That is, participants who are censored have the same underlying survival curve after their censoring time as patients who are not censored.

Source

Created based on Table 14.12 on page 787 of Rosner (2006).

References

Berson, E.L., Rosner, B., Sandberg, M.A., Hayes, K.C., Nicholson, B.W., Weigel-DiFranco, C., and Willett, W.C. (1993). A randomized trial of vitamin A and vitamin E supplementation for retinitis pigmentosa. Archives of Ophthalmology. 111:761-772.

Rosner B. (2006). Fundamentals of Biostatistics. (6-th edition). Thomson Brooks/Cole.

Examples

data(Oph)

Calculate Number of Deaths Required for Cox Proportional Hazards Regression with Two Covariates for Epidemiological Studies

Description

Calculate number of deaths required for Cox proportional hazards regression with two covariates for epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates. Some parameters will be estimated based on a pilot data set.

Usage

numDEpi(X1, 
	X2, 
	power, 
	theta, 
	alpha = 0.05)

Arguments

Details

This is an implementation of the calculation of the number of required deaths derived by Latouche et al. (2004) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2),

where the covariate X_1 is of our interest. The covariate X_1 should be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of deaths required to achieve a power of 1-\beta is

D=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{ [\log(\theta)]^2 p (1-p) (1-\rho^2), }

where z_{a} is the 100 a-th percentile of the standard normal distribution,

\rho=corr(X_1, X_2)=(p_1-p_0)\times \sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1).

p and rho will be estimated from a pilot data set.

Value

Note

(1) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

(2) When rho2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio \theta measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio \theta measures the difference of effect on the cause-specific hazard.

References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

See Also

numDEpi.default

Examples

  # generate a toy pilot data set
  X1 <- c(rep(1, 39), rep(0, 61))
  set.seed(123456)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  res <- numDEpi(X1 = X1, 
		 X2 = X2, 
		 power = 0.8, 
		 theta = 2, 
		 alpha = 0.05)
  print(res)

  # proportion of subjects died of the disease of interest.
  psi <- 0.505

  # total number of subjects required to achieve the desired power
  ceiling(res$D / psi)

Calculate Number of Deaths Required for Cox Proportional Hazards Regression with Two Covariates for Epidemiological Studies

Description

Calculate number of deaths required for Cox proportional hazards regression with two covariates for epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates.

Usage

numDEpi.default(power, 
		theta, 
		p, 
		rho2, 
		alpha = 0.05)

Arguments

Details

This is an implementation of the calculation of the number of required deaths derived by Latouche et al. (2004) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2),

where the covariate X_1 is of our interest. The covariate X_1 should be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of deaths required to achieve a power of 1-\beta is

D=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{ [\log(\theta)]^2 p (1-p) (1-\rho^2), }

where z_{a} is the 100 a-th percentile of the standard normal distribution,

\rho=corr(X_1, X_2)=(p_1-p_0)\times \sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1).

Value

The number of deaths required to achieve the desired power with given type I error rate.

Note

(1) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

(2) When rho2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio \theta measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio \theta measures the difference of effect on the cause-specific hazard.

References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

See Also

numDEpi

Examples

  # Example at the end of Section 5.2 of Latouche et al. (2004)
  # for a cohort study.
  D <- numDEpi.default(power = 0.8, 
		       theta = 2, 
		       p = 0.39, 
                       rho2 = 0.132^2, 
		       alpha = 0.05)

  # proportion of subjects died of the disease of interest.
  psi <- 0.505

  # total number of subjects required to achieve the desired power
  ceiling(D / psi)

Power Calculation for Survival Analysis with Binary Predictor and Exponential Survival Function

Description

Power calculation for survival analysis with binary predictor and exponential survival function.

Usage

power.stratify(
    n, 
    timeUnit, 
    gVec, 
    PVec, 
    HR, 
    lambda0Vec, 
    power.ini = 0.8, 
    power.low = 0.001, 
    power.upp = 0.999, 
    alpha = 0.05, 
    verbose = TRUE)

Arguments

Details

We assume (1) there is only one predictor and no covariates in the survival model (exponential survival function); (2) there are m strata; (3) the predictor x is a binary variable indicating treatment group 1 (x=1) or treatment group 0 (x=0); (3) the treatment effect is constant over time (proportional hazards); (4) the hazard ratio is the same in all strata, and (5) the data will be analyzed by the stratified log rank test.

The sample size formula is Formula (1) on page 801 of Palta M and Amini SB (1985):

n=(Z_{\alpha}+Z_{\beta})^2/\mu^2

where \alpha is the Type I error rate, \beta is the Type II error rate (power=1-\beta), Z_{\alpha} is the 100(1-\alpha)-th percentile of standard normal distribution, and

\mu=\log(\delta)\sqrt{ \sum_{s=1}^{m} g_s P_s (1 - P_s) V_s }

and

V_s=P_s\left[1-\frac{1}{\lambda_{1s}} \left\{ \exp\left[-\lambda_{1s}(T-1)\right] -\exp(-\lambda_{1s}T) \right\} \right] +(1-P_s)\left[ 1-\frac{1}{\lambda_{0s}} \left\{ \exp\left[-\lambda_{0s}(T-1)\right] -\exp(-\lambda_{0s}T \right\} \right]

In the above formulas, m is the number of strata, T is the total study length, \delta is the hazard ratio, g_s is the proportion of the total sample size in stratum s, P_s is the proportion of stratum s, which is in treatment group 1, and \lambda_{is} is the hazard for the i-th treatment group in stratum s.

Value

A list of 2 elments.

References

Palta M and Amini SB. (1985). Consideration of covariates and stratification in sample size determination for survival time studies. Journal of Chronic Diseases. 38(9):801-809.

See Also

ssize.stratify

Examples

# example on page 803 of Palta M and Amini SB. (1985). 
res.power <- power.stratify(
  n = 146, 
  timeUnit = 1.25, 
  gVec = c(0.5, 0.5),
  PVec = c(0.5, 0.5), 
  HR = 1 / 1.91, 
  lambda0Vec = c(2.303, 1.139),
  power.ini = 0.8, 
  power.low = 0.001, 
  power.upp = 0.999,
  alpha = 0.05, 
  verbose = TRUE
  )

Power Calculation in the Analysis of Survival Data for Clinical Trials

Description

Power calculation for the Comparison of Survival Curves Between Two Groups under the Cox Proportional-Hazards Model for clinical trials. Some parameters will be estimated based on a pilot data set.

Usage

powerCT(formula, 
	dat, 
	nE, 
	nC, 
	RR, 
	alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation method described in Section 14.12 (page 807) of Rosner (2006). The method was proposed by Freedman (1982).

The movitation of this function is that some times we do not have information about m or p_E and p_C available, but we have a pilot data set that can be used to estimate p_E and p_C hence m, where m=n_E p_E + n_C p_C is the expected total number of events over both groups, n_E and n_C are numbers of participants in group E (experimental group) and group C (control group), respectively. p_E is the probability of failure in group E (experimental group) over the maximum time period of the study (t years). p_C is the probability of failure in group C (control group) over the maximum time period of the study (t years).

Suppose we want to compare the survival curves between an experimental group (E) and a control group (C) in a clinical trial with a maximum follow-up of t years. The Cox proportional hazards regression model is assumed to have the form:

h(t|X_1)=h_0(t)\exp(\beta_1 X_1).

Let n_E be the number of participants in the E group and n_C be the number of participants in the C group. We wish to test the hypothesis H0: RR=1 versus H1: RR not equal to 1, where RR=\exp(\beta_1)=underlying hazard ratio for the E group versus the C group. Let RR be the postulated hazard ratio, \alpha be the significance level. Assume that the test is a two-sided test. If the ratio of participants in group E compared to group C = n_E/n_C=k, then the power of the test is

power=\Phi(\sqrt{k*m}*|RR-1|/(k*RR+1)-z_{1-\alpha/2}),

where

m=n_E p_E+n_C p_C,

and z_{1-\alpha/2} is the 100 (1-\alpha/2)-th percentile of the standard normal distribution N(0, 1), \Phi is the cumulative distribution function (CDF) of N(0, 1).

p_C and p_E can be calculated from the following formulaes:

p_C=\sum_{i=1}^{t}D_i, p_E=\sum_{i=1}^{t}E_i,

where D_i=\lambda_i A_i C_i, E_i=RR\lambda_i B_i C_i, A_i=\prod_{j=0}^{i-1}(1-\lambda_j), B_i=\prod_{j=0}^{i-1}(1-RR\lambda_j), C_i=\prod_{j=0}^{i-1}(1-\delta_j). And \lambda_i is the probability of failure at time i among participants in the control group, given that a participant has survived to time i-1 and is not censored at time i-1, i.e., the approximate hazard time i in the control group, i=1,...,t; RRlambda_i is the probability of failure at time i among participants in the experimental group, given that a participant has survived to time i-1 and is not censored at time i-1, i.e., the approximate hazard time i in the experimental group, i=1,...,t; delta is the prbability that a participant is censored at time i given that he was followed up to time i and has not failed, i=0, 1, ..., t, which is assumed the same in each group.

Value

Note

(1) The estimates of RRlambda_i=RR*\lambda_i. That is, RRlambda is not directly estimated based on data from the experimental group; (2) The power formula assumes that the central-limit theorem is valid and hence is appropriate for large samples.

References

Freedman, L.S. (1982). Tables of the number of patients required in clinical trials using the log-rank test. Statistics in Medicine. 1: 121-129

Rosner B. (2006). Fundamentals of Biostatistics. (6-th edition). Thomson Brooks/Cole.

See Also

powerCT.default0, powerCT.default

Examples

  # Example 14.42 in Rosner B. Fundamentals of Biostatistics. 
  # (6-th edition). (2006) page 809

  library(survival)

  data(Oph)
  res <- powerCT(formula = Surv(times, status) ~ group, 
		 dat = Oph, 
                 nE = 200, 
		 nC = 200, 
		 RR = 0.7, 
		 alpha = 0.05)

  # Table 14.24 on page 809 of Rosner (2006)
  print(round(res$mat.lambda, 4))

  # Table 14.12 on page 787 of Rosner (2006)
  print(round(res$mat.event, 4))

  # the power
  print(round(res$power, 2))

Power Calculation in the Analysis of Survival Data for Clinical Trials

Description

Power calculation for the Comparison of Survival Curves Between Two Groups under the Cox Proportional-Hazards Model for clinical trials.

Usage

powerCT.default(nE, 
		nC, 
		pE, 
		pC, 
		RR, 
		alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation method described in Section 14.12 (page 807) of Rosner (2006). The method was proposed by Freedman (1982).

Suppose we want to compare the survival curves between an experimental group (E) and a control group (C) in a clinical trial with a maximum follow-up of t years. The Cox proportional hazards regression model is assumed to have the form:

h(t|X_1)=h_0(t)\exp(\beta_1 X_1).

Let n_E be the number of participants in the E group and n_C be the number of participants in the C group. We wish to test the hypothesis H0: RR=1 versus H1: RR not equal to 1, where RR=\exp(\beta_1)=underlying hazard ratio for the E group versus the C group. Let RR be the postulated hazard ratio, \alpha be the significance level. Assume that the test is a two-sided test. If the ratio of participants in group E compared to group C = n_E/n_C=k, then the power of the test is

power=\Phi(\sqrt{k*m}*|RR-1|/(k*RR+1)-z_{1-\alpha/2}),

where

m=n_E p_E+n_C p_C,

and z_{1-\alpha/2} is the 100 (1-\alpha/2)-th percentile of the standard normal distribution N(0, 1), \Phi is the cumulative distribution function (CDF) of N(0, 1).

Value

The power of the test.

Note

The power formula assumes that the central-limit theorem is valid and hence is appropriate for large samples.

References

Freedman, L.S. (1982). Tables of the number of patients required in clinical trials using the log-rank test. Statistics in Medicine. 1: 121-129

Rosner B. (2006). Fundamentals of Biostatistics. (6-th edition). Thomson Brooks/Cole.

See Also

powerCT.default0, powerCT

Examples

  # Example 14.42 in Rosner B. Fundamentals of Biostatistics. 
  # (6-th edition). (2006) page 809
  powerCT.default(nE = 200, 
		  nC = 200, 
		  pE = 0.3707, 
		  pC = 0.4890, 
                  RR = 0.7, 
		  alpha = 0.05)

Power Calculation in the Analysis of Survival Data for Clinical Trials

Description

Power calculation for the Comparison of Survival Curves Between Two Groups under the Cox Proportional-Hazards Model for clinical trials.

Usage

powerCT.default0(k, 
		 m, 
		 RR, 
		 alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation method described in Section 14.12 (page 807) of Rosner (2006). The method was proposed by Freedman (1982).

Suppose we want to compare the survival curves between an experimental group (E) and a control group (C) in a clinical trial with a maximum follow-up of t years. The Cox proportional hazards regression model is assumed to have the form:

h(t|X_1)=h_0(t)\exp(\beta_1 X_1).

Let n_E be the number of participants in the E group and n_C be the number of participants in the C group. We wish to test the hypothesis H0: RR=1 versus H1: RR not equal to 1, where RR=\exp(\beta_1)=underlying hazard ratio for the E group versus the C group. Let RR be the postulated hazard ratio, \alpha be the significance level. Assume that the test is a two-sided test. If the ratio of participants in group E compared to group C = n_E/n_C=k, then the power of the test is

power=\Phi(\sqrt{k*m}*|RR-1|/(k*RR+1)-z_{1-\alpha/2}),

where z_{1-\alpha/2} is the 100 (1-\alpha/2)-th percentile of the standard normal distribution N(0, 1), \Phi is the cumulative distribution function (CDF) of N(0, 1).

Value

The power of the test.

Note

The power formula assumes that the central-limit theorem is valid and hence is appropriate for large samples.

References

Freedman, L.S. (1982). Tables of the number of patients required in clinical trials using the log-rank test. Statistics in Medicine. 1: 121-129

Rosner B. (2006). Fundamentals of Biostatistics. (6-th edition). Thomson Brooks/Cole.

See Also

powerCT.default, powerCT

Examples

  # Example 14.42 in Rosner B. Fundamentals of Biostatistics. 
  # (6-th edition). (2006) page 809
  powerCT.default0(k = 1, 
		   m = 171.9, 
		   RR = 0.7, 
		   alpha = 0.05)

Sample Size Calculation for Conditional Logistic Regression with Binary Covariate

Description

Sample Size Calculation for Conditional Logistic Regression with Binary Covariate, such as matched logistic regression or nested case-control study.

Usage

powerConLogistic.bin(
  N = NULL, 
  power = 0.8, 
  OR, 
  pE, 
  nD, 
  nH, 
  R2 = 0, 
  alpha = 0.05, 
  nTests = 1,
  OR.low = 1.01,
  OR.upp = 100
)

Arguments

Details

The power and sample size calculation formulas are provided by Lachin (2008, Section 3.3, Formula (38))

power = \Phi\left( \sqrt{N c} - z_{\alpha/(2 nTests)}\right)

and

N = (z_{power} + z_{\alpha/(2 nTests)})^2/ c

where \Phi is the cumulative distribution function of the standard normal distribution N(0, 1), z_{a} is the upper 100 a-th percentile of N(0, 1),

c = \theta^2 pE (1-pE) (1-R^2)nD*nH/(nD+nH)

and R^2 is the coefficient of determination for linear regression linking the exposure with other covariates.

Value

If the inputs is.null(N) = TRUE and is.null(power) = FALSE, then the function returns the number N of sets.

If the inputs is.null(N) = FALSE and is.null(power) = TRUE, then the function returns the power.

Otherwise, an error message is output.

References

Lachin, JM Sample Size Evaluation for a Multiply Matched Case-Control Study Using the Score Test From a Conditional Logistic (Discrete Cox PH) Regression Model. Stat Med. 2008 27(14): 2509-2523

Examples

# estimate power
power = powerConLogistic.bin(
  N = 59, 
  power = NULL, 
  OR = 3.5, 
  pE = 0.15, 
  nD = 1, 
  nH = 2, 
  R2 = 0, 
  alpha = 0.05, 
  nTests = 1)

print(power) # 0.80

# estimate N (number of sets)
N = powerConLogistic.bin(
  N = NULL, 
  power = 0.80, 
  OR = 3.5, 
  pE = 0.15, 
  nD = 1, 
  nH = 2, 
  R2 = 0, 
  alpha = 0.05, 
  nTests = 1)

print(ceiling(N)) # 59

# estimate OR
OR = powerConLogistic.bin(
  N = 59, 
  power = 0.80, 
  OR = NULL, 
  pE = 0.15, 
  nD = 1, 
  nH = 2, 
  R2 = 0, 
  alpha = 0.05, 
  nTests = 1,
  OR.low = 1.01,
  OR.upp = 100)

print(OR) # 3.49

Sample Size Calculation for Conditional Logistic Regression with Continuous Covariate

Description

Sample Size Calculation for Conditional Logistic Regression with Continuous Covariate, such as matched logistic regression or nested case-control study.

Usage

powerConLogistic.con(
  N = NULL, 
  power = 0.8, 
  OR, 
  sigma, 
  nD, 
  nH, 
  R2 = 0, 
  alpha = 0.05, 
  nTests = 1,
  OR.low = 1.01,
  OR.upp = 100
)

Arguments

Details

The power and sample size calculation formulas are provided by Lachin (2008, Section 3.1, Formulas (24) and (25))

power = \Phi\left( \sqrt{N c} - z_{\alpha/(2 nTests)}\right)

and

N = (z_{power} + z_{\alpha/(2 nTests)})^2/ c

where \Phi is the cumulative distribution function of the standard normal distribution N(0, 1), z_{a} is the upper 100 a-th percentile of N(0, 1),

c = \theta^2 \sigma^2 nD (1-1/b) (1-R^2)

and b is the Binomial coefficient (n chooses nD), n = nD + nH, and R^2 is the coefficient of determination for linear regression linking the exposure with other covariates.

Value

If the inputs is.null(N) = TRUE and is.null(power) = FALSE, then the function returns the number N of sets.

If the inputs is.null(N) = FALSE and is.null(power) = TRUE, then the function returns the power.

Otherwise, an error message is output.

References

Lachin, JM Sample Size Evaluation for a Multiply Matched Case-Control Study Using the Score Test From a Conditional Logistic (Discrete Cox PH) Regression Model. Stat Med. 2008 27(14): 2509-2523

Examples

library(pracma)

# Section 4.1 in Lachin (2008) 

# estimate number of sets 
N = powerConLogistic.con(N = NULL,
                                power = 0.85,
                                OR = 1.39,
                                sigma = 1,
                                nD = 1,
                                nH = 2,
                                R2 = 0,
                                alpha = 0.05,
                                nTests = 1)
print(ceiling(N)) # 125

# estimate power 
power = powerConLogistic.con(N = 125,
                                power = NULL,
                                OR = 1.39,
                                sigma = 1,
                                nD = 1,
                                nH = 2,
                                R2 = 0,
                                alpha = 0.05,
                                nTests = 1)
print(power) # 0.85

# estimate OR 
OR = powerConLogistic.con(N = 125,
                                power = 0.85,
                                OR = NULL,
                                sigma = 1,
                                nD = 1,
                                nH = 2,
                                R2 = 0,
                                alpha = 0.05,
                                nTests = 1)
print(OR) # 1.39

Power Calculation for Cox Proportional Hazards Regression with Two Covariates for Epidemiological Studies

Description

Power calculation for Cox proportional hazards regression with two covariates for epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates. Some parameters will be estimated based on a pilot data set.

Usage

powerEpi(X1, X2, failureFlag, n, theta, alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Latouche et al. (2004) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2),

where the covariate X_1 is of our interest. The covariate X_1 should be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\beta_1)=\theta is

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{n[\log(\theta)]^2 p (1-p) \psi (1-\rho^2)}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times \sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1).

p, \rho^2, and \psi will be estimated from a pilot data set.

Value

Note

(1) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

(2) When \rho^2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio \theta measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio \theta measures the difference of effect on the cause-specific hazard.

References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

See Also

powerEpi.default

Examples

  # generate a toy pilot data set
  X1 <- c(rep(1, 39), rep(0, 61))
  set.seed(123456)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  failureFlag <- sample(c(0, 1), 100, prob = c(0.5, 0.5), replace = TRUE)

  powerEpi(X1 = X1, X2 = X2, failureFlag = failureFlag, 
    n = 139, theta = 2, alpha = 0.05)

Power Calculation for Cox Proportional Hazards Regression with Two Covariates for Epidemiological Studies

Description

Power calculation for Cox proportional hazards regression with two covariates for epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates.

Usage

powerEpi.default(n, 
		 theta, 
		 p, 
		 psi, 
		 rho2, 
		 alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Latouche et al. (2004) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2),

where the covariate X_1 is of our interest. The covariate X_1 should be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\beta_1)=\theta is

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{n[\log(\theta)]^2 p (1-p) \psi (1-\rho^2)}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times \sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1).

Value

The power of the test.

Note

(1) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

(2) When rho2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio \theta measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio \theta measures the difference of effect on the cause-specific hazard.

References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

See Also

powerEpi

Examples

  # Example at the end of Section 5.2 of Latouche et al. (2004)
  # for a cohort study.
  powerEpi.default(n = 139, 
		   theta = 2, 
		   p = 0.39, 
		   psi = 0.505,
                   rho2 = 0.132^2, 
		   alpha = 0.05)

Power Calculation for Cox Proportional Hazards Regression with Nonbinary Covariates for Epidemiological Studies

Description

Power calculation for Cox proportional hazards regression with nonbinary covariates for Epidemiological Studies. Some parameters will be estimated based on a pilot data set.

Usage

powerEpiCont(formula, 
	     dat, 
	     var.X1, 
	     var.failureFlag, 
	     n, 
	     theta, 
	     alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Hsieh and Lavori (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, \boldsymbol{x}_2)=h_0(t)\exp(\beta_1 x_1+\boldsymbol{\beta}_2 \boldsymbol{x}_2),

where the covariate X_1 is a nonbinary variable and \boldsymbol{X}_2 is a vector of other covariates.

Suppose we want to check if the hazard ratio of the main effect X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\beta_1)=\theta is

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{n[\log(\theta)]^2 \sigma^2 \psi (1-\rho^2)}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution, \sigma^2=Var(X_1), \psi is the proportion of subjects died of the disease of interest, and \rho is the multiple correlation coefficient of the following linear regression:

x_1=b_0+\boldsymbol{b}^T\boldsymbol{x}_2.

That is, \rho^2=R^2, where R^2 is the proportion of variance explained by the regression of X_1 on the vector of covriates \boldsymbol{X}_2.

rho will be estimated from a pilot study.

Value

Note

(1) Hsieh and Lavori (2000) assumed one-sided test, while this implementation assumed two-sided test. (2) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

References

Hsieh F.Y. and Lavori P.W. (2000). Sample-size calculation for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials. 21:552-560.

See Also

powerEpiCont.default

Examples

  # generate a toy pilot data set
  set.seed(123456)
  X1 <- rnorm(100, mean = 0, sd = 0.3126)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  failureFlag <- sample(c(0, 1), 100, prob = c(0.25, 0.75), replace = TRUE)
  dat <- data.frame(X1 = X1, X2 = X2, failureFlag = failureFlag)

  powerEpiCont(formula = X1 ~ X2, 
	       dat = dat, 
	       var.X1 = "X1", 
	       var.failureFlag = "failureFlag", 
               n = 107, 
	       theta = exp(1), 
	       alpha = 0.05)

Power Calculation for Cox Proportional Hazards Regression with Nonbinary Covariates for Epidemiological Studies

Description

Power calculation for Cox proportional hazards regression with nonbinary covariates for Epidemiological Studies.

Usage

powerEpiCont.default(n, 
		     theta, 
		     sigma2, 
		     psi, 
		     rho2, 
		     alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Hsieh and Lavori (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, \boldsymbol{x}_2)=h_0(t)\exp(\beta_1 x_1+\boldsymbol{\beta}_2 \boldsymbol{x}_2),

where the covariate X_1 is a nonbinary variable and \boldsymbol{X}_2 is a vector of other covariates.

Suppose we want to check if the hazard ratio of the main effect X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\beta_1)=\theta is

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{n[\log(\theta)]^2 \sigma^2 \psi (1-\rho^2)}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution, \sigma^2=Var(X_1), \psi is the proportion of subjects died of the disease of interest, and \rho is the multiple correlation coefficient of the following linear regression:

x_1=b_0+\boldsymbol{b}^T\boldsymbol{x}_2.

That is, \rho^2=R^2, where R^2 is the proportion of variance explained by the regression of X_1 on the vector of covriates \boldsymbol{X}_2.

Value

The power of the test.

Note

(1) Hsieh and Lavori (2000) assumed one-sided test, while this implementation assumed two-sided test. (2) The formula can be used to calculate power for a randomized trial study by setting rho2=0.

References

Hsieh F.Y. and Lavori P.W. (2000). Sample-size calculation for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials. 21:552-560.

See Also

powerEpiCont

Examples

  # example in the EXAMPLE section (page 557) of Hsieh and Lavori (2000).
  # Hsieh and Lavori (2000) assumed one-sided test, 
  # while this implementation assumed two-sided test. 
  # Hence alpha=0.1 here (two-sided test) will correspond
  # to alpha=0.05 of one-sided test in Hsieh and Lavori's (2000) example.
  powerEpiCont.default(n = 107, 
		       theta = exp(1), 
		       sigma2 = 0.3126^2, 
                       psi = 0.738, 
		       rho2 = 0.1837, 
		       alpha = 0.1)

Power Calculation Testing Interaction Effect for Cox Proportional Hazards Regression with two covariates for Epidemiological Studies (Both covariates should be binary)

Description

Power calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates. Some parameters will be estimated based on a pilot study.

Usage

powerEpiInt(X1, 
	    X2, 
	    failureFlag, 
	    n, 
	    theta, 
	    alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemoilogical studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\gamma)=\theta is:

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{\frac{n}{\delta}[\log(\theta)]^2 \psi}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution,

\delta=\frac{1}{p_{00}}+\frac{1}{p_{01}}+\frac{1}{p_{10}} +\frac{1}{p_{11}},

\psi is the proportion of subjects died of the disease of interest, and p_{00}=Pr(X_1=0,\mbox{and}, X_2=0), p_{01}=Pr(X_1=0,\mbox{and}, X_2=1), p_{10}=Pr(X_1=1,\mbox{and}, X_2=0), p_{11}=Pr(X_1=1,\mbox{and}, X_2=1).

p_{00}, p_{01}, p_{10}, p_{11}, and \psi will be estimated from the pilot data.

Value

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

powerEpiInt.default0, powerEpiInt2

Examples

  # generate a toy pilot data set
  X1 <- c(rep(1, 39), rep(0, 61))
  set.seed(123456)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  failureFlag <- sample(c(0, 1), 100, prob = c(0.25, 0.75), replace = TRUE)

  powerEpiInt(X1 = X1, 
	      X2 = X2, 
	      failureFlag = failureFlag, 
	      n = 184, 
	      theta = 3, 
	      alpha = 0.05)
  

Power Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

Description

Power calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

Usage

powerEpiInt.default0(n, 
		     theta, 
		     p, 
		     psi, 
		     G, 
		     rho2, 
		     alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\gamma)=\theta is

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{\frac{n}{G}[\log(\theta)]^2 p (1-p) \psi (1-\rho^2)}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times \sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1), and

G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1}.

If X_1 and X_2 are uncorrelated, we have p_0=p_1=p leading to 1/[(1-q)q]. For q=0.5, we have G=4.

Value

The power of the test.

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

powerEpiInt.default1, powerEpiInt2

Examples

  # Example at the end of Section 4 of Schmoor et al. (2000).
  powerEpiInt.default0(n = 184, 
		       theta = 3, 
		       p = 0.61, 
		       psi = 139 / 184, 
                       G = 4.79177, 
		       rho2 = 0.015^2, 
		       alpha = 0.05)
  

Power Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

Description

Power calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

Usage

powerEpiInt.default1(n, 
		     theta, 
		     psi, 
		     p00, 
		     p01, 
		     p10, 
		     p11, 
		     alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemoilogical studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\gamma)=\theta is:

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{\frac{n}{\delta}[\log(\theta)]^2 \psi}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution,

\delta=\frac{1}{p_{00}}+\frac{1}{p_{01}}+\frac{1}{p_{10}} +\frac{1}{p_{11}},

\psi is the proportion of subjects died of the disease of interest, and p_{00}=Pr(X_1=0,\mbox{and}, X_2=0), p_{01}=Pr(X_1=0,\mbox{and}, X_2=1), p_{10}=Pr(X_1=1,\mbox{and}, X_2=0), p_{11}=Pr(X_1=1,\mbox{and}, X_2=1).

Value

The power of the test.

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

powerEpiInt.default0, powerEpiInt2

Examples

  # Example at the end of Section 4 of Schmoor et al. (2000).
  # p00, p01, p10, and p11 are calculated based on Table III on page 448
  # of Schmoor et al. (2000).
  powerEpiInt.default1(n = 184, 
		       theta = 3, 
		       psi = 139 / 184,
                       p00 = 50 / 184, 
		       p01 = 21 / 184, 
		       p10 = 78 / 184, 
		       p11 = 35 / 184,
                       alpha = 0.05)
  

Power Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

Description

Power calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

Usage

powerEpiInt2(n, 
	     theta, 
	     psi, 
	     mya, 
	     myb, 
	     myc, 
	     myd, 
	     alpha = 0.05)

Arguments

Details

This is an implementation of the power calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the power required to detect a hazard ratio as small as \exp(\gamma)=\theta is

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{\frac{n}{G}[\log(\theta)]^2 p (1-p) \psi (1-\rho^2)}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times\sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1), and

G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1},

and p0=Pr(X_1=1 | X_2=0)=myc/(mya+myc), p1=Pr(X_1=1 | X_2=1)=myd/(myb+myd), p=Pr(X_1=1)=(myc+myd)/n_{obs}, q=Pr(X_2=1)=(myb+myd)/n_{obs}, n_{obs}=mya+myb+myc+myd.

p_{00}=Pr(X_1=0,\mbox{and}, X_2=0), p_{01}=Pr(X_1=0,\mbox{and}, X_2=1), p_{10}=Pr(X_1=1,\mbox{and}, X_2=0), p_{11}=Pr(X_1=1,\mbox{and}, X_2=1).

Value

The power of the test.

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

powerEpiInt.default0, powerEpiInt.default1

Examples

  # Example at the end of Section 4 of Schmoor et al. (2000).
  # mya, myb, myc, and myd are obtained from Table III on page 448
  # of Schmoor et al. (2000).
  powerEpiInt2(n = 184, 
	       theta = 3, 
	       psi = 139 / 184,
               mya = 50, 
	       myb = 21, 
	       myc = 78, 
	       myd = 35, 
	       alpha = 0.05)
  

Power of Two-Sided Two Sample T Test With Unequal Variances And Unequal Sample Sizes

Description

Power of two-sided 2 sample t test with unequal variances and unequal sample sizes.

Usage

powerWelchT(
  n1, 
  n2, 
  meanDiff, 
  sd1, 
  sd2, 
  alpha = 0.05)

Arguments

Details

The power formula is

power = Pr\left(|T| > t_{1-\alpha/2, \nu} | T \sim t_{\nu, \lambda}\right),

where \lambda is the noncentrality parameter of the t distribution with degree of freedom \nu. t_{1-\alpha/2, \nu} is the upper 100\alpha/2 percentile of the t distribution with degree of freedom \nu. \alpha is the significance level. The noncentrality parameter \lambda is defined as

\lambda = \frac{|\mu_1 - \mu_2|}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}.

The degree \nu of freedom is the Satterthwaite approximation and is defined as

\nu = \frac{\left(\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}\right)^2}{ \frac{\left(\frac{\sigma_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{\sigma_2^2}{n_2}\right)^2}{n_2-1} }

Value

power

Examples

powerWelchT(
    n1 = 64, # sample size for group 1 
    n2 = 30, # sample size for group 2
    meanDiff = 1, # mean difference between 2 groups
    sd1 = 2, # SD of group 1
    sd2 = 1, # SD of group 2
    alpha = 0.05 # type I error rate
)
# 0.8918191

Sample size calculation for Survival Analysis with Binary Predictor and Exponential Survival Function

Description

Sample size calculation for survival analysis with binary predictor and exponential survival function.

Usage

ssize.stratify(
    power, 
    timeUnit, 
    gVec, 
    PVec, 
    HR, 
    lambda0Vec, 
    alpha = 0.05, 
    verbose = TRUE)

Arguments

Details

We assume (1) there is only one predictor and no covariates in the survival model (exponential survival function); (2) there are m strata; (3) the predictor x is a binary variable indicating treatment group 1 (x=1) or treatment group 0 (x=0); (3) the treatment effect is constant over time (proportional hazards); (4) the hazard ratio is the same in all strata, and (5) the data will be analyzed by the stratified log rank test.

The sample size formula is Formula (1) on page 801 of Palta M and Amini SB (1985):

n=(Z_{\alpha}+Z_{\beta})^2/\mu^2

where \alpha is the Type I error rate, \beta is the Type II error rate (power=1-\beta), Z_{\alpha} is the 100(1-\alpha)-th percentile of standard normal distribution, and

\mu=\log(\delta)\sqrt{ \sum_{s=1}^{m} g_s P_s (1 - P_s) V_s }

and

V_s=P_s\left[1-\frac{1}{\lambda_{1s}} \left\{ \exp\left[-\lambda_{1s}(T-1)\right] -\exp(-\lambda_{1s}T) \right\} \right] +(1-P_s)\left[ 1-\frac{1}{\lambda_{0s}} \left\{ \exp\left[-\lambda_{0s}(T-1)\right] -\exp(-\lambda_{0s}T \right\} \right]

In the above formulas, m is the number of strata, T is the total study length, \delta is the hazard ratio, g_s is the proportion of the total sample size in stratum s, P_s is the proportion of stratum s, which is in treatment group 1, and \lambda_{is} is the hazard for the i-th treatment group in stratum s.

Value

The sample size.

References

Palta M and Amini SB. (1985). Consideration of covariates and stratification in sample size determination for survival time studies. Journal of Chronic Diseases. 38(9):801-809.

See Also

power.stratify

Examples

# example on page 803 of Palta M and Amini SB. (1985). 
n <- ssize.stratify(
  power = 0.9, 
  timeUnit = 1.25, 
  gVec = c(0.5, 0.5),
  PVec = c(0.5, 0.5), 
  HR = 1 / 1.91, 
  lambda0Vec = c(2.303, 1.139),
  alpha = 0.05, 
  verbose = TRUE
)

Sample Size Calculation in the Analysis of Survival Data for Clinical Trials

Description

Sample size calculation for the Comparison of Survival Curves Between Two Groups under the Cox Proportional-Hazards Model for clinical trials. Some parameters will be estimated based on a pilot data set.

Usage

ssizeCT(formula, 
	dat, 
	power, 
	k, 
	RR, 
	alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation method described in Section 14.12 (page 807) of Rosner (2006). The method was proposed by Freedman (1982).

The movitation of this function is that some times we do not have information about m or p_E and p_C available, but we have a pilot data set that can be used to estimate p_E and p_C hence m, where m=n_E p_E + n_C p_C is the expected total number of events over both groups, n_E and n_C are numbers of participants in group E (experimental group) and group C (control group), respectively. p_E is the probability of failure in group E (experimental group) over the maximum time period of the study (t years). p_C is the probability of failure in group C (control group) over the maximum time period of the study (t years).

Suppose we want to compare the survival curves between an experimental group (E) and a control group (C) in a clinical trial with a maximum follow-up of t years. The Cox proportional hazards regression model is assumed to have the form:

h(t|X_1)=h_0(t)\exp(\beta_1 X_1).

Let n_E be the number of participants in the E group and n_C be the number of participants in the C group. We wish to test the hypothesis H0: RR=1 versus H1: RR not equal to 1, where RR=\exp(\beta_1)=underlying hazard ratio for the E group versus the C group. Let RR be the postulated hazard ratio, \alpha be the significance level. Assume that the test is a two-sided test. If the ratio of participants in group E compared to group C = n_E/n_C=k, then the number of participants needed in each group to achieve a power of 1-\beta is

n_E=\frac{m k}{k p_E + p_C}, n_C=\frac{m}{k p_E + p_C}

where

m=\frac{1}{k}\left(\frac{k RR + 1}{RR - 1}\right)^2\left( z_{1-\alpha/2}+z_{1-\beta} \right)^2,

and z_{1-\alpha/2} is the 100 (1-\alpha/2)-th percentile of the standard normal distribution N(0, 1).

p_C and p_E can be calculated from the following formulaes:

p_C=\sum_{i=1}^{t}D_i, p_E=\sum_{i=1}^{t}E_i,

where D_i=\lambda_i A_i C_i, E_i=RR\lambda_i B_i C_i, A_i=\prod_{j=0}^{i-1}(1-\lambda_j), B_i=\prod_{j=0}^{i-1}(1-RR\lambda_j), C_i=\prod_{j=0}^{i-1}(1-\delta_j). And \lambda_i is the probability of failure at time i among participants in the control group, given that a participant has survived to time i-1 and is not censored at time i-1, i.e., the approximate hazard time i in the control group, i=1,...,t; RRlambda_i is the probability of failure at time i among participants in the experimental group, given that a participant has survived to time i-1 and is not censored at time i-1, i.e., the approximate hazard time i in the experimental group, i=1,...,t; delta is the prbability that a participant is censored at time i given that he was followed up to time i and has not failed, i=0, 1, ..., t, which is assumed the same in each group.

Value

Note

(1) The estimates of RRlambda_i=RR*\lambda_i. That is, RRlambda is not directly estimated based on data from the experimental group; (2) The sample size formula assumes that the central-limit theorem is valid and hence is appropriate for large samples. (3) n_E and n_C will be rounded up to integers.

References

Freedman, L.S. (1982). Tables of the number of patients required in clinical trials using the log-rank test. Statistics in Medicine. 1: 121-129

Rosner B. (2006). Fundamentals of Biostatistics. (6-th edition). Thomson Brooks/Cole.

See Also

ssizeCT.default

Examples

  # Example 14.42 in Rosner B. Fundamentals of Biostatistics. 
  # (6-th edition). (2006) page 809

  library(survival)

  data(Oph)
  res <- ssizeCT(formula = Surv(times, status) ~ group, 
		 dat = Oph, 
                 power = 0.8, 
		 k = 1, 
		 RR = 0.7, 
		 alpha = 0.05)

  # Table 14.24 on page 809 of Rosner (2006)
  print(round(res$mat.lambda, 4))

  # Table 14.12 on page 787 of Rosner (2006)
  print(round(res$mat.event, 4))

  # the sample size
  print(res$ssize)

Sample Size Calculation in the Analysis of Survival Data for Clinical Trials

Description

Sample size calculation for the Comparison of Survival Curves Between Two Groups under the Cox Proportional-Hazards Model for clinical trials.

Usage

ssizeCT.default(power, 
		k, 
		pE, 
		pC, 
		RR, 
		alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation method described in Section 14.12 (page 807) of Rosner (2006). The method was proposed by Freedman (1982).

Suppose we want to compare the survival curves between an experimental group (E) and a control group (C) in a clinical trial with a maximum follow-up of t years. The Cox proportional hazards regression model is assumed to have the form:

h(t|X_1)=h_0(t)\exp(\beta_1 X_1).

Let n_E be the number of participants in the E group and n_C be the number of participants in the C group. We wish to test the hypothesis H0: RR=1 versus H1: RR not equal to 1, where RR=\exp(\beta_1)=underlying hazard ratio for the E group versus the C group. Let RR be the postulated hazard ratio, \alpha be the significance level. Assume that the test is a two-sided test. If the ratio of participants in group E compared to group C = n_E/n_C=k, then the number of participants needed in each group to achieve a power of 1-\beta is

n_E=\frac{m k}{k p_E + p_C}, n_C=\frac{m}{k p_E + p_C}

where

m=\frac{1}{k}\left(\frac{k RR + 1}{RR - 1}\right)^2\left( z_{1-\alpha/2}+z_{1-\beta} \right)^2,

and z_{1-\alpha/2} is the 100 (1-\alpha/2)-th percentile of the standard normal distribution N(0, 1).

Value

A two-element vector. The first element is n_E and the second element is n_C.

Note

(1) The sample size formula assumes that the central-limit theorem is valid and hence is appropriate for large samples. (2) n_E and n_C will be rounded up to integers.

References

Freedman, L.S. (1982). Tables of the number of patients required in clinical trials using the log-rank test. Statistics in Medicine. 1: 121-129

Rosner B. (2006). Fundamentals of Biostatistics. (6-th edition). Thomson Brooks/Cole.

See Also

ssizeCT

Examples

  # Example 14.42 in Rosner B. Fundamentals of Biostatistics. 
  # (6-th edition). (2006) page 809
  ssizeCT.default(power = 0.8, 
		  k = 1, 
		  pE = 0.3707, 
		  pC = 0.4890, 
                  RR = 0.7, 
		  alpha = 0.05)

Sample Size Calculation for Cox Proportional Hazards Regression

Description

Sample size calculation for Cox proportional hazards regression with two covariates for Epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates.

Usage

ssizeEpi(X1, 
	 X2, 
	 failureFlag, 
	 power, 
	 theta, 
	 alpha = 0.05)

Arguments

Details

This is an implementation of the sample size formula derived by Latouche et al. (2004) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2),

where the covariate X_1 is of our interest. The covariate X_1 has to be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve a power of 1-\beta is

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{ [\log(\theta)]^2 p (1-p) \psi (1-\rho^2)},

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times\sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1).

p, \rho^2, and \psi will be estimated from a pilot study.

Value

Note

(1) The calculated sample size will be round up to an integer.

(2) The formula can be used to calculate sample size required for a randomized trial study by setting rho2=0.

(3) When rho2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio \exp(\beta_1)=\theta measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio \theta measures the difference of effect on the cause-specific hazard.

References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

See Also

ssizeEpi.default

Examples

  # generate a toy pilot data set
  X1 <- c(rep(1, 39), rep(0, 61))
  set.seed(123456)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  failureFlag <- sample(c(0, 1), 100, prob = c(0.5, 0.5), replace = TRUE)

  ssizeEpi(X1 = X1, 
	   X2 = X2, 
	   failureFlag = failureFlag, 
           power = 0.80, 
	   theta = 2, 
	   alpha = 0.05)

Sample Size Calculation for Cox Proportional Hazards Regression

Description

Sample size calculation for Cox proportional hazards regression with two covariates for Epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates.

Usage

ssizeEpi.default(power, 
		 theta, 
		 p, 
		 psi, 
		 rho2, 
		 alpha = 0.05)

Arguments

Details

This is an implementation of the sample size formula derived by Latouche et al. (2004) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2),

where the covariate X_1 is of our interest. The covariate X_1 has to be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve a power of 1-\beta is

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{ [\log(\theta)]^2 p (1-p) \psi (1-\rho^2)},

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times\sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1).

Value

The required sample size to achieve the specified power with the given type I error rate.

Note

(1) The calculated sample size will be round up to an integer.

(2) The formula can be used to calculate sample size required for a randomized trial study by setting rho2=0.

(3) When rho2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio \exp(\beta_1)=\theta measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio \theta measures the difference of effect on the cause-specific hazard.

References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

See Also

ssizeEpi

Examples

  # Examples at the end of Section 5.2 of Latouche et al. (2004)
  # for a cohort study.
  ssizeEpi.default(power = 0.80, 
		   theta = 2, 
		   p = 0.39, 
		   psi = 0.505,
                   rho2 = 0.132^2, 
		   alpha = 0.05)

Sample Size Calculation for Cox Proportional Hazards Regression with Nonbinary Covariates for Epidemiological Studies

Description

Sample size calculation for Cox proportional hazards regression with nonbinary covariates for Epidemiological Studies.

Usage

ssizeEpiCont(formula, 
	     dat, 
	     var.X1, 
	     var.failureFlag, 
	     power, 
	     theta, 
	     alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation formula derived by Hsieh and Lavori (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, \boldsymbol{x}_2)=h_0(t)\exp(\beta_1 x_1+\boldsymbol{\beta}_2 \boldsymbol{x}_2,

where the covariate X_1 is a nonbinary variable and \boldsymbol{X}_2 is a vector of other covariates.

Suppose we want to check if the hazard ratio of the main effect X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve a sample size of 1-\beta is

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{ [\log(\theta)]^2 \sigma^2 \psi (1-\rho^2) },

where z_{a} is the 100 a-th percentile of the standard normal distribution, \sigma^2=Var(X_1), \psi is the proportion of subjects died of the disease of interest, and \rho is the multiple correlation coefficient of the following linear regression:

x_1=b_0+\boldsymbol{b}^T\boldsymbol{x}_2.

That is, \rho^2=R^2, where R^2 is the proportion of variance explained by the regression of X_1 on the vector of covriates \boldsymbol{X}_2.

rho^2, \sigma^2, and \psi will be estimated from a pilot study.

Value

Note

(1) Hsieh and Lavori (2000) assumed one-sided test, while this implementation assumed two-sided test. (2) The formula can be used to calculate ssize for a randomized trial study by setting rho2=0.

References

Hsieh F.Y. and Lavori P.W. (2000). Sample-size calculation for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials. 21:552-560.

See Also

ssizeEpiCont.default

Examples

  # generate a toy pilot data set
  set.seed(123456)
  X1 <- rnorm(100, mean = 0, sd = 0.3126)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  failureFlag <- sample(c(0, 1), 100, prob = c(0.25, 0.75), replace = TRUE)
  dat <- data.frame(X1 = X1, X2 = X2, failureFlag = failureFlag)

  ssizeEpiCont(formula = X1 ~ X2, 
	       dat = dat, 
	       var.X1 = "X1", 
	       var.failureFlag = "failureFlag", 
               power = 0.806, 
	       theta = exp(1), 
	       alpha = 0.05)

Sample Size Calculation for Cox Proportional Hazards Regression with Nonbinary Covariates for Epidemiological Studies

Description

Sample size calculation for Cox proportional hazards regression with nonbinary covariates for Epidemiological Studies.

Usage

ssizeEpiCont.default(power, 
		     theta, 
		     sigma2, 
		     psi, 
		     rho2, 
		     alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation formula derived by Hsieh and Lavori (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, \boldsymbol{x}_2)=h_0(t)\exp(\beta_1 x_1+\boldsymbol{\beta}_2 \boldsymbol{x}_2,

where the covariate X_1 is a nonbinary variable and \boldsymbol{X}_2 is a vector of other covariates.

Suppose we want to check if the hazard ratio of the main effect X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve a sample size of 1-\beta is

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{ [\log(\theta)]^2 \sigma^2 \psi (1-\rho^2) },

where z_{a} is the 100 a-th percentile of the standard normal distribution, \sigma^2=Var(X_1), \psi is the proportion of subjects died of the disease of interest, and \rho is the multiple correlation coefficient of the following linear regression:

x_1=b_0+\boldsymbol{b}^T\boldsymbol{x}_2.

That is, \rho^2=R^2, where R^2 is the proportion of variance explained by the regression of X_1 on the vector of covriates \boldsymbol{X}_2.

Value

The total number of subjects required.

Note

(1) Hsieh and Lavori (2000) assumed one-sided test, while this implementation assumed two-sided test. (2) The formula can be used to calculate ssize for a randomized trial study by setting rho2=0.

References

Hsieh F.Y. and Lavori P.W. (2000). Sample-size calculation for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials. 21:552-560.

See Also

ssizeEpiCont

Examples

  # example in the EXAMPLE section (page 557) of Hsieh and Lavori (2000).
  # Hsieh and Lavori (2000) assumed one-sided test, 
  # while this implementation assumed two-sided test. 
  # Hence alpha=0.1 here (two-sided test) will correspond
  # to alpha=0.05 of one-sided test in Hsieh and Lavori's (2000) example.
  ssizeEpiCont.default(power = 0.806, 
		       theta = exp(1), 
		       sigma2 = 0.3126^2, 
                       psi = 0.738, 
		       rho2 = 0.1837, 
		       alpha = 0.1)

Sample Size Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

Description

Sample size calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

Usage

ssizeEpiInt(X1, 
	    X2, 
	    failureFlag, 
	    power, 
	    theta, 
	    alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemoilogical studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve the desired power 1-\beta is:

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2 G}{ [\log(\theta)]^2 \psi (1-p) p (1-\rho^2) },

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times\sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1), and

G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1},

and p0=Pr(X_1=1 | X_2=0)=myc/(mya+myc), p1=Pr(X_1=1 | X_2=1)=myd/(myb+myd), p=Pr(X_1=1)=(myc+myd)/n, q=Pr(X_2=1)=(myb+myd)/n, n=mya+myb+myc+myd.

p_{00}=Pr(X_1=0,\mbox{and}, X_2=0), p_{01}=Pr(X_1=0,\mbox{and}, X_2=1), p_{10}=Pr(X_1=1,\mbox{and}, X_2=0), p_{11}=Pr(X_1=1,\mbox{and}, X_2=1).

p_{00}, p_{01}, p_{10}, p_{11}, and \psi will be estimated from the pilot data.

Value

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

ssizeEpiInt.default0, ssizeEpiInt2

Examples

  # generate a toy pilot data set
  X1 <- c(rep(1, 39), rep(0, 61))
  set.seed(123456)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  failureFlag <- sample(c(0, 1), 100, prob = c(0.25, 0.75), replace = TRUE)

  ssizeEpiInt(X1 = X1, 
	      X2 = X2, 
	      failureFlag = failureFlag, 
	      power = 0.88, 
	      theta = 3, 
	      alpha = 0.05)
  

Sample Size Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

Description

Sample size calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

Usage

ssizeEpiInt.default0(power, 
		     theta, 
		     p, 
		     psi, 
		     G, 
		     rho2, 
		     alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve a power of 1-\beta is

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2 G}{ [\log(\theta)]^2 \psi (1-p) p (1-\rho^2) },

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times\sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1), and

G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1}

.

If X_1 and X_2 are uncorrelated, we have p_0=p_1=p leading to 1/[(1-q)q]. For q=0.5, we have G=4.

Value

The total number of subjects required.

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

ssizeEpiInt.default1, ssizeEpiInt2

Examples

  # Example at the end of Section 4 of Schmoor et al. (2000).
  ssizeEpiInt.default0(power = 0.8227, 
		       theta = 3, 
		       p = 0.61, 
		       psi = 139 / 184, 
                       G = 4.79177, 
		       rho2 = 0.015^2, 
		       alpha = 0.05)
  

Sample Size Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

Description

Sample size calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

Usage

ssizeEpiInt.default1(power, 
		     theta, 
		     psi, 
		     p00, 
		     p01, 
		     p10, 
		     p11, 
		     alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemoilogical studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve a power of 1-\beta is

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2\delta}{[\log(\theta)]^2 \psi},

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest,

\delta=\frac{1}{p_{00}}+\frac{1}{p_{01}}+\frac{1}{p_{10}} +\frac{1}{p_{11}},

and p_{00}=Pr(X_1=0,\mbox{and}, X_2=0), p_{01}=Pr(X_1=0,\mbox{and}, X_2=1), p_{10}=Pr(X_1=1,\mbox{and}, X_2=0), p_{11}=Pr(X_1=1,\mbox{and}, X_2=1).

Value

The ssize of the test.

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

ssizeEpiInt.default0, ssizeEpiInt2

Examples

  # Example at the end of Section 4 of Schmoor et al. (2000).
  # p00, p01, p10, and p11 are calculated based on Table III on page 448
  # of Schmoor et al. (2000).
  ssizeEpiInt.default1(power = 0.8227, 
		       theta = 3, 
		       psi = 139 / 184,
                       p00 = 50/184, 
		       p01 = 21 / 184, 
		       p10 = 78 / 184, 
		       p11 = 35 / 184,
                       alpha = 0.05)
  

Sample Size Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

Description

Sample size calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

Usage

ssizeEpiInt2(power, 
	     theta, 
	     psi, 
	     mya, 
	     myb, 
	     myc, 
	     myd, 
	     alpha = 0.05)

Arguments

Details

This is an implementation of the sample size calculation formula derived by Schmoor et al. (2000) for the following Cox proportional hazards regression in the epidemiological studies:

h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),

where both covariates X_1 and X_2 are binary variables.

Suppose we want to check if the hazard ratio of the interaction effect X_1 X_2=1 to X_1 X_2=0 is equal to 1 or is equal to \exp(\gamma)=\theta. Given the type I error rate \alpha for a two-sided test, the total number of subjects required to achieve a power of 1-\beta is

n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2 G}{ [\log(\theta)]^2 \psi (1-p) p (1-\rho^2) },

where z_{a} is the 100 a-th percentile of the standard normal distribution, \psi is the proportion of subjects died of the disease of interest, and

\rho=corr(X_1, X_2)=(p_1-p_0)\times\sqrt{\frac{q(1-q)}{p(1-p)}},

and p=Pr(X_1=1), q=Pr(X_2=1), p_0=Pr(X_1=1|X_2=0), and p_1=Pr(X_1=1 | X_2=1), and

G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1},

and p0=Pr(X_1=1 | X_2=0)=myc/(mya+myc), p1=Pr(X_1=1 | X_2=1)=myd/(myb+myd), p=Pr(X_1=1)=(myc+myd)/n, q=Pr(X_2=1)=(myb+myd)/n, n=mya+myb+myc+myd.

p_{00}=Pr(X_1=0,\mbox{and}, X_2=0), p_{01}=Pr(X_1=0,\mbox{and}, X_2=1), p_{10}=Pr(X_1=1,\mbox{and}, X_2=0), p_{11}=Pr(X_1=1,\mbox{and}, X_2=1).

Value

The total number of subjects required.

References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

See Also

ssizeEpiInt.default0, ssizeEpiInt.default1

Examples

  # Example at the end of Section 4 of Schmoor et al. (2000).
  # mya, myb, myc, and myd are obtained from Table III on page 448
  # of Schmoor et al. (2000).
  ssizeEpiInt2(power = 0.8227, 
	       theta = 3, 
	       psi = 139 / 184,
               mya = 50, 
	       myb = 21, 
	       myc = 78, 
	       myd = 35, 
	       alpha = 0.05)
  

Sample Size Calculation For Two-Sided Two Sample T Test With Unequal Variances And Unequal Sample Sizes

Description

Sample size calculation for two-sided two sample t test with unequal variances and unequal sample sizes

Usage

ssizeWelchT(
  ratioN2toN1, 
  meanDiff, 
  sd1, 
  sd2, 
  power = 0.8, 
  alpha = 0.05, 
  minN1 = 3)

Arguments

Details

The power formula is

power = Pr\left(|T| > t_{1-\alpha/2, \nu} | T \sim t_{\nu, \lambda}\right),

where \lambda is the noncentrality parameter of the t distribution with degree of freedom \nu. t_{1-\alpha/2, \nu} is the upper 100\alpha/2 percentile of the t distribution with degree of freedom \nu. \alpha is the significance level. The noncentrality parameter \lambda is defined as

\lambda = \frac{|\mu_1 - \mu_2|}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}.

The degree \nu of freedom is the Satterthwaite approximation and is defined as

\nu = \frac{\left(\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}\right)^2}{ \frac{\left(\frac{\sigma_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{\sigma_2^2}{n_2}\right)^2}{n_2-1} }

Value

A list with 2 elements

Examples

ssizeWelchT(
    ratioN2toN1=30/64, # ratio of sample size for group 2 to sample size for group 1
    meanDiff = 1, # mean difference between 2 groups
    sd1 = 2, # SD of group 1
    sd2 = 1, # SD of group 2
    power = 0.8918191, # power
    alpha = 0.05, # type I error rate
    minN1 = 3 # minimu possible sample size for group 1
)
# n1 = 64 and n2 = 30