pwrss.t.reg: Linear Regression: Single Coefficient (t or z Test)

Description

Calculates statistical power or minimum required sample size (only one can be NULL at a time) to test a single coefficient in multiple linear regression. The predictor is assumed to be continuous by default. However, one can find statistical power or minimum required sample size for a binary predictor (such as treatment and control groups in an experimental design) by specifying sdx = sqrt(p*(1-p)) where p is the proportion of subjects in one of the groups. The sample size in each group would be n*p and n*(1-p). pwrss.t.regression() and pwrss.t.reg() are the same functions.

When HIGHER values of an outcome is a good thing, beta1 is expected to be greater than beta0 + margin for non-inferiority and superiority tests. In this case, margin is NEGATIVE for the non-inferiority test but it is POSITIVE for the superiority test.

When LOWER values of an outcome is a good thing, beta1 is expected to be less than beta0 + margin for non-inferiority and superiority tests. In this case, margin is POSITIVE for the non-inferiority test but it is NEGATIVE for the superiority test.

For equivalence tests the value of beta0 shifts both to the left and right as beta0 - margin and beta0 + margin. For equivalence tests margin is stated as the absolute value and beta1 is expected to fall between beta0 - margin and beta0 + margin.

Formulas are validated using Monte Carlo simulation, G*Power, tables in PASS documentation, and tables in Bulus (2021).

Usage

pwrss.t.reg(beta1 = 0.25, beta0 = 0, margin = 0,
            sdx = 1, sdy = 1,
            k = 1, r2 = (beta1 * sdx / sdy)^2,
            alpha = 0.05, n = NULL, power = NULL,
            alternative = c("not equal", "less", "greater",
                            "non-inferior", "superior", "equivalent"),
            verbose = TRUE)
pwrss.z.reg(beta1 = 0.25, beta0 = 0, margin = 0,
            sdx = 1, sdy = 1,
            k = 1, r2 = (beta1 * sdx / sdy)^2,
            alpha = 0.05, n = NULL, power = NULL,
            alternative = c("not equal", "less", "greater",
                            "non-inferior", "superior", "equivalent"),
            verbose = TRUE)

Value

parms: list of parameters used in calculation
test: type of the statistical test (z or t test)
df: numerator degrees of freedom
ncp: non-centrality parameter
power: statistical power \((1-\beta)\)
n: total sample size

Arguments

beta1: expected regression coefficient. One can use standardized regression coefficient, but should keep sdx = 1 and sdy = 1 or leave them out as they are default specifications
beta0: regression coefficient under null hypothesis (usually zero). Not to be confused with the intercept. One can use standardized regression coefficient, but should keep sdx = 1 and sdy = 1 or leave them out as they are default specifications
margin: non-inferiority, superiority, or equivalence margin (margin: boundry of beta1 - beta0 that is practically insignificant)
sdx: expected standard deviation of the predictor. For a binary predictor, sdx = sqrt(p*(1-p)) wherep is the proportion of subjects in one of the groups
sdy: expected standard deviation of the outcome
k: (total) number of predictors
r2: expected model R-squared. The default is r2 = (beta * sdx / sdy)^2 assuming a linear regression with one predictor. Thus, an r2 below this value will throw a warning. To consider other covariates in the model provide a value greater than the default r2 along with the argument k>1.
n: total sample size
power: statistical power \((1-\beta)\)
alpha: probability of type I error
alternative: direction or type of the hypothesis test: "not equal", "greater", "less", "non-inferior", "superior", or "equivalent"
verbose: if FALSE no output is printed on the console

References

Bulus, M. (2021). Sample size determination and optimal design of randomized/non-equivalent pretest-posttest control-group designs. Adiyaman Univesity Journal of Educational Sciences, 11(1), 48-69.

Phillips, K. F. (1990). Power of the two one-Sided tests procedure in bioequivalence. Journal of Pharmacokinetics and Biopharmaceutics, 18(2), 137-144.

Dupont, W. D., and Plummer, W. D. (1998). Power and sample size calculations for studies involving linear regression. Controlled Clinical Trials, 19(6), 589-601.

Examples

Run this code

# continuous predictor x (and 4 covariates)
pwrss.t.reg(beta1 = 0.20, alpha = 0.05,
            alternative = "not equal",
            k = 5, r2 = 0.30,
            power = 0.80)

pwrss.t.reg(beta1 = 0.20, alpha = 0.05,
            alternative = "not equal",
            k = 5, r2 = 0.30,
            n = 138)

# binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
pwrss.t.reg(beta1 = 0.20, alpha = 0.05,
            alternative = "not equal",
            sdx = sqrt(p*(1-p)),
            k = 5, r2 = 0.30,
            power = 0.80)

pwrss.t.reg(beta1 = 0.20, alpha = 0.05,
            alternative = "not equal",
            sdx = sqrt(p*(1-p)) ,
            k = 5, r2 = 0.30,
            n = 550)

# non-inferiority test with binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
pwrss.t.reg(beta1 = 0.20, beta0 = 0.10, margin = -0.05,
            alpha = 0.05, alternative = "non-inferior",
            sdx = sqrt(p*(1-p)),
            k = 5, r2 = 0.30,
            power = 0.80)

pwrss.t.reg(beta1 = 0.20, beta0 = 0.10, margin = -0.05,
            alpha = 0.05, alternative = "non-inferior",
            sdx = sqrt(p*(1-p)),
            k = 5, r2 = 0.30,
            n = 770)

# superiority test with binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
pwrss.t.reg(beta1 = 0.20, beta0 = 0.10, margin = 0.01,
            alpha = 0.05, alternative = "superior",
            sdx = sqrt(p*(1-p)),
            k = 5, r2 = 0.30,
            power = 0.80)

pwrss.t.reg(beta1 = 0.20, beta0 = 0.10, margin = 0.01,
            alpha = 0.05, alternative = "superior",
            sdx = sqrt(p*(1-p)),
            k = 5, r2 = 0.30,
            n = 2138)

# equivalence test with binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
pwrss.t.reg(beta1 = 0.20, beta0 = 0.20, margin = 0.05,
            alpha = 0.05, alternative = "equivalent",
            sdx = sqrt(p*(1-p)),
            k = 5, r2 = 0.30,
            power = 0.80)

pwrss.t.reg(beta1 = 0.20, beta0 = 0.20, margin = 0.05,
            alpha = 0.05, alternative = "equivalent",
            sdx = sqrt(p*(1-p)),
            k = 5, r2 = 0.30,
            n = 9592)

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