4-Linear-Regression-Beta: Linear Regression: Single Coefficient (t or z Test)

In pwrss: Statistical Power and Sample Size Calculation Tools

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)

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

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

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

# 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)

pwrss documentation built on April 12, 2023, 12:34 p.m.