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Power Analysis for Linear Regression: Single Coefficient (T-Test)

regression.linear.t.Rd

Calculates power or 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 calculate power or sample size for a binary predictor (such as treatment and control groups in an experimental design) by specifying sd.predictor = 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). power.t.regression()pwrss.t.regression() are the same functions, as well as power.t.reg() and pwrss.t.reg().

Minimal effect and equivalence tests are implemented in line with Hodges and Lehmann (1954), Kim and Robinson (2019), Phillips (1990), and Dupont and Plummer (1998).

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

NOTE: The pwrss.t.regression() function and its alias pwrss.z.reg() are deprecated, but they will remain available as a wrapper for power.t.regression() during the transition period.

Usage

power.t.regression(beta, null.beta = 0, margin = 0,
                   sd.predictor = 1, sd.outcome = 1,
                   r.squared = (beta * sd.predictor / sd.outcome)^2,
                   k.total = 1, n = NULL, power = NULL, alpha = 0.05,
                   alternative = c("two.sided", "one.sided", "two.one.sided"),
                   ceiling = TRUE, verbose = TRUE, pretty = FALSE)

Arguments

beta

regression coefficient. One can use standardized regression coefficient, but should keep sd.predictor = 1 and sd.outcome = 1 or leave them out as they are default specifications.

null.beta

regression coefficient under null hypothesis (typically zero). One can use standardized regression coefficient, but should keep sd.predictor = 1 and sd.outcome = 1 or leave them out as they are default specifications.

margin

margin - ignorable beta - null.beta difference.

sd.predictor

standard deviation of the predictor. For a binary predictor, sd.predictor = sqrt(p * (1 - p)) where p is the proportion of subjects in one of the groups.

sd.outcome

standard deviation of the outcome.

k.total

integer; total number of predictors, including the predictor of interest.

r.squared

model R-squared. The default is r.squared = (beta * sd.predictor / sd.outcome)^2 assuming a linear regression with one predictor. Thus, an r.squared below this value will throw a warning. To consider other covariates in the model provide a value greater than the default r.squared along with the argument k.total > 1.

n

integer; sample size.

power

statistical power, defined as the probability of correctly rejecting a false null hypothesis, denoted as \(1 - \beta\).

alpha

type 1 error rate, defined as the probability of incorrectly rejecting a true null hypothesis, denoted as \(\alpha\).

alternative

character; the direction or type of the hypothesis test: "two.sided", "one.sided", or "two.one.sided".

ceiling

logical; whether sample size should be rounded up. TRUE by default.

verbose

logical; whether the output should be printed on the console. TRUE by default.

pretty

logical; whether the output should show Unicode characters (if encoding allows for it). FALSE by default.

Value

parms

list of parameters used in calculation.

test

type of the statistical test (T-Test).

df

degrees of freedom.

ncp

non-centrality parameter for the alternative.

null.ncp

non-centrality parameter for the null.

t.alpha

critical value(s).

power

statistical power \((1-\beta)\).

n

sample size.

References

Bulus, M. (2021). Sample size determination and optimal design of randomized/non-equivalent pretest-post-test control-group designs. Adiyaman University Journal of Educational Sciences, 11(1), 48-69. doi:10.17984/adyuebd.941434

Hodges Jr, J. L., & Lehmann, E. L. (1954). Testing the approximate validity of statistical hypotheses. Journal of the Royal Statistical Society Series B: Statistical Methodology, 16(2), 261-268. doi:10.1111/j.2517-6161.1954.tb00169.x

Kim, J. H., & Robinson, A. P. (2019). Interval-based hypothesis testing and its applications to economics and finance. Econometrics, 7(2), 21. doi:10.3390/econometrics7020021

Phillips, K. F. (1990). Power of the two one-sided tests procedure in bioequivalence. Journal of Pharmacokinetics and Biopharmaceutics, 18(2), 137-144. doi:10.1007/bf01063556

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. doi:10.1016/s0197-2456(98)00037-3

Examples

# continuous predictor x (and 4 covariates)
power.t.regression(beta = 0.20,
            k.total = 5,
            r.squared = 0.30,
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Linear Regression Coefficient (T-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : beta - null.beta = 0 
#>   H1 (Alt. Claim) : beta - null.beta != 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 140  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error           = 0.198
#>   Statistical Power      = 0.802
#> 

# binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
power.t.regression(beta = 0.20,
            sd.predictor = sqrt(p*(1-p)),
            k.total = 5,
            r.squared = 0.30,
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Linear Regression Coefficient (T-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : beta - null.beta = 0 
#>   H1 (Alt. Claim) : beta - null.beta != 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 552  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error           = 0.200
#>   Statistical Power      = 0.8
#> 

# non-inferiority test with binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
power.t.regression(beta = 0.20, # Cohen's d
            margin = -0.05, # non-inferiority margin in Cohen's d unit
            alternative = "one.sided",
            sd.predictor = sqrt(p*(1-p)),
            k.total = 5,
            r.squared = 0.30,
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Linear Regression Coefficient (T-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : beta - null.beta <= margin 
#>   H1 (Alt. Claim) : beta - null.beta >  margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 278  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error           = 0.200
#>   Statistical Power      = 0.8
#> 

# superiority test with binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
power.t.regression(beta = 0.20, # Cohen's d
            margin = 0.05, # superiority margin in Cohen's d unit
            alternative = "one.sided",
            sd.predictor = sqrt(p*(1-p)),
            k.total = 5,
            r.squared = 0.30,
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Linear Regression Coefficient (T-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : beta - null.beta <= margin 
#>   H1 (Alt. Claim) : beta - null.beta >  margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 773  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error           = 0.200
#>   Statistical Power      = 0.8
#> 

# equivalence test with binary predictor x (and 4 covariates)
p <- 0.50 # proportion of subjects in one group
power.t.regression(beta = 0, # Cohen's d
            margin = c(-0.05, 0.05), # equivalence bounds in Cohen's d unit
            alternative = "two.one.sided",
            sd.predictor = sqrt(p*(1 - p)),
            k.total = 5,
            r.squared = 0.30,
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Linear Regression Coefficient (T-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : beta - null.beta <= min(margin) or 
#>                     beta - null.beta >= max(margin) 
#>   H1 (Alt. Claim) : beta - null.beta > min(margin) and 
#>                     beta - null.beta < max(margin)
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 9593  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error           = 0.200
#>   Statistical Power      = 0.8
#>