Challenge: Many Ways to Find the Sample Size

Installing and loading the {pwrss} package in R:

install.packages("pwrss")
library(pwrss)

Effect Size Category	Cohen’s d Range
Very Small Effect	d < 0.20
Small Effect	0.20 ≤ d < 0.50
Medium Effect	0.50 ≤ d < 0.80
Large Effect	d ≥ 0.80

In the calculations below, 0.35 will be used for a SMALL effect size (which lies between 0.20 and 0.50), and 0.65 for a MEDIUM effect size (which lies between 0.50 and 0.80). Standardized regression coefficients, R-squared values, and eta-squared values should be obtained from these values using conversion formulas.

Assume balanced allocation, α = 0.05, power = 0.80, and a two‑tailed test.

1 Posttest-Only Control Group Design

Group	Randomization	Pretest Measured	Treatment Applied	Posttest Measured
Experimental	✓	—	✓	✓
Control	✓	—	—	✓

1.1 t Test Approach

Using the t test approach, a total of at least 260 participants are required to detect a SMALL effect, and a total of at least 78 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

power.t.student(d = 0.35,  
                power = 0.80, alpha = 0.05,
                design = "independent",
                alternative = "two.sided")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Student's T-Test (Independent Samples)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : d - null.d  = 0
  H1 (Alternative) : d - null.d != 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (d)      = 0.350 (vs. null.d = 0)
  Sample Size          = 130 and 130  <<
  Type 1 Error (alpha) = 0.050
  Type 2 Error (beta)  = 0.197
  Statistical Power    = 0.803

If the expected or minimum meaningful effect can be characterized as MEDIUM:

power.t.student(d = 0.65,  
                power = 0.80, alpha = 0.05,
                design = "independent",
                alternative = "two.sided")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Student's T-Test (Independent Samples)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : d - null.d  = 0
  H1 (Alternative) : d - null.d != 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (d)      = 0.650 (vs. null.d = 0)
  Sample Size          = 39 and 39  <<
  Type 1 Error (alpha) = 0.050
  Type 2 Error (beta)  = 0.191
  Statistical Power    = 0.809

1.2 Regression Approach

1.2.1 Regression Omnibus F Test

Using the regression omnibus F test, at least 259 participants are required to detect a SMALL effect, and at least 77 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

# convert d to eta-squared 
means.to.etasq(mu = c(0.35, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.02971498

power.f.regression(r.squared = 0.02971498, 
                   power = 0.80, alpha = 0.05,
                   k.total = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Linear Regression (F-Test) - Fixed Predictors

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : R-squared = 0
  H1 (Alternative) : R-squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (R-squared) = 0.030
  Sample Size             = 259  <<
  Type 1 Error (alpha)    = 0.050
  Type 2 Error (beta)     = 0.199
  Statistical Power       = 0.801

If the expected or minimum meaningful effect can be characterized as MEDIUM:

# convert d to eta-squared 
means.to.etasq(mu = c(0.65, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.0955342

power.f.regression(r.squared = 0.0955342, 
                   power = 0.80, alpha = 0.05,
                   k.total = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Linear Regression (F-Test) - Fixed Predictors

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : R-squared = 0
  H1 (Alternative) : R-squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (R-squared) = 0.096
  Sample Size             = 77  <<
  Type 1 Error (alpha)    = 0.050
  Type 2 Error (beta)     = 0.196
  Statistical Power       = 0.804

1.2.2 Regression Coefficient t Test

Using the regression coefficient t test, at least 259 participants are required to detect a SMALL effect, and at least 77 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

p <- 0.50

power.t.regression(beta = 0.35, 
                   sd.predictor = sqrt(p*(1-p)),
                   power = 0.80, alpha = 0.05,
                   k.total = 1, r.squared = 0,
                   alternative = "two.sided")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Linear Regression Coefficient (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : beta - null.beta  = 0
  H1 (Alternative) : beta - null.beta != 0

----------------------------------------------------
Results
----------------------------------------------------
  Target Effect (Std. beta) = 0.175
  Sample Size               = 259  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.199
  Statistical Power         = 0.801

If the expected or minimum meaningful effect can be characterized as MEDIUM:

p <- 0.50

power.t.regression(beta = 0.65, 
                   sd.predictor = sqrt(p*(1-p)),
                   power = 0.80, alpha = 0.05,
                   k.total = 1, r.squared = 0,
                   alternative = "two.sided")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Linear Regression Coefficient (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : beta - null.beta  = 0
  H1 (Alternative) : beta - null.beta != 0

----------------------------------------------------
Results
----------------------------------------------------
  Target Effect (Std. beta) = 0.325
  Sample Size               = 77  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.196
  Statistical Power         = 0.804

1.3 ANCOVA Approach

Using the ANCOVA approach, at least 260 participants are required to detect a SMALL effect, and at least 78 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

# convert d to eta-squared 
means.to.etasq(mu = c(0.35, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.02971498

power.f.ancova(eta.squared = 0.02971498,
               factor.levels = 2,
               power = 0.80, alpha = 0.05,
               k.covariates = 0)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Variance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.030
  Total Sample Size         = 260  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.197
  Statistical Power         = 0.803

# or 

power.f.ancova.keppel(mu.vector = c(0.35, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.030
  Total Sample Size         = 260  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.197
  Statistical Power         = 0.803

# or

power.f.ancova.shieh(mu.vector = c(0.35, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.030
  Total Sample Size         = 260  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.199
  Statistical Power         = 0.801

# or 

power.t.contrast(mu.vector = c(0.35, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               contrast.vector = c(1, -1),
               power = 0.80, alpha = 0.05,
               r.squared = 0,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Single Contrast Analysis (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : psi  = 0
  H1 (Alternative) : psi != 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (d)      = 0.350
  Total Sample Size    = 260  <<
  Type 1 Error (alpha) = 0.050
  Type 2 Error (beta)  = 0.199
  Statistical Power    = 0.801

If the expected or minimum meaningful effect can be characterized as MEDIUM:

# convert d to eta-squared 
means.to.etasq(mu = c(0.65, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.0955342

power.f.ancova(eta.squared = 0.0955342,
               factor.levels = 2,
               power = 0.80, alpha = 0.05,
               k.covariates = 0)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Variance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.096
  Total Sample Size         = 78  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.191
  Statistical Power         = 0.809

# or 

power.f.ancova.keppel(mu.vector = c(0.65, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.096
  Total Sample Size         = 78  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.191
  Statistical Power         = 0.809

# or

power.f.ancova.shieh(mu.vector = c(0.65, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.094
  Total Sample Size         = 78  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.196
  Statistical Power         = 0.804

# or 

power.t.contrast(mu.vector = c(0.65, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               contrast.vector = c(1, -1),
               power = 0.80, alpha = 0.05,
               r.squared = 0,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Single Contrast Analysis (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : psi  = 0
  H1 (Alternative) : psi != 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (d)      = 0.650
  Total Sample Size    = 78  <<
  Type 1 Error (alpha) = 0.050
  Type 2 Error (beta)  = 0.196
  Statistical Power    = 0.804

1.4 Repeated-Measures ANOVA Approach

Using the repeated-measures ANOVA approach, at least 260 participants are required to detect a SMALL effect, and at least 78 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

# convert d to eta-squared 
means.to.etasq(mu = c(0.35, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.02971498

power.f.mixed.anova(eta.squared = 0.02971498,
                    factor.levels = c(2, 1), # c(between, within)
                    power = 0.80, alpha = 0.05,
                    rho.within = NA, 
                    effect = "between")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Analysis of Variance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.030
  Total Sample Size         = 260  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.197
  Statistical Power         = 0.803

If the expected or minimum meaningful effect can be characterized as MEDIUM:

# convert d to eta-squared 
means.to.etasq(mu = c(0.65, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.0955342

power.f.mixed.anova(eta.squared = 0.0955342,
                    factor.levels = c(2, 1), # c(between, within)
                    power = 0.80, alpha = 0.05,
                    rho.within = NA, 
                    effect = "between")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Analysis of Variance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.096
  Total Sample Size         = 78  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.191
  Statistical Power         = 0.809

Bonus question: Why do regression approaches differ by a unit?

2 Pretest–Posttest Control Group Design

Group	Randomization	Pretest Measured	Treatment Applied	Posttest Measured
Experimental	✓	✓	✓	✓
Control	✓	✓	—	✓

Treatment and pretest explains 50% of the variance in the posttest.

2.1 Regression Approach

2.1.1 Regression Omnibus F Test

Using the regression omnibus F test, at least 131 participants are required to detect a SMALL effect, and at least 40 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

# convert d to eta-squared 
means.to.etasq(mu = c(0.35, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0.50,
               k.covariates = 2,
               verbose = FALSE)$eta.squared

[1] 0.05771496

power.f.regression(r.squared = 0.05771496, 
                   power = 0.80, alpha = 0.05,
                   k.total = 2, k.tested = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Hierarchical Linear Regression (F-Test) - Fixed Pred

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : Change in R-squared = 0
  H1 (Alternative) : Change in R-squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (R-squared) = 0.058
  Sample Size             = 131  <<
  Type 1 Error (alpha)    = 0.050
  Type 2 Error (beta)     = 0.197
  Statistical Power       = 0.803

If the expected or minimum meaningful effect can be characterized as MEDIUM:

# convert d to eta-squared 
means.to.etasq(mu = c(0.65, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0.50,
               k.covariates = 2,
               verbose = FALSE)$eta.squared

[1] 0.1744066

power.f.regression(r.squared = 0.1744066, 
                   power = 0.80, alpha = 0.05,
                   k.total = 2, k.tested = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Hierarchical Linear Regression (F-Test) - Fixed Pred

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : Change in R-squared = 0
  H1 (Alternative) : Change in R-squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (R-squared) = 0.174
  Sample Size             = 40  <<
  Type 1 Error (alpha)    = 0.050
  Type 2 Error (beta)     = 0.192
  Statistical Power       = 0.808

2.1.2 Regression Coefficient t Test

Using the regression coefficient t test, at least 131 participants are required to detect a SMALL effect, and at least 40 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

p <- 0.50

power.t.regression(beta = 0.35, 
                   sd.outcome = 1,
                   sd.predictor = sqrt(p*(1-p)),
                   power = 0.80, alpha = 0.05,
                   k.total = 2, r.squared = 0.50,
                   alternative = "two.sided")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Linear Regression Coefficient (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : beta - null.beta  = 0
  H1 (Alternative) : beta - null.beta != 0

----------------------------------------------------
Results
----------------------------------------------------
  Target Effect (Std. beta) = 0.175
  Sample Size               = 131  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.197
  Statistical Power         = 0.803

If the expected or minimum meaningful effect can be characterized as MEDIUM:

p <- 0.50

power.t.regression(beta = 0.65,
                   sd.outcome = 1,
                   sd.predictor = sqrt(p*(1-p)),
                   power = 0.80, alpha = 0.05,
                   k.total = 2, r.squared = 0.50,
                   alternative = "two.sided")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Linear Regression Coefficient (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : beta - null.beta  = 0
  H1 (Alternative) : beta - null.beta != 0

----------------------------------------------------
Results
----------------------------------------------------
  Target Effect (Std. beta) = 0.325
  Sample Size               = 40  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.192
  Statistical Power         = 0.808

2.2 ANCOVA Approach

Using the ANCOVA approach, at least 132 participants are required to detect a SMALL effect, and at least 40 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

# convert d to eta-squared 
means.to.etasq(mu = c(0.35, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0.50,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.05771496

power.f.ancova(eta.squared = 0.05771496,
               factor.levels = 2,
               power = 0.80, alpha = 0.05,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.058
  Total Sample Size         = 132  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.194
  Statistical Power         = 0.806

# or 

power.f.ancova.keppel(mu.vector = c(0.35, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0.50,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.058
  Total Sample Size         = 132  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.194
  Statistical Power         = 0.806

# or

power.f.ancova.shieh(mu.vector = c(0.35, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0.50,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.057
  Total Sample Size         = 132  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.197
  Statistical Power         = 0.803

# or

power.t.contrast(mu.vector = c(0.35, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               contrast.vector = c(1, -1),
               power = 0.80, alpha = 0.05,
               r.squared = 0.50,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Single Contrast Analysis (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : psi  = 0
  H1 (Alternative) : psi != 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (d)      = 0.495
  Total Sample Size    = 132  <<
  Type 1 Error (alpha) = 0.050
  Type 2 Error (beta)  = 0.197
  Statistical Power    = 0.803

If the expected or minimum meaningful effect can be characterized as MEDIUM:

# convert d to eta-squared 
means.to.etasq(mu = c(0.65, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0.50,
               k.covariates = 1,
               verbose = FALSE)$eta.squared

[1] 0.1744066

power.f.ancova(eta.squared = 0.1744066,
               factor.levels = 2,
               power = 0.80, alpha = 0.05,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.174
  Total Sample Size         = 40  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.192
  Statistical Power         = 0.808

# or 

power.f.ancova.keppel(mu.vector = c(0.65, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0.50,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.174
  Total Sample Size         = 40  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.192
  Statistical Power         = 0.808

# or

power.f.ancova.shieh(mu.vector = c(0.65, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               power = 0.80, alpha = 0.05,
               r.squared = 0.50,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

One-Way Analysis of Covariance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.171
  Total Sample Size         = 42  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.182
  Statistical Power         = 0.818

# or 

power.t.contrast(mu.vector = c(0.65, 0), 
               sd.vector = c(1, 1), 
               p.vector = c(0.50, 0.50),
               contrast.vector = c(1, -1),
               power = 0.80, alpha = 0.05,
               r.squared = 0.50,
               k.covariates = 1)

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Single Contrast Analysis (T-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : psi  = 0
  H1 (Alternative) : psi != 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (d)      = 0.919
  Total Sample Size    = 42  <<
  Type 1 Error (alpha) = 0.050
  Type 2 Error (beta)  = 0.182
  Statistical Power    = 0.818

2.3 Repeated-Measures ANOVA Approach

Using the repeated-measures ANOVA approach, at least 132 participants are required to detect a SMALL effect, and at least 40 participants are required to detect a MEDIUM effect.

Find out how:

If the expected or minimum meaningful effect can be characterized as SMALL:

# convert d to eta-squared 
means.to.etasq(mu = c(0.35, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0.50,
               k.covariates = 2,
               verbose = FALSE)$eta.squared

[1] 0.05771496

power.f.mixed.anova(eta.squared = 0.05771496,
                    factor.levels = c(2, 2), # c(between, within)
                    power = 0.80, alpha = 0.05,
                    rho.within = NA, 
                    effect = "between")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Mixed-Effects Analysis of Variance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.058
  Total Sample Size         = 132  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.194
  Statistical Power         = 0.806

If the expected or minimum meaningful effect can be characterized as MEDIUM:

# convert d to eta-squared 
means.to.etasq(mu = c(0.65, 0), 
               sd = c(1, 1), 
               n = c(1e10, 1e10),
               r.squared = 0.50,
               k.covariates = 2,
               verbose = FALSE)$eta.squared

[1] 0.1744066

power.f.mixed.anova(eta.squared = 0.1744066,
                    factor.levels = c(2, 2), # c(between, within)
                    power = 0.80, alpha = 0.05,
                    rho.within = NA, 
                    effect = "between")

+--------------------------------------------------+
|             SAMPLE SIZE CALCULATION              |
+--------------------------------------------------+

Mixed-Effects Analysis of Variance (F-Test)

----------------------------------------------------
Hypotheses
----------------------------------------------------
  H0 (Null)        : eta.squared = 0
  H1 (Alternative) : eta.squared > 0

----------------------------------------------------
Results
----------------------------------------------------
  Effect Size (eta-squared) = 0.174
  Total Sample Size         = 40  <<
  Type 1 Error (alpha)      = 0.050
  Type 2 Error (beta)       = 0.191
  Statistical Power         = 0.809

References

Bulus, M., & Jentschke, S. (2026). pwrss: Statistical Power, Sample Size, and Detectable Effect Calculations. R package version 1.2.0. https://doi.org/10.32614/CRAN.package.pwrss

Bulus, M., & Polat, C. (2023). pwrss R paketi ile istatistiksel guc analizi [Statistical power analysis with pwrss R package]. Ahi Evran Universitesi Kirsehir Egitim Fakultesi Dergisi, 24(3), 2207-2328. https://doi.org/10.29299/kefad.1209913

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.

Keppel, G., & Wickens, T. D. (2004). Design and analysis: A researcher’s handbook (4th ed.). Pearson.

Shieh, G. (2020). Power analysis and sample size planning in ANCOVA designs. Psychometrika, 85(1), 101-120. https://doi.org/10.1007/s11336-019-09692-3

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