Quick Template: Moderation with Observed Variables: Two Moderators
2025-08-31
Source:vignettes/articles/template_n_from_power_moderation_obs_two_ws.Rmd
template_n_from_power_moderation_obs_two_ws.RmdIntroduction
This and other “Quick Template” articles are examples of R code to determine the range of sample sizes for a target level of power in typical models using power4mome. Users can quickly adapt them for their scenarios. A summary of the code examples can be found in the section Code Template at the end of this document.
Prerequisite
Basic knowledge about fitting models by lavaan and
power4mome is required.
This file is not intended to be an introduction on how to use
functions in power4mome. For details on how to use
power4test(), refer to the Get-Started
article. Please also refer to the help page of
n_region_from_power(), and the article
on n_from_power(), which is called twice by
n_region_from_power() to find the regions described
below.
Scope
This file is for moderation models with two or more moderators, and only two-way interaction effects are involved.
Functions Used in This Template
-
- Set up the model and the population values, generate the data, and generate the Monte Carlo simulated estimates for Monte Carlo confidence interval.
-
- Find the regions of sample sizes based on the target power.
-
- Test selected parameters. Used by
power4test()to test a selected product term (interaction term).
- Test selected parameters. Used by
Common Flow
The following chart summarizes the steps covered below.
Common Workflow
In practice, steps can be repeated, and population values changed, until the desired goal is achieved (e.g., the region of sample sizes with power close to the target power is found).
Set Up The Model and Test
Load the package first:
Estimate the power for a sample size.
The case of two moderators is illustrated but the code can be easily extended to any number of moderators.
The code for the model:
# ====== Model: Form ======
# Make sure all 1st order terms (w1, w2, w3, etc.) are included
model <-
"
y ~ x + w1 + w2 + x:w1 + x:w2
"
# ====== Model: Population Values ======
# For a regression coefficient
# l: large (.50 by default)
# m: medium (.30 by default)
# s: small (.10 by default)
# n: nil (.00 by default)
# For the product term:
# l: large (.15 by default)
# m: medium (.10 by default)
# s: small (.05 by default)
# -l, -m, and -s denote negative values
# Omitted paths are zero by default
# Can also set to a number directly
# Set each path to the hypothesized magnitude
# For a path moderated, the coefficient
# of a predictor is its standardized
# effect when the moderator equal to
# its mean.
model_es <-
"
y ~ x: s
y ~ w1: s
y ~ w2: s
y ~ x:w1: l
y ~ x:w2: m
"
The Model
# ====== Test the Model Specification ======
# Fit the model by regression using lm()
# Add: fit_model_args = list(fit_function = "lm")
out <- power4test(nrep = 2,
model = model,
pop_es = model_es,
n = 50000,
fit_model_args = list(fit_function = "lm"),
iseed = 1234)
# ====== Check the Data Generated ======
print(out,
data_long = TRUE)
# ====== Estimate the Power ======
# For n = 100.
# Find the power with *both* x:w1 and x:w2 significant.
# in the regression results of lm().
# The test by CI is equivalent to the two-tailed t-test.
# Add omnibus = "all_sig" to find this power.
out <- power4test(nrep = 400,
model = model,
pop_es = model_es,
n = 100,
fit_model_args = list(fit_function = "lm"),
test_fun = test_parameters,
test_args = list(pars = c("y~x:w1",
"y~x:w2"),
omnibus = "all_sig"),
iseed = 1234,
parallel = TRUE)
# ====== Compute the Rejection Rate ======
rejection_rates(out)The results:
print(out,
data_long = TRUE)
#>
#> ====================== Model Information ======================
#>
#> == Model on Factors/Variables ==
#>
#> y ~ x + w1 + w2 + x:w1 + x:w2
#>
#> == Model on Variables/Indicators ==
#>
#> y ~ x + w1 + w2 + x:w1 + x:w2
#>
#> ====== Population Values ======
#>
#> Regressions:
#> Population
#> y ~
#> x 0.100
#> w1 0.100
#> w2 0.100
#> x:w1 0.150
#> x:w2 0.100
#>
#> Covariances:
#> Population
#> x ~~
#> w1 0.000
#> w2 0.000
#> x:w1 0.000
#> x:w2 0.000
#> w1 ~~
#> w2 0.000
#> x:w1 0.000
#> x:w2 0.000
#> w2 ~~
#> x:w1 0.000
#> x:w2 0.000
#> x:w1 ~~
#> x:w2 0.000
#>
#> Variances:
#> Population
#> .y 0.937
#> x 1.000
#> w1 1.000
#> w2 1.000
#> x:w1 1.000
#> x:w2 1.000
#>
#> (Computing conditional effects for 3 paths ...)
#>
#> == Population Conditional/Indirect Effect(s) ==
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1, w2
#> Moderator(s) represented by: w1, w2
#>
#> [w1] [w2] (w1) (w2) ind
#> 1 M+1.0SD M+1.0SD 1 1 0.350
#> 2 M+1.0SD M-1.0SD 1 -1 0.150
#> 3 M-1.0SD M+1.0SD -1 1 0.050
#> 4 M-1.0SD M-1.0SD -1 -1 -0.150
#>
#> - The 'ind' column shows the conditional effects.
#>
#>
#> == Conditional effects ==
#>
#> Path: w1 -> y
#> Conditional on moderator(s): x
#> Moderator(s) represented by: x
#>
#> [x] (x) ind
#> 1 M+1.0SD 1 0.250
#> 2 Mean 0 0.100
#> 3 M-1.0SD -1 -0.050
#>
#> - The 'ind' column shows the conditional effects.
#>
#>
#> == Conditional effects ==
#>
#> Path: w2 -> y
#> Conditional on moderator(s): x
#> Moderator(s) represented by: x
#>
#> [x] (x) ind
#> 1 M+1.0SD 1 0.200
#> 2 Mean 0 0.100
#> 3 M-1.0SD -1 0.000
#>
#> - The 'ind' column shows the conditional effects.
#>
#>
#> ======================= Data Information =======================
#>
#> Number of Replications: 400
#> Sample Sizes: 100
#>
#> ==== Descriptive Statistics ====
#>
#> vars n mean sd skew kurtosis se
#> y 1 40000 0.00 1 0.03 0.04 0.01
#> x 2 40000 -0.01 1 0.00 0.02 0.01
#> w1 3 40000 0.01 1 0.01 -0.03 0.01
#> w2 4 40000 0.00 1 0.00 0.01 0.01
#> x:w1 5 40000 0.00 1 0.04 5.73 0.01
#> x:w2 6 40000 0.00 1 0.02 6.02 0.01
#>
#> ==== Parameter Estimates Based on All 400 Samples Combined ====
#>
#> Total Sample Size: 40000
#>
#> ==== Standardized Estimates ====
#>
#> Variances and error variances omitted.
#>
#> Regressions:
#> est.std
#> y ~
#> x 0.094
#> w1 0.095
#> w2 0.106
#> x:w1 0.149
#> x:w2 0.099
#>
#> Covariances:
#> est.std
#> x ~~
#> w1 -0.001
#> w2 0.002
#> x:w1 0.022
#> x:w2 0.007
#> w1 ~~
#> w2 0.004
#> x:w1 -0.000
#> x:w2 0.004
#> w2 ~~
#> x:w1 0.004
#> x:w2 -0.009
#> x:w1 ~~
#> x:w2 0.002
#>
#>
#> ==================== Extra Element(s) Found ====================
#>
#> - fit
#>
#> === Element(s) of the First Dataset ===
#>
#> ============ <fit> ============
#>
#>
#> The models:
#> y ~ x + w1 + w2 + x:w1 + x:w2
#> <environment: 0x000001e71ca5bbd8>
#>
#>
#> ====================== Test(s) Conducted ======================
#>
#> - test_parameters: CIs (pars: y~x:w1,y~x:w2)
#>
#> Call print() and set 'test_long = TRUE' for a detailed report.
rejection_rates(out)
#> [test]: test_parameters: CIs (pars: y~x:w1,y~x:w2)
#> [test_label]: All sig
#> est p.v reject r.cilo r.cihi
#> 1 NaN 1.000 0.062 0.039 0.086
#> Notes:
#> - p.v: The proportion of valid replications.
#> - est: The mean of the estimates in a test across replications.
#> - reject: The proportion of 'significant' replications, that is, the
#> rejection rate. If the null hypothesis is true, this is the Type I
#> error rate. If the null hypothesis is false, this is the power.
#> - r.cilo,r.cihi: The confidence interval of the rejection rate, based
#> on normal approximation.
#> - Refer to the tests for the meanings of other columns.Find the Regions of N Based on the Target Power
Search, by simulation, the following two regions of sample sizes:
Sample sizes with estimated levels of power significantly below the target level (e.g., .80), tested by the confidence interval (95% by default).
Sample sizes with estimated levels of power significantly above the target level (e.g., .80), tested by the confidence interval (95% by default).
In practice, we rarely need high precision for these regions for sample size planning. Therefore, we only need to find the two sample sizes with the corresponding confidence bounds close enough to the target power, defined by a tolerance value. In the function below, this value is .02 by default.
It can take some time to run if the estimated power of the sample size is too different from the target power.
We can find the two regions by
n_region_from_power().
The code:
#
# ===== Reuse the output of power4test() =====
#
# Call n_region_from_power()
# - Set target power: target_power = .80 (Default, can be omitted)
# - Set the seed for the simulation: Integer. Should always be set.
# To set desired precision:
# - Set final number of R: final_R = 1000 (Default, can be omitted)
# - Set final number of replications: final_nrep = 400 (Default, can be omitted)
n_power_region <- n_region_from_power(out,
seed = 1357)
# ===== Basic Results =====
n_power_region
# ===== Plot the (Crude) Power Curve and the Regions =====
plot(n_power_region)The results:
# ===== Basic Results =====
n_power_region
#> Call:
#> n_region_from_power(object = out, seed = 1357)
#>
#> Setting
#> Predictor(x) Sample Size
#> Goal: Power significantly below or above the target
#> algorithm: bisection
#> Level of confidence: 95.00%
#> Target Power: 0.800
#>
#> Solution:
#>
#> Approximate region of sample sizes with power:
#> - not significantly different from 0.800: 709 to 838
#> - significantly lower than 0.800: 709
#> - significantly higher than 0.800: 838
#>
#> Confidence intervals of the estimated power:
#> - for the lower bound (709): [0.721, 0.804]
#> - for the upper bound (838): [0.796, 0.869]
#>
#> Call `summary()` for detailed results.
# ===== Plot the (Crude) Power Curve and the Regions =====
plot(n_power_region)
Power Curve
As shown above, approximately:
sample sizes lower than 709 have power significantly lower than .80, and
sample sizes higher than 838 have power significantly higher than .80.
In other words, sample sizes between 709 and 838 have power not significantly different from .80.
If necessary, detailed results can be printed by
summary():
# ===== Detailed Results =====
summary(n_power_region)
#>
#> ======<< Summary for the Lower Region >>======
#>
#>
#> ====== x_from_power Results ======
#>
#> Call:
#> power4mome::x_from_power(object = out, x = "n", what = "ub",
#> goal = "close_enough", seed = 1357)
#>
#> Predictor (x): Sample Size
#>
#> - Target Power: 0.800
#> - Goal: Find 'x' with estimated upper confidence bound close enough to
#> the target power.
#>
#> === Major Results ===
#>
#> - Final Value (Sample Size): 709
#>
#> - Final Estimated Power: 0.762
#> - Confidence Interval: [0.721; 0.804]
#> - Level of confidence: 95.0%
#> - Based on 400 replications.
#>
#> === Technical Information ===
#>
#> - Algorithm: bisection
#> - Tolerance for 'close enough': Within 0.02000 of 0.800
#> - The range of values explored: 100 to 985
#> - Time spent in the search: 58.95 secs
#> - The final crude model for the power-predictor relation:
#>
#> Model Type: Logistic Regression
#>
#> Call:
#> power_curve(object = by_x_1, formula = power_model, start = power_curve_start,
#> lower_bound = lower_bound, upper_bound = upper_bound, nls_args = nls_args,
#> nls_control = nls_control, verbose = progress)
#>
#> Predictor: n (Sample Size)
#>
#> Model:
#>
#> Call: stats::glm(formula = reject ~ x, family = "binomial", data = reject1)
#>
#> Coefficients:
#> (Intercept) x
#> -2.576927 0.005038
#>
#> Degrees of Freedom: 3199 Total (i.e. Null); 3198 Residual
#> Null Deviance: 3753
#> Residual Deviance: 2765 AIC: 2769
#>
#> - Detailed Results:
#>
#> [test]: test_parameters: CIs (pars: y~x:w1,y~x:w2)
#> [test_label]: All sig
#> n est p.v reject r.cilo r.cihi
#> 1 100 NaN 1.000 0.062 0.039 0.086
#> 2 690 NaN 1.000 0.715 0.671 0.759
#> 3 709 NaN 1.000 0.762 0.721 0.804
#> 4 727 NaN 1.000 0.787 0.747 0.828
#> 5 764 NaN 1.000 0.830 0.793 0.867
#> 6 838 NaN 1.000 0.833 0.796 0.869
#> 7 985 NaN 1.000 0.890 0.859 0.921
#> 8 1280 NaN 1.000 0.935 0.911 0.959
#> Notes:
#> - n: The sample size in a trial.
#> - p.v: The proportion of valid replications.
#> - est: The mean of the estimates in a test across replications.
#> - reject: The proportion of 'significant' replications, that is, the
#> rejection rate. If the null hypothesis is true, this is the Type I
#> error rate. If the null hypothesis is false, this is the power.
#> - r.cilo,r.cihi: The confidence interval of the rejection rate, based
#> on normal approximation.
#> - Refer to the tests for the meanings of other columns.
#>
#>
#>
#> ======<< Summary for the Upper Region >>======
#>
#>
#> ====== x_from_power Results ======
#>
#> Call:
#> power4mome::x_from_power(object = out, seed = 1357, x = "n",
#> what = "lb", goal = "close_enough")
#>
#> Predictor (x): Sample Size
#>
#> - Target Power: 0.800
#> - Goal: Find 'x' with estimated lower confidence bound close enough to
#> the target power.
#>
#> === Major Results ===
#>
#> - Final Value (Sample Size): 838
#>
#> - Final Estimated Power: 0.833
#> - Confidence Interval: [0.796; 0.869]
#> - Level of confidence: 95.0%
#> - Based on 400 replications.
#>
#> === Technical Information ===
#>
#> - Algorithm: bisection
#> - Tolerance for 'close enough': Within of 0.800
#> - The range of values explored: 100 to 985
#> - Time spent in the search: 0.572 secs
#> - The final crude model for the power-predictor relation:
#>
#> Model Type: Logistic Regression
#>
#> Call:
#> power_curve(object = by_x_1, formula = power_curve_args$power_model,
#> start = power_curve_args$start, lower_bound = power_curve_args$lower_bound,
#> upper_bound = power_curve_args$upper_bound, nls_args = power_curve_args$nls_args,
#> nls_control = power_curve_args$nls_control, verbose = progress)
#>
#> Predictor: n (Sample Size)
#>
#> Model:
#>
#> Call: stats::glm(formula = reject ~ x, family = "binomial", data = reject1)
#>
#> Coefficients:
#> (Intercept) x
#> -2.576927 0.005038
#>
#> Degrees of Freedom: 3199 Total (i.e. Null); 3198 Residual
#> Null Deviance: 3753
#> Residual Deviance: 2765 AIC: 2769
#>
#> - Detailed Results:
#>
#> [test]: test_parameters: CIs (pars: y~x:w1,y~x:w2)
#> [test_label]: All sig
#> n est p.v reject r.cilo r.cihi
#> 1 100 NaN 1.000 0.062 0.039 0.086
#> 2 690 NaN 1.000 0.715 0.671 0.759
#> 3 709 NaN 1.000 0.762 0.721 0.804
#> 4 727 NaN 1.000 0.787 0.747 0.828
#> 5 764 NaN 1.000 0.830 0.793 0.867
#> 6 838 NaN 1.000 0.833 0.796 0.869
#> 7 985 NaN 1.000 0.890 0.859 0.921
#> 8 1280 NaN 1.000 0.935 0.911 0.959
#> Notes:
#> - n: The sample size in a trial.
#> - p.v: The proportion of valid replications.
#> - est: The mean of the estimates in a test across replications.
#> - reject: The proportion of 'significant' replications, that is, the
#> rejection rate. If the null hypothesis is true, this is the Type I
#> error rate. If the null hypothesis is false, this is the power.
#> - r.cilo,r.cihi: The confidence interval of the rejection rate, based
#> on normal approximation.
#> - Refer to the tests for the meanings of other columns.Code Template
This is the code used above:
library(power4mome)
# ====== Model and Effect Size (Population Values) ======
model <-
"
y ~ x + w1 + w2 + x:w1 + x:w2
"
model_es <-
"
y ~ x: s
y ~ w1: s
y ~ w2: s
y ~ x:w1: l
y ~ x:w2: m
"
# Test the Model Specification
out <- power4test(nrep = 2,
model = model,
pop_es = model_es,
n = 50000,
fit_model_args = list(fit_function = "lm"),
iseed = 1234)
# Check the Data Generated
print(out,
data_long = TRUE)
# ====== Try One N and Estimate the Power ======
# For n = 100.
# Find the power with *both* x:w1 and x:w2 significant.
# in the regression results of lm().
# The test by CI is equivalent to the two-tailed t-test.
# Add omnibus = "all_sig" to find this power.
out <- power4test(nrep = 400,
model = model,
pop_es = model_es,
n = 100,
fit_model_args = list(fit_function = "lm"),
test_fun = test_parameters,
test_args = list(pars = c("y~x:w1",
"y~x:w2"),
omnibus = "all_sig"),
iseed = 1234,
parallel = TRUE)
rejection_rates(out)
# ====== Regions of Ns ======
# Call n_region_from_power()
# - Set target power: target_power = .80 (Default, can be omitted)
# - Set the seed for the simulation: Integer. Should always be set.
# To set desired precision:
# - Set final number of R: final_R = 1000 (Default, can be omitted)
# - Set final number of replications: final_nrep = 400 (Default, can be omitted)
n_power_region <- n_region_from_power(out,
seed = 1357)
n_power_region
plot(n_power_region)
summary(n_power_region)Final Remarks
For other options of power4test() and
n_region_from_power(), please refer to their help pages, as
well as the Get-Started
article and this article
for n_from_power(), which is the function to find one of
the regions, called twice by n_region_from_power().