Quick Template: Moderation with Observed Variables: Two Moderators
2026-02-07
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 = 200.
# 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 = 600,
model = model,
pop_es = model_es,
n = 200,
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: 600
#> Sample Sizes: 200
#>
#> ==== Descriptive Statistics ====
#>
#> vars n mean sd skew kurtosis se
#> y 1 120000 0.00 1 0.02 0.02 0
#> x 2 120000 0.00 1 0.00 -0.01 0
#> w1 3 120000 0.00 1 0.00 -0.01 0
#> w2 4 120000 0.00 1 0.00 0.00 0
#> x:w1 5 120000 0.00 1 0.03 5.84 0
#> x:w2 6 120000 -0.01 1 -0.06 5.90 0
#>
#> ==== Parameter Estimates Based on All 600 Samples Combined ====
#>
#> Total Sample Size: 120000
#>
#> ==== Standardized Estimates ====
#>
#> Variances and error variances omitted.
#>
#> Regressions:
#> est.std
#> y ~
#> x 0.098
#> w1 0.101
#> w2 0.102
#> x:w1 0.148
#> x:w2 0.105
#>
#> Covariances:
#> est.std
#> x ~~
#> w1 -0.002
#> w2 -0.006
#> x:w1 -0.005
#> x:w2 -0.004
#> w1 ~~
#> w2 0.008
#> x:w1 -0.007
#> x:w2 0.002
#> w2 ~~
#> x:w1 0.002
#> x:w2 -0.002
#> x:w1 ~~
#> x:w2 0.005
#>
#>
#> ==================== Extra Element(s) Found ====================
#>
#> - fit
#>
#> === Element(s) of the First Dataset ===
#>
#> ============ <fit> ============
#>
#>
#> The models:
#> y ~ x + w1 + w2 + x:w1 + x:w2
#> <environment: 0x00000226b7872428>
#>
#>
#> ===== <test_parameters: CIs (pars: y~x:w1,y~x:w2)> =====
#>
#> Mean(s) across replication:
#> test_label lhs op rhs est se pvalue cilo cihi sig
#> 1 All sig <NA> <NA> <NA> NaN NaN NaN NaN NaN 0.187
#>
#> - The column 'sig' shows the rejection rates.
#> - If the null hypothesis is false, the rate is the power.
#> - Number of valid replications for rejection rate(s): 600
#> - Proportion of valid replications for rejection rate(s): 1.000
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 0.187 0.158 0.220
#> 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 Wilson's (1927) method.
#> - 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. If omitted,
# it is set to 1000 or set to R in the original object.
# - Set final number of replications: final_nrep. If omitted,
# it is set to 400 or set to nrep in the original object.
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: 703 to 782
#> - significantly lower than 0.800: 703
#> - significantly higher than 0.800: 782
#>
#> Confidence intervals of the estimated power:
#> - for the lower bound (703): [0.745, 0.811]
#> - for the upper bound (782): [0.782, 0.844]
#>
#> 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 703 have power significantly lower than .80, and
sample sizes higher than 782 have power significantly higher than .80.
In other words, sample sizes between 703 and 782 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:
#> x_from_power(object = out, x = "n", what = "ub", goal = "close_enough",
#> final_nrep = 600, final_R = 1000, 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): 703
#>
#> - Final Estimated Power: 0.780
#> - Confidence Interval: [0.745; 0.811]
#> - Level of confidence: 95.0%
#> - Based on 600 replications.
#>
#> === Technical Information ===
#>
#> - Algorithm: bisection
#> - Tolerance for 'close enough': Within 0.02000 of 0.800
#> - The range of values explored: 200 to 857
#> - Time spent in the search: 1.481 mins
#> - 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
#> -2.191888 0.004842
#>
#> Degrees of Freedom: 4199 Total (i.e. Null); 4198 Residual
#> Null Deviance: 5451
#> Residual Deviance: 4613 AIC: 4617
#>
#> - 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
#> 1 200 NaN 1.000 0.187 0.158 0.220
#> 2 400 NaN 1.000 0.492 0.452 0.532
#> 3 629 NaN 1.000 0.727 0.690 0.761
#> 4 687 NaN 1.000 0.742 0.705 0.775
#> 5 696 NaN 1.000 0.745 0.709 0.778
#> 6 703 NaN 1.000 0.780 0.745 0.811
#> 7 857 NaN 1.000 0.862 0.832 0.887
#> 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 Wilson's (1927) method.
#> - Refer to the tests for the meanings of other columns.
#>
#>
#>
#> ======<< Summary for the Upper Region >>======
#>
#>
#> ====== x_from_power Results ======
#>
#> Call:
#> x_from_power(object = out, seed = 1357, final_nrep = 600, final_R = 1000,
#> 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): 782
#>
#> - Final Estimated Power: 0.815
#> - Confidence Interval: [0.782; 0.844]
#> - Level of confidence: 95.0%
#> - Based on 600 replications.
#>
#> === Technical Information ===
#>
#> - Algorithm: bisection
#> - Tolerance for 'close enough': Within 0.02000 of 0.800
#> - The range of values explored: 200 to 857
#> - Time spent in the search: 32.98 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
#> -2.159859 0.004778
#>
#> Degrees of Freedom: 5399 Total (i.e. Null); 5398 Residual
#> Null Deviance: 6918
#> Residual Deviance: 5997 AIC: 6001
#>
#> - 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
#> 1 200 NaN 1.000 0.187 0.158 0.220
#> 2 400 NaN 1.000 0.492 0.452 0.532
#> 3 529 NaN 1.000 0.598 0.559 0.637
#> 4 629 NaN 1.000 0.727 0.690 0.761
#> 5 687 NaN 1.000 0.742 0.705 0.775
#> 6 696 NaN 1.000 0.745 0.709 0.778
#> 7 703 NaN 1.000 0.780 0.745 0.811
#> 8 782 NaN 1.000 0.815 0.782 0.844
#> 9 857 NaN 1.000 0.862 0.832 0.887
#> 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 Wilson's (1927) method.
#> - 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 = 200.
# 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 = 600,
model = model,
pop_es = model_es,
n = 200,
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. If omitted,
# it is set to 1000 or set to R in the original object.
# - Set final number of replications: final_nrep. If omitted,
# it is set to 400 or set to nrep in the original object.
n_power_region <- n_region_from_power(out,
seed = 1357)
n_power_region
plot(n_power_region)
summary(n_power_region)Final Remarks
For a moderate to small nrep, the results may be
sensitive to the seed. It is advised to do a final check of
the sample size to be used using power4test() and an
nrep of 1000 or 2000.
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().