Determine the necessary sample size for a structural equation model study
so that every parameter of interest is statistically significant in the
same study with a desired probability, or, given a sample size, return
that probability. Composite power is the probability that all of the named
parameters are significant jointly, the quantity a design must be planned
against when its conclusion requires more than one result to hold at once:
a study can have adequate power for each hypothesis on its own and still
be underpowered for the conclusion that rests on all of them together
(Maxwell, 2004). The parameters of interest are any labeled parameters of
a lavaan analysis model, structural paths, loadings, covariances, or
quantities defined with := such as an indirect effect, and any
subset of them can make up the composite.
ss_power_composite_sem(
model,
Sigma = NULL,
pop_model = NULL,
mu = NULL,
parameters = NULL,
desired_power = 0.85,
alpha_level = 0.05,
N = NULL,
G = 1000,
seed = NULL,
...
)A data.frame with term and value columns: the
necessary_N (or supplied specified_N), the
composite_power and its simulation standard error
composite_power_mc_se, then for each parameter its marginal
power_<label> and purported population_<label> value under
the analysis model, followed by alpha_level, the requested
replications, the converged_replications the summary is
based on, and, when a size was planned, the desired_power. The
result carries the dmar_ss_power class, so
summarize the sample size and the composite power in broom convention.
A single character string giving the free analysis model in
lavaan model syntax (see model.syntax), the model
that would be fit to the data. Each parameter of interest must carry a
parameter label so it can be referred to by name, for example
"f2 ~ b*f1" labels the structural path b, and
"ab := a*b" defines an indirect effect from the labeled paths
a and b.
Population covariance matrix of the observed variables, with
row and column names matching the observed variables in model. It
is typically obtained from a fully fixed population model via
cov_sem. Supply exactly one of Sigma or
pop_model.
A single character string giving the population model in
lavaan model syntax with every parameter fixed to its population value,
from which cov_sem derives Sigma (and the
population means, when the model has a mean structure). This is where
the purported population values of the parameters of interest are
chosen; they are values the researcher posits (from theory, prior
studies, or pilot data), never sample estimates. Supply exactly one of
Sigma or pop_model.
Optional population means of the observed variables, used with
Sigma: a named numeric vector with one entry per observed
variable, or an unnamed vector in the row order of Sigma. The
default NULL is zero means. Means matter only when the analysis
model has a mean structure (an intercept term such as s ~ 1, as
in a latent growth curve model); when pop_model is supplied its
mean structure provides the means and mu must not also be given.
Character vector of the parameter labels that make up
the composite. The default NULL uses every user-labeled parameter
in model, in order of appearance, so labeling exactly the
parameters of interest is the simplest way to state the set.
Desired composite statistical power (default 0.85).
Used only when N is NULL.
Type I error rate for each individual two-sided Wald z test (default 0.05), the per-test rate, not a rate for the composite event.
Sample size; if supplied, the realized composite power at that N is returned rather than a sample size planned.
Number of converged Monte Carlo replications per evaluated sample
size (default 1000). The simulation error of each estimated power is
about \(\sqrt{p(1 - p)/G}\); raise G for a sharper answer.
Optional integer seed for reproducibility. The default
NULL uses the current state of the random number generator; a
supplied seed is set internally and the prior state restored on exit.
Additional arguments passed to sem,
both when the population values are resolved and for every Monte Carlo
fit (for example std.lv = TRUE or missing = "listwise").
A robust estimator such as "MLM" cannot be used here: the same
arguments reach the setup fit, which is always from summary statistics,
and lavaan refuses a robust estimator there. The Monte Carlo data are
multivariate normal by construction, so a robust estimator would buy
nothing.
Each reported power is a proportion of G replications, with
simulation standard error about \(\sqrt{p(1 - p)/G}\); the
composite_power_mc_se row reports it for the composite. The
necessary sample size inherits that uncertainty: near the target the power
curve is flat enough that neighboring N are separated by less than
the simulation error, so repeated calls with different seeds return
slightly different sizes. Raising G narrows the spread; reporting
the seed makes a plan reproducible. A proportion of G
replications takes only the values \(0, 1/G, \ldots, 1\), so a
desired_power above \(1 - 1/G\) is refused with a message
saying how large G must be for that target; the same
resolution guard applies to ss_aipe_composite_sem's
assurance.
Ken Kelley [email protected]
Analytic sample size planning methods in SEM exist for a single targeted
parameter (Satorra & Saris, 1985; Lai & Kelley, 2011) or for overall model
fit (MacCallum, Browne, & Sugawara, 1996; ss_power_sem), but
most studies state several hypotheses and support their conclusion only
when all of them hold. This function plans for that case by a priori Monte
Carlo simulation (Muthén & Muthén, 2002; Maxwell,
Kelley, & Rausch, 2008): for a candidate N, G data sets are
drawn from the multivariate normal population with covariance matrix
Sigma, the analysis model is fit to each, and each parameter of
interest is tested with its two-sided Wald z test at
alpha_level. The proportion of replications in which every
parameter is significant estimates the composite power, and the
per-parameter proportions estimate the marginal powers. Because the
estimates share one fitted model, the tests are dependent; the simulation
reflects that dependence exactly, at the stated N, with no
asymptotic shortcut.
The composite event is contained in each marginal event, so composite
power is at most the smallest marginal power: the weakest parameter
governs the design, and the marginal power_<label> rows show which
parameter that is.
When N is NULL the necessary sample size is searched for.
The search starts where the product of the marginal Wald powers (an
independence approximation computed from the asymptotic variances, spent
before any simulation) reaches desired_power, brackets the crossing
geometrically, and bisects to adjacent integers, each candidate evaluated
with its own G replications. A planning call therefore fits the
analysis model several thousand times at the default G, and even
at the smallest admissible G the search runs for several seconds,
so the example below evaluates a stated N, which is the cheap half
of the method. A planning call is the same call with N left out
and desired_power stated or left at its default, and the first row
of the result is then necessary_N rather than
specified_N. The vignette
vignette("composite_sem_planning", package = "DMAR") works through
the planning calls for a mediation model and a latent growth curve model,
with reference values computed at G = 10000.
Lai, K., & Kelley, K. (2011). Accuracy in parameter estimation for targeted effects in structural equation modeling: Sample size planning for narrow confidence intervals. Psychological Methods, 16(2), 127--148. tools:::Rd_expr_doi("10.1037/a0021764")
MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1(2), 130--149. tools:::Rd_expr_doi("10.1037/1082-989X.1.2.130")
Maxwell, S. E. (2004). The persistence of underpowered studies in psychological research: Causes, consequences, and remedies. Psychological Methods, 9(2), 147--163. tools:::Rd_expr_doi("10.1037/1082-989X.9.2.147")
Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3 on statistical power.)
Maxwell, S. E., Kelley, K., & Rausch, J. R. (2008). Sample size planning for statistical power and accuracy in parameter estimation. Annual Review of Psychology, 59, 537--563. tools:::Rd_expr_doi("10.1146/annurev.psych.59.103006.093735")
Muthén, L. K., & Muthén, B. O. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling, 9(4), 599--620. tools:::Rd_expr_doi("10.1207/S15328007SEM0904_8")
Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1--36. tools:::Rd_expr_doi("10.18637/jss.v048.i02")
Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50(1), 83--90.
ss_aipe_composite_sem for the same set of
parameters planned for accuracy in parameter estimation (AIPE) instead
of significance; cov_sem for deriving Sigma from a
fully fixed population model; ss_power_sem for overall
model fit; ss_aipe_sem_path for a single targeted path;
ss_power_composite_anova and its siblings for composite
power in ANOVA and ANCOVA designs, where the composite is evaluated by
quadrature rather than simulation.
Other sample size for power:
power_fisher_exact(),
ss_aipe_mixed_effects(),
ss_power_R2(),
ss_power_R2_sensitivity(),
ss_power_c(),
ss_power_c_ancova(),
ss_power_composite_ancova(),
ss_power_composite_ancova_2group(),
ss_power_composite_anova(),
ss_power_composite_factorial_ancova(),
ss_power_composite_factorial_ancova_het(),
ss_power_composite_factorial_anova(),
ss_power_contrast(),
ss_power_equivalence_c(),
ss_power_factorial_ancova(),
ss_power_factorial_anova(),
ss_power_indirect_effect(),
ss_power_mixed_effects(),
ss_power_one_way_anova(),
ss_power_pcm(),
ss_power_r(),
ss_power_rc(),
ss_power_reg_coef(),
ss_power_reg_coef_sensitivity(),
ss_power_rm_anova(),
ss_power_sc(),
ss_power_sem(),
ss_power_smd(),
ss_power_split_plot_anova()
Other composite power:
ss_power_composite_ancova(),
ss_power_composite_ancova_2group(),
ss_power_composite_anova(),
ss_power_composite_factorial_ancova(),
ss_power_composite_factorial_ancova_het(),
ss_power_composite_factorial_anova()
# A three-factor model whose conclusion rests on three structural paths
# at once: f1 predicting f2, f2 predicting f3, and f1 predicting f3
# directly. The population model fixes every parameter to its purported
# population value.
pop_model <- "
f1 =~ 1*y1 + 0.8*y2 + 0.8*y3
f2 =~ 1*y4 + 0.8*y5 + 0.8*y6
f3 =~ 1*y7 + 0.8*y8 + 0.8*y9
f2 ~ 0.4*f1
f3 ~ 0.3*f2 + 0.25*f1
f1 ~~ 1*f1
f2 ~~ 0.84*f2
f3 ~~ 0.8*f3
y1 ~~ 0.5*y1; y2 ~~ 0.5*y2; y3 ~~ 0.5*y3
y4 ~~ 0.5*y4; y5 ~~ 0.5*y5; y6 ~~ 0.5*y6
y7 ~~ 0.5*y7; y8 ~~ 0.5*y8; y9 ~~ 0.5*y9
"
# The analysis model is free; the labels name the parameters of interest.
analysis_model <- "
f1 =~ y1 + y2 + y3
f2 =~ y4 + y5 + y6
f3 =~ y7 + y8 + y9
f2 ~ a*f1
f3 ~ b*f2 + c*f1
"
# Realized composite power at N = 200. The probability that all three
# paths come out significant in the same study is lower than the marginal
# power of any one of them: the composite event sits inside each marginal
# event, so the weakest parameter governs the design. G = 20 keeps the
# example quick; a reported plan deserves the default G = 1000 or more.
ss_power_composite_sem(model = analysis_model, pop_model = pop_model,
N = 200, G = 20, seed = 113)
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