This function employs an iterative algorithm to pick the number of random
item-to-parcel allocations needed to meet user-defined stability criteria
for a fitted structural equation model (SEM) (see Details below for
more information). Pooled point and standard-error estimates from this SEM
can be outputted at this final selected number of allocations (however, it
is more efficient to save the allocations and treat them as multiple
imputations using runMI; see See Also for links with
examples). Additionally, new indices (see Sterba & Rights, 2016) are
outputted for assessing the relative contributions of parcel-allocation
variability vs. sampling variability in each estimate. At each iteration,
this function generates a given number of random item-to-parcel allocations,
fits a SEM to each allocation, pools estimates across allocations from that
iteration, and then assesses whether stopping criteria are met. If stopping
criteria are not met, the algorithm increments the number of allocations
used (generating all new allocations).
poolMAlloc(nPerPar, facPlc, nAllocStart, nAllocAdd = 0,
parceloutput = NULL, syntax, dataset, stopProp, stopValue,
selectParam = NULL, indices = "default", double = FALSE,
checkConv = FALSE, names = "default", leaveout = 0,
useTotalAlloc = FALSE, ...)A list in which each element is a vector, corresponding to each factor, indicating sizes of parcels. If variables are left out of parceling, they should not be accounted for here (i.e., there should not be parcels of size "1").
A list of vectors, each corresponding to a factor, specifying the item indicators of that factor (whether included in parceling or not). Either variable names or column numbers. Variables not listed will not be modeled or included in output datasets.
The number of random allocations of items to parcels to generate in the first iteration of the algorithm.
The number of allocations to add with each iteration of the
algorithm. Note that if only one iteration is desired, nAllocAdd can
be set to \(0\) and results will be output for nAllocStart
allocationsonly.
Optional character. Path (folder/directory) where
M (the final selected number of allocations) parceled data sets will
be outputted from the iteration where the algorithm met stopping criteria.
Note for Windows users: file path must be specified using forward slashes
(/), not backslashes (\). See path.expand
for details. If NULL (default), nothing is saved to disk.
lavaan syntax that defines the model.
Item-level dataset
Value used in defining stopping criteria of the algorithm
(\(\delta_a\) in Sterba & Rights, 2016). This is the minimum proportion of
change (in any pooled parameter or pooled standard error estimate listed in
selectParam) that is allowable from one iteration of the algorithm to
the next. That is, change in pooled estimates and pooled standard errors
from one iteration to the next must all be less than (stopProp) x
(value from former iteration). Note that stopValue can override this
criterion (see below). Also note that values less than .01 are unlikely to
lead to more substantively meaningful precision. Also note that if only
stopValue is a desired criterion, stopProp can be set to 0.
Value used in defining stopping criteria of the algorithm
(\(\delta_b\) in Sterba & Rights, 2016). stopValue is a minimum
allowable amount of absolute change (in any pooled parameter or pooled
standard error estimate listed in selectParam) from one iteration of
the algorithm to the next. For a given pooled estimate or pooled standard
error, stopValue is only invoked as a stopping criteria when the
minimum change required by stopProp is less than stopValue.
Note that values less than .01 are unlikely to lead to more substantively
meaningful precision. Also note that if only stopProp is a desired
criterion, stopValue can be set to 0.
(Optional) A list of the pooled parameters to be used in
defining stopping criteria (i.e., stopProp and stopValue).
These parameters should appear in the order they are listed in the lavaan
syntax. By default, all pooled parameters are used. Note that
selectParam should only contain freely-estimated parameters. In one
example from Sterba & Rights (2016) selectParam included all free
parameters except item intercepts and in another example selectParam
included only structural parameters.
Optional character vector indicating the names of
available fitMeasures to be included in the output.
The first and second elements should be a chi-squared test statistic and its
associated degrees of freedom, both of which will be added if missing. If
"default", the indices will be c("chisq", "df", "cfi", "tli",
"rmsea","srmr"). If a robust test statistic is requested (see
lavOptions), c("chisq","df") will be replaced
by c("chisq.scaled","df.scaled"). For the output to include both the
naive and robust test statistics, indices should include both, but
put the scaled test statistics first, as in indices =
c("chisq.scaled", "df.scaled", "chisq", "df")
(Optional) If set to TRUE, requires stopping criteria
(stopProp and stopValue) to be met for all parameters (in
selectParam) for two consecutive iterations of the algorithm. By
default, this is set to FALSE, meaning stopping criteria need only be
met at one iteration of the algorithm.
(Optional) If set to TRUE, function will output pooled estimates and standard errors from 10 iterations post-convergence.
(Optional) A character vector containing the names of parceled variables.
(Optional) A vector of variables to be left out of randomized parceling. Either variable names or column numbers are allowed.
(Optional) If set to TRUE, function will output
a separate set of results that uses all allocations created by the
algorithm, rather than M allocations (see "Allocations needed for
stability" below). This distinction is further discussed in Sterba and
Rights (2016).
Additional arguments to be passed to
lavaan. See also lavOptions
A table containing pooled results across M allocations at the iteration where stopping criteria were met. Columns correspond to individual parameter name, pooled estimate, pooled standard error, p-value for a z-test of the parameter, z-based 95% confidence interval, p-value for a t-test of the parameter (using degrees of freedom described in Sterba & Rights, 2016), and t-based 95% confidence interval for the parameter.
A table containing results related to model fit from the M allocations at the iteration where stopping criteria were met. Columns correspond to fit index names, the average of each index across allocations, the standard deviation of each fit index across allocations, the maximum of each fit index across allocations, the minimum of each fit index across allocations, the range of each fit index across allocations, and the percent of the M allocations where the chi-square test of absolute fit was significant.
A table containing the proportion of the final M allocations that converged (using a maximum likelihood estimator) and the proportion of allocations that converged to proper solutions. Note that pooled estimates, pooled standard errors, and other results are computed using only the converged, proper allocations.
The number of allocations (M) at which the algorithm's stopping criteria (defined above) were met.
A table containing individual parameter names, an estimate of the proportion of total variance of a pooled parameter estimate that is attributable to parcel-allocation variability (PPAV), and an estimate of the ratio of the between-allocation variance of a pooled parameter estimate to the within-allocation variance (RPAV). See Sterba & Rights (2016) for more detail.
The total runtime of the function, in minutes.
Note that the total runtime will be greater when the specified model
encounters convergence problems for some allocations, as is the case with the
simParcel dataset used below.
For further details on the benefits of the random allocation of items to parcels, see Sterba (2011) and Sterba & MacCallum (2010).
This function implements an algorithm for choosing the number of allocations (M; described in Sterba & Rights, 2016), pools point and standard-error estimates across these M allocations, and produces indices for assessing the relative contributions of parcel-allocation variability vs. sampling variability in each estimate.
To obtain pooled test statistics for model fit or model comparison, the
list or parcel allocations can be passed to runMI
(find Examples on the help pages for parcelAllocation
and PAVranking).
This function randomly generates a given number (nAllocStart) of
item-to-parcel allocations, fits a SEM to each allocation, and then
increments the number of allocations used (by nAllocAdd) until the
pooled point and standard-error estimates fulfill stopping criteria
(stopProp and stopValue, defined above). A summary of results
from the model that was fit to the M allocations are returned.
Additionally, this function outputs the proportion of allocations with solutions that converged (using a maximum likelihood estimator) as well as the proportion of allocations with solutions that were converged and proper. The converged and proper solutions among the final M allocations are used in computing pooled results.
Additionally, after each iteration of the algorithm, information useful in monitoring the algorithm is outputted. The number of allocations used at that iteration, the proportion of pooled parameter estimates meeting stopping criteria at the previous iteration, the proportion of pooled standard errors meeting stopping criteria at the previous iteration, and the runtime of that iteration are outputted. When stopping criteria are satisfied, the full set of results are outputted.
Sterba, S. K. (2011). Implications of parcel-allocation variability for comparing fit of item-solutions and parcel-solutions. Structural Equation Modeling, 18(4), 554--577. 10.1080/10705511.2011.607073
Sterba, S. K., & MacCallum, R. C. (2010). Variability in parameter estimates and model fit across random allocations of items to parcels. Multivariate Behavioral Research, 45(2), 322--358. 10.1080/00273171003680302
Sterba, S. K., & Rights, J. D. (2016). Accounting for parcel-allocation variability in practice: Combining sources of uncertainty and choosing the number of allocations. Multivariate Behavioral Research, 51(2--3), 296--313. 10.1080/00273171.2016.1144502
Sterba, S. K., & Rights, J. D. (2017). Effects of parceling on model selection: Parcel-allocation variability in model ranking. Psychological Methods, 22(1), 47--68. 10.1037/met0000067
runMI for treating allocations as multiple imputations to
pool results across allocations. See Examples on help pages for:
parcelAllocation for fitting a single model
PAVranking for comparing 2 models
# NOT RUN {
# }
# NOT RUN {
## lavaan syntax: A 2 Correlated
## factor CFA model to be fit to parceled data
parmodel <- '
f1 =~ NA*p1f1 + p2f1 + p3f1
f2 =~ NA*p1f2 + p2f2 + p3f2
p1f1 ~ 1
p2f1 ~ 1
p3f1 ~ 1
p1f2 ~ 1
p2f2 ~ 1
p3f2 ~ 1
p1f1 ~~ p1f1
p2f1 ~~ p2f1
p3f1 ~~ p3f1
p1f2 ~~ p1f2
p2f2 ~~ p2f2
p3f2 ~~ p3f2
f1 ~~ 1*f1
f2 ~~ 1*f2
f1 ~~ f2
'
## specify items for each factor
f1name <- colnames(simParcel)[1:9]
f2name <- colnames(simParcel)[10:18]
## run function
poolMAlloc(nPerPar = list(c(3,3,3), c(3,3,3)),
facPlc = list(f1name, f2name), nAllocStart = 10, nAllocAdd = 10,
syntax = parmodel, dataset = simParcel, stopProp = .03,
stopValue = .03, selectParam = c(1:6, 13:18, 21),
names = list("p1f1","p2f1","p3f1","p1f2","p2f2","p3f2"),
double = FALSE, useTotalAlloc = FALSE)
# }
# NOT RUN {
## See examples on ?parcelAllocation and ?PAVranking for how to obtain
## pooled test statistics and other pooled lavaan output.
## Details provided in Sterba & Rights (2016).
# }
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