Automatic Short Form Creation for scales. Currently, the Ant Colony Optimization (ACO) Algorithm and the Tabu search are implemented. The original R implementation for the ACO algorithm is from Leite, Huang, & Marcoulides (2008), while the Tabu search function was taken from Marcoulides & Falk (2018). There does not yet seem to be an application of Simulated Annealing (SA) within psychometrics, but Drezner & Marcoulides, 1999 (in Multiple Linear Regression Viewpoints, Volume 25(2); not available online) used SA for multiple regression model selection; this package appears to be the first to implement SA for psychometric models.
This document was created on 2024-05-22.
install.packages("ShortForm") # the CRAN-approved version
require("devtools")
devtools::install_github("AnthonyRaborn/ShortForm", branch = "devel") # the developmental version
Here are some (slightly modified) examples from the help documentation using lavaan. Be warned, the algorithms may take some time to converge, particularly with large forms, multiple dimensions, and different settings. The time for these examples to converge on a laptop with an Intel Core i7 8th Gen processor is printed at the bottom. See the sessionInfo() below.
sessionInfo() ## R version 4.3.3 (2024-02-29 ucrt) ## Platform: x86_64-w64-mingw32/x64 (64-bit) ## Running under: Windows 11 x64 (build 22621) ## ## Matrix products: default ## ## ## locale: ## [1] LC_COLLATE=English_United States.utf8 ## [2] LC_CTYPE=English_United States.utf8 ## [3] LC_MONETARY=English_United States.utf8 ## [4] LC_NUMERIC=C ## [5] LC_TIME=English_United States.utf8 ## ## time zone: America/Phoenix ## tzcode source: internal ## ## attached base packages: ## [1] stats graphics grDevices utils datasets methods base ## ## loaded via a namespace (and not attached): ## [1] compiler_4.3.3 fastmap_1.1.1 cli_3.6.2 tools_4.3.3 ## [5] htmltools_0.5.7 rstudioapi_0.15.0 yaml_2.3.8 rmarkdown_2.26 ## [9] knitr_1.45 xfun_0.42 digest_0.6.35 rlang_1.1.3 ## [13] evaluate_0.23
start.time.ACO <- Sys.time()
library(ShortForm, quietly = T)
## Package 'ShortForm' version 0.5.4
# using simulated test data and the default values for lavaan.model.specs
set.seed(1)
# create simulation data from the `psych` package
# four factors, 12 items each, 48 total items
# factor loading matrix - not quite simple structure
fxMatrix <-
matrix(data = c(rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4*3), # first factor loadings
rep(0.2, times = 3*4),
rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4*2), # second factor loadings
rep(0.2, times = 3*4*2),
rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4), # third factor loadings
rep(0.2, times = 3*4*3),
rep(x = c(.8, .8, .4, .3), times = 3) # fourth factor loadings
),
ncol = 4)
# factor correlation matrix - all factors uncorrelated
PhiMatrix <-
matrix(data = c(1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1), ncol = 4)
antData <-
psych::sim(
fx = fxMatrix,
Phi = PhiMatrix,
n = 600,
mu = c(-2, -1, 1, 2),
raw = TRUE
)$observed # observed is the simulated observed data
colnames(antData) = paste0("Item", 1:48)
antModel <- '
Trait1 =~ Item1 + Item2 + Item3 + Item4 + Item5 + Item6 + Item7 + Item8 + Item9 + Item10 + Item11 + Item12
Trait2 =~ Item13 + Item14 + Item15 + Item16 + Item17 + Item18 + Item19 + Item20 + Item21 + Item22 + Item23 + Item24
Trait3 =~ Item25 + Item26 + Item27 + Item28 + Item29 + Item30 + Item31 + Item32 + Item33 + Item34 + Item35 + Item36
Trait4 =~ Item37 + Item38 + Item39 + Item40 + Item41 + Item42 + Item43 + Item44 + Item45 + Item46 + Item47 + Item48
'
# then, create the list of the items by the factors
list.items <-
list(
paste0("Item", 1:12),
paste0("Item", 13:24),
paste0("Item", 25:36),
paste0("Item", 37:48)
)
# finally, call the function with some minor changes to the default values.
abilityShortForm =
antcolony.lavaan(data = antData,
ants = 10, evaporation = 0.9, antModel = antModel,
list.items = list.items, full = 48, i.per.f = c(6,6,6,6),
lavaan.model.specs =
list(model.type = "cfa", auto.var = T, estimator = "default",
ordered = NULL, int.ov.free = TRUE,
int.lv.free = FALSE, auto.fix.first = TRUE,
auto.fix.single = TRUE, std.lv = FALSE, auto.cov.lv.x = TRUE,
auto.th = TRUE, auto.delta = TRUE,
auto.cov.y = TRUE),
factors = c("Trait1", "Trait2", "Trait3", "Trait4"), steps = 100,
max.run = 100,
parallel = T)
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abilityShortForm # print the results of the final short form
## Algorithm: Ant Colony Optimization
## Total Run Time: 1.015 mins
##
## Function call:
## antcolony.lavaan(data = antData, ants = 10, evaporation = 0.9, antModel =
## antModel, list.items = list.items, full = 48, i.per.f = c(6, 6, 6, 6), factors
## = c("Trait1", "Trait2", "Trait3", "Trait4"), steps = 100, lavaan.model.specs
## = list(model.type = "cfa", auto.var = T, estimator = "default", ordered
## = NULL, int.ov.free = TRUE, int.lv.free = FALSE, auto.fix.first = TRUE,
## auto.fix.single = TRUE, std.lv = FALSE, auto.cov.lv.x = TRUE, auto.th = TRUE,
## auto.delta = TRUE, auto.cov.y = TRUE), max.run = 100, parallel = T)
##
## Final Model Syntax:
##
## Trait1 =~ Item9 + Item2 + Item10 + Item5 + Item11 + Item6
## Trait2 =~ Item21 + Item14 + Item13 + Item22 + Item18 + Item17
## Trait3 =~ Item29 + Item34 + Item25 + Item30 + Item33 + Item26
## Trait4 =~ Item45 + Item46 + Item41 + Item38 + Item37 + Item42
plot(abilityShortForm, type = 'pheromone') # the pheromone plot for class "antcolony"
A similar example can be found in the antcolony.mplus function, but requires you to have a valid Mplus installation on the computer. It took a total of 1.06 mins to run this example.
This example demonstrates how to use the Tabu search for model specification searches when the original model may be misspecified in some way.
start.time.Tabu <- Sys.time()
library(ShortForm, quietly = T)
set.seed(2)
# create simulation data from the `psych` package
# two factors, 12 items total
# factor loading matrix - not quite simple structure
fxMatrix <-
matrix(data = c(
# first factor loadings
rep(x = c(.8, .8, .6, .6), times = 3),
# second factor loadings
rep(x = c(.2), times = 12)
),
ncol = 2)
# factor correlation matrix - all factors uncorrelated
PhiMatrix <-
matrix(data = c(1,0,
0,1
), ncol = 2)
tabuData <-
psych::sim(
fx = fxMatrix,
Phi = PhiMatrix,
n = 600,
raw = TRUE
)$observed # observed is the simulated observed data
colnames(tabuData) = paste0("Item", 1:12)
tabuModel <- '
Trait1 =~ Item1 + Item2 + Item3 + Item4 + Item5 + Item6 + 0*Item7 + 0*Item8 + 0*Item9 + 0*Item10 + 0*Item11 + 0*Item12
Trait2 =~ 0*Item1 + 0*Item2 + 0*Item3 + 0*Item4 + 0*Item5 + 0*Item6 + Item7 + Item8 + Item9 + Item10 + Item11 + Item12
'
# fit the initial misspecified model for Tabu
init.model <- lavaan::lavaan(model = tabuModel, data = tabuData,
auto.var=TRUE, auto.fix.first=FALSE, std.lv=TRUE,
auto.cov.lv.x=FALSE)
# use search.prep to prepare for the Tabu search
ptab <-
search.prep(fitted.model = init.model,
loadings=TRUE,
fcov=FALSE,
errors=FALSE)
Tabu_example <-
suppressWarnings(
tabu.sem(init.model = init.model,
ptab = ptab,
obj = AIC,
niter = 20,
tabu.size = 10)
) # the suppressWarning wrapping hides the lavaan WARNING output from improper models
## Running iteration 1 of 20. Running iteration 2 of 20. Running iteration 3 of 20. Running iteration 4 of 20. Running iteration 5 of 20. Running iteration 6 of 20. Running iteration 7 of 20. Running iteration 8 of 20. Running iteration 9 of 20. Running iteration 10 of 20. Running iteration 11 of 20. Running iteration 12 of 20. Running iteration 13 of 20. Running iteration 14 of 20. Running iteration 15 of 20. Running iteration 16 of 20. Running iteration 17 of 20. Running iteration 18 of 20. Running iteration 19 of 20. Running iteration 20 of 20.
# check the final model
summary(Tabu_example)
## Algorithm: Tabu Search
## Total Run Time: 1.908 mins
##
## lavaan 0.6.17 ended normally after 32 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 29
##
## Number of observations 600
##
## Model Test User Model:
##
## Test statistic 42.131
## Degrees of freedom 49
## P-value (Chi-square) 0.746
##
##
## Final Model Syntax:
## Trait1 =~ Item1 + Item2 + Item3 + Item4 + Item5 + Item6 + Item7 + Item8 + Item9
## + Item10 + Item11 + Item12
## Trait2 =~ Item1 + Item3 + Item5 + Item10 + Item11
# plot the change in the objective/criterion function over each run
plot(Tabu_example)
It took a total of 1.92 mins to run this example.
The next Tabu example demonstrates how to use it to find a short form of a prespecified length with different data.
start.time.Tabu <- Sys.time()
library(ShortForm, quietly = T)
# set the seed to reproduce this example
set.seed(3)
# create simulation data from the `psych` package
# four factors, 12 items each, 48 total items
# factor loading matrix - not quite simple structure
fxMatrix <-
matrix(data = c(rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4*3), # first factor loadings
rep(0.2, times = 3*4),
rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4*2), # second factor loadings
rep(0.2, times = 3*4*2),
rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4), # third factor loadings
rep(0.2, times = 3*4*3),
rep(x = c(.8, .8, .4, .3), times = 3) # fourth factor loadings
),
ncol = 4)
# factor correlation matrix - all factors uncorrelated
PhiMatrix <-
matrix(data = c(1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1), ncol = 4)
tabuData <-
psych::sim(
fx = fxMatrix,
Phi = PhiMatrix,
n = 600,
mu = c(-2, -1, 1, 2),
raw = TRUE
)$observed # observed is the simulated observed data
colnames(tabuData) = paste0("Item", 1:48)
tabuModel <- '
Trait1 =~ Item1 + Item2 + Item3 + Item4 + Item5 + Item6 + Item7 + Item8 + Item9 + Item10 + Item11 + Item12
Trait2 =~ Item13 + Item14 + Item15 + Item16 + Item17 + Item18 + Item19 + Item20 + Item21 + Item22 + Item23 + Item24
Trait3 =~ Item25 + Item26 + Item27 + Item28 + Item29 + Item30 + Item31 + Item32 + Item33 + Item34 + Item35 + Item36
Trait4 =~ Item37 + Item38 + Item39 + Item40 + Item41 + Item42 + Item43 + Item44 + Item45 + Item46 + Item47 + Item48
'
# specify the criterion function that the Tabu Search minimizes
# wrap this in a tryCatch in case a model does not converge!
# specify an appropriate error value: since we're minimizing, error value must be large
tabuCriterion = function(x) {
tryCatch(lavaan::fitmeasures(object = x, fit.measures = 'chisq'),
error = function(e) Inf)
}
# use the tabuShortForm function
# reduce form to the best 12 items, 3 per factor
tabuShort <-
tabuShortForm(initialModel = tabuModel, originalData = tabuData,
numItems = c(5,5,5,5), criterion = tabuCriterion,
niter = 20, tabu.size = 10, verbose = FALSE
)
## Running iteration 1 of 20. Running iteration 2 of 20. Running iteration 3 of 20. Running iteration 4 of 20. Running iteration 5 of 20. Running iteration 6 of 20. Running iteration 7 of 20. Running iteration 8 of 20. Running iteration 9 of 20. Running iteration 10 of 20. Running iteration 11 of 20. Running iteration 12 of 20. Running iteration 13 of 20. Running iteration 14 of 20. Running iteration 15 of 20. Running iteration 16 of 20. Running iteration 17 of 20. Running iteration 18 of 20. Running iteration 19 of 20. Running iteration 20 of 20.
# check the chosen model
summary(tabuShort)
## Algorithm: Tabu Search
## Total Run Time: 2.137 mins
##
## lavaan 0.6.17 ended normally after 32 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 46
##
## Number of observations 600
##
## Model Test User Model:
##
## Test statistic 129.734
## Degrees of freedom 164
## P-value (Chi-square) 0.978
##
##
## Final Model Syntax:
## Trait1 =~ Item1 + Item2 + Item5 + Item6 + Item9
## Trait2 =~ Item13 + Item14 + Item17 + Item18 + Item22
## Trait3 =~ Item26 + Item29 + Item30 + Item33 + Item34
## Trait4 =~ Item39 + Item43 + Item44 + Item47 + Item40
# plot the changes in the objective function over each iteration
plot(tabuShort)
It took a total of 2.14 mins to run this example.
This example demonstrates the use of simulated annealing for creating short forms.
start.time.SA <- Sys.time()
library(ShortForm, quietly = T)
# create simulation data from the `psych` package
# four factors, 12 items each, 48 total items
# factor loading matrix - not quite simple structure
set.seed(4)
fxMatrix <-
matrix(data = c(rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4*3), # first factor loadings
rep(0.2, times = 3*4),
rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4*2), # second factor loadings
rep(0.2, times = 3*4*2),
rep(x = c(.8, .8, .4, .3), times = 3),
rep(0.2, times = 3*4), # third factor loadings
rep(0.2, times = 3*4*3),
rep(x = c(.8, .8, .4, .3), times = 3) # fourth factor loadings
),
ncol = 4)
# factor correlation matrix - all factors uncorrelated
PhiMatrix <-
matrix(data = c(1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1), ncol = 4)
annealData <-
psych::sim(
fx = fxMatrix,
Phi = PhiMatrix,
n = 600,
mu = c(-2, -1, 1, 2),
raw = TRUE
)$observed # observed is the simulated observed data
colnames(annealData) = paste0("Item", 1:48)
annealModel <- '
Trait1 =~ Item1 + Item2 + Item3 + Item4 + Item5 + Item6 + Item7 + Item8 + Item9 + Item10 + Item11 + Item12
Trait2 =~ Item13 + Item14 + Item15 + Item16 + Item17 + Item18 + Item19 + Item20 + Item21 + Item22 + Item23 + Item24
Trait3 =~ Item25 + Item26 + Item27 + Item28 + Item29 + Item30 + Item31 + Item32 + Item33 + Item34 + Item35 + Item36
Trait4 =~ Item37 + Item38 + Item39 + Item40 + Item41 + Item42 + Item43 + Item44 + Item45 + Item46 + Item47 + Item48
'
lavaan.model.specs <-
list(model.type = "cfa",
auto.var = TRUE, estimator = "default", ordered = NULL,
int.ov.free = TRUE, int.lv.free = FALSE, std.lv = TRUE, auto.fix.first = FALSE,
auto.fix.single = TRUE, auto.cov.lv.x = TRUE, auto.th = TRUE,
auto.delta = TRUE, auto.cov.y = TRUE)
# perform the SA algorithm
set.seed(1)
SA_example <-
simulatedAnnealing(initialModel = annealModel, originalData = annealData, maxSteps = 200,
fitStatistic = 'cfi', maximize = TRUE,
temperature = "logistic", items = paste0("Item", 1:48),
lavaan.model.specs = lavaan.model.specs,
maxChanges = 3, maxItems = c(6,6,6,6), setChains = 4)
## Initializing short form creation.
## The initial short form is:
## Trait1 =~ Item9 + Item4 + Item7 + Item1 + Item2 + Item5
## Trait2 =~ Item19 + Item23 + Item14 + Item15 + Item13 + Item17
## Trait3 =~ Item29 + Item34 + Item30 + Item31 + Item25 + Item33
## Trait4 =~ Item41 + Item48 + Item45 + Item46 + Item42 + Item47
## Using the short form randomNeighbor function.
## Finished initializing short form options.
## Current Progress:
## Chain number 1 complete.
## Chain number 2 complete.
## Chain number 3 complete.
## Chain number 4 complete.
summary(SA_example)
## Algorithm: Simulated Annealing
## Total Run Time: 41.456 secs
##
## lavaan 0.6.17 ended normally after 33 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 54
##
## Number of observations 600
##
## Model Test User Model:
##
## Test statistic 367.147
## Degrees of freedom 246
## P-value (Chi-square) 0.000
##
##
## Final Model Syntax:
## Trait1 =~ Item11 + Item2 + Item3 + Item6 + Item4 + Item9
## Trait2 =~ Item17 + Item21 + Item13 + Item24 + Item16 + Item23
## Trait3 =~ Item26 + Item35 + Item33 + Item34 + Item30 + Item25
## Trait4 =~ Item43 + Item37 + Item42 + Item40 + Item38 + Item46
plot(SA_example) # plot showing how the fit value changes at each step
It took a total of 42.09 secs to run the SA example, and a total of 5.82 mins to run all four together.
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