An analysis based on structural equation modeling may suffer from the impact of a potential missing confounder that may change its conclusions (Harring, McNeish, & Hancock, 2017). This package is a tool to evaluate the sensitivity of structural equation models to potential missing confounders using the Ant Colony Optimization (ACO; Colorni, Dorigo, & Maniezzo, 1992; Dorigo & Stützle, 2004; Socha & Dorigo, 2008) algorithm. The reference for the current package is: Leite, W., Shen, Z., Marcoulides, K., Fish, C., & Harring, J. (2022). Using ant colony optimization for sensitivity analysis in structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 29 (1), 47-56.
The current version includes three main functions and they are
gen.sens.pars function: This function generates a set of required sensitivity parameters according to the rank of object function, with which the method has been developed by Socha & Dorigo (2008).
sa.aco function: This function automatically performs the search for an omitted confounder in structural equation modeling using the ACO algorithm
sens.table function: This function provides five summary tables of results produced in the sa.aco function.
Next, we use an example to illustrate how to perform sensitivity analysis in structural equation modeling.
Given a data set (or variance covariance matrix) and an analytic model, this function searches an omitted confounder to optimize the objective function. sa.aco function calls gen.sens.pars function to perform the algorithm together.
We need the following information/arguments to perform the search
To see more arguments, please run ?sa.aco in R or RStudio.
The example is from: Kim, Y. S. G., & Schatschneider, C. (2017). Expanding the developmental models of writing: A direct and indirect effects model of developmental writing (DIEW). Journal of Educational psychology, 109(1), 35-50.
# STEP 1: Prepare data:
# Lower diagonal correlation matrix in the study by Kim & Schatschneider (2017)
lower = '
1.00
.40 1.00
.40 .64 1.00
.41 .66 .61 1.00
.42 .52 .53 .61 1.00
.34 .50 .46 .53 .48 1.00
.42 .47 .41 .43 .47 .55 1.00
.39 .46 .39 .30 .21 .30 .37 1.00
.24 .31 .30 .31 .26 .32 .27 .56 1.00
.33 .35 .35 .40 .31 .25 .35 .51 .42 1.00
.30 .42 .36 .32 .24 .37 .43 .44 .37 .49 1.00'
# Convert to full covariance matrix
sample.cov = getCov(lower, sds = c(5.64,14.68,6.57,6.07,3.39,10.16,6.11,4.91,15.59,0.96,0.99),
names = c("Working_memory",
"Vocabulary",
"Grammar",
"Inference",
"ToM",
"TNL",
"Expository",
"Spelling",
"Sentence_copying",
"One_day",
"Castle"))
# STEP 2: Set up analytic model and sensitivity anaysis model
# The original analytic model
model <-'Vocabulary~Working_memory
Grammar~Working_memory
Inference~Vocabulary+Grammar+Working_memory
ToM~Vocabulary+Grammar+Working_memory
Spelling~Working_memory
Sentence_copying~Working_memory
Discourse~Inference+ToM+Vocabulary+Grammar+Working_memory
Writing~Spelling+Sentence_copying+Discourse
Discourse=~TNL+Expository
Writing=~One_day+Castle
Vocabulary~~Grammar
Grammar~~Sentence_copying
Vocabulary~~Sentence_copying
Grammar~~Spelling
Vocabulary~~Spelling
Inference~~ToM
Discourse~~Sentence_copying
Discourse~~Spelling
Spelling~~Sentence_copying'
# A sensitivity analysis model template, which additionally includes paths
# from a phantom variable to a set of variables (= number of sensitivity parameters)
# in the analytic model.
sens.model <- 'Vocabulary~Working_memory
Grammar~Working_memory
Inference~Vocabulary+Grammar+Working_memory
ToM~Vocabulary+Grammar+Working_memory
Spelling~Working_memory
Sentence_copying~Working_memory
Discourse~Inference+ToM+Vocabulary+Grammar+Working_memory
Writing~Spelling+Sentence_copying+Discourse
Discourse=~TNL+Expository
Writing=~One_day+Castle
Vocabulary~~Grammar
Grammar~~Sentence_copying
Vocabulary~~Sentence_copying
Grammar~~Spelling
Vocabulary~~Spelling
Inference~~ToM
Discourse~~Sentence_copying
Discourse~~Spelling
Spelling~~Sentence_copying
Working_memory ~ phantom1*phantom
Grammar ~ phantom2*phantom
Vocabulary ~ phantom3*phantom
ToM ~ phantom4*phantom
Inference ~ phantom5*phantom
Spelling ~ phantom6*phantom
Sentence_copying ~ phantom7*phantom
Discourse ~ phantom8*phantom
Writing ~ phantom9*phantom
phantom =~ 0 # added for mean of zero
phantom ~~ 1*phantom'
# STEP 3: Set up the paths of interest to be evaluated in sensitivity analysis.
paths <- 'Vocabulary~Working_memory
Grammar~Working_memory
Inference~Vocabulary+Grammar+Working_memory
ToM~Vocabulary+Grammar+Working_memory
Spelling~Working_memory
Sentence_copying~Working_memory
Discourse~Inference+ToM+Vocabulary+Grammar+Working_memory
Writing~Spelling+Sentence_copying+Discourse'
# STEP 4: Perform sensitivity analysis.
my.sa <-sa.aco(model = model, sens.model = sens.model, sample.cov = sample.cov,
sample.nobs = 193, k = 50, max.value= 2000, max.iter = 100,
opt.fun = 4, ## from significant to just significant
paths = paths, seed = 1, verbose = FALSE)
# We set up a max iteration of 100 and solution archive length of 50 for
# illustration purpose. Please specify a larger number of iteration (e.g., 1000),
# and a larger k (e.g., 100).sens.tables function can help summarize the sensitivity analysis results. Beyond this function, investigators always can summarize the results by accessing to the results in sa.aco function.
my.table <- sens.tables(my.sa)
# Table 1: Summary of the sensitivity analysis for each path
my.table[[1]]## model.est model.pvalue mean.est.sens
## Discourse~Grammar 0.1003703 2.544519e-01 0.1052304
## Inference~Working_memory 0.1245455 2.598216e-02 0.1232723
## Writing~Sentence_copying 0.1719948 4.222710e-02 0.1633071
## Discourse~Working_memory 0.1888038 7.941220e-03 0.1910799
## Discourse~Inference 0.1943264 3.608645e-02 0.1920310
## ToM~Working_memory 0.2036364 1.096483e-03 0.2030261
## Sentence_copying~Working_memory 0.2399999 3.306180e-04 0.2401460
## ToM~Vocabulary 0.2565656 6.016058e-04 0.2540932
## Discourse~ToM 0.2610613 1.360093e-03 0.2545997
## Discourse~Vocabulary 0.2589070 4.345386e-03 0.2615377
## ToM~Grammar 0.2843434 1.345400e-04 0.2878169
## Inference~Grammar 0.2873737 1.336915e-05 0.2894558
## Spelling~Working_memory 0.3899999 2.976197e-11 0.3883753
## Writing~Spelling 0.3759830 3.442787e-05 0.3886019
## Grammar~Working_memory 0.3999999 5.308198e-12 0.3979180
## Vocabulary~Working_memory 0.3999999 5.308198e-12 0.3996722
## Inference~Vocabulary 0.4262626 3.005818e-11 0.4261592
## Writing~Discourse 0.4504663 3.396553e-07 0.4355599
## min.est.sens max.est.sens
## Discourse~Grammar 0.04912704 0.1774544
## Inference~Working_memory 0.10477421 0.1485761
## Writing~Sentence_copying 0.05199882 0.2657367
## Discourse~Working_memory 0.14529692 0.2331204
## Discourse~Inference 0.15029004 0.2484007
## ToM~Working_memory 0.17420102 0.2505132
## Sentence_copying~Working_memory 0.23129571 0.2588857
## ToM~Vocabulary 0.22454770 0.3059992
## Discourse~ToM 0.15889535 0.3087243
## Discourse~Vocabulary 0.23408924 0.3120485
## ToM~Grammar 0.21758721 0.3477721
## Inference~Grammar 0.23936198 0.3261455
## Spelling~Working_memory 0.36602307 0.4141131
## Writing~Spelling 0.19253896 0.6603765
## Grammar~Working_memory 0.36081953 0.4178667
## Vocabulary~Working_memory 0.38836436 0.4096268
## Inference~Vocabulary 0.40745780 0.4641450
## Writing~Discourse 0.16872738 0.5776849## mean.phan min.phan max.phan
## Grammar~phantom -0.0255901298 -0.19855864 0.14923981
## Working_memory~phantom -0.0116100689 -0.21192127 0.16501694
## Inference~phantom -0.0059654784 -0.15852757 0.19673890
## Vocabulary~phantom 0.0009162615 -0.07989683 0.06497973
## Sentence_copying~phantom 0.0040785197 -0.05962908 0.11357489
## Discourse~phantom 0.0302226186 -0.12647184 0.24583688
## Spelling~phantom 0.0366678782 -0.19658642 0.24672188
## ToM~phantom 0.0561965145 -0.23155307 0.49606813
## Writing~phantom 0.1132356278 -0.52889818 0.61827105## Working_memory~phantom Grammar~phantom
## Vocabulary~Working_memory -0.211921268 -0.18415235
## Grammar~Working_memory -0.211921268 -0.18415235
## Inference~Vocabulary 0.002184957 -0.19855864
## Inference~Grammar -0.110365605 -0.15220596
## Inference~Working_memory -0.167003237 -0.04988583
## ToM~Vocabulary -0.030085236 -0.12832087
## ToM~Grammar -0.110365605 -0.15220596
## ToM~Working_memory -0.167003237 -0.04988583
## Spelling~Working_memory 0.152931730 -0.01137375
## Sentence_copying~Working_memory -0.155173551 0.08682558
## Discourse~Inference 0.044223231 -0.06027779
## Discourse~ToM -0.034499544 -0.13024128
## Discourse~Vocabulary 0.051240582 -0.06690173
## Discourse~Grammar -0.120907356 0.14923981
## Discourse~Working_memory 0.165016936 -0.01754155
## Writing~Spelling 0.048004382 -0.02181512
## Writing~Sentence_copying 0.072626707 0.11784237
## Writing~Discourse 0.072626707 0.11784237
## Vocabulary~phantom ToM~phantom
## Vocabulary~Working_memory -0.054220461 0.22938668
## Grammar~Working_memory -0.054220461 0.22938668
## Inference~Vocabulary -0.030296701 0.20987620
## Inference~Grammar 0.051535666 -0.21725287
## Inference~Working_memory -0.030364651 -0.22577041
## ToM~Vocabulary 0.003831129 0.27028636
## ToM~Grammar 0.051535666 -0.21725287
## ToM~Working_memory -0.030364651 -0.22577041
## Spelling~Working_memory 0.049251697 -0.11155276
## Sentence_copying~Working_memory -0.011157983 0.28069048
## Discourse~Inference -0.035560127 -0.08384902
## Discourse~ToM -0.079896827 0.49606813
## Discourse~Vocabulary 0.062516527 -0.20132409
## Discourse~Grammar -0.013843073 -0.18486429
## Discourse~Working_memory -0.017716484 -0.19524466
## Writing~Spelling -0.064244453 0.14765876
## Writing~Sentence_copying -0.023845171 0.27678824
## Writing~Discourse -0.023845171 0.27678824
## Inference~phantom Spelling~phantom
## Vocabulary~Working_memory 0.11169709 0.11546808
## Grammar~Working_memory 0.11169709 0.11546808
## Inference~Vocabulary 0.12562746 0.15355763
## Inference~Grammar -0.15852757 0.18009840
## Inference~Working_memory -0.15021823 -0.05282820
## ToM~Vocabulary 0.06902098 0.02181626
## ToM~Grammar -0.15852757 0.18009840
## ToM~Working_memory -0.15021823 -0.05282820
## Spelling~Working_memory -0.09672352 0.17233148
## Sentence_copying~Working_memory -0.07186973 0.14099539
## Discourse~Inference 0.12930466 0.19994834
## Discourse~ToM 0.14159692 0.24672188
## Discourse~Vocabulary -0.02696729 -0.10090585
## Discourse~Grammar -0.00735754 0.01807200
## Discourse~Working_memory -0.14682888 0.06457722
## Writing~Spelling -0.09555416 0.20318504
## Writing~Sentence_copying 0.16155105 -0.18908332
## Writing~Discourse 0.16155105 -0.18908332
## Sentence_copying~phantom Discourse~phantom
## Vocabulary~Working_memory 0.08826528 0.24583688
## Grammar~Working_memory 0.08826528 0.24583688
## Inference~Vocabulary 0.05705421 0.10077127
## Inference~Grammar -0.02665071 -0.08985596
## Inference~Working_memory -0.02098849 -0.08802815
## ToM~Vocabulary -0.03303541 0.07442881
## ToM~Grammar -0.02665071 -0.08985596
## ToM~Working_memory -0.02098849 -0.08802815
## Spelling~Working_memory -0.05756797 -0.02387311
## Sentence_copying~Working_memory -0.05369555 0.10337791
## Discourse~Inference 0.02990099 0.13270063
## Discourse~ToM 0.11357489 0.17226044
## Discourse~Vocabulary -0.05219069 0.08786791
## Discourse~Grammar -0.02210510 0.13456352
## Discourse~Working_memory 0.01005041 0.13126727
## Writing~Spelling 0.05236181 0.05720109
## Writing~Sentence_copying 0.05069574 0.08586779
## Writing~Discourse 0.05069574 0.08586779
## Writing~phantom
## Vocabulary~Working_memory -0.12163204
## Grammar~Working_memory -0.12163204
## Inference~Vocabulary 0.08132341
## Inference~Grammar -0.28691915
## Inference~Working_memory 0.36057655
## ToM~Vocabulary 0.41217884
## ToM~Grammar -0.28691915
## ToM~Working_memory 0.36057655
## Spelling~Working_memory -0.12680418
## Sentence_copying~Working_memory -0.08211608
## Discourse~Inference 0.11588441
## Discourse~ToM 0.21492898
## Discourse~Vocabulary 0.07410964
## Discourse~Grammar 0.08140368
## Discourse~Working_memory 0.30589957
## Writing~Spelling 0.61827105
## Writing~Sentence_copying 0.60389707
## Writing~Discourse 0.60389707## Working_memory~phantom Grammar~phantom
## Vocabulary~Working_memory -0.176061341 0.03160994
## Grammar~Working_memory -0.120907356 0.14923981
## Inference~Vocabulary -0.110365605 -0.15220596
## Inference~Grammar 0.002184957 -0.19855864
## Inference~Working_memory 0.165016936 -0.01754155
## ToM~Vocabulary -0.110365605 -0.15220596
## ToM~Grammar 0.002184957 -0.19855864
## ToM~Working_memory -0.155173551 0.08682558
## Spelling~Working_memory -0.176061341 0.03160994
## Sentence_copying~Working_memory -0.211921268 -0.18415235
## Discourse~Inference -0.014368362 -0.02012266
## Discourse~ToM -0.120907356 0.14923981
## Discourse~Vocabulary -0.110365605 -0.15220596
## Discourse~Grammar -0.211921268 -0.18415235
## Discourse~Working_memory -0.211921268 -0.18415235
## Writing~Spelling 0.072626707 0.11784237
## Writing~Sentence_copying -0.138008251 -0.05072568
## Writing~Discourse -0.167003237 -0.04988583
## Vocabulary~phantom ToM~phantom
## Vocabulary~Working_memory 0.055384298 0.1217345
## Grammar~Working_memory -0.013843073 -0.1848643
## Inference~Vocabulary 0.051535666 -0.2172529
## Inference~Grammar -0.030296701 0.2098762
## Inference~Working_memory -0.017716484 -0.1952447
## ToM~Vocabulary 0.051535666 -0.2172529
## ToM~Grammar -0.030296701 0.2098762
## ToM~Working_memory -0.011157983 0.2806905
## Spelling~Working_memory 0.055384298 0.1217345
## Sentence_copying~Working_memory -0.054220461 0.2293867
## Discourse~Inference 0.002323414 0.2029898
## Discourse~ToM -0.013843073 -0.1848643
## Discourse~Vocabulary 0.051535666 -0.2172529
## Discourse~Grammar -0.054220461 0.2293867
## Discourse~Working_memory -0.054220461 0.2293867
## Writing~Spelling -0.023845171 0.2767882
## Writing~Sentence_copying 0.046036288 -0.1319380
## Writing~Discourse -0.030364651 -0.2257704
## Inference~phantom Spelling~phantom
## Vocabulary~Working_memory -0.07860246 0.14301228
## Grammar~Working_memory -0.00735754 0.01807200
## Inference~Vocabulary -0.15852757 0.18009840
## Inference~Grammar 0.12562746 0.15355763
## Inference~Working_memory -0.14682888 0.06457722
## ToM~Vocabulary -0.15852757 0.18009840
## ToM~Grammar 0.12562746 0.15355763
## ToM~Working_memory -0.07186973 0.14099539
## Spelling~Working_memory -0.07860246 0.14301228
## Sentence_copying~Working_memory 0.11169709 0.11546808
## Discourse~Inference -0.14558684 0.02211563
## Discourse~ToM -0.00735754 0.01807200
## Discourse~Vocabulary -0.15852757 0.18009840
## Discourse~Grammar 0.11169709 0.11546808
## Discourse~Working_memory 0.11169709 0.11546808
## Writing~Spelling 0.16155105 -0.18908332
## Writing~Sentence_copying 0.02871137 0.13761525
## Writing~Discourse -0.15021823 -0.05282820
## Sentence_copying~phantom Discourse~phantom
## Vocabulary~Working_memory -0.02140238 0.02093407
## Grammar~Working_memory -0.02210510 0.13456352
## Inference~Vocabulary -0.02665071 -0.08985596
## Inference~Grammar 0.05705421 0.10077127
## Inference~Working_memory 0.01005041 0.13126727
## ToM~Vocabulary -0.02665071 -0.08985596
## ToM~Grammar 0.05705421 0.10077127
## ToM~Working_memory -0.05369555 0.10337791
## Spelling~Working_memory -0.02140238 0.02093407
## Sentence_copying~Working_memory 0.08826528 0.24583688
## Discourse~Inference 0.02421053 0.04225371
## Discourse~ToM -0.02210510 0.13456352
## Discourse~Vocabulary -0.02665071 -0.08985596
## Discourse~Grammar 0.08826528 0.24583688
## Discourse~Working_memory 0.08826528 0.24583688
## Writing~Spelling 0.05069574 0.08586779
## Writing~Sentence_copying -0.05513577 0.05556959
## Writing~Discourse -0.02098849 -0.08802815
## Writing~phantom
## Vocabulary~Working_memory -0.19619587
## Grammar~Working_memory 0.08140368
## Inference~Vocabulary -0.28691915
## Inference~Grammar 0.08132341
## Inference~Working_memory 0.30589957
## ToM~Vocabulary -0.28691915
## ToM~Grammar 0.08132341
## ToM~Working_memory -0.08211608
## Spelling~Working_memory -0.19619587
## Sentence_copying~Working_memory -0.12163204
## Discourse~Inference 0.44354405
## Discourse~ToM 0.08140368
## Discourse~Vocabulary -0.28691915
## Discourse~Grammar -0.12163204
## Discourse~Working_memory -0.12163204
## Writing~Spelling 0.60389707
## Writing~Sentence_copying 0.57441944
## Writing~Discourse 0.36057655## p.value p.changed Working_memory~phantom
## Discourse~Grammar 2.544519e-01 0.04361667 -0.21192127
## Writing~Sentence_copying 4.222710e-02 0.05085436 -0.08329305
## Discourse~Inference 3.608645e-02 0.05138185 -0.05379439
## Inference~Working_memory 2.598216e-02 0.05154444 -0.17606134
## Writing~Discourse 3.396553e-07 0.06126915 0.07262671
## Vocabulary~Working_memory 5.308198e-12 NA NA
## Grammar~Working_memory 5.308198e-12 NA NA
## Inference~Vocabulary 3.005818e-11 NA NA
## Inference~Grammar 1.336915e-05 NA NA
## ToM~Vocabulary 6.016058e-04 NA NA
## ToM~Grammar 1.345400e-04 NA NA
## ToM~Working_memory 1.096483e-03 NA NA
## Spelling~Working_memory 2.976197e-11 NA NA
## Sentence_copying~Working_memory 3.306180e-04 NA NA
## Discourse~ToM 1.360093e-03 NA NA
## Discourse~Vocabulary 4.345386e-03 NA NA
## Discourse~Working_memory 7.941220e-03 NA NA
## Writing~Spelling 3.442787e-05 NA NA
## Grammar~phantom Vocabulary~phantom ToM~phantom
## Discourse~Grammar -0.184152355 -0.05422046 0.22938668
## Writing~Sentence_copying 0.009402667 0.02524991 -0.06890207
## Discourse~Inference -0.113046928 -0.01528202 0.25107119
## Inference~Working_memory 0.031609938 0.05538430 0.12173446
## Writing~Discourse 0.117842375 -0.02384517 0.27678824
## Vocabulary~Working_memory NA NA NA
## Grammar~Working_memory NA NA NA
## Inference~Vocabulary NA NA NA
## Inference~Grammar NA NA NA
## ToM~Vocabulary NA NA NA
## ToM~Grammar NA NA NA
## ToM~Working_memory NA NA NA
## Spelling~Working_memory NA NA NA
## Sentence_copying~Working_memory NA NA NA
## Discourse~ToM NA NA NA
## Discourse~Vocabulary NA NA NA
## Discourse~Working_memory NA NA NA
## Writing~Spelling NA NA NA
## Inference~phantom Spelling~phantom
## Discourse~Grammar 0.11169709 0.1154681
## Writing~Sentence_copying 0.15064797 -0.1555506
## Discourse~Inference 0.10060722 0.1050860
## Inference~Working_memory -0.07860246 0.1430123
## Writing~Discourse 0.16155105 -0.1890833
## Vocabulary~Working_memory NA NA
## Grammar~Working_memory NA NA
## Inference~Vocabulary NA NA
## Inference~Grammar NA NA
## ToM~Vocabulary NA NA
## ToM~Grammar NA NA
## ToM~Working_memory NA NA
## Spelling~Working_memory NA NA
## Sentence_copying~Working_memory NA NA
## Discourse~ToM NA NA
## Discourse~Vocabulary NA NA
## Discourse~Working_memory NA NA
## Writing~Spelling NA NA
## Sentence_copying~phantom Discourse~phantom
## Discourse~Grammar 0.08826528 0.245836878
## Writing~Sentence_copying -0.05921337 0.001210933
## Discourse~Inference 0.05885676 0.132617696
## Inference~Working_memory -0.02140238 0.020934066
## Writing~Discourse 0.05069574 0.085867787
## Vocabulary~Working_memory NA NA
## Grammar~Working_memory NA NA
## Inference~Vocabulary NA NA
## Inference~Grammar NA NA
## ToM~Vocabulary NA NA
## ToM~Grammar NA NA
## ToM~Working_memory NA NA
## Spelling~Working_memory NA NA
## Sentence_copying~Working_memory NA NA
## Discourse~ToM NA NA
## Discourse~Vocabulary NA NA
## Discourse~Working_memory NA NA
## Writing~Spelling NA NA
## Writing~phantom
## Discourse~Grammar -0.1216320
## Writing~Sentence_copying 0.2289633
## Discourse~Inference 0.2951056
## Inference~Working_memory -0.1961959
## Writing~Discourse 0.6038971
## Vocabulary~Working_memory NA
## Grammar~Working_memory NA
## Inference~Vocabulary NA
## Inference~Grammar NA
## ToM~Vocabulary NA
## ToM~Grammar NA
## ToM~Working_memory NA
## Spelling~Working_memory NA
## Sentence_copying~Working_memory NA
## Discourse~ToM NA
## Discourse~Vocabulary NA
## Discourse~Working_memory NA
## Writing~Spelling NA