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 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 object 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.
## This is lavaan 0.6-6## lavaan is BETA software! Please report any bugs.# 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"))
# 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'
# Perform sensitivity analysis
my.sa <-sa.aco(model = model, sens.model = sens.model, sample.cov = sample.cov,
sample.nobs = 193, n.of.sens.pars = 9, k = 5, max.value= 2000, max.iter = 10,
opt.fun = 4, ## to just significant
paths = c(1:18), seed = 1, verbose = FALSE)
# Note: We run with k = 5 and max.iter = 10 for illustration purpose in 5 seconds,
# please specify them as larger numbers (e.g., default value of k = 50 and mat.iter = 1000)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.
## model.est model.pvalue mean.est.sens
## Vocabulary~Working_memory 0.3999999 5.308198e-12 0.39872261
## Grammar~Working_memory 0.3999999 5.308198e-12 0.39806685
## Inference~Vocabulary 0.4262626 3.005818e-11 0.42224054
## Inference~Grammar 0.2873737 1.336915e-05 0.29423449
## Inference~Working_memory 0.1245455 2.598216e-02 0.12304797
## ToM~Vocabulary 0.2565656 6.016058e-04 0.25315240
## ToM~Grammar 0.2843434 1.345400e-04 0.28882779
## ToM~Working_memory 0.2036364 1.096483e-03 0.20934769
## Spelling~Working_memory 0.3899999 2.976197e-11 0.38871442
## Sentence_copying~Working_memory 0.2399999 3.306180e-04 0.24200482
## Discourse~Inference 0.1943264 3.608645e-02 0.19676153
## Discourse~ToM 0.2610613 1.360093e-03 0.25092806
## Discourse~Vocabulary 0.2589070 4.345386e-03 0.26482507
## Discourse~Grammar 0.1003703 2.544519e-01 0.09706123
## Discourse~Working_memory 0.1888038 7.941219e-03 0.19341256
## Writing~Spelling 0.3759830 3.442786e-05 0.35241630
## Writing~Sentence_copying 0.1719948 4.222710e-02 0.18537904
## Writing~Discourse 0.4504663 3.396553e-07 0.44664294
## Discourse=~TNL 0.7350592 0.000000e+00 0.73380354
## Discourse=~Expository 0.7368861 0.000000e+00 0.73567089
## Writing=~One_day 0.7143560 0.000000e+00 0.71893482
## Writing=~Castle 0.6892360 0.000000e+00 0.69041317
## Vocabulary~~Grammar 0.5714286 0.000000e+00 0.57466718
## Grammar~~Sentence_copying 0.2292836 7.736456e-04 0.23527298
## Vocabulary~~Sentence_copying 0.2405229 3.901946e-04 0.24076833
## Grammar~~Spelling 0.2772707 3.009244e-05 0.27944127
## Vocabulary~~Spelling 0.3602149 8.902825e-09 0.35998171
## Inference~~ToM 0.3298078 2.732152e-07 0.33269903
## Discourse~~Sentence_copying 0.1967324 4.827032e-02 0.19925413
## Discourse~~Spelling 0.1957359 4.141040e-02 0.19941635
## Spelling~~Sentence_copying 0.5217574 0.000000e+00 0.52335331
## TNL~~TNL 0.4596879 6.614931e-12 0.46152452
## Expository~~Expository 0.4569988 8.863354e-12 0.45878036
## One_day~~One_day 0.4896954 5.159029e-11 0.48301356
## Castle~~Castle 0.5249537 6.514789e-13 0.52332680
## Vocabulary~~Vocabulary 0.8400001 0.000000e+00 0.83943125
## Grammar~~Grammar 0.8400001 0.000000e+00 0.83124247
## Inference~~Inference 0.4923051 0.000000e+00 0.48449346
## ToM~~ToM 0.6303566 0.000000e+00 0.61048439
## Spelling~~Spelling 0.8479001 0.000000e+00 0.84211992
## Sentence_copying~~Sentence_copying 0.9424000 0.000000e+00 0.94029028
## Discourse~~Discourse 0.3721435 4.576088e-08 0.36432851
## Writing~~Writing 0.3288733 1.130810e-04 0.25199373
## Working_memory~~Working_memory 1.0000000 NA 0.99081837
## min.est.sens max.est.sens
## Vocabulary~Working_memory 0.39599955 0.4012074
## Grammar~Working_memory 0.38105672 0.4124640
## Inference~Vocabulary 0.40816232 0.4307239
## Inference~Grammar 0.28150851 0.3123527
## Inference~Working_memory 0.11053360 0.1304513
## ToM~Vocabulary 0.24231216 0.2655951
## ToM~Grammar 0.27231808 0.3060838
## ToM~Working_memory 0.20350945 0.2143759
## Spelling~Working_memory 0.38347049 0.3976111
## Sentence_copying~Working_memory 0.23612845 0.2482389
## Discourse~Inference 0.17578823 0.2306160
## Discourse~ToM 0.22393591 0.2771862
## Discourse~Vocabulary 0.25034078 0.2897950
## Discourse~Grammar 0.06420101 0.1180993
## Discourse~Working_memory 0.17964729 0.2022682
## Writing~Spelling 0.32352355 0.3809975
## Writing~Sentence_copying 0.17027667 0.2052543
## Writing~Discourse 0.40783811 0.4958486
## Discourse=~TNL 0.72838185 0.7360754
## Discourse=~Expository 0.73008047 0.7377431
## Writing=~One_day 0.70795320 0.7395872
## Writing=~Castle 0.68857482 0.6933282
## Vocabulary~~Grammar 0.57201775 0.5806482
## Grammar~~Sentence_copying 0.23071073 0.2399883
## Vocabulary~~Sentence_copying 0.23935846 0.2428235
## Grammar~~Spelling 0.27357356 0.2879136
## Vocabulary~~Spelling 0.35897399 0.3609897
## Inference~~ToM 0.29481153 0.3624987
## Discourse~~Sentence_copying 0.19242238 0.2061836
## Discourse~~Spelling 0.19062790 0.2129985
## Spelling~~Sentence_copying 0.52047792 0.5282968
## TNL~~TNL 0.45819305 0.4694599
## Expository~~Expository 0.45573511 0.4669825
## One_day~~One_day 0.45301071 0.4988023
## Castle~~Castle 0.51929601 0.5258647
## Vocabulary~~Vocabulary 0.83884942 0.8400195
## Grammar~~Grammar 0.82560232 0.8375977
## Inference~~Inference 0.47753689 0.4917001
## ToM~~ToM 0.57357690 0.6302682
## Spelling~~Spelling 0.82649124 0.8474983
## Sentence_copying~~Sentence_copying 0.93958271 0.9411195
## Discourse~~Discourse 0.35803750 0.3721311
## Writing~~Writing 0.09107708 0.3212527
## Working_memory~~Working_memory 0.97668330 0.9982559## mean.phan min.phan max.phan
## Working_memory~phantom -0.045513735 -0.15269808 0.10463497
## Grammar~phantom -0.043759615 -0.12231028 0.11282895
## Vocabulary~phantom -0.001922026 -0.02509978 0.03056035
## ToM~phantom 0.083706147 -0.05250043 0.23789872
## Inference~phantom 0.042920720 -0.09825143 0.12036290
## Spelling~phantom 0.027187140 -0.04500182 0.16158107
## Sentence_copying~phantom 0.028406158 -0.04035309 0.05716587
## Discourse~phantom 0.042789464 -0.05532141 0.12343835
## Writing~phantom 0.122684660 -0.19266126 0.48562736## Working_memory~phantom Grammar~phantom
## Vocabulary~Working_memory -0.15269808 -0.12231028
## Grammar~Working_memory -0.15269808 -0.12231028
## Inference~Vocabulary 0.10463497 -0.11959369
## Inference~Grammar -0.04176254 0.11282895
## Inference~Working_memory 0.10463497 -0.11959369
## ToM~Vocabulary -0.08322882 -0.03898185
## ToM~Grammar 0.10463497 -0.11959369
## ToM~Working_memory -0.04176254 0.11282895
## Spelling~Working_memory 0.10463497 -0.11959369
## Sentence_copying~Working_memory 0.10463497 -0.11959369
## Discourse~Inference -0.04176254 0.11282895
## Discourse~ToM -0.08322882 -0.03898185
## Discourse~Vocabulary -0.08322882 -0.03898185
## Discourse~Grammar -0.04176254 0.11282895
## Discourse~Working_memory -0.15269808 -0.12231028
## Writing~Spelling -0.08322882 -0.03898185
## Writing~Sentence_copying -0.08322882 -0.03898185
## Writing~Discourse -0.04176254 0.11282895
## Discourse=~TNL -0.04176254 0.11282895
## Discourse=~Expository -0.08322882 -0.03898185
## Writing=~One_day -0.05451420 -0.05074120
## Writing=~Castle 0.10463497 -0.11959369
## Vocabulary~~Grammar -0.08322882 -0.03898185
## Grammar~~Sentence_copying -0.08322882 -0.03898185
## Vocabulary~~Sentence_copying 0.10463497 -0.11959369
## Grammar~~Spelling -0.05451420 -0.05074120
## Vocabulary~~Spelling 0.10463497 -0.11959369
## Inference~~ToM -0.05451420 -0.05074120
## Discourse~~Sentence_copying -0.05451420 -0.05074120
## Discourse~~Spelling -0.08322882 -0.03898185
## Spelling~~Sentence_copying 0.10463497 -0.11959369
## TNL~~TNL -0.05451420 -0.05074120
## Expository~~Expository -0.05451420 -0.05074120
## One_day~~One_day -0.08322882 -0.03898185
## Castle~~Castle -0.05451420 -0.05074120
## Vocabulary~~Vocabulary -0.04176254 0.11282895
## Grammar~~Grammar -0.15269808 -0.12231028
## Inference~~Inference -0.05451420 -0.05074120
## ToM~~ToM -0.05451420 -0.05074120
## Spelling~~Spelling -0.08322882 -0.03898185
## Sentence_copying~~Sentence_copying -0.15269808 -0.12231028
## Discourse~~Discourse -0.04176254 0.11282895
## Writing~~Writing -0.04176254 0.11282895
## Working_memory~~Working_memory -0.15269808 -0.12231028
## Vocabulary~phantom ToM~phantom
## Vocabulary~Working_memory -0.025099776 0.01102614
## Grammar~Working_memory -0.025099776 0.01102614
## Inference~Vocabulary 0.030560353 -0.05250043
## Inference~Grammar -0.021790195 -0.01060198
## Inference~Working_memory 0.030560353 -0.05250043
## ToM~Vocabulary 0.009949832 0.23789872
## ToM~Grammar 0.030560353 -0.05250043
## ToM~Working_memory -0.021790195 -0.01060198
## Spelling~Working_memory 0.030560353 -0.05250043
## Sentence_copying~Working_memory 0.030560353 -0.05250043
## Discourse~Inference -0.021790195 -0.01060198
## Discourse~ToM 0.009949832 0.23789872
## Discourse~Vocabulary 0.009949832 0.23789872
## Discourse~Grammar -0.021790195 -0.01060198
## Discourse~Working_memory -0.025099776 0.01102614
## Writing~Spelling 0.009949832 0.23789872
## Writing~Sentence_copying 0.009949832 0.23789872
## Writing~Discourse -0.021790195 -0.01060198
## Discourse=~TNL -0.021790195 -0.01060198
## Discourse=~Expository 0.009949832 0.23789872
## Writing=~One_day -0.003230345 0.23270829
## Writing=~Castle 0.030560353 -0.05250043
## Vocabulary~~Grammar 0.009949832 0.23789872
## Grammar~~Sentence_copying 0.009949832 0.23789872
## Vocabulary~~Sentence_copying 0.030560353 -0.05250043
## Grammar~~Spelling -0.003230345 0.23270829
## Vocabulary~~Spelling 0.030560353 -0.05250043
## Inference~~ToM -0.003230345 0.23270829
## Discourse~~Sentence_copying -0.003230345 0.23270829
## Discourse~~Spelling 0.009949832 0.23789872
## Spelling~~Sentence_copying 0.030560353 -0.05250043
## TNL~~TNL -0.003230345 0.23270829
## Expository~~Expository -0.003230345 0.23270829
## One_day~~One_day 0.009949832 0.23789872
## Castle~~Castle -0.003230345 0.23270829
## Vocabulary~~Vocabulary -0.021790195 -0.01060198
## Grammar~~Grammar -0.025099776 0.01102614
## Inference~~Inference -0.003230345 0.23270829
## ToM~~ToM -0.003230345 0.23270829
## Spelling~~Spelling 0.009949832 0.23789872
## Sentence_copying~~Sentence_copying -0.025099776 0.01102614
## Discourse~~Discourse -0.021790195 -0.01060198
## Writing~~Writing -0.021790195 -0.01060198
## Working_memory~~Working_memory -0.025099776 0.01102614
## Inference~phantom Spelling~phantom
## Vocabulary~Working_memory 0.05354083 -0.038059487
## Grammar~Working_memory 0.05354083 -0.038059487
## Inference~Vocabulary 0.10603215 0.060053190
## Inference~Grammar 0.03291916 -0.002637251
## Inference~Working_memory 0.10603215 0.060053190
## ToM~Vocabulary -0.09825143 0.161581071
## ToM~Grammar 0.10603215 0.060053190
## ToM~Working_memory 0.03291916 -0.002637251
## Spelling~Working_memory 0.10603215 0.060053190
## Sentence_copying~Working_memory 0.10603215 0.060053190
## Discourse~Inference 0.03291916 -0.002637251
## Discourse~ToM -0.09825143 0.161581071
## Discourse~Vocabulary -0.09825143 0.161581071
## Discourse~Grammar 0.03291916 -0.002637251
## Discourse~Working_memory 0.05354083 -0.038059487
## Writing~Spelling -0.09825143 0.161581071
## Writing~Sentence_copying -0.09825143 0.161581071
## Writing~Discourse 0.03291916 -0.002637251
## Discourse=~TNL 0.03291916 -0.002637251
## Discourse=~Expository -0.09825143 0.161581071
## Writing=~One_day 0.12036290 -0.045001821
## Writing=~Castle 0.10603215 0.060053190
## Vocabulary~~Grammar -0.09825143 0.161581071
## Grammar~~Sentence_copying -0.09825143 0.161581071
## Vocabulary~~Sentence_copying 0.10603215 0.060053190
## Grammar~~Spelling 0.12036290 -0.045001821
## Vocabulary~~Spelling 0.10603215 0.060053190
## Inference~~ToM 0.12036290 -0.045001821
## Discourse~~Sentence_copying 0.12036290 -0.045001821
## Discourse~~Spelling -0.09825143 0.161581071
## Spelling~~Sentence_copying 0.10603215 0.060053190
## TNL~~TNL 0.12036290 -0.045001821
## Expository~~Expository 0.12036290 -0.045001821
## One_day~~One_day -0.09825143 0.161581071
## Castle~~Castle 0.12036290 -0.045001821
## Vocabulary~~Vocabulary 0.03291916 -0.002637251
## Grammar~~Grammar 0.05354083 -0.038059487
## Inference~~Inference 0.12036290 -0.045001821
## ToM~~ToM 0.12036290 -0.045001821
## Spelling~~Spelling -0.09825143 0.161581071
## Sentence_copying~~Sentence_copying 0.05354083 -0.038059487
## Discourse~~Discourse 0.03291916 -0.002637251
## Writing~~Writing 0.03291916 -0.002637251
## Working_memory~~Working_memory 0.05354083 -0.038059487
## Sentence_copying~phantom Discourse~phantom
## Vocabulary~Working_memory 0.05312295 -0.05532141
## Grammar~Working_memory 0.05312295 -0.05532141
## Inference~Vocabulary 0.03640137 0.01422618
## Inference~Grammar -0.04035309 0.08844906
## Inference~Working_memory 0.03640137 0.01422618
## ToM~Vocabulary 0.05716587 0.04315514
## ToM~Grammar 0.03640137 0.01422618
## ToM~Working_memory -0.04035309 0.08844906
## Spelling~Working_memory 0.03640137 0.01422618
## Sentence_copying~Working_memory 0.03640137 0.01422618
## Discourse~Inference -0.04035309 0.08844906
## Discourse~ToM 0.05716587 0.04315514
## Discourse~Vocabulary 0.05716587 0.04315514
## Discourse~Grammar -0.04035309 0.08844906
## Discourse~Working_memory 0.05312295 -0.05532141
## Writing~Spelling 0.05716587 0.04315514
## Writing~Sentence_copying 0.05716587 0.04315514
## Writing~Discourse -0.04035309 0.08844906
## Discourse=~TNL -0.04035309 0.08844906
## Discourse=~Expository 0.05716587 0.04315514
## Writing=~One_day 0.03569370 0.12343835
## Writing=~Castle 0.03640137 0.01422618
## Vocabulary~~Grammar 0.05716587 0.04315514
## Grammar~~Sentence_copying 0.05716587 0.04315514
## Vocabulary~~Sentence_copying 0.03640137 0.01422618
## Grammar~~Spelling 0.03569370 0.12343835
## Vocabulary~~Spelling 0.03640137 0.01422618
## Inference~~ToM 0.03569370 0.12343835
## Discourse~~Sentence_copying 0.03569370 0.12343835
## Discourse~~Spelling 0.05716587 0.04315514
## Spelling~~Sentence_copying 0.03640137 0.01422618
## TNL~~TNL 0.03569370 0.12343835
## Expository~~Expository 0.03569370 0.12343835
## One_day~~One_day 0.05716587 0.04315514
## Castle~~Castle 0.03569370 0.12343835
## Vocabulary~~Vocabulary -0.04035309 0.08844906
## Grammar~~Grammar 0.05312295 -0.05532141
## Inference~~Inference 0.03569370 0.12343835
## ToM~~ToM 0.03569370 0.12343835
## Spelling~~Spelling 0.05716587 0.04315514
## Sentence_copying~~Sentence_copying 0.05312295 -0.05532141
## Discourse~~Discourse -0.04035309 0.08844906
## Writing~~Writing -0.04035309 0.08844906
## Working_memory~~Working_memory 0.05312295 -0.05532141
## Writing~phantom
## Vocabulary~Working_memory -0.11958035
## Grammar~Working_memory -0.11958035
## Inference~Vocabulary 0.08666854
## Inference~Grammar 0.48562736
## Inference~Working_memory 0.08666854
## ToM~Vocabulary 0.35336901
## ToM~Grammar 0.08666854
## ToM~Working_memory 0.48562736
## Spelling~Working_memory 0.08666854
## Sentence_copying~Working_memory 0.08666854
## Discourse~Inference 0.48562736
## Discourse~ToM 0.35336901
## Discourse~Vocabulary 0.35336901
## Discourse~Grammar 0.48562736
## Discourse~Working_memory -0.11958035
## Writing~Spelling 0.35336901
## Writing~Sentence_copying 0.35336901
## Writing~Discourse 0.48562736
## Discourse=~TNL 0.48562736
## Discourse=~Expository 0.35336901
## Writing=~One_day -0.19266126
## Writing=~Castle 0.08666854
## Vocabulary~~Grammar 0.35336901
## Grammar~~Sentence_copying 0.35336901
## Vocabulary~~Sentence_copying 0.08666854
## Grammar~~Spelling -0.19266126
## Vocabulary~~Spelling 0.08666854
## Inference~~ToM -0.19266126
## Discourse~~Sentence_copying -0.19266126
## Discourse~~Spelling 0.35336901
## Spelling~~Sentence_copying 0.08666854
## TNL~~TNL -0.19266126
## Expository~~Expository -0.19266126
## One_day~~One_day 0.35336901
## Castle~~Castle -0.19266126
## Vocabulary~~Vocabulary 0.48562736
## Grammar~~Grammar -0.11958035
## Inference~~Inference -0.19266126
## ToM~~ToM -0.19266126
## Spelling~~Spelling 0.35336901
## Sentence_copying~~Sentence_copying -0.11958035
## Discourse~~Discourse 0.48562736
## Writing~~Writing 0.48562736
## Working_memory~~Working_memory -0.11958035## Working_memory~phantom Grammar~phantom
## Vocabulary~Working_memory -0.08322882 -0.03898185
## Grammar~Working_memory 0.10463497 -0.11959369
## Inference~Vocabulary -0.04176254 0.11282895
## Inference~Grammar 0.10463497 -0.11959369
## Inference~Working_memory -0.15269808 -0.12231028
## ToM~Vocabulary 0.10463497 -0.11959369
## ToM~Grammar -0.05451420 -0.05074120
## ToM~Working_memory -0.08322882 -0.03898185
## Spelling~Working_memory -0.08322882 -0.03898185
## Sentence_copying~Working_memory -0.15269808 -0.12231028
## Discourse~Inference -0.08322882 -0.03898185
## Discourse~ToM -0.04176254 0.11282895
## Discourse~Vocabulary -0.04176254 0.11282895
## Discourse~Grammar -0.05451420 -0.05074120
## Discourse~Working_memory -0.08322882 -0.03898185
## Writing~Spelling -0.04176254 0.11282895
## Writing~Sentence_copying -0.04176254 0.11282895
## Writing~Discourse -0.05451420 -0.05074120
## Discourse=~TNL -0.05451420 -0.05074120
## Discourse=~Expository -0.05451420 -0.05074120
## Writing=~One_day -0.08322882 -0.03898185
## Writing=~Castle -0.05451420 -0.05074120
## Vocabulary~~Grammar 0.10463497 -0.11959369
## Grammar~~Sentence_copying -0.15269808 -0.12231028
## Vocabulary~~Sentence_copying -0.15269808 -0.12231028
## Grammar~~Spelling 0.10463497 -0.11959369
## Vocabulary~~Spelling -0.05451420 -0.05074120
## Inference~~ToM -0.08322882 -0.03898185
## Discourse~~Sentence_copying -0.04176254 0.11282895
## Discourse~~Spelling -0.05451420 -0.05074120
## Spelling~~Sentence_copying -0.05451420 -0.05074120
## TNL~~TNL -0.04176254 0.11282895
## Expository~~Expository -0.08322882 -0.03898185
## One_day~~One_day -0.05451420 -0.05074120
## Castle~~Castle 0.10463497 -0.11959369
## Vocabulary~~Vocabulary -0.05451420 -0.05074120
## Grammar~~Grammar -0.08322882 -0.03898185
## Inference~~Inference -0.04176254 0.11282895
## ToM~~ToM -0.15269808 -0.12231028
## Spelling~~Spelling -0.04176254 0.11282895
## Sentence_copying~~Sentence_copying 0.10463497 -0.11959369
## Discourse~~Discourse 0.10463497 -0.11959369
## Writing~~Writing 0.10463497 -0.11959369
## Working_memory~~Working_memory -0.04176254 0.11282895
## Vocabulary~phantom ToM~phantom
## Vocabulary~Working_memory 0.009949832 0.23789872
## Grammar~Working_memory 0.030560353 -0.05250043
## Inference~Vocabulary -0.021790195 -0.01060198
## Inference~Grammar 0.030560353 -0.05250043
## Inference~Working_memory -0.025099776 0.01102614
## ToM~Vocabulary 0.030560353 -0.05250043
## ToM~Grammar -0.003230345 0.23270829
## ToM~Working_memory 0.009949832 0.23789872
## Spelling~Working_memory 0.009949832 0.23789872
## Sentence_copying~Working_memory -0.025099776 0.01102614
## Discourse~Inference 0.009949832 0.23789872
## Discourse~ToM -0.021790195 -0.01060198
## Discourse~Vocabulary -0.021790195 -0.01060198
## Discourse~Grammar -0.003230345 0.23270829
## Discourse~Working_memory 0.009949832 0.23789872
## Writing~Spelling -0.021790195 -0.01060198
## Writing~Sentence_copying -0.021790195 -0.01060198
## Writing~Discourse -0.003230345 0.23270829
## Discourse=~TNL -0.003230345 0.23270829
## Discourse=~Expository -0.003230345 0.23270829
## Writing=~One_day 0.009949832 0.23789872
## Writing=~Castle -0.003230345 0.23270829
## Vocabulary~~Grammar 0.030560353 -0.05250043
## Grammar~~Sentence_copying -0.025099776 0.01102614
## Vocabulary~~Sentence_copying -0.025099776 0.01102614
## Grammar~~Spelling 0.030560353 -0.05250043
## Vocabulary~~Spelling -0.003230345 0.23270829
## Inference~~ToM 0.009949832 0.23789872
## Discourse~~Sentence_copying -0.021790195 -0.01060198
## Discourse~~Spelling -0.003230345 0.23270829
## Spelling~~Sentence_copying -0.003230345 0.23270829
## TNL~~TNL -0.021790195 -0.01060198
## Expository~~Expository 0.009949832 0.23789872
## One_day~~One_day -0.003230345 0.23270829
## Castle~~Castle 0.030560353 -0.05250043
## Vocabulary~~Vocabulary -0.003230345 0.23270829
## Grammar~~Grammar 0.009949832 0.23789872
## Inference~~Inference -0.021790195 -0.01060198
## ToM~~ToM -0.025099776 0.01102614
## Spelling~~Spelling -0.021790195 -0.01060198
## Sentence_copying~~Sentence_copying 0.030560353 -0.05250043
## Discourse~~Discourse 0.030560353 -0.05250043
## Writing~~Writing 0.030560353 -0.05250043
## Working_memory~~Working_memory -0.021790195 -0.01060198
## Inference~phantom Spelling~phantom
## Vocabulary~Working_memory -0.09825143 0.161581071
## Grammar~Working_memory 0.10603215 0.060053190
## Inference~Vocabulary 0.03291916 -0.002637251
## Inference~Grammar 0.10603215 0.060053190
## Inference~Working_memory 0.05354083 -0.038059487
## ToM~Vocabulary 0.10603215 0.060053190
## ToM~Grammar 0.12036290 -0.045001821
## ToM~Working_memory -0.09825143 0.161581071
## Spelling~Working_memory -0.09825143 0.161581071
## Sentence_copying~Working_memory 0.05354083 -0.038059487
## Discourse~Inference -0.09825143 0.161581071
## Discourse~ToM 0.03291916 -0.002637251
## Discourse~Vocabulary 0.03291916 -0.002637251
## Discourse~Grammar 0.12036290 -0.045001821
## Discourse~Working_memory -0.09825143 0.161581071
## Writing~Spelling 0.03291916 -0.002637251
## Writing~Sentence_copying 0.03291916 -0.002637251
## Writing~Discourse 0.12036290 -0.045001821
## Discourse=~TNL 0.12036290 -0.045001821
## Discourse=~Expository 0.12036290 -0.045001821
## Writing=~One_day -0.09825143 0.161581071
## Writing=~Castle 0.12036290 -0.045001821
## Vocabulary~~Grammar 0.10603215 0.060053190
## Grammar~~Sentence_copying 0.05354083 -0.038059487
## Vocabulary~~Sentence_copying 0.05354083 -0.038059487
## Grammar~~Spelling 0.10603215 0.060053190
## Vocabulary~~Spelling 0.12036290 -0.045001821
## Inference~~ToM -0.09825143 0.161581071
## Discourse~~Sentence_copying 0.03291916 -0.002637251
## Discourse~~Spelling 0.12036290 -0.045001821
## Spelling~~Sentence_copying 0.12036290 -0.045001821
## TNL~~TNL 0.03291916 -0.002637251
## Expository~~Expository -0.09825143 0.161581071
## One_day~~One_day 0.12036290 -0.045001821
## Castle~~Castle 0.10603215 0.060053190
## Vocabulary~~Vocabulary 0.12036290 -0.045001821
## Grammar~~Grammar -0.09825143 0.161581071
## Inference~~Inference 0.03291916 -0.002637251
## ToM~~ToM 0.05354083 -0.038059487
## Spelling~~Spelling 0.03291916 -0.002637251
## Sentence_copying~~Sentence_copying 0.10603215 0.060053190
## Discourse~~Discourse 0.10603215 0.060053190
## Writing~~Writing 0.10603215 0.060053190
## Working_memory~~Working_memory 0.03291916 -0.002637251
## Sentence_copying~phantom Discourse~phantom
## Vocabulary~Working_memory 0.05716587 0.04315514
## Grammar~Working_memory 0.03640137 0.01422618
## Inference~Vocabulary -0.04035309 0.08844906
## Inference~Grammar 0.03640137 0.01422618
## Inference~Working_memory 0.05312295 -0.05532141
## ToM~Vocabulary 0.03640137 0.01422618
## ToM~Grammar 0.03569370 0.12343835
## ToM~Working_memory 0.05716587 0.04315514
## Spelling~Working_memory 0.05716587 0.04315514
## Sentence_copying~Working_memory 0.05312295 -0.05532141
## Discourse~Inference 0.05716587 0.04315514
## Discourse~ToM -0.04035309 0.08844906
## Discourse~Vocabulary -0.04035309 0.08844906
## Discourse~Grammar 0.03569370 0.12343835
## Discourse~Working_memory 0.05716587 0.04315514
## Writing~Spelling -0.04035309 0.08844906
## Writing~Sentence_copying -0.04035309 0.08844906
## Writing~Discourse 0.03569370 0.12343835
## Discourse=~TNL 0.03569370 0.12343835
## Discourse=~Expository 0.03569370 0.12343835
## Writing=~One_day 0.05716587 0.04315514
## Writing=~Castle 0.03569370 0.12343835
## Vocabulary~~Grammar 0.03640137 0.01422618
## Grammar~~Sentence_copying 0.05312295 -0.05532141
## Vocabulary~~Sentence_copying 0.05312295 -0.05532141
## Grammar~~Spelling 0.03640137 0.01422618
## Vocabulary~~Spelling 0.03569370 0.12343835
## Inference~~ToM 0.05716587 0.04315514
## Discourse~~Sentence_copying -0.04035309 0.08844906
## Discourse~~Spelling 0.03569370 0.12343835
## Spelling~~Sentence_copying 0.03569370 0.12343835
## TNL~~TNL -0.04035309 0.08844906
## Expository~~Expository 0.05716587 0.04315514
## One_day~~One_day 0.03569370 0.12343835
## Castle~~Castle 0.03640137 0.01422618
## Vocabulary~~Vocabulary 0.03569370 0.12343835
## Grammar~~Grammar 0.05716587 0.04315514
## Inference~~Inference -0.04035309 0.08844906
## ToM~~ToM 0.05312295 -0.05532141
## Spelling~~Spelling -0.04035309 0.08844906
## Sentence_copying~~Sentence_copying 0.03640137 0.01422618
## Discourse~~Discourse 0.03640137 0.01422618
## Writing~~Writing 0.03640137 0.01422618
## Working_memory~~Working_memory -0.04035309 0.08844906
## Writing~phantom
## Vocabulary~Working_memory 0.35336901
## Grammar~Working_memory 0.08666854
## Inference~Vocabulary 0.48562736
## Inference~Grammar 0.08666854
## Inference~Working_memory -0.11958035
## ToM~Vocabulary 0.08666854
## ToM~Grammar -0.19266126
## ToM~Working_memory 0.35336901
## Spelling~Working_memory 0.35336901
## Sentence_copying~Working_memory -0.11958035
## Discourse~Inference 0.35336901
## Discourse~ToM 0.48562736
## Discourse~Vocabulary 0.48562736
## Discourse~Grammar -0.19266126
## Discourse~Working_memory 0.35336901
## Writing~Spelling 0.48562736
## Writing~Sentence_copying 0.48562736
## Writing~Discourse -0.19266126
## Discourse=~TNL -0.19266126
## Discourse=~Expository -0.19266126
## Writing=~One_day 0.35336901
## Writing=~Castle -0.19266126
## Vocabulary~~Grammar 0.08666854
## Grammar~~Sentence_copying -0.11958035
## Vocabulary~~Sentence_copying -0.11958035
## Grammar~~Spelling 0.08666854
## Vocabulary~~Spelling -0.19266126
## Inference~~ToM 0.35336901
## Discourse~~Sentence_copying 0.48562736
## Discourse~~Spelling -0.19266126
## Spelling~~Sentence_copying -0.19266126
## TNL~~TNL 0.48562736
## Expository~~Expository 0.35336901
## One_day~~One_day -0.19266126
## Castle~~Castle 0.08666854
## Vocabulary~~Vocabulary -0.19266126
## Grammar~~Grammar 0.35336901
## Inference~~Inference 0.48562736
## ToM~~ToM -0.11958035
## Spelling~~Spelling 0.48562736
## Sentence_copying~~Sentence_copying 0.08666854
## Discourse~~Discourse 0.08666854
## Writing~~Writing 0.08666854
## Working_memory~~Working_memory 0.48562736## p.value p.changed
## Vocabulary~Working_memory 5.308198e-12 NA
## Grammar~Working_memory 5.308198e-12 NA
## Inference~Vocabulary 3.005818e-11 NA
## Inference~Grammar 1.336915e-05 NA
## Inference~Working_memory 2.598216e-02 NA
## ToM~Vocabulary 6.016058e-04 NA
## ToM~Grammar 1.345400e-04 NA
## ToM~Working_memory 1.096483e-03 NA
## Spelling~Working_memory 2.976197e-11 NA
## Sentence_copying~Working_memory 3.306180e-04 NA
## Discourse~Inference 3.608645e-02 0.06205715
## Discourse~ToM 1.360093e-03 NA
## Discourse~Vocabulary 4.345386e-03 NA
## Discourse~Grammar 2.544519e-01 NA
## Discourse~Working_memory 7.941219e-03 NA
## Writing~Spelling 3.442786e-05 NA
## Writing~Sentence_copying 4.222710e-02 NA
## Writing~Discourse 3.396553e-07 NA
## Discourse=~TNL 0.000000e+00 NA
## Discourse=~Expository 0.000000e+00 NA
## Writing=~One_day 0.000000e+00 NA
## Writing=~Castle 0.000000e+00 NA
## Vocabulary~~Grammar 0.000000e+00 NA
## Grammar~~Sentence_copying 7.736456e-04 NA
## Vocabulary~~Sentence_copying 3.901946e-04 NA
## Grammar~~Spelling 3.009244e-05 NA
## Vocabulary~~Spelling 8.902825e-09 NA
## Inference~~ToM 2.732152e-07 NA
## Discourse~~Sentence_copying 4.827032e-02 0.05119927
## Discourse~~Spelling 4.141040e-02 0.05360831
## Spelling~~Sentence_copying 0.000000e+00 NA
## TNL~~TNL 6.614931e-12 NA
## Expository~~Expository 8.863354e-12 NA
## One_day~~One_day 5.159029e-11 NA
## Castle~~Castle 6.514789e-13 NA
## Vocabulary~~Vocabulary 0.000000e+00 NA
## Grammar~~Grammar 0.000000e+00 NA
## Inference~~Inference 0.000000e+00 NA
## ToM~~ToM 0.000000e+00 NA
## Spelling~~Spelling 0.000000e+00 NA
## Sentence_copying~~Sentence_copying 0.000000e+00 NA
## Discourse~~Discourse 4.576088e-08 NA
## Writing~~Writing 1.130810e-04 0.37188286
## Working_memory~~Working_memory NA NA
## Working_memory~phantom Grammar~phantom
## Vocabulary~Working_memory NA NA
## Grammar~Working_memory NA NA
## Inference~Vocabulary NA NA
## Inference~Grammar NA NA
## Inference~Working_memory 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~Inference -0.04176254 0.11282895
## Discourse~ToM NA NA
## Discourse~Vocabulary NA NA
## Discourse~Grammar NA NA
## Discourse~Working_memory NA NA
## Writing~Spelling NA NA
## Writing~Sentence_copying NA NA
## Writing~Discourse NA NA
## Discourse=~TNL NA NA
## Discourse=~Expository NA NA
## Writing=~One_day NA NA
## Writing=~Castle NA NA
## Vocabulary~~Grammar NA NA
## Grammar~~Sentence_copying NA NA
## Vocabulary~~Sentence_copying NA NA
## Grammar~~Spelling NA NA
## Vocabulary~~Spelling NA NA
## Inference~~ToM NA NA
## Discourse~~Sentence_copying -0.08322882 -0.03898185
## Discourse~~Spelling -0.08322882 -0.03898185
## Spelling~~Sentence_copying NA NA
## TNL~~TNL NA NA
## Expository~~Expository NA NA
## One_day~~One_day NA NA
## Castle~~Castle NA NA
## Vocabulary~~Vocabulary NA NA
## Grammar~~Grammar NA NA
## Inference~~Inference NA NA
## ToM~~ToM NA NA
## Spelling~~Spelling NA NA
## Sentence_copying~~Sentence_copying NA NA
## Discourse~~Discourse NA NA
## Writing~~Writing -0.04176254 0.11282895
## Working_memory~~Working_memory NA NA
## Vocabulary~phantom ToM~phantom
## Vocabulary~Working_memory NA NA
## Grammar~Working_memory NA NA
## Inference~Vocabulary NA NA
## Inference~Grammar NA NA
## Inference~Working_memory 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~Inference -0.021790195 -0.01060198
## Discourse~ToM NA NA
## Discourse~Vocabulary NA NA
## Discourse~Grammar NA NA
## Discourse~Working_memory NA NA
## Writing~Spelling NA NA
## Writing~Sentence_copying NA NA
## Writing~Discourse NA NA
## Discourse=~TNL NA NA
## Discourse=~Expository NA NA
## Writing=~One_day NA NA
## Writing=~Castle NA NA
## Vocabulary~~Grammar NA NA
## Grammar~~Sentence_copying NA NA
## Vocabulary~~Sentence_copying NA NA
## Grammar~~Spelling NA NA
## Vocabulary~~Spelling NA NA
## Inference~~ToM NA NA
## Discourse~~Sentence_copying 0.009949832 0.23789872
## Discourse~~Spelling 0.009949832 0.23789872
## Spelling~~Sentence_copying NA NA
## TNL~~TNL NA NA
## Expository~~Expository NA NA
## One_day~~One_day NA NA
## Castle~~Castle NA NA
## Vocabulary~~Vocabulary NA NA
## Grammar~~Grammar NA NA
## Inference~~Inference NA NA
## ToM~~ToM NA NA
## Spelling~~Spelling NA NA
## Sentence_copying~~Sentence_copying NA NA
## Discourse~~Discourse NA NA
## Writing~~Writing -0.021790195 -0.01060198
## Working_memory~~Working_memory NA NA
## Inference~phantom Spelling~phantom
## Vocabulary~Working_memory NA NA
## Grammar~Working_memory NA NA
## Inference~Vocabulary NA NA
## Inference~Grammar NA NA
## Inference~Working_memory 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~Inference 0.03291916 -0.002637251
## Discourse~ToM NA NA
## Discourse~Vocabulary NA NA
## Discourse~Grammar NA NA
## Discourse~Working_memory NA NA
## Writing~Spelling NA NA
## Writing~Sentence_copying NA NA
## Writing~Discourse NA NA
## Discourse=~TNL NA NA
## Discourse=~Expository NA NA
## Writing=~One_day NA NA
## Writing=~Castle NA NA
## Vocabulary~~Grammar NA NA
## Grammar~~Sentence_copying NA NA
## Vocabulary~~Sentence_copying NA NA
## Grammar~~Spelling NA NA
## Vocabulary~~Spelling NA NA
## Inference~~ToM NA NA
## Discourse~~Sentence_copying -0.09825143 0.161581071
## Discourse~~Spelling -0.09825143 0.161581071
## Spelling~~Sentence_copying NA NA
## TNL~~TNL NA NA
## Expository~~Expository NA NA
## One_day~~One_day NA NA
## Castle~~Castle NA NA
## Vocabulary~~Vocabulary NA NA
## Grammar~~Grammar NA NA
## Inference~~Inference NA NA
## ToM~~ToM NA NA
## Spelling~~Spelling NA NA
## Sentence_copying~~Sentence_copying NA NA
## Discourse~~Discourse NA NA
## Writing~~Writing 0.03291916 -0.002637251
## Working_memory~~Working_memory NA NA
## Sentence_copying~phantom Discourse~phantom
## Vocabulary~Working_memory NA NA
## Grammar~Working_memory NA NA
## Inference~Vocabulary NA NA
## Inference~Grammar NA NA
## Inference~Working_memory 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~Inference -0.04035309 0.08844906
## Discourse~ToM NA NA
## Discourse~Vocabulary NA NA
## Discourse~Grammar NA NA
## Discourse~Working_memory NA NA
## Writing~Spelling NA NA
## Writing~Sentence_copying NA NA
## Writing~Discourse NA NA
## Discourse=~TNL NA NA
## Discourse=~Expository NA NA
## Writing=~One_day NA NA
## Writing=~Castle NA NA
## Vocabulary~~Grammar NA NA
## Grammar~~Sentence_copying NA NA
## Vocabulary~~Sentence_copying NA NA
## Grammar~~Spelling NA NA
## Vocabulary~~Spelling NA NA
## Inference~~ToM NA NA
## Discourse~~Sentence_copying 0.05716587 0.04315514
## Discourse~~Spelling 0.05716587 0.04315514
## Spelling~~Sentence_copying NA NA
## TNL~~TNL NA NA
## Expository~~Expository NA NA
## One_day~~One_day NA NA
## Castle~~Castle NA NA
## Vocabulary~~Vocabulary NA NA
## Grammar~~Grammar NA NA
## Inference~~Inference NA NA
## ToM~~ToM NA NA
## Spelling~~Spelling NA NA
## Sentence_copying~~Sentence_copying NA NA
## Discourse~~Discourse NA NA
## Writing~~Writing -0.04035309 0.08844906
## Working_memory~~Working_memory NA NA
## Writing~phantom
## Vocabulary~Working_memory NA
## Grammar~Working_memory NA
## Inference~Vocabulary NA
## Inference~Grammar NA
## Inference~Working_memory NA
## ToM~Vocabulary NA
## ToM~Grammar NA
## ToM~Working_memory NA
## Spelling~Working_memory NA
## Sentence_copying~Working_memory NA
## Discourse~Inference 0.4856274
## Discourse~ToM NA
## Discourse~Vocabulary NA
## Discourse~Grammar NA
## Discourse~Working_memory NA
## Writing~Spelling NA
## Writing~Sentence_copying NA
## Writing~Discourse NA
## Discourse=~TNL NA
## Discourse=~Expository NA
## Writing=~One_day NA
## Writing=~Castle NA
## Vocabulary~~Grammar NA
## Grammar~~Sentence_copying NA
## Vocabulary~~Sentence_copying NA
## Grammar~~Spelling NA
## Vocabulary~~Spelling NA
## Inference~~ToM NA
## Discourse~~Sentence_copying 0.3533690
## Discourse~~Spelling 0.3533690
## Spelling~~Sentence_copying NA
## TNL~~TNL NA
## Expository~~Expository NA
## One_day~~One_day NA
## Castle~~Castle NA
## Vocabulary~~Vocabulary NA
## Grammar~~Grammar NA
## Inference~~Inference NA
## ToM~~ToM NA
## Spelling~~Spelling NA
## Sentence_copying~~Sentence_copying NA
## Discourse~~Discourse NA
## Writing~~Writing 0.4856274
## Working_memory~~Working_memory NA