9 Acquiescence Bias
Questionnaires usually assume that respondents select a response category because it best represents the content of the item. In practice, some people also differ in their tendency to agree with statements independently of what those statements say. This response style is called acquiescence (Cronbach, 1942; Robinson et al., 1991).
Acquiescence is therefore not simply “high responding” or extreme responding. It is a directional tendency toward endorsement that can operate across items with different substantive content.
NoteChapter map
This chapter develops acquiescence from three perspectives:
- What is acquiescence? — a response style with both stable and situational components.
- Why is it difficult to control? — reverse-keyed items can reveal acquiescence but can also introduce wording and comprehension effects.
- How can it be modeled? — factor-analytic response-bias models, random-intercept CFA, and random-intercept EGA.
The final section implements these approaches in R.
9.1 What Does Acquiescence Look Like?
Cronbach (1942) observed that respondents differed in how often they selected the affirmative response in true–false tests. The same logic extends to modern Likert-type questionnaires: an acquiescent respondent has a greater tendency to endorse statements regardless of their substantive polarity.
Consider a simplified Extraversion questionnaire:
| Item | Totally disagree | Partially disagree | Partially agree | Totally agree |
|---|---|---|---|---|
| I am a communicative person | 1 | 2 | 3 | 4 |
| I like interacting with people | 1 | 2 | 3 | 4 |
| I do not feel energized after large social interactions | 1 | 2 | 3 | 4 |
The important pattern is not that the respondent chooses extreme categories. The third response is substantively inconsistent with the first two after accounting for item polarity, yet the respondent still endorses the statement. This illustrates why balanced keying can provide information about acquiescence.
ImportantAcquiescence is different from extreme responding
An extreme responder tends to choose endpoints such as 1 or 5. An acquiescent responder tends to choose the agreement side of the scale. A person can show one response style without showing the other.
9.2 Is Acquiescence a Trait or a State?
Acquiescence is associated with individual and contextual characteristics. Studies have reported relations with age, education, cognitive characteristics, and cultural context (Chen et al., 1995; Hinz et al., 2007; Soto et al., 2008). Twin research also suggests that acquiescence cannot be reduced to a simple inherited individual difference (Kam et al., 2013).
At the same time, some portion of acquiescence appears stable across measurement occasions (Billiet & Davidov, 2008). Latent state–trait analyses indicate that acquiescence contains both a relatively stable component and occasion-specific variation (Danner et al., 2015).
A useful interpretation is therefore:
| Component | Examples |
|---|---|
| Trait-like | Stable response habits, cognitive style, enduring person characteristics |
| State-like | Fatigue, motivation, interview context, momentary attention |
| Instrument-related | Item wording, response labels, balance of keyed directions |
This helps explain why acquiescence can generalize across content domains without necessarily being a psychological personality trait in the same sense as Extraversion or Neuroticism (Ferrando et al., 2004).
9.3 Why Acquiescence Matters
Even a modest acquiescence component can change the covariance structure of questionnaire data. Simulation and empirical studies show that it can distort factor recovery, affect regression or validity coefficients, and alter the apparent dimensionality of scales (Kam et al., 2012; Savalei & Falk, 2014; Valentini, 2017; Valentini & Hauck Filho, 2020).
One intuitive reason is that acquiescence creates shared variance among items that point in the same response direction. If this shared variance is not modeled, a factor analysis can interpret response-style covariance as substantive psychological covariance.
TipControl response style before interpreting covariance
If acquiescence is expected to contribute meaningfully to the item covariance matrix, it should be considered before interpreting reliability, factor structure, or structural relations among latent variables (Billiet & McClendon, 2000; Cambré et al., 2002).
10 Reverse-Keyed Items: Solution and New Problem
A common strategy is to write some items in the positive direction of the construct and others in the opposite direction (Baumgartner & Steenkamp, 2001). For example:
- positive direction: “I often feel sad.”
- opposite direction: “I generally feel cheerful.”
After reverse scoring, both items are intended to point in the same substantive direction. If a respondent simply agrees with both, inconsistent substantive responses provide information about acquiescence.
This logic is useful, but reversing items is not psychometrically neutral.
10.1 1. Opposite wording may change cognitive processing
Negated and reverse-keyed items can require additional reading and interpretation. Respondents may misunderstand the negation, miss the key word, or use a different response process than they use for positively worded items (Van Sonderen et al., 2013; Weijters & Baumgartner, 2012).
These problems can be especially important when reading ability varies across respondents (Marsh, 1996).
10.2 2. Positive and negative items may not be substantively equivalent
Reversing wording assumes that the two item formulations represent opposite locations on one psychological continuum. That assumption is not always defensible (Chang, 1995; Suárez-Alvarez et al., 2018).
Agreeing with “I feel resilient” is not necessarily psychologically equivalent to disagreeing with “I give up easily.” The two statements may activate different content even if they are scored as opposites.
10.3 3. Reverse-keyed items can create method structure
Positive and reverse-keyed items sometimes form separate clusters or factors, particularly when a small subset of respondents answers the reverse-keyed items carelessly (Hughes, 2009; Knight et al., 1988; Woods, 2006).
Research has also found that scales containing only positively worded items can sometimes display cleaner covariance structures than mixed-key versions (Salazar, 2015; Schriesheim & Hill, 1981). This does not imply that acquiescence has disappeared; it may simply become harder to identify because all items share the same response direction.
WarningBetter fit after removing reversed items is not proof that acquiescence is gone
If all items are positively keyed, acquiescence can contribute covariance in the same direction across the entire scale. A cleaner factor solution may therefore reflect less visible response-style conflict, not necessarily less response-style variance.
10.4 Should scales still contain oppositely keyed items?
There is no universal answer. The design involves a tradeoff:
| Potential advantage | Potential cost |
|---|---|
| Provides information for separating agreement tendency from content | Can introduce negation or wording-method effects |
| Helps identify random-intercept/acquiescence factors | Can increase respondent confusion |
| Makes uniform agreement patterns observable | Can create artificial positive-vs-negative item factors |
| Supports some balanced-scale methods | May be inappropriate when the construct has no natural opposite |
The key is to design substantively opposite items rather than simply adding grammatical negation whenever possible.
11 Statistical and Design-Based Control
Acquiescence can be addressed through both the measurement design and the statistical model.
11.1 Statistical control
Many models treat acquiescence as a latent response-style component that operates across items. This generally requires two substantive assumptions:
- the acquiescence component is distinguishable from the substantive latent traits; and
- the same response tendency contributes systematically across multiple items (Ferrando et al., 2003; Savalei & Falk, 2014).
Random-intercept item-factor models are a common example. Each item loads equally on a response-style factor, which captures a respondent’s general tendency to endorse higher or lower categories across the scale (Billiet & McClendon, 2000; Kam et al., 2012).
11.2 Questionnaire design
Scale format can also influence response styles. Research has examined response-category labeling, midpoint availability, keying direction, and alternative formats (Weijters et al., 2010).
Semantic differential scales are one alternative. In some applications, they show cleaner factor structure than agreement-style Likert items (Friborg et al., 2006). Another approach is the expanded format, where each response option is replaced by a substantive statement representing a different position on the construct (Zhang & Savalei, 2016).
NoteDesign and modeling are complementary
A good statistical model cannot fully repair poorly written items, and good item writing cannot guarantee that response styles are absent. Response-process considerations should inform both the questionnaire design and the analysis.
12 Controlling Acquiescence in R
12.1 Approach 1: vampyr
The vampyr package implements the factor-analytic response-bias framework discussed in the previous chapter (Ferrando et al., 2009; Navarro-Gonzalez et al., 2021).
V2 V8 V13 V21
Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
1st Qu.:3.000 1st Qu.:2.000 1st Qu.:1.000 1st Qu.:2.000
Median :4.000 Median :4.000 Median :2.000 Median :3.000
Mean :3.667 Mean :3.263 Mean :2.317 Mean :2.947
3rd Qu.:5.000 3rd Qu.:4.000 3rd Qu.:3.000 3rd Qu.:4.000
Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
V1 V6 V17 V19 V20
Min. :1.000 Min. :1.000 Min. :1.00 Min. :1.000 Min. :1.000
1st Qu.:3.000 1st Qu.:1.000 1st Qu.:3.00 1st Qu.:3.000 1st Qu.:1.000
Median :4.000 Median :2.000 Median :4.00 Median :4.000 Median :2.000
Mean :3.643 Mean :2.467 Mean :3.71 Mean :3.493 Mean :1.997
3rd Qu.:5.000 3rd Qu.:3.000 3rd Qu.:5.00 3rd Qu.:5.000 3rd Qu.:3.000
Max. :5.000 Max. :5.000 Max. :5.00 Max. :5.000 Max. :5.000
V25
Min. :1.000
1st Qu.:1.000
Median :1.000
Mean :1.687
3rd Qu.:2.000
Max. :5.000 The example dataset contains 300 respondents, four social-desirability markers, and six physical-aggression items with both positively and negatively keyed indicators.
DETAILS OF ANALYSIS
Number of participants : 300
Number of items : 10
Items selected as SD items : 1, 2, 3, 4
Dispersion Matrix : Polychoric Correlations
Method for factor extraction : Unweighted Least Squares (ULS)
Rotation Method : none
-----------------------------------------------------------------------
Univariate item descriptives
Item Mean Variance Skewness Kurtosis (Zero centered)
Item 1 3.667 1.260 -0.555 -0.566
Item 2 3.263 1.760 -0.379 -1.005
Item 3 2.317 1.695 0.601 -0.880
Item 4 2.947 1.924 -0.033 -1.284
Item 5 3.643 1.374 -0.565 -0.535
Item 6 2.467 1.802 0.487 -0.967
Item 7 3.710 1.678 -0.652 -0.716
Item 8 3.493 1.629 -0.411 -0.862
Item 9 1.997 1.515 1.041 -0.011
Item 10 1.687 0.925 1.293 0.838
Polychoric correlation is advised when the univariate distributions of ordinal items are
asymmetric or with excess of kurtosis. If both indices are lower than one in absolute value,
then Pearson correlation is advised. You can read more about this subject in:
Muthen, B., & Kaplan D. (1985). A Comparison of Some Methodologies for the Factor Analysis of
Non-Normal Likert Variables. British Journal of Mathematical and Statistical Psychology, 38, 171-189.
Muthen, B., & Kaplan D. (1992). A Comparison of Some Methodologies for the Factor Analysis of
Non-Normal Likert Variables: A Note on the Size of the Model. British Journal of Mathematical
and Statistical Psychology, 45, 19-30.
-----------------------------------------------------------------------
Adequacy of the dispersion matrix
Determinant of the matrix = 0.047816437916936
Bartlett's statistic = 896.4 (df = 45; P = 0.000000)
Kaiser-Meyer-Olkin (KMO) test = 0.76664 (fair)
-----------------------------------------------------------------------
EXPLORATORY FACTOR ANALYSIS CONTROLLING SOCIAL DESIRABILITY AND ACQUIESCENCE
-----------------------------------------------------------------------
Robust Goodness of Fit statistics
Root Mean Square Error of Approximation (RMSEA) = 0.032
Robust Mean-Scaled Chi Square with 23 degrees of freedom = 30.146
Non-Normed Fit Index (NNFI; Tucker & Lewis) = 0.989
Comparative Fit Index (CFI) = 0.994
Goodness of Fit Index (GFI) = 0.977
-----------------------------------------------------------------------
Root Mean Square Residuals (RMSR) = 0.0452
Expected mean value of RMSR for an acceptable model = 0.0578 (Kelley's criterion)
-----------------------------------------------------------------------
Unrotated loading matrix
Factor SD Factor AC Factor 1
Item 1 0.60257 0.00000 0.00000
Item 2 0.51526 0.00000 0.00000
Item 3 0.72710 0.00000 0.00000
Item 4 0.71131 0.00000 0.00000
Item 5 -0.07851 0.23762 -0.54831
Item 6 0.27520 0.00235 0.49059
Item 7 -0.16412 0.57405 -0.70143
Item 8 -0.14320 0.54055 -0.59111
Item 9 0.26559 0.19617 0.66808
Item 10 0.31732 0.06252 0.68219contAC = TRUE requests control for acquiescence. The SD_items argument additionally identifies the social-desirability markers. Therefore:
- to model both social desirability and acquiescence, retain both
SD_itemsandcontAC = TRUE; - to model acquiescence without a social-desirability marker factor, omit the
SD_itemsargument if permitted by the intended workflow; - to disable acquiescence control, use
contAC = FALSE.
If the item set is not balanced, the positions of items that do not have an opposite-key counterpart can be passed through unbalanced_items.
Factor scores can be requested by setting factor_scores = TRUE:
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DETAILS OF ANALYSIS
Number of participants : 300
Number of items : 10
Items selected as SD items : 1, 2, 3, 4
Dispersion Matrix : Polychoric Correlations
Method for factor extraction : Unweighted Least Squares (ULS)
Rotation Method : none
-----------------------------------------------------------------------
Univariate item descriptives
Item Mean Variance Skewness Kurtosis (Zero centered)
Item 1 3.667 1.260 -0.555 -0.566
Item 2 3.263 1.760 -0.379 -1.005
Item 3 2.317 1.695 0.601 -0.880
Item 4 2.947 1.924 -0.033 -1.284
Item 5 3.643 1.374 -0.565 -0.535
Item 6 2.467 1.802 0.487 -0.967
Item 7 3.710 1.678 -0.652 -0.716
Item 8 3.493 1.629 -0.411 -0.862
Item 9 1.997 1.515 1.041 -0.011
Item 10 1.687 0.925 1.293 0.838
Polychoric correlation is advised when the univariate distributions of ordinal items are
asymmetric or with excess of kurtosis. If both indices are lower than one in absolute value,
then Pearson correlation is advised. You can read more about this subject in:
Muthen, B., & Kaplan D. (1985). A Comparison of Some Methodologies for the Factor Analysis of
Non-Normal Likert Variables. British Journal of Mathematical and Statistical Psychology, 38, 171-189.
Muthen, B., & Kaplan D. (1992). A Comparison of Some Methodologies for the Factor Analysis of
Non-Normal Likert Variables: A Note on the Size of the Model. British Journal of Mathematical
and Statistical Psychology, 45, 19-30.
-----------------------------------------------------------------------
Adequacy of the dispersion matrix
Determinant of the matrix = 0.047816437916936
Bartlett's statistic = 896.4 (df = 45; P = 0.000000)
Kaiser-Meyer-Olkin (KMO) test = 0.76664 (fair)
-----------------------------------------------------------------------
EXPLORATORY FACTOR ANALYSIS CONTROLLING SOCIAL DESIRABILITY AND ACQUIESCENCE
-----------------------------------------------------------------------
Robust Goodness of Fit statistics
Root Mean Square Error of Approximation (RMSEA) = 0.032
Robust Mean-Scaled Chi Square with 23 degrees of freedom = 30.146
Non-Normed Fit Index (NNFI; Tucker & Lewis) = 0.989
Comparative Fit Index (CFI) = 0.994
Goodness of Fit Index (GFI) = 0.977
-----------------------------------------------------------------------
Root Mean Square Residuals (RMSR) = 0.0452
Expected mean value of RMSR for an acceptable model = 0.0578 (Kelley's criterion)
-----------------------------------------------------------------------
Unrotated loading matrix
Factor SD Factor AC Factor 1
Item 1 0.60253 0.00000 0.00000
Item 2 0.51526 0.00000 0.00000
Item 3 0.72704 0.00000 0.00000
Item 4 0.71131 0.00000 0.00000
Item 5 -0.07850 0.23753 0.54832
Item 6 0.27518 0.00260 -0.49070
Item 7 -0.16411 0.57389 0.70153
Item 8 -0.14319 0.54036 0.59123
Item 9 0.26558 0.19654 -0.66825
Item 10 0.31730 0.06258 -0.68213
-----------------------------------------------------------------------
RELIABILITY OF EAP SCORES
Factor EAP Reliability estimate
SD : 0.6740
Acquiescence : 0.4113
Factor 1 : 0.6594
PARTICIPANTS'S SCORES ON FACTORS:
Rescaled to mean = 50 and standard deviation = 10 in the sample
Factor SD Factor AC Factor 1
[1,] 44.30693 48.53333 51.44554
[2,] 59.62589 38.14346 50.57165
[3,] 61.19054 42.28909 57.66120
[4,] 55.02773 43.35121 56.66889
[5,] 48.23123 38.65266 39.70465
[6,] 44.51619 64.27199 54.13617
[7,] 44.73218 65.06108 43.76337
[8,] 39.62591 50.47004 27.99325
[9,] 61.35089 36.07724 54.53989
[10,] 52.16276 54.34207 45.29344
[11,] 55.76837 44.66200 43.94625
[12,] 53.40863 58.25979 41.60500
[13,] 46.66442 51.23092 48.62351
[14,] 78.49640 78.23464 44.76011
[15,] 55.73234 54.95433 44.28975
[16,] 44.79371 64.49245 62.46085
[17,] 45.01493 46.09226 43.67809
[18,] 44.88499 59.50155 44.67008
[19,] 43.98230 35.50952 32.96192
[20,] 56.02719 34.92454 45.45263
[21,] 43.26996 33.50331 56.80886
[22,] 55.84810 64.01859 33.23316
[23,] 28.11587 58.09739 66.63817
[24,] 49.33801 42.43831 49.35854
[25,] 39.88321 45.29596 52.73939
[26,] 37.03512 57.52852 57.22556
[27,] 55.01006 47.35659 52.52035
[28,] 55.91189 62.54358 49.19137
[29,] 43.96927 45.20916 43.43612
[30,] 55.62030 44.38434 44.27717
[31,] 37.85433 65.94432 58.47460
[32,] 44.88425 43.32807 54.77204
[33,] 46.71557 40.36808 37.19440
[34,] 65.13345 49.63985 50.15156
[35,] 47.81356 48.87639 49.12380
[36,] 67.06698 41.07036 59.82670
[37,] 44.91543 53.52469 46.46942
[38,] 54.26559 57.86662 32.03135
[39,] 55.62404 44.10899 43.98672
[40,] 43.18532 54.51220 43.88604
[41,] 52.53533 46.38576 41.35634
[42,] 64.91994 50.81471 51.36486
[43,] 65.34059 53.50234 57.04835
[44,] 54.85163 42.89256 45.58115
[45,] 55.73981 55.70850 44.26563
[46,] 52.50882 44.32568 44.69953
[47,] 34.77887 41.68987 47.26047
[48,] 45.26524 45.00766 43.73164
[49,] 32.23291 36.27044 49.27663
[50,] 55.73959 43.90253 55.99082
[51,] 32.22156 56.28429 43.71595
[52,] 49.72514 51.33507 58.72944
[53,] 46.29025 38.95594 48.76390
[54,] 50.73533 58.75385 69.74790
[55,] 56.36102 50.67899 32.93594
[56,] 34.49192 41.64343 61.91512
[57,] 44.14591 50.13327 54.24789
[58,] 44.05135 53.41648 46.56807
[59,] 78.32611 61.80584 54.36919
[60,] 46.94619 53.90221 45.60982
[61,] 78.53930 71.25219 56.53518
[62,] 55.64357 47.46930 44.28069
[63,] 43.98645 53.49732 44.31058
[64,] 57.69061 47.71755 35.95774
[65,] 55.68321 42.59590 46.20676
[66,] 55.60401 43.40642 44.26971
[67,] 55.75838 56.92367 43.60277
[68,] 64.71961 52.13575 43.68025
[69,] 45.40345 60.14602 66.26557
[70,] 48.54799 62.54567 62.93461
[71,] 43.40130 44.68513 54.75671
[72,] 43.79504 58.59898 68.77227
[73,] 37.82186 58.41796 67.94224
[74,] 72.69024 68.24316 59.96968
[75,] 38.54384 60.71244 53.83940
[76,] 44.39554 77.52443 33.66295
[77,] 34.98427 43.00733 33.98936
[78,] 42.29033 37.15494 52.27701
[79,] 55.43507 33.83137 35.09011
[80,] 41.18887 67.26499 56.56664
[81,] 44.31976 50.68758 44.59956
[82,] 78.46775 61.64153 69.53114
[83,] 53.24704 71.48802 27.67509
[84,] 41.16995 50.34212 56.71693
[85,] 55.38692 62.01918 38.04596
[86,] 44.96077 74.37443 21.44334
[87,] 54.44571 39.64716 32.48738
[88,] 51.48805 52.15982 54.15668
[89,] 45.36171 43.59959 40.06175
[90,] 55.82176 45.90922 54.09556
[91,] 37.66055 71.91785 51.35576
[92,] 44.12155 53.92820 46.05766
[93,] 34.84139 66.23853 38.44514
[94,] 54.82210 53.86959 45.82189
[95,] 66.64004 51.15018 60.31911
[96,] 67.81468 78.46926 44.26293
[97,] 56.12442 51.66110 44.52901
[98,] 33.52131 58.28471 67.34880
[99,] 48.00178 51.28965 55.13201
[100,] 54.35116 53.20460 43.29480
[101,] 44.91259 68.10615 55.67961
[102,] 66.49296 44.37002 43.99023
[103,] 49.86741 66.59882 49.96871
[104,] 44.95405 65.13864 32.76418
[105,] 33.06061 48.67840 49.85371
[106,] 75.49319 42.91926 53.95907
[107,] 56.44580 55.56875 34.04476
[108,] 70.56629 54.84289 46.26655
[109,] 56.47430 66.37101 52.40816
[110,] 54.42747 50.12035 40.50429
[111,] 44.33562 46.81305 53.41868
[112,] 55.87860 50.70268 49.30562
[113,] 40.23125 46.74339 53.50564
[114,] 60.99905 46.10467 55.92798
[115,] 55.65068 34.94991 45.25854
[116,] 66.93604 44.63417 56.36218
[117,] 55.99772 56.97442 42.33407
[118,] 55.54234 58.38975 41.49610
[119,] 44.58544 49.90701 50.06975
[120,] 75.70712 57.31583 66.39735
[121,] 53.27660 48.42651 51.57274
[122,] 46.07506 37.19541 44.31314
[123,] 67.24756 55.06597 44.71137
[124,] 38.76309 50.00996 44.56689
[125,] 64.49795 52.48493 47.50387
[126,] 41.72542 58.53579 68.48474
[127,] 43.12616 46.42547 21.98973
[128,] 41.30243 45.13707 32.88256
[129,] 42.51628 62.30525 56.25193
[130,] 31.75905 35.57259 52.85979
[131,] 37.35715 75.96149 33.55345
[132,] 46.13451 58.64768 69.10241
[133,] 43.81907 41.37528 47.29721
[134,] 66.99948 55.75837 43.00825
[135,] 44.32081 36.68745 32.89073
[136,] 42.22120 58.55133 68.55349
[137,] 43.89334 48.57296 51.26597
[138,] 65.19329 49.36649 62.12677
[139,] 42.66200 71.27133 43.43428
[140,] 33.85796 58.29630 67.39340
[141,] 42.68680 46.67931 63.53368
[142,] 44.93812 52.99551 57.75789
[143,] 55.74570 46.77641 44.82189
[144,] 49.66252 58.72872 69.59873
[145,] 32.17716 47.61180 43.31868
[146,] 47.70755 40.15131 59.87204
[147,] 29.64881 58.01817 54.69417
[148,] 33.28783 40.07510 46.30938
[149,] 55.81802 46.33249 42.35236
[150,] 48.65142 58.70883 69.45463
[151,] 47.96954 55.06233 43.82333
[152,] 43.65859 38.22523 38.05658
[153,] 54.17576 46.08657 42.22037
[154,] 61.57690 49.03793 62.95040
[155,] 43.32867 58.58478 68.70746
[156,] 55.71130 58.85820 70.45301
[157,] 53.68410 55.24076 44.77955
[158,] 44.74147 57.75218 58.49819
[159,] 55.04700 47.04627 49.42422
[160,] 44.80917 58.62540 68.91409
[161,] 55.75041 63.31818 56.49360
[162,] 60.59537 46.32220 54.69732
[163,] 68.80053 42.99100 57.15267
[164,] 56.44475 64.90036 35.18837
[165,] 43.77049 43.79944 57.75513
[166,] 55.74922 52.55974 44.48193
[167,] 55.72438 45.39851 44.36239
[168,] 55.67117 49.90305 48.65478
[169,] 56.73583 68.12086 57.22961
[170,] 68.26663 40.70238 53.44398
[171,] 62.50599 67.14426 60.03550
[172,] 53.84318 47.51506 51.77765
[173,] 67.01732 55.69438 44.41920
[174,] 42.47457 57.65714 58.39502
[175,] 67.06228 47.50713 44.30888
[176,] 50.88671 63.05566 47.41429
[177,] 46.05298 60.55398 55.56958
[178,] 54.32736 55.21106 55.86801
[179,] 56.59291 46.34890 55.89768
[180,] 44.87827 49.55050 61.39963
[181,] 58.32160 44.61133 55.67777
[182,] 54.62206 47.86318 51.29531
[183,] 60.13668 58.59063 44.30436
[184,] 34.34853 58.31159 67.46095
[185,] 47.20400 58.67753 69.25098
[186,] 60.62814 46.21580 65.73782
[187,] 78.42786 58.69172 65.24703
[188,] 37.86733 57.55291 57.35776
[189,] 55.41221 42.20911 45.18345
[190,] 47.64655 54.34091 35.66668
[191,] 55.69738 44.73492 55.27301
[192,] 62.74181 44.59657 53.30229
[193,] 40.99746 49.14638 60.60332
[194,] 56.91781 47.76767 23.83859
[195,] 41.66001 40.42141 31.09670
[196,] 55.68602 39.82666 44.87841
[197,] 68.74878 61.65220 44.16419
[198,] 55.57637 56.17315 55.18519
[199,] 55.71428 69.80068 58.16348
[200,] 44.21216 56.42316 43.54466
[201,] 55.18579 47.83881 43.23304
[202,] 44.77232 69.69548 53.72343
[203,] 51.10243 41.47817 52.24413
[204,] 56.46084 77.80894 44.34424
[205,] 39.54510 60.63639 54.09999
[206,] 44.14082 60.15513 66.02529
[207,] 61.72457 43.40962 44.31042
[208,] 54.37954 46.29849 33.90631
[209,] 50.24051 46.51842 53.44186
[210,] 44.67903 48.55167 56.27340
[211,] 58.58784 57.01544 34.89663
[212,] 55.73781 58.86172 70.45222
[213,] 54.03654 32.77720 46.52539
[214,] 55.87161 58.86217 70.47479
[215,] 41.35863 50.65896 49.76773
[216,] 47.11890 52.41924 25.46364
[217,] 50.97341 67.98192 57.65033
[218,] 55.74759 51.40449 57.71154
[219,] 58.51399 43.21493 32.87924
[220,] 51.73471 42.11551 54.14053
[221,] 77.03195 50.62836 20.87354
[222,] 55.68614 66.10544 55.31267
[223,] 45.47022 48.19087 51.84919
[224,] 29.00478 62.70793 52.91521
[225,] 67.09957 44.87906 55.32408
[226,] 44.36715 66.30755 32.77596
[227,] 39.95301 36.14798 41.45555
[228,] 56.17832 36.70424 55.84522
[229,] 45.74690 60.14354 66.33147
[230,] 67.13419 46.00870 54.04177
[231,] 44.94682 55.42967 44.43491
[232,] 55.60031 44.88354 55.04719
[233,] 55.71751 43.85485 44.28774
[234,] 67.10088 44.62123 55.41465
[235,] 67.23702 35.23461 53.41365
[236,] 48.48319 58.70514 69.43099
[237,] 55.69196 44.54694 55.46071
[238,] 43.46558 45.45591 63.94042
[239,] 43.86880 33.04691 55.88409
[240,] 66.99831 44.58163 55.47372
[241,] 56.14206 44.02883 44.84155
[242,] 61.02549 38.22987 54.37303
[243,] 42.84727 51.91398 35.71997
[244,] 55.70342 59.35807 52.03293
[245,] 55.73563 55.37076 55.97727
[246,] 58.90692 58.03032 62.31620
[247,] 55.73359 44.57381 43.58576
[248,] 44.26833 38.82980 47.31009
[249,] 36.53438 75.32455 48.18661
[250,] 78.49796 54.79313 55.80569
[251,] 65.86789 45.78176 24.04964
[252,] 55.80809 44.30993 44.76013
[253,] 71.62196 44.53799 67.72502
[254,] 70.38684 51.52423 46.95256
[255,] 66.84969 36.94340 32.94678
[256,] 66.65261 60.64741 50.32906
[257,] 45.58429 45.56195 64.33035
[258,] 55.70542 47.15318 49.67167
[259,] 48.96152 51.76851 47.60459
[260,] 75.67421 46.47265 53.94876
[261,] 45.70155 46.84392 64.07378
[262,] 51.98984 43.83698 56.16502
[263,] 55.51901 71.16579 56.05550
[264,] 34.67720 29.95117 53.80839
[265,] 44.45824 67.55811 56.35699
[266,] 41.74108 58.53668 68.48679
[267,] 33.97296 58.63711 55.37404
[268,] 50.61076 58.92899 48.50386
[269,] 35.30877 63.29761 60.27346
[270,] 34.33379 34.60294 31.01479
[271,] 53.10675 42.90331 54.11253
[272,] 67.41024 45.39661 44.23895
[273,] 53.08258 60.09076 67.70576
[274,] 55.68563 72.88740 45.77337
[275,] 55.47507 57.59952 53.76175
[276,] 45.34361 58.63728 68.98923
[277,] 36.99583 70.18279 44.38520
[278,] 40.28448 53.76930 44.62264
[279,] 44.44515 56.11669 32.63906
[280,] 45.25829 59.73503 49.26898
[281,] 55.31328 60.07462 68.12385
[282,] 44.27962 62.84108 61.71772
[283,] 55.54118 47.68386 33.87229
[284,] 55.28901 38.82994 42.58729
[285,] 51.51234 58.77026 69.85777
[286,] 43.30875 57.65718 58.54995
[287,] 67.66592 69.67636 43.87207
[288,] 54.75927 65.90884 53.41881
[289,] 46.98778 37.41784 50.10966
[290,] 55.54860 44.00472 44.43358
[291,] 45.58042 46.66428 44.76454
[292,] 62.80568 55.74188 56.24008
[293,] 45.74401 51.35739 48.56243
[294,] 52.38182 44.70606 65.27921
[295,] 57.57962 50.80088 47.36610
[296,] 75.39490 65.89827 62.33904
[297,] 67.16584 45.44543 44.00016
[298,] 58.57901 36.36588 53.30822
[299,] 55.86654 42.66499 55.50919
[300,] 73.41983 45.74091 42.92351
NOTE: The precision matrices for the 3 factors were not printed for preventing console spacing issues.
These matrices are stored in $Precision_matrix in the output variable.As in the social-desirability chapter, these scores are model-dependent estimates rather than directly observed traits.
12.2 Approach 2: Random-Intercept CFA
A random-intercept item-factor model represents acquiescence as a common factor with equal loadings across items (Billiet & McClendon, 2000; Savalei & Falk, 2014). The substantive factors capture content, while the random-intercept factor captures a person’s general tendency to use higher or lower response categories.
We use lavaan (Rosseel, 2012) and the wmt2 dataset distributed with EGAnet (Golino & Christensen, 2023).
This is lavaan 0.7-2
lavaan is FREE software! Please report any bugs.[1;m[4;m
EGAnet (version 2.4.1)[0m[0m
For help getting started, see <https://r-ega.net>
For bugs and errors, submit an issue to <https://github.com/hfgolino/EGAnet/issues>12.2.1 Specify the model
Show random-intercept CFA syntax
model_RI <- '
factor1 =~ NA*wmt1 + wmt2 + wmt3 + wmt5 + wmt11 +
wmt12 + wmt13 + wmt15 + wmt16 + wmt17 + wmt18
factor2 =~ NA*wmt4 + wmt6 + wmt7 + wmt8 +
wmt9 + wmt10 + wmt14
# Random Intercepts
acquiescence =~ 1*wmt1 + 1*wmt2 + 1*wmt3 + 1*wmt5 +
1*wmt11 + 1*wmt12 + 1*wmt13 + 1*wmt15 + 1*wmt16 +
1*wmt17 + 1*wmt18 + 1*wmt4 + 1*wmt6 + 1*wmt7 +
1*wmt8 + 1*wmt9 + 1*wmt10 + 1*wmt14
factor1 ~~ 0*acquiescence
factor2 ~~ 0*acquiescence
acquiescence ~~ acquiescence
factor1 ~~ 1*factor1
factor2 ~~ 1*factor2
'The important part is:
Every indicator has a loading fixed to 1 on the acquiescence factor. This factor therefore represents a common person-specific response tendency rather than a content factor defined by differential item loadings.
The substantive factors are constrained to be orthogonal to the acquiescence factor. That is a strong but useful identification and interpretation assumption; it should be justified rather than treated as a purely technical detail.
12.2.2 Fit the ordinal CFA
lavaan 0.7-2 ended normally after 43 iterations
Estimator DWLS
Optimization method NLMINB
Number of model parameters 38
Number of observations 1185
Model Test User Model:
Standard Scaled
Test Statistic 232.896 285.231
Degrees of freedom 133 133
P-value (Unknown) NA 0.000
Scaling correction factor 0.873
Shift parameter 18.557
simple second-order correction
Model Test Baseline Model:
Test statistic 12385.490 7849.254
Degrees of freedom 153 153
P-value NA 0.000
Scaling correction factor 1.589
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.992 0.980
Tucker-Lewis Index (TLI) 0.991 0.977
Robust Comparative Fit Index (CFI) 0.922
Robust Tucker-Lewis Index (TLI) 0.910
Root Mean Square Error of Approximation:
RMSEA 0.025 0.031
90 Percent confidence interval - lower 0.020 0.026
90 Percent confidence interval - upper 0.030 0.036
P-value H_0: RMSEA <= 0.050 1.000 1.000
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.065
90 Percent confidence interval - lower 0.054
90 Percent confidence interval - upper 0.076
P-value H_0: Robust RMSEA <= 0.050 0.011
P-value H_0: Robust RMSEA >= 0.080 0.012
Standardized Root Mean Square Residual:
SRMR 0.052 0.052
Goodness of Fit Index:
Goodness of Fit Index (GFI) 0.991
90 Percent confidence interval - lower 0.986
90 Percent confidence interval - upper 0.994
Robust GFI 0.941
90 Percent confidence interval - lower 0.921
90 Percent confidence interval - upper 0.958
Parameter Estimates:
Parameterization Delta
Standard errors Robust.sem
Information Expected
Information saturated (h1) model Unstructured
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
factor1 =~
wmt1 0.233 0.065 3.580 0.000 0.233 0.233
wmt2 0.607 0.066 9.213 0.000 0.607 0.607
wmt3 0.449 0.061 7.410 0.000 0.449 0.449
wmt5 0.314 0.062 5.054 0.000 0.314 0.314
wmt11 0.019 0.074 0.260 0.795 0.019 0.019
wmt12 0.061 0.074 0.828 0.408 0.061 0.061
wmt13 0.110 0.069 1.603 0.109 0.110 0.110
wmt15 0.136 0.070 1.947 0.052 0.136 0.136
wmt16 0.126 0.071 1.772 0.076 0.126 0.126
wmt17 -0.044 0.078 -0.568 0.570 -0.044 -0.044
wmt18 -0.339 0.098 -3.452 0.001 -0.339 -0.339
factor2 =~
wmt4 0.300 0.056 5.328 0.000 0.300 0.300
wmt6 0.504 0.052 9.750 0.000 0.504 0.504
wmt7 0.352 0.055 6.452 0.000 0.352 0.352
wmt8 0.269 0.057 4.695 0.000 0.269 0.269
wmt9 0.393 0.054 7.292 0.000 0.393 0.393
wmt10 0.477 0.054 8.910 0.000 0.477 0.477
wmt14 0.227 0.060 3.817 0.000 0.227 0.227
acquiescence =~
wmt1 1.000 0.580 0.580
wmt2 1.000 0.580 0.580
wmt3 1.000 0.580 0.580
wmt5 1.000 0.580 0.580
wmt11 1.000 0.580 0.580
wmt12 1.000 0.580 0.580
wmt13 1.000 0.580 0.580
wmt15 1.000 0.580 0.580
wmt16 1.000 0.580 0.580
wmt17 1.000 0.580 0.580
wmt18 1.000 0.580 0.580
wmt4 1.000 0.580 0.580
wmt6 1.000 0.580 0.580
wmt7 1.000 0.580 0.580
wmt8 1.000 0.580 0.580
wmt9 1.000 0.580 0.580
wmt10 1.000 0.580 0.580
wmt14 1.000 0.580 0.580
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
factor1 ~~
acquiescence 0.000 0.000 0.000
factor2 ~~
acquiescence 0.000 0.000 0.000
factor1 ~~
factor2 0.591 0.078 7.602 0.000 0.591 0.591
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
wmt1|t1 -0.475 0.038 -12.521 0.000 -0.475 -0.475
wmt2|t1 -0.881 0.042 -20.956 0.000 -0.881 -0.881
wmt3|t1 -0.651 0.039 -16.544 0.000 -0.651 -0.651
wmt5|t1 -0.475 0.038 -12.521 0.000 -0.475 -0.475
wmt11|t1 0.447 0.038 11.833 0.000 0.447 0.447
wmt12|t1 0.471 0.038 12.406 0.000 0.471 0.471
wmt13|t1 0.195 0.037 5.311 0.000 0.195 0.195
wmt15|t1 0.445 0.038 11.776 0.000 0.445 0.445
wmt16|t1 0.412 0.038 10.972 0.000 0.412 0.412
wmt17|t1 0.815 0.041 19.787 0.000 0.815 0.815
wmt18|t1 0.641 0.039 16.320 0.000 0.641 0.641
wmt4|t1 -0.158 0.037 -4.325 0.000 -0.158 -0.158
wmt6|t1 -0.355 0.037 -9.533 0.000 -0.355 -0.355
wmt7|t1 -0.208 0.037 -5.658 0.000 -0.208 -0.208
wmt8|t1 0.116 0.037 3.164 0.002 0.116 0.116
wmt9|t1 -0.158 0.037 -4.325 0.000 -0.158 -0.158
wmt10|t1 -0.280 0.037 -7.569 0.000 -0.280 -0.280
wmt14|t1 0.128 0.037 3.513 0.000 0.128 0.128
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
acquiescence 0.337 0.016 21.029 0.000 1.000 1.000
factor1 1.000 1.000 1.000
factor2 1.000 1.000 1.000
.wmt1 0.609 0.609 0.609
.wmt2 0.295 0.295 0.295
.wmt3 0.462 0.462 0.462
.wmt5 0.565 0.565 0.565
.wmt11 0.663 0.663 0.663
.wmt12 0.659 0.659 0.659
.wmt13 0.651 0.651 0.651
.wmt15 0.645 0.645 0.645
.wmt16 0.647 0.647 0.647
.wmt17 0.661 0.661 0.661
.wmt18 0.548 0.548 0.548
.wmt4 0.573 0.573 0.573
.wmt6 0.409 0.409 0.409
.wmt7 0.539 0.539 0.539
.wmt8 0.591 0.591 0.591
.wmt9 0.508 0.508 0.508
.wmt10 0.436 0.436 0.436
.wmt14 0.611 0.611 0.611Inspect:
- standardized substantive loadings;
- the variance of the acquiescence factor;
- global fit;
- residuals and modification diagnostics; and
- whether the substantive solution is interpretable after the response-style factor is included.
12.2.3 Estimate factor scores
When downstream analyses use these scores, remember that both the substantive scores and the acquiescence score inherit the assumptions of the fitted model.
12.3 Approach 3: Random-Intercept EGA
EGAnet also implements random-intercept Exploratory Graph Analysis (riEGA), allowing dimensionality to be explored after separating a general response-style component (Golino & Christensen, 2023).
Warning: Some variables did not belong to a dimension: wmt16
Use caution: These variables have been removed from the TEFI calculationThe random-intercept model converged. Wording effects likely. Results are only valid if data are [4;munrecoded[0m.
The resulting structure can be bootstrapped:
Show bootstrap riEGA code
Warning: Some variables did not belong to a dimension: wmt16
Use caution: These variables have been removed from the TEFI calculationThe random-intercept model converged. Wording effects likely. Results are only valid if data are [4;munrecoded[0m.Warning in order(as.numeric(names(dimension_stability))): NAs introduzidos por
coerção
Inspect the distribution of recovered dimensions:
Model: GLASSO (EBIC)
Correlations: auto
Algorithm: Walktrap
Unidimensional Method: Louvain
----
EGA Type: riEGA
Bootstrap Samples: 500 (Parametric)
1 2 3 4 5 6
Frequency: 0.002 0.212 0.358 0.298 0.108 0.022
Median dimensions: 3 [1.02, 4.98] 95% CIand item stability:
EGA Type: riEGA
Bootstrap Samples: 500 (Parametric)
Proportion Replicated in Dimensions:
wmt1 wmt2 wmt3 wmt4 wmt5 wmt6 wmt7 wmt8 wmt9 wmt10 wmt11 wmt12 wmt13
0.372 0.930 0.866 0.380 0.914 0.550 0.716 0.678 0.582 0.470 0.546 0.570 0.534
wmt14 wmt15 wmt16 wmt17 wmt18
0.638 0.302 NA 0.366 0.824 Network loadings can also be inspected:
The default 'loading.method' has changed to "revised" in {EGAnet} version >= 2.0.7.
For the previous default (version <= 2.0.6), use `loading.method = "original"` 1 2 3 4 NA
wmt1 0.2444576992 0.247258716 -0.099319925 -0.06883984 NaN
wmt17 0.1370056212 -0.074162443 -0.002957388 -0.07293878 NaN
wmt4 0.1259691396 0.047716919 0.000000000 0.03862790 NaN
wmt13 -0.3415948996 0.135160655 -0.027042546 0.06328015 NaN
wmt2 0.0375993097 0.429380347 0.030746520 0.06192070 -Inf
wmt3 -0.0525405946 0.347886178 0.081803378 -0.09782837 NaN
wmt5 0.0009795657 0.169016832 0.000000000 0.00000000 NaN
wmt12 -0.0199842313 -0.014839834 0.000000000 -0.02866109 -Inf
wmt15 -0.0426694270 -0.052333808 0.000000000 0.02031100 NaN
wmt18 -0.0662899852 -0.222850866 -0.098300041 -0.10194983 NaN
wmt7 -0.0291471025 0.037039191 0.285159251 0.01822694 NaN
wmt6 0.0000000000 0.066980959 0.188532984 0.10200772 NaN
wmt8 -0.0206784252 0.000000000 0.166287494 0.03269760 NaN
wmt9 -0.0626554022 0.049519634 0.102199182 0.29389997 NaN
wmt14 -0.0135862414 -0.072701567 0.026287115 0.22635876 NaN
wmt10 -0.0664420920 0.058653950 0.092074489 0.14410384 NaN
wmt11 0.0162793907 -0.017457853 0.000000000 -0.14361692 NaN
wmt16 0.0000000000 0.007163448 0.000000000 0.00000000 NaNLow item stability or unexpected network loadings should trigger substantive investigation. They should not automatically initiate an iterative delete-and-rerun procedure until every statistic exceeds a fixed cutoff.
WarningStability thresholds are diagnostic, not deletion algorithms
An unstable item can represent ambiguous wording, construct overlap, sampling uncertainty, or a genuine boundary between dimensions. Removing it solely to increase stability can narrow the construct and capitalize on the current sample.
12.3.1 Optional CFA follow-up
A factor model based on the EGA structure can be fitted with EGAnet:
[1] "wmt1" "wmt4" "wmt13" "wmt17"
[1] "wmt2" "wmt3" "wmt5" "wmt12" "wmt15" "wmt18"
[1] "wmt6" "wmt7" "wmt8"
[1] "wmt9" "wmt10" "wmt11" "wmt14"Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 1') is deprecated at line 1, pos 1
Fat 1 =~ wmt1 + wmt4 + wmt13 + wmt17
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 1') is deprecated at line 1, pos 1
Fat 1 =~ wmt1 + wmt4 + wmt13 + wmt17
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 1') is deprecated at line 1, pos 1
Fat 1 =~ wmt1 + wmt4 + wmt13 + wmt17
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 1') is deprecated at line 1, pos 1
Fat 1 =~ wmt1 + wmt4 + wmt13 + wmt17
^Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 2') is deprecated at line 2, pos 2
Fat 2 =~ wmt2 + wmt3 + wmt5 + wmt12 + wmt15 + wmt18
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 2') is deprecated at line 2, pos 2
Fat 2 =~ wmt2 + wmt3 + wmt5 + wmt12 + wmt15 + wmt18
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 2') is deprecated at line 2, pos 2
Fat 2 =~ wmt2 + wmt3 + wmt5 + wmt12 + wmt15 + wmt18
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 2') is deprecated at line 2, pos 2
Fat 2 =~ wmt2 + wmt3 + wmt5 + wmt12 + wmt15 + wmt18
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 2') is deprecated at line 2, pos 2
Fat 2 =~ wmt2 + wmt3 + wmt5 + wmt12 + wmt15 + wmt18
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 2') is deprecated at line 2, pos 2
Fat 2 =~ wmt2 + wmt3 + wmt5 + wmt12 + wmt15 + wmt18
^Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 3') is deprecated at line 3, pos 2
Fat 3 =~ wmt6 + wmt7 + wmt8
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 3') is deprecated at line 3, pos 2
Fat 3 =~ wmt6 + wmt7 + wmt8
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 3') is deprecated at line 3, pos 2
Fat 3 =~ wmt6 + wmt7 + wmt8
^Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 4') is deprecated at line 4, pos 2
Fat 4 =~ wmt9 + wmt10 + wmt11 + wmt14
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 4') is deprecated at line 4, pos 2
Fat 4 =~ wmt9 + wmt10 + wmt11 + wmt14
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 4') is deprecated at line 4, pos 2
Fat 4 =~ wmt9 + wmt10 + wmt11 + wmt14
^
Warning: lavaan->lav_parse_check_relational():
having identifiers with spaces ('Fat 4') is deprecated at line 4, pos 2
Fat 4 =~ wmt9 + wmt10 + wmt11 + wmt14
^Warning: lavaan->lav_object_post_check():
covariance matrix of latent variables is not positive definite ; use
lavInspect(fit, "cov.lv") to investigate.
Warning: lavaan->lav_object_post_check():
covariance matrix of latent variables is not positive definite ; use
lavInspect(fit, "cov.lv") to investigate.
Warning: lavaan->lav_object_post_check():
covariance matrix of latent variables is not positive definite ; use
lavInspect(fit, "cov.lv") to investigate.
and inspected with lavaan:
Warning: lavaan->lav_object_post_check():
covariance matrix of latent variables is not positive definite ; use
lavInspect(fit, "cov.lv") to investigate.
Warning: lavaan->lav_object_post_check():
covariance matrix of latent variables is not positive definite ; use
lavInspect(fit, "cov.lv") to investigate.
Warning: lavaan->lav_object_post_check():
covariance matrix of latent variables is not positive definite ; use
lavInspect(fit, "cov.lv") to investigate. npar fmin
40.000 0.098
chisq df
231.983 113.000
pvalue chisq.scaled
NA 323.795
df.scaled pvalue.scaled
113.000 0.000
chisq.scaling.factor baseline.chisq
0.741 11168.058
baseline.df baseline.pvalue
136.000 NA
baseline.chisq.scaled baseline.df.scaled
7229.787 136.000
baseline.pvalue.scaled baseline.chisq.scaling.factor
0.000 1.555
cfi tli
0.989 0.987
cfi.scaled tli.scaled
0.970 0.964
cfi.robust tli.robust
0.888 0.865
nnfi rfi
0.987 0.975
nfi pnfi
0.979 0.814
ifi rni
0.989 0.989
nnfi.scaled rfi.scaled
0.964 0.946
nfi.scaled pnfi.scaled
0.955 0.794
ifi.scaled rni.scaled
0.970 0.970
nnfi.robust rni.robust
0.865 0.888
rmsea rmsea.ci.lower
0.030 0.024
rmsea.ci.upper rmsea.ci.level
0.035 0.900
rmsea.pvalue rmsea.close.h0
1.000 0.050
rmsea.notclose.pvalue rmsea.notclose.h0
0.000 0.080
rmsea.scaled rmsea.ci.lower.scaled
0.040 0.035
rmsea.ci.upper.scaled rmsea.pvalue.scaled
0.045 1.000
rmsea.notclose.pvalue.scaled rmsea.robust
0.000 0.082
rmsea.ci.lower.robust rmsea.ci.upper.robust
0.072 0.093
rmsea.pvalue.robust rmsea.notclose.pvalue.robust
0.000 0.660
rmr rmr_nomean
0.053 0.056
srmr srmr_bentler
0.056 0.053
srmr_bentler_nomean crmr
0.056 0.056
crmr_nomean srmr_mplus
0.059 NA
srmr_mplus_nomean gfi
NA 0.988
gfi.ci.lower gfi.ci.upper
0.984 0.992
gfi.ci.level gfi.robust
0.900 0.917
gfi.ci.lower.robust gfi.ci.upper.robust
0.897 0.936
cn_05 cn_01
709.467 771.074
gfi_lisrel agfi_lisrel
0.983 0.977
pgfi mfi
0.726 0.951
wrmr
1.231
attr(,"scaled.test")
[1] "scaled.shifted"If the structure was discovered and evaluated in the same sample, the CFA is a follow-up description rather than independent confirmation. Cross-validation in a separate sample provides stronger evidence.
Latent scores from the fitted CFA can be estimated as follows:
13 A Practical Acquiescence Checklist
| Stage | Question |
|---|---|
| Item design | Do the positive and opposite-key items represent genuinely opposite content? |
| Response format | Could the category labels themselves encourage agreement? |
| Data structure | Do positive and reverse-keyed items form separate method clusters? |
| Model | Is acquiescence represented independently from substantive factors? |
| Magnitude | Does the response-style component explain nontrivial variance? |
| Stability | Are item assignments stable when acquiescence is controlled? |
| Interpretation | Does control improve substantive interpretability rather than only numerical fit? |
14 Concluding Remarks
Acquiescence is deceptively simple to define but difficult to control. The basic tendency—agreement regardless of content—can create systematic covariance across questionnaire items. Oppositely keyed items make this tendency observable, but they can also introduce their own wording and comprehension effects.
The strongest approach is therefore not to treat reverse wording as a universal remedy. Item design, response format, and statistical modeling should work together. Random-intercept models and related response-bias methods can separate a general endorsement tendency from substantive traits, but the separation is only as defensible as the assumptions that identify the response-style component.
In practical work, the objective is not to obtain a questionnaire that is statistically free of every response tendency. It is to understand which parts of the observed covariance can plausibly be attributed to the target construct and which may arise from the way respondents use the response scale.