8 Social Desirability Bias
Self-report measures are central to research in psychology, education, health, and the social sciences. They are efficient and often provide access to behaviors, evaluations, feelings, and experiences that are not directly observable (Lange & Dewitte, 2019; Peterson & Kerin, 1981). At the same time, responses to self-report items can reflect more than the construct the researcher intends to measure.
Among these additional influences are response styles and response biases. Social desirability, acquiescence, and extreme responding are common examples. These influences matter because they can change item means, associations among items, factor structures, and relations with external variables.
NoteChapter map
This chapter distinguishes faking from socially desirable responding, examines how social desirability can enter a psychometric model, and reviews several strategies for controlling it:
- social-desirability scales;
- factor-analytic response-bias models;
- factor-mixture models;
- common-method-factor models; and
- Peabody-style balanced quadruplets and MIMIC modeling.
The final section implements two of these approaches in R.
8.1 Faking: Context, Motivation, and Opportunity
Faking is best understood as a context-dependent response behavior. A respondent deliberately changes answers in order to produce a self-presentation that serves some objective (Ziegler et al., 2011). The behavior is therefore not determined only by the person or only by the questionnaire. It emerges from an interaction among individual characteristics, situational demands, and features of the assessment itself.
Two broad forms are often distinguished:
| Form | General purpose |
|---|---|
| Faking good | Presenting oneself more favorably than one would under ordinary responding |
| Faking bad | Fabricating or exaggerating undesirable characteristics or symptoms for a desired outcome |
Faking good has been studied extensively in personality and selection contexts (Zickar & Robie, 1999). Faking bad is especially relevant when the respondent may benefit from appearing more impaired or symptomatic (Ziegler et al., 2011).
8.1.1 Why would a person fake?
The expectancy-based account summarized by Ziegler et al. (2011) emphasizes three kinds of beliefs:
- Can I do it? — the respondent believes that changing the intended impression is possible;
- Will it matter? — the respondent believes that altered responses can affect the outcome; and
- Do I value that outcome? — the resulting benefit is sufficiently important.
These beliefs can be influenced by personality, cognitive ability, knowledge of the construct being assessed, item transparency, previous testing experience, and characteristics of the testing situation (McFarland & Ryan, 2000; Raymark & Tafero, 2009; Riggio et al., 1988; Snell et al., 1999).
ImportantFaking is not identical to social desirability
Faking refers to an intentional response process in a particular context. Social desirability is broader: it concerns the tendency for responses to be related to the social evaluation of the content. A high score on a social-desirability scale should therefore not automatically be interpreted as proof that a respondent lied.
8.6 Peabody Quadruplets: Separating Descriptive and Evaluative Content
A different strategy addresses social desirability during item construction. Peabody (1967) distinguished the descriptive content of a trait term from its evaluative desirability. By crossing low versus high descriptive content with low versus high evaluative content, researchers can construct balanced quadruplets.
| Low desirability | High desirability | |
|---|---|---|
| Low descriptive pole | Withdrawn | Introspective |
| High descriptive pole | Chatty | Communicative |
The design can also help separate acquiescence because positive and negative descriptive poles are represented within the same balanced set (Mirowsky & Ross, 1991).
For a four-item quadruplet, a simplified two-factor representation is
\[ \begin{bmatrix} x_1\\ x_2\\ x_3\\ x_4 \end{bmatrix} = \begin{bmatrix} -\lambda_{1c} & +\lambda_{1d}\\ -\lambda_{2c} & -\lambda_{2d}\\ +\lambda_{3c} & +\lambda_{3d}\\ +\lambda_{4c} & -\lambda_{4d} \end{bmatrix} \begin{bmatrix} \eta_c\\ \eta_d \end{bmatrix} + \begin{bmatrix} \epsilon_1\\ \epsilon_2\\ \epsilon_3\\ \epsilon_4 \end{bmatrix}, \]
where \(\eta_c\) represents descriptive content and \(\eta_d\) represents evaluative desirability. Related adjective-based approaches have used this logic to distinguish evaluative and descriptive personality variance (Pettersson et al., 2012, 2014; Saucier et al., 2001).
8.6.1 Extending the desirability factor with a MIMIC model
Degobi & Valentini (2023) evaluated whether a desirability factor identified from manipulated quadruplets could predict desirability effects in additional items through a Multiple Indicators Multiple Causes (MIMIC) model. Their simulation studies considered Likert-type and forced-choice versions of the approach and evaluated recovery using bias and coverage criteria.
The practical idea is:
- identify the desirability factor using items in which evaluative content was deliberately manipulated;
- estimate the substantive content factors from all relevant items; and
- regress additional, non-quadruplet items on the desirability factor.
This avoids requiring every item in a scale to be written as a quadruplet, although the method still depends on the quality and number of the manipulated quadruplets.
NoteDesign tools associated with the MIMIC-quadruplet approach
Two simulation tools were developed for planning these models: quadSimple, intended for early-stage design when little parameter information is available, and quadSim, intended for settings in which researchers already have more detailed information about the expected model parameters.
9 Controlling Social Desirability in R
9.1 Ferrando et al.’s Procedure with vampyr
The vampyr package implements factor-analytic procedures for controlling response bias (Navarro-Gonzalez et al., 2021).
Installation is required only once:
Load the package and inspect its example dataset:
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 contains 10 variables from 300 respondents. Four variables are desirability markers, while the remaining six represent physical aggression. The content items include 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
Items selected as unbalanced : 0
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 0.60252 0.00000 0.00000
Item 2 0.51526 0.00000 0.00000
Item 3 0.72702 0.00000 0.00000
Item 4 0.71131 0.00000 0.00000
Item 5 -0.07850 0.23751 -0.54832
Item 6 0.27517 0.00263 0.49072
Item 7 -0.16411 0.57387 -0.70155
Item 8 -0.14318 0.54034 -0.59125
Item 9 0.26557 0.19660 0.66828
Item 10 0.31730 0.06258 0.68212Here, contAC = TRUE requests simultaneous control for acquiescence. If acquiescence is not part of the analysis, change it to FALSE.
The output should be interpreted at three levels:
- the content-factor solution after response-bias components are taken into account;
- loadings associated with the social-desirability and acquiescence components; and
- the adequacy and interpretability of the resulting factor solution.
Factor scores can also be requested:
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DETAILS OF ANALYSIS
Number of participants : 300
Number of items : 10
Items selected as SD items : 1, 2, 3, 4
Items selected as unbalanced : 0
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 0.51525 0.00000 0.00000
Item 3 0.72709 0.00000 0.00000
Item 4 0.71129 0.00000 0.00000
Item 5 -0.07851 0.23762 -0.54830
Item 6 0.27519 0.00217 0.49050
Item 7 -0.16412 0.57419 -0.70137
Item 8 -0.14320 0.54067 -0.59104
Item 9 0.26559 0.19597 0.66799
Item 10 0.31732 0.06246 0.68221
-----------------------------------------------------------------------
RELIABILITY OF EAP SCORES
Factor EAP Reliability estimate
SD : 0.6741
Acquiescence : 0.4110
Factor 1 : 0.6589
PARTICIPANTS'S SCORES ON FACTORS:
Rescaled to mean = 50 and standard deviation = 10 in the sample
Factor SD Factor AC Factor 1
[1,] 44.30701 48.53108 48.55202
[2,] 59.62138 38.14741 49.43079
[3,] 61.18625 42.28635 42.33527
[4,] 55.02636 43.35037 43.33025
[5,] 48.22922 38.64151 60.28440
[6,] 44.51591 64.27684 45.87749
[7,] 44.73255 65.04926 56.23803
[8,] 39.63917 50.43811 71.99098
[9,] 61.34344 36.08764 45.46285
[10,] 52.16717 54.33677 54.70079
[11,] 55.76827 44.66385 56.05595
[12,] 53.40486 58.26886 58.40459
[13,] 46.67115 51.21173 51.35699
[14,] 78.49690 78.23462 55.24198
[15,] 55.73240 54.95498 55.71045
[16,] 44.79377 64.48834 37.53640
[17,] 45.01397 46.08958 56.32517
[18,] 44.88619 59.48709 55.32836
[19,] 43.98181 35.51016 67.03736
[20,] 56.02792 34.92565 54.53968
[21,] 43.26937 33.50092 43.18549
[22,] 55.84771 64.01367 66.76828
[23,] 28.12220 58.07185 33.26633
[24,] 49.33772 42.43312 50.63642
[25,] 39.89137 45.29797 47.25504
[26,] 37.03565 57.53473 42.78019
[27,] 55.01197 47.35786 47.47995
[28,] 55.91160 62.54166 50.80955
[29,] 43.96945 45.20953 56.56501
[30,] 55.62014 44.38368 55.72272
[31,] 37.85532 65.93751 41.52338
[32,] 44.88456 43.34268 45.22780
[33,] 46.71661 40.35834 62.79386
[34,] 65.13218 49.65353 49.86109
[35,] 47.81878 48.87813 50.87092
[36,] 67.06675 41.07778 40.17719
[37,] 44.91480 53.52407 53.52993
[38,] 54.25462 57.87583 67.98566
[39,] 55.62406 44.10900 56.01380
[40,] 43.18239 54.50944 56.11575
[41,] 52.53007 46.39364 58.65074
[42,] 64.91130 50.83205 48.65427
[43,] 65.34185 53.50412 42.94909
[44,] 54.85252 42.88793 54.41311
[45,] 55.73982 55.70866 55.73461
[46,] 52.50584 44.32355 55.29879
[47,] 34.77558 41.69290 52.74273
[48,] 45.26303 45.00804 56.26994
[49,] 32.23582 36.27421 50.71474
[50,] 55.73954 43.90114 44.00651
[51,] 32.22493 56.28736 56.28759
[52,] 49.72712 51.33979 41.26882
[53,] 46.28992 38.95460 51.22886
[54,] 50.74997 58.68553 30.04858
[55,] 56.36378 50.67707 67.06379
[56,] 34.49170 41.64022 38.07818
[57,] 44.14537 50.13263 45.75367
[58,] 44.05198 53.41548 53.43084
[59,] 78.32488 61.79794 45.64526
[60,] 46.94949 53.90589 54.39186
[61,] 78.53937 71.24588 43.46483
[62,] 55.64334 47.45841 55.71923
[63,] 43.98731 53.49975 55.68979
[64,] 57.69312 47.71475 64.03548
[65,] 55.68321 42.58519 53.78265
[66,] 55.60386 43.40730 55.72978
[67,] 55.75809 56.92097 56.40342
[68,] 64.71213 52.12510 56.32229
[69,] 45.40363 60.13967 33.71294
[70,] 48.54707 62.54332 37.06513
[71,] 43.40332 44.68100 45.23626
[72,] 43.79681 58.55066 31.07520
[73,] 37.82992 58.37995 31.92960
[74,] 72.68739 68.25381 40.04592
[75,] 38.54145 60.71585 46.16882
[76,] 44.39734 77.52058 66.33631
[77,] 34.98962 43.00002 66.00357
[78,] 42.29032 37.14529 47.71327
[79,] 55.43490 33.82119 64.89849
[80,] 41.19005 67.25730 43.43257
[81,] 44.31976 50.67432 55.39842
[82,] 78.46787 61.62999 30.43687
[83,] 53.24780 71.49634 72.34327
[84,] 41.17021 50.35090 43.28430
[85,] 55.38927 62.00664 61.94293
[86,] 44.96839 74.33731 78.48550
[87,] 54.44504 39.63063 67.51161
[88,] 51.48323 52.16362 45.84701
[89,] 45.36002 43.61214 59.94967
[90,] 55.82145 45.90406 45.89933
[91,] 37.66105 71.90680 48.64857
[92,] 44.12224 53.92678 53.94084
[93,] 34.83375 66.23422 61.57234
[94,] 54.82501 53.86442 54.17176
[95,] 66.64026 51.15448 39.68492
[96,] 67.81858 78.46801 55.73766
[97,] 56.12518 51.65467 55.46954
[98,] 33.52509 58.25343 32.54090
[99,] 48.00185 51.29224 44.86860
[100,] 54.34950 53.19997 56.70573
[101,] 44.91288 68.09801 44.32200
[102,] 66.49211 44.36762 56.00912
[103,] 49.86799 66.59253 50.03713
[104,] 44.95695 65.11863 67.23711
[105,] 33.06385 48.68058 50.14241
[106,] 75.49482 42.91828 46.03487
[107,] 56.44246 55.56866 65.95981
[108,] 70.56547 54.84812 53.73933
[109,] 56.47425 66.36626 47.59875
[110,] 54.43004 50.10789 59.49633
[111,] 44.33457 46.82482 46.59368
[112,] 55.87862 50.68531 50.67717
[113,] 40.23216 46.74074 46.49343
[114,] 60.99477 46.11130 44.07640
[115,] 55.65058 34.95109 54.73447
[116,] 66.93588 44.63601 43.63745
[117,] 55.99766 56.98890 57.68375
[118,] 55.54200 58.41736 58.53135
[119,] 44.58476 49.91317 49.93617
[120,] 75.70628 57.32543 33.60860
[121,] 53.27321 48.43855 48.43924
[122,] 46.07408 37.19317 55.68566
[123,] 67.24767 55.06490 55.28679
[124,] 38.76287 50.00618 55.43151
[125,] 64.49811 52.49385 52.50490
[126,] 41.73244 58.49105 31.37057
[127,] 43.14170 46.41235 77.99035
[128,] 41.30294 45.13437 67.11730
[129,] 42.51574 62.30447 43.75108
[130,] 31.75776 35.56369 47.12999
[131,] 37.36628 75.93414 66.44332
[132,] 46.14688 58.59219 30.72566
[133,] 43.81891 41.37265 52.70027
[134,] 66.99872 55.76471 57.00761
[135,] 44.32075 36.68566 67.10887
[136,] 42.22617 58.50578 31.30020
[137,] 43.89432 48.57712 48.73682
[138,] 65.19128 49.37094 37.87734
[139,] 42.66219 71.28397 56.58937
[140,] 33.86136 58.26464 32.49538
[141,] 42.68994 46.67950 36.45908
[142,] 44.93871 52.98705 42.23349
[143,] 55.74581 46.76498 55.17295
[144,] 49.67967 58.66335 30.20444
[145,] 32.17281 47.61256 56.68694
[146,] 47.70305 40.14519 40.12169
[147,] 29.64896 58.02735 45.31533
[148,] 33.28932 40.06241 53.67718
[149,] 55.81780 46.34863 57.66354
[150,] 48.66224 58.64674 30.35768
[151,] 47.97415 55.06853 56.18060
[152,] 43.65891 38.22845 61.94341
[153,] 54.17812 46.08165 57.77608
[154,] 61.57328 49.03333 37.04562
[155,] 43.33181 58.53729 31.14177
[156,] 55.71442 58.77401 29.30729
[157,] 53.68776 55.24015 55.22009
[158,] 44.74144 57.76006 41.50823
[159,] 55.04712 47.06459 50.58313
[160,] 44.81192 58.57472 30.92741
[161,] 55.75063 63.31866 43.50675
[162,] 60.59029 46.32817 45.30926
[163,] 68.79728 42.98774 42.84378
[164,] 56.44139 64.91748 64.83057
[165,] 43.76941 43.79634 42.24331
[166,] 55.74944 52.56716 55.52064
[167,] 55.72438 45.39506 55.63701
[168,] 55.67130 49.90725 51.34229
[169,] 56.73583 68.11540 42.76953
[170,] 68.26755 40.71651 46.55638
[171,] 62.50534 67.15730 39.97988
[172,] 53.84667 47.51465 48.21877
[173,] 67.01734 55.69224 55.57967
[174,] 42.47363 57.66482 41.61226
[175,] 67.06214 47.50433 55.69087
[176,] 50.89323 63.06702 52.59677
[177,] 46.05382 60.55515 44.43494
[178,] 54.32947 55.21161 44.13109
[179,] 56.59253 46.35268 44.10420
[180,] 44.87849 49.54493 38.59122
[181,] 58.31664 44.61238 44.32336
[182,] 54.62445 47.85768 48.69596
[183,] 60.12665 58.57206 55.69575
[184,] 34.35325 58.27920 32.42570
[185,] 47.21502 58.61982 30.57190
[186,] 60.62144 46.21341 34.25969
[187,] 78.42770 58.70180 34.76222
[188,] 37.86695 57.55925 42.64818
[189,] 55.41232 42.21178 54.81368
[190,] 47.63988 54.35476 64.35083
[191,] 55.69734 44.73767 44.72971
[192,] 62.74632 44.58665 46.68503
[193,] 41.00049 49.14216 39.38732
[194,] 56.91885 47.76392 76.15910
[195,] 41.66044 40.41011 68.90212
[196,] 55.68583 39.83030 55.11935
[197,] 68.74465 61.61819 55.83638
[198,] 55.57550 56.17652 44.81807
[199,] 55.71430 69.79323 41.83343
[200,] 44.21235 56.42914 56.46142
[201,] 55.18636 47.82623 56.76818
[202,] 44.77243 69.69359 46.28913
[203,] 51.10428 41.49555 47.76118
[204,] 56.46256 77.80215 55.65609
[205,] 39.54438 60.63929 45.90737
[206,] 44.14108 60.14961 33.95550
[207,] 61.72461 43.41597 55.69178
[208,] 54.38081 46.29163 66.08591
[209,] 50.23607 46.52342 46.56268
[210,] 44.67939 48.55557 43.72594
[211,] 58.58177 57.01626 65.11569
[212,] 55.73808 58.77796 29.30964
[213,] 54.03617 32.77291 53.46468
[214,] 55.87395 58.77757 29.28467
[215,] 41.35666 50.67146 50.24655
[216,] 47.10619 52.42834 74.55462
[217,] 50.97880 67.98246 42.35222
[218,] 55.74765 51.41262 42.28641
[219,] 58.51372 43.21230 67.12063
[220,] 51.74153 42.13171 45.85932
[221,] 77.23746 50.45728 78.38560
[222,] 55.68603 66.10306 44.69160
[223,] 45.46759 48.19078 48.15096
[224,] 29.00415 62.70807 47.09119
[225,] 67.09954 44.87643 44.67446
[226,] 44.36776 66.29706 67.22509
[227,] 39.95165 36.14293 58.53979
[228,] 56.17736 36.70008 44.14978
[229,] 45.74829 60.13696 33.64622
[230,] 67.13420 46.00472 45.95450
[231,] 44.94890 55.42788 55.56346
[232,] 55.60048 44.88343 44.95203
[233,] 55.71750 43.85535 55.71209
[234,] 67.10077 44.62075 44.58494
[235,] 67.23646 35.23233 46.58278
[236,] 48.49785 58.64362 30.38196
[237,] 55.69186 44.54974 44.54204
[238,] 43.46603 45.45763 36.05378
[239,] 43.86774 33.04188 44.11042
[240,] 66.99800 44.58227 44.52703
[241,] 56.14267 44.02456 55.15464
[242,] 61.02502 38.24302 45.62884
[243,] 42.85201 51.89690 64.26466
[244,] 55.70343 59.37584 47.98454
[245,] 55.73565 55.37287 44.02408
[246,] 58.90467 58.04529 37.69489
[247,] 55.73357 44.57461 56.41493
[248,] 44.26878 38.82406 52.68050
[249,] 36.53861 75.31109 51.80845
[250,] 78.49793 54.79512 44.19334
[251,] 65.90953 45.74340 75.74219
[252,] 55.80844 44.30286 55.23457
[253,] 71.61781 44.53133 32.26731
[254,] 70.39185 51.52310 53.04271
[255,] 66.84941 36.94742 67.05290
[256,] 66.65350 60.65353 49.67673
[257,] 45.58459 45.56319 35.66377
[258,] 55.70545 47.17249 50.33604
[259,] 48.96286 51.77813 52.40321
[260,] 75.67159 46.46903 46.04864
[261,] 45.70171 46.84214 35.91939
[262,] 51.98463 43.83745 43.83529
[263,] 55.51929 71.16138 43.94496
[264,] 34.67687 29.94087 46.18497
[265,] 44.45839 67.56616 43.65206
[266,] 41.74766 58.49198 31.36866
[267,] 33.97261 58.64133 44.63196
[268,] 50.61057 58.91399 51.48937
[269,] 35.30987 63.30393 39.73258
[270,] 34.33509 34.58958 68.97580
[271,] 53.11267 42.91709 45.88642
[272,] 67.40973 45.39324 55.76110
[273,] 53.08524 60.07884 32.25711
[274,] 55.68584 72.87877 54.23249
[275,] 55.47554 57.60425 46.24301
[276,] 45.34846 58.58506 30.84816
[277,] 37.00517 70.17388 55.61616
[278,] 40.28614 53.76325 55.37406
[279,] 44.44572 56.11214 67.36096
[280,] 45.25826 59.73420 50.73335
[281,] 55.31350 60.06111 31.83511
[282,] 44.27975 62.83794 38.28146
[283,] 55.54177 47.69208 66.12952
[284,] 55.28961 38.82365 57.40962
[285,] 51.52534 58.69948 29.93308
[286,] 43.30860 57.66488 41.45712
[287,] 67.66624 69.66626 56.13217
[288,] 54.75810 65.92594 46.60027
[289,] 46.98766 37.41970 49.88570
[290,] 55.54859 44.00506 55.56584
[291,] 45.58112 46.66336 55.23592
[292,] 62.80624 55.73447 43.75547
[293,] 45.74769 51.33571 51.41575
[294,] 52.38584 44.70768 34.71360
[295,] 57.57734 50.81225 52.64297
[296,] 75.39585 65.89203 37.66219
[297,] 67.16580 45.44132 55.99977
[298,] 58.57378 36.37319 46.69537
[299,] 55.86659 42.67761 44.49157
[300,] 73.41784 45.75003 57.08362
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.
WarningFactor scores are model-dependent estimates
The resulting scores are not bias-free observed traits. They are estimates conditional on the response-bias model and its assumptions, including the treatment of the desirability markers.
9.2 MIMIC Modeling with Peabody Quadruplets
The second example follows the MIMIC-quadruplet logic evaluated by Degobi & Valentini (2023). We use lavaan (Rosseel, 2012) for model simulation and estimation and semPlot (Epskamp, 2022) for a path diagram.
This is lavaan 0.7-2
lavaan is FREE software! Please report any bugs.9.2.1 1. Simulate a design with quadruplets
The following code creates two substantive factors, a social-desirability factor identified by 16 quadruplet items, and 10 additional items whose responses are also influenced by social desirability.
Show data-simulation code
#Quadruple Factor Loadings on Social Desirability
FactorLoadingsSDQ<- rep(0.3, 16)*c(1,-1,1,-1)
#Quadruple Factor Loadings on the Target Construct
RandomFactorLoadingsQ<-rep(0.7, 16)*c(-1,-1,1,1)
# Factor Loads of the extra item in the Target Construct
set.seed(2021)
RandomFactorLoadings <- round(runif((10), min = .3, max = .8), 3)
# Desirability Regressions for Target Construct items
set.seed(2021)
RandomSDregression <- round(runif((10), min = .1, max = .5), 3)
# Item Thresholds
set.seed(2020)
thld1Vet<-round(runif(26, min=-2, max=.5),3)
thld2Vet<-round(thld1Vet +.5,3)
thld3Vet<-round(thld1Vet + 1,3)
thld4Vet<-round(thld1Vet + 1.5,3)
# Simulated Model
simModel <- paste0("fator1 =~",RandomFactorLoadings[1],"*it1 +",
RandomFactorLoadings[2],"*it2 +",
RandomFactorLoadings[3],"*it3 +",
RandomFactorLoadings[4],"*it4 +",
RandomFactorLoadings[5],"*it5 +",
RandomFactorLoadingsQ[1],"*sd1 +",
RandomFactorLoadingsQ[2],"*sd2 +",
RandomFactorLoadingsQ[3],"*sd3 +",
RandomFactorLoadingsQ[4],"*sd4 +",
RandomFactorLoadingsQ[5],"*sd5 +",
RandomFactorLoadingsQ[6],"*sd6 +",
RandomFactorLoadingsQ[7],"*sd7 +",
RandomFactorLoadingsQ[8],"*sd8\n",
"fator2 =~", RandomFactorLoadingsQ[6],"*it6 +",
RandomFactorLoadingsQ[7],"*it7 +",
RandomFactorLoadingsQ[8],"*it8 +",
RandomFactorLoadingsQ[9],"*it9 +",
RandomFactorLoadingsQ[10],"*it10 +",
RandomFactorLoadingsQ[9],"*sd9 +",
RandomFactorLoadingsQ[10],"*sd10 +",
RandomFactorLoadingsQ[11],"*sd11 +",
RandomFactorLoadingsQ[12],"*sd12 +",
RandomFactorLoadingsQ[13],"*sd13 +",
RandomFactorLoadingsQ[14],"*sd14 +",
RandomFactorLoadingsQ[15],"*sd15 +",
RandomFactorLoadingsQ[16],"*sd16\n",
"SD =~", FactorLoadingsSDQ[1], "*sd1 +",
FactorLoadingsSDQ[2],"*sd2 +",
FactorLoadingsSDQ[3],"*sd3 +",
FactorLoadingsSDQ[4],"*sd4 +",
FactorLoadingsSDQ[5], "*sd5 +",
FactorLoadingsSDQ[6],"*sd6 +",
FactorLoadingsSDQ[7],"*sd7 +",
FactorLoadingsSDQ[8],"*sd8 +",
FactorLoadingsSDQ[9], "*sd9 +",
FactorLoadingsSDQ[10],"*sd10 +",
FactorLoadingsSDQ[11],"*sd11 +",
FactorLoadingsSDQ[12],"*sd12 +",
FactorLoadingsSDQ[13], "*sd13 +",
FactorLoadingsSDQ[14],"*sd14 +",
FactorLoadingsSDQ[15],"*sd15 +",
FactorLoadingsSDQ[16],"*sd16\n",
"SD ~~ 1*SD\n",
"fator1 ~~ 1*fator1\n",
"fator2 ~~ 1*fator2\n",
"fator1 ~~ 0*SD\n",
"fator2 ~~ 0*SD\n",
"fator1 ~~ .3*fator2\n",
"it1 ~",RandomSDregression[1],"*SD\n",
"it2 ~",RandomSDregression[2],"*SD\n",
"it3 ~",RandomSDregression[3],"*SD\n",
"it4 ~",RandomSDregression[4],"*SD\n",
"it5 ~",RandomSDregression[5],"*SD\n",
"it6 ~",RandomSDregression[6],"*SD\n",
"it7 ~",RandomSDregression[7],"*SD\n",
"it8 ~",RandomSDregression[8],"*SD\n",
"it9 ~",RandomSDregression[9],"*SD\n",
"it10 ~",RandomSDregression[10],"*SD\n",
"sd1 |",thld1Vet[1],"*t1 +", thld2Vet[1], "*t2 +",
thld3Vet[1],"*t3 +",thld4Vet[1],"*t4\n",
"sd2 |",thld1Vet[2],"*t1 +", thld2Vet[2], "*t2 +",
thld3Vet[2],"*t3 +",thld4Vet[2],"*t4\n",
"sd3 |",thld1Vet[3],"*t1 +", thld2Vet[3], "*t2 +",
thld3Vet[3],"*t3 +",thld4Vet[3],"*t4\n",
"sd4 |",thld1Vet[4],"*t1 +", thld2Vet[4], "*t2 +",
thld3Vet[4],"*t3 +",thld4Vet[4],"*t4\n",
"it1 |",thld1Vet[5],"*t1 +", thld2Vet[5], "*t2 +",
thld3Vet[5],"*t3 +",thld4Vet[5],"*t4\n",
"it2 |",thld1Vet[6],"*t1 +", thld2Vet[6], "*t2 +",
thld3Vet[6],"*t3 +",thld4Vet[6],"*t4\n",
"it3 |",thld1Vet[7],"*t1 +", thld2Vet[7], "*t2 +",
thld3Vet[7],"*t3 +",thld4Vet[7],"*t4\n",
"it4 |",thld1Vet[8],"*t1 +", thld2Vet[8], "*t2 +",
thld3Vet[8],"*t3 +",thld4Vet[8],"*t4\n",
"it5 |",thld1Vet[9],"*t1 +", thld2Vet[9], "*t2 +",
thld3Vet[9],"*t3 +",thld4Vet[9],"*t4\n",
"it6 |",thld1Vet[10],"*t1 +", thld2Vet[10], "*t2 +",
thld3Vet[10],"*t3 +",thld4Vet[10],"*t4\n",
"it7 |",thld1Vet[11],"*t1 +", thld2Vet[11], "*t2 +",
thld3Vet[11],"*t3 +",thld4Vet[11],"*t4\n",
"it8 |",thld1Vet[12],"*t1 +", thld2Vet[12], "*t2 +",
thld3Vet[12],"*t3 +",thld4Vet[12],"*t4\n",
"it9 |",thld1Vet[13],"*t1 +", thld2Vet[13], "*t2 +",
thld3Vet[13],"*t3 +",thld4Vet[13],"*t4\n",
"it10 |",thld1Vet[14],"*t1 +", thld2Vet[14], "*t2 +",
thld3Vet[14],"*t3 +",thld4Vet[14],"*t4\n",
"sd5 |",thld1Vet[15],"*t1 +", thld2Vet[15], "*t2 +",
thld3Vet[15],"*t3 +",thld4Vet[15],"*t4\n",
"sd6 |",thld1Vet[16],"*t1 +", thld2Vet[16], "*t2 +",
thld3Vet[16],"*t3 +",thld4Vet[16],"*t4\n",
"sd7 |",thld1Vet[17],"*t1 +", thld2Vet[17], "*t2 +",
thld3Vet[17],"*t3 +",thld4Vet[17],"*t4\n",
"sd8 |",thld1Vet[18],"*t1 +", thld2Vet[18], "*t2 +",
thld3Vet[18],"*t3 +",thld4Vet[18],"*t4\n",
"sd9 |",thld1Vet[19],"*t1 +", thld2Vet[19], "*t2 +",
thld3Vet[19],"*t3 +",thld4Vet[19],"*t4\n",
"sd10 |",thld1Vet[20],"*t1 +", thld2Vet[20], "*t2 +",
thld3Vet[20],"*t3 +",thld4Vet[20],"*t4\n",
"sd11 |",thld1Vet[21],"*t1 +", thld2Vet[21], "*t2 +",
thld3Vet[21],"*t3 +",thld4Vet[21],"*t4\n",
"sd12 |",thld1Vet[22],"*t1 +", thld2Vet[22], "*t2 +",
thld3Vet[22],"*t3 +",thld4Vet[22],"*t4\n",
"sd13 |",thld1Vet[23],"*t1 +", thld2Vet[23], "*t2 +",
thld3Vet[23],"*t3 +",thld4Vet[23],"*t4\n",
"sd14 |",thld1Vet[24],"*t1 +", thld2Vet[24], "*t2 +",
thld3Vet[24],"*t3 +",thld4Vet[24],"*t4\n",
"sd15 |",thld1Vet[25],"*t1 +", thld2Vet[25], "*t2 +",
thld3Vet[25],"*t3 +",thld4Vet[25],"*t4\n",
"sd16 |",thld1Vet[26],"*t1 +", thld2Vet[26], "*t2 +",
thld3Vet[26],"*t3 +",thld4Vet[26],"*t4")
#Simulating the Data
simulatedData <- lavaan::simulateData(model = simModel,
model.type = "sem",
sample.nobs = 4000,
seed = 2024,
return.type = "data.frame",
standardized = TRUE
)The simulated dataset contains 26 five-category items: 16 items arranged as four quadruplets and 10 additional items outside the quadruplet format.
it1 it2 it3 it4
Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
1st Qu.:3.000 1st Qu.:4.000 1st Qu.:4.000 1st Qu.:2.000
Median :5.000 Median :5.000 Median :5.000 Median :4.000
Mean :4.142 Mean :4.297 Mean :4.151 Mean :3.356
3rd Qu.:5.000 3rd Qu.:5.000 3rd Qu.:5.000 3rd Qu.:5.000
Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
it5 sd1 sd2 sd3
Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
1st Qu.:4.000 1st Qu.:1.000 1st Qu.:2.000 1st Qu.:1.000
Median :5.000 Median :2.000 Median :4.000 Median :2.000
Mean :4.436 Mean :2.507 Mean :3.344 Mean :2.606
3rd Qu.:5.000 3rd Qu.:4.000 3rd Qu.:5.000 3rd Qu.:4.000
Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
sd4 sd5 sd6 sd7 sd8
Min. :1.000 Min. :1.00 Min. :1.000 Min. :1.000 Min. :1.000
1st Qu.:2.000 1st Qu.:2.00 1st Qu.:1.000 1st Qu.:1.000 1st Qu.:1.000
Median :3.000 Median :3.00 Median :3.000 Median :1.000 Median :2.000
Mean :3.082 Mean :3.31 Mean :2.857 Mean :1.639 Mean :2.485
3rd Qu.:4.000 3rd Qu.:5.00 3rd Qu.:4.000 3rd Qu.:2.000 3rd Qu.:4.000
Max. :5.000 Max. :5.00 Max. :5.000 Max. :5.000 Max. :5.000
it6 it7 it8 it9 it10
Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.00
1st Qu.:1.000 1st Qu.:1.000 1st Qu.:1.000 1st Qu.:1.000 1st Qu.:2.00
Median :2.000 Median :2.000 Median :2.000 Median :1.000 Median :3.00
Mean :2.589 Mean :2.133 Mean :2.196 Mean :1.961 Mean :3.26
3rd Qu.:4.000 3rd Qu.:3.000 3rd Qu.:3.000 3rd Qu.:3.000 3rd Qu.:5.00
Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.00
sd9 sd10 sd11 sd12
Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
1st Qu.:1.000 1st Qu.:3.000 1st Qu.:3.000 1st Qu.:4.000
Median :3.000 Median :4.000 Median :5.000 Median :5.000
Mean :2.835 Mean :3.772 Mean :3.957 Mean :4.281
3rd Qu.:4.000 3rd Qu.:5.000 3rd Qu.:5.000 3rd Qu.:5.000
Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
sd13 sd14 sd15 sd16
Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
1st Qu.:1.000 1st Qu.:1.000 1st Qu.:1.000 1st Qu.:3.000
Median :1.000 Median :1.000 Median :1.000 Median :5.000
Mean :1.982 Mean :1.679 Mean :1.813 Mean :4.061
3rd Qu.:3.000 3rd Qu.:2.000 3rd Qu.:2.000 3rd Qu.:5.000
Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000 9.2.2 2. Specify the empirical MIMIC model
All content-relevant items define their substantive factor. The desirability factor is identified only by the manipulated quadruplet items, and the additional items are regressed on that desirability factor.
The substantive factors are constrained to be orthogonal to the desirability factor for identification in this demonstration. That constraint is a modeling assumption and should be considered carefully in empirical applications.
Show MIMIC-quadruplet model syntax
empiricalModel <- "
factor1 =~ NA*it1 + it2 + it3 + it4 + it5 + sd1 + sd2 + sd3 +
sd4 + sd5 + sd6 + sd7 + sd8
factor2 =~ NA*it6 + it7 + it8 + it9 + it10 + sd9 + sd10 + sd11 +
sd12 + sd13 + sd14 + sd15 + sd16
SD =~ NA*sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 +
sd10 + sd11 + sd12 + sd13 + sd14 + sd15 + sd16
SD ~~ 1*SD
factor1 ~~ 1*factor1
factor2 ~~ 1*factor2
factor1 ~~ 0*SD
factor2 ~~ 0*SD
factor1 ~~ factor2
it1 ~ SD
it2 ~ SD
it3 ~ SD
it4 ~ SD
it5 ~ SD
it6 ~ SD
it7 ~ SD
it8 ~ SD
it9 ~ SD
it10 ~SD"9.2.3 3. Fit the model for ordinal responses
Because the simulated items are ordinal, the model is estimated with WLSMV.
lavaan 0.7-2 ended normally after 43 iterations
Estimator DWLS
Optimization method NLMINB
Number of model parameters 157
Number of observations 4000
Model Test User Model:
Standard Scaled
Test Statistic 134.313 262.042
Degrees of freedom 272 272
P-value (Unknown) NA 0.657
Scaling correction factor 0.829
Shift parameter 100.035
simple second-order correction
Model Test Baseline Model:
Test statistic 182910.056 68572.439
Degrees of freedom 325 325
P-value NA 0.000
Scaling correction factor 2.675
User Model versus Baseline Model:
Comparative Fit Index (CFI) 1.000 1.000
Tucker-Lewis Index (TLI) 1.001 1.000
Robust Comparative Fit Index (CFI) 1.000
Robust Tucker-Lewis Index (TLI) 1.000
Root Mean Square Error of Approximation:
RMSEA 0.000 0.000
90 Percent confidence interval - lower 0.000 0.000
90 Percent confidence interval - upper 0.000 0.005
P-value H_0: RMSEA <= 0.050 1.000 1.000
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.000
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.009
P-value H_0: Robust RMSEA <= 0.050 1.000
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.011 0.011
Goodness of Fit Index:
Goodness of Fit Index (GFI) 1.000
90 Percent confidence interval - lower 1.000
90 Percent confidence interval - upper 1.000
Robust GFI 1.000
90 Percent confidence interval - lower 0.998
90 Percent confidence interval - upper 1.000
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 =~
it1 0.533 0.015 34.817 0.000 0.533 0.533
it2 0.710 0.013 54.436 0.000 0.710 0.710
it3 0.657 0.013 49.952 0.000 0.657 0.657
it4 0.469 0.015 31.176 0.000 0.469 0.469
it5 0.629 0.015 42.507 0.000 0.629 0.629
sd1 -0.702 0.011 -63.668 0.000 -0.702 -0.702
sd2 -0.710 0.011 -66.074 0.000 -0.710 -0.710
sd3 0.702 0.011 64.968 0.000 0.702 0.702
sd4 0.695 0.011 62.271 0.000 0.695 0.695
sd5 -0.686 0.011 -60.281 0.000 -0.686 -0.686
sd6 -0.711 0.011 -66.961 0.000 -0.711 -0.711
sd7 0.691 0.013 51.760 0.000 0.691 0.691
sd8 0.684 0.012 58.123 0.000 0.684 0.684
factor2 =~
it6 0.706 0.011 64.652 0.000 0.706 0.706
it7 -0.693 0.012 -57.861 0.000 -0.693 -0.693
it8 -0.703 0.011 -63.503 0.000 -0.703 -0.703
it9 0.694 0.012 56.075 0.000 0.694 0.694
it10 0.701 0.012 59.331 0.000 0.701 0.701
sd9 0.674 0.011 59.334 0.000 0.674 0.674
sd10 0.697 0.011 61.714 0.000 0.697 0.697
sd11 -0.720 0.012 -62.435 0.000 -0.720 -0.720
sd12 -0.688 0.013 -54.044 0.000 -0.688 -0.688
sd13 0.709 0.011 61.910 0.000 0.709 0.709
sd14 0.702 0.013 55.003 0.000 0.702 0.702
sd15 -0.692 0.013 -54.627 0.000 -0.692 -0.692
sd16 -0.688 0.012 -58.482 0.000 -0.688 -0.688
SD =~
sd1 0.313 0.019 16.640 0.000 0.313 0.313
sd2 -0.259 0.019 -13.382 0.000 -0.259 -0.259
sd3 0.298 0.019 16.043 0.000 0.298 0.298
sd4 -0.307 0.019 -16.431 0.000 -0.307 -0.307
sd5 0.303 0.019 15.838 0.000 0.303 0.303
sd6 -0.318 0.018 -17.525 0.000 -0.318 -0.318
sd7 0.310 0.021 14.747 0.000 0.310 0.310
sd8 -0.309 0.019 -16.377 0.000 -0.309 -0.309
sd9 0.356 0.018 19.303 0.000 0.356 0.356
sd10 -0.292 0.020 -14.681 0.000 -0.292 -0.292
sd11 0.308 0.020 15.377 0.000 0.308 0.308
sd12 -0.331 0.020 -16.296 0.000 -0.331 -0.331
sd13 0.290 0.020 14.403 0.000 0.290 0.290
sd14 -0.302 0.021 -14.363 0.000 -0.302 -0.302
sd15 0.312 0.021 15.141 0.000 0.312 0.312
sd16 -0.295 0.020 -14.743 0.000 -0.295 -0.295
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
it1 ~
SD 0.278 0.020 14.008 0.000 0.278 0.278
it2 ~
SD 0.413 0.020 21.023 0.000 0.413 0.413
it3 ~
SD 0.371 0.020 18.908 0.000 0.371 0.371
it4 ~
SD 0.280 0.019 15.075 0.000 0.280 0.280
it5 ~
SD 0.364 0.020 18.041 0.000 0.364 0.364
it6 ~
SD 0.394 0.018 21.530 0.000 0.394 0.394
it7 ~
SD 0.382 0.019 19.824 0.000 0.382 0.382
it8 ~
SD 0.213 0.020 10.436 0.000 0.213 0.213
it9 ~
SD 0.424 0.019 22.368 0.000 0.424 0.424
it10 ~
SD 0.502 0.017 29.643 0.000 0.502 0.502
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
factor1 ~~
SD 0.000 0.000 0.000
factor2 ~~
SD 0.000 0.000 0.000
factor1 ~~
factor2 -0.289 0.017 -17.396 0.000 -0.289 -0.289
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
it1|t1 -1.655 0.034 -49.192 0.000 -1.655 -1.655
it1|t2 -1.141 0.025 -45.092 0.000 -1.141 -1.141
it1|t3 -0.665 0.021 -30.938 0.000 -0.665 -0.665
it1|t4 -0.180 0.020 -9.007 0.000 -0.180 -0.180
it2|t1 -1.780 0.037 -48.455 0.000 -1.780 -1.780
it2|t2 -1.314 0.027 -47.808 0.000 -1.314 -1.314
it2|t3 -0.818 0.022 -36.464 0.000 -0.818 -0.818
it2|t4 -0.347 0.020 -17.134 0.000 -0.347 -0.347
it3|t1 -1.672 0.034 -49.127 0.000 -1.672 -1.672
it3|t2 -1.144 0.025 -45.163 0.000 -1.144 -1.144
it3|t3 -0.685 0.022 -31.696 0.000 -0.685 -0.685
it3|t4 -0.180 0.020 -9.007 0.000 -0.180 -0.180
it4|t1 -0.997 0.024 -41.791 0.000 -0.997 -0.997
it4|t2 -0.515 0.021 -24.758 0.000 -0.515 -0.515
it4|t3 -0.012 0.020 -0.601 0.548 -0.012 -0.012
it4|t4 0.486 0.021 23.483 0.000 0.486 0.486
it5|t1 -1.986 0.043 -46.008 0.000 -1.986 -1.986
it5|t2 -1.497 0.030 -49.192 0.000 -1.497 -1.497
it5|t3 -0.995 0.024 -41.764 0.000 -0.995 -0.995
it5|t4 -0.485 0.021 -23.452 0.000 -0.485 -0.485
sd1|t1 -0.383 0.020 -18.828 0.000 -0.383 -0.383
sd1|t2 0.113 0.020 5.690 0.000 0.113 0.113
sd1|t3 0.622 0.021 29.229 0.000 0.622 0.622
sd1|t4 1.100 0.025 44.246 0.000 1.100 1.100
sd2|t1 -0.997 0.024 -41.791 0.000 -0.997 -0.997
sd2|t2 -0.475 0.021 -22.985 0.000 -0.475 -0.475
sd2|t3 -0.019 0.020 -0.980 0.327 -0.019 -0.019
sd2|t4 0.486 0.021 23.483 0.000 0.486 0.486
sd3|t1 -0.456 0.021 -22.144 0.000 -0.456 -0.456
sd3|t2 0.035 0.020 1.771 0.077 0.035 0.035
sd3|t3 0.537 0.021 25.718 0.000 0.537 0.537
sd3|t4 1.044 0.024 42.979 0.000 1.044 1.044
sd4|t1 -0.803 0.022 -35.966 0.000 -0.803 -0.803
sd4|t2 -0.325 0.020 -16.097 0.000 -0.325 -0.325
sd4|t3 0.182 0.020 9.133 0.000 0.182 0.182
sd4|t4 0.713 0.022 32.752 0.000 0.713 0.713
sd5|t1 -0.974 0.024 -41.196 0.000 -0.974 -0.974
sd5|t2 -0.466 0.021 -22.580 0.000 -0.466 -0.466
sd5|t3 0.033 0.020 1.644 0.100 0.033 0.033
sd5|t4 0.500 0.021 24.105 0.000 0.500 0.500
sd6|t1 -0.647 0.021 -30.238 0.000 -0.647 -0.647
sd6|t2 -0.129 0.020 -6.512 0.000 -0.129 -0.129
sd6|t3 0.346 0.020 17.071 0.000 0.346 0.346
sd6|t4 0.843 0.023 37.308 0.000 0.843 0.843
sd7|t1 0.404 0.020 19.768 0.000 0.404 0.404
sd7|t2 0.895 0.023 38.915 0.000 0.895 0.895
sd7|t3 1.405 0.029 48.691 0.000 1.405 1.405
sd7|t4 1.881 0.040 47.440 0.000 1.881 1.881
sd8|t1 -0.374 0.020 -18.389 0.000 -0.374 -0.374
sd8|t2 0.135 0.020 6.765 0.000 0.135 0.135
sd8|t3 0.633 0.021 29.688 0.000 0.633 0.633
sd8|t4 1.129 0.025 44.854 0.000 1.129 1.129
it6|t1 -0.440 0.021 -21.426 0.000 -0.440 -0.440
it6|t2 0.058 0.020 2.909 0.004 0.058 0.058
it6|t3 0.548 0.021 26.152 0.000 0.548 0.548
it6|t4 1.036 0.024 42.797 0.000 1.036 1.036
it7|t1 -0.073 0.020 -3.699 0.000 -0.073 -0.073
it7|t2 0.421 0.020 20.551 0.000 0.421 0.421
it7|t3 0.890 0.023 38.773 0.000 0.890 0.890
it7|t4 1.403 0.029 48.678 0.000 1.403 1.403
it8|t1 -0.148 0.020 -7.428 0.000 -0.148 -0.148
it8|t2 0.348 0.020 17.165 0.000 0.348 0.348
it8|t3 0.872 0.023 38.230 0.000 0.872 0.872
it8|t4 1.390 0.029 48.573 0.000 1.390 1.390
it9|t1 0.056 0.020 2.814 0.005 0.056 0.056
it9|t2 0.564 0.021 26.831 0.000 0.564 0.564
it9|t3 1.069 0.025 43.570 0.000 1.069 1.069
it9|t4 1.603 0.033 49.310 0.000 1.603 1.603
it10|t1 -0.929 0.023 -39.928 0.000 -0.929 -0.929
it10|t2 -0.434 0.021 -21.176 0.000 -0.434 -0.434
it10|t3 0.045 0.020 2.276 0.023 0.045 0.045
it10|t4 0.564 0.021 26.831 0.000 0.564 0.564
sd9|t1 -0.655 0.021 -30.542 0.000 -0.655 -0.655
sd9|t2 -0.121 0.020 -6.069 0.000 -0.121 -0.121
sd9|t3 0.364 0.020 17.919 0.000 0.364 0.364
sd9|t4 0.896 0.023 38.943 0.000 0.896 0.896
sd10|t1 -1.339 0.028 -48.095 0.000 -1.339 -1.339
sd10|t2 -0.840 0.023 -37.192 0.000 -0.840 -0.840
sd10|t3 -0.330 0.020 -16.317 0.000 -0.330 -0.330
sd10|t4 0.167 0.020 8.375 0.000 0.167 0.167
sd11|t1 -1.467 0.030 -49.069 0.000 -1.467 -1.467
sd11|t2 -0.979 0.024 -41.332 0.000 -0.979 -0.979
sd11|t3 -0.485 0.021 -23.452 0.000 -0.485 -0.485
sd11|t4 -0.014 0.020 -0.727 0.467 -0.014 -0.014
sd12|t1 -1.881 0.040 -47.440 0.000 -1.881 -1.881
sd12|t2 -1.323 0.028 -47.912 0.000 -1.323 -1.323
sd12|t3 -0.798 0.022 -35.790 0.000 -0.798 -0.798
sd12|t4 -0.297 0.020 -14.745 0.000 -0.297 -0.297
sd13|t1 0.054 0.020 2.719 0.007 0.054 0.054
sd13|t2 0.532 0.021 25.502 0.000 0.532 0.532
sd13|t3 1.047 0.024 43.057 0.000 1.047 1.047
sd13|t4 1.565 0.032 49.324 0.000 1.565 1.565
sd14|t1 0.352 0.020 17.385 0.000 0.352 0.352
sd14|t2 0.857 0.023 37.742 0.000 0.857 0.857
sd14|t3 1.356 0.028 48.269 0.000 1.356 1.356
sd14|t4 1.835 0.038 47.947 0.000 1.835 1.835
sd15|t1 0.206 0.020 10.301 0.000 0.206 0.206
sd15|t2 0.717 0.022 32.903 0.000 0.717 0.717
sd15|t3 1.198 0.026 46.141 0.000 1.198 1.198
sd15|t4 1.728 0.035 48.838 0.000 1.728 1.728
sd16|t1 -1.616 0.033 -49.290 0.000 -1.616 -1.616
sd16|t2 -1.090 0.025 -44.047 0.000 -1.090 -1.090
sd16|t3 -0.579 0.021 -27.479 0.000 -0.579 -0.579
sd16|t4 -0.083 0.020 -4.205 0.000 -0.083 -0.083
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
SD 1.000 1.000 1.000
factor1 1.000 1.000 1.000
factor2 1.000 1.000 1.000
.it1 0.639 0.639 0.639
.it2 0.326 0.326 0.326
.it3 0.431 0.431 0.431
.it4 0.701 0.701 0.701
.it5 0.471 0.471 0.471
.sd1 0.409 0.409 0.409
.sd2 0.429 0.429 0.429
.sd3 0.419 0.419 0.419
.sd4 0.422 0.422 0.422
.sd5 0.438 0.438 0.438
.sd6 0.392 0.392 0.392
.sd7 0.427 0.427 0.427
.sd8 0.437 0.437 0.437
.it6 0.346 0.346 0.346
.it7 0.374 0.374 0.374
.it8 0.460 0.460 0.460
.it9 0.339 0.339 0.339
.it10 0.257 0.257 0.257
.sd9 0.419 0.419 0.419
.sd10 0.428 0.428 0.428
.sd11 0.387 0.387 0.387
.sd12 0.417 0.417 0.417
.sd13 0.413 0.413 0.413
.sd14 0.415 0.415 0.415
.sd15 0.424 0.424 0.424
.sd16 0.439 0.439 0.439Inspect global fit, standardized content and desirability loadings, and the regressions from SD to it1–it10. In a simulation, recovery can also be evaluated against the parameter values used to generate the data.
9.2.4 4. Estimate latent scores
The resulting object contains model-based latent-score estimates appended to the data.
9.2.5 5. Visualize the model
For large models, path diagrams can become visually dense. They are most useful for checking the intended parameterization rather than replacing numerical inspection of the fitted model.
10 Choosing a Strategy
No social-desirability control is assumption-free. A useful choice depends on what information the study design can provide.
| Situation | Particularly relevant strategy |
|---|---|
| Existing dataset with a separate SD scale | Sensitivity analyses; factor-based control if marker assumptions are plausible |
| Hypothesis that only some respondents fake | Factor-mixture model |
| Experimental manipulation of response context | Common-method-factor / repeated-measures design |
| New instrument under development | Peabody-style balanced content or related item-design strategies |
| Desire to extend an identified evaluative factor to ordinary items | MIMIC-quadruplet model |
ImportantControl is not automatically better than no control
A response-bias adjustment is useful only when the model separates nuisance variance from substantive variance credibly. A poorly identified desirability factor can remove meaningful construct variance and produce scores that appear cleaner while being less valid.
11 Concluding Remarks
Social desirability is not a single statistical nuisance with a universal correction. It can reflect intentional self-presentation, self-favoring beliefs, socially valued substantive traits, item evaluation, or shared method variance. Different control procedures target different parts of that problem.
The practical lesson is therefore to make the assumed response process explicit. If social desirability is represented as a latent method factor, the researcher should explain why the factor is identifiable and why it is distinguishable from substantive content. If it is addressed through item design, the evaluative and descriptive manipulations should be defensible. If a separate desirability scale is used, its score should not be treated automatically as a pure measure of dishonesty.
The next chapter turns to another response tendency—acquiescence—and shows why simply reversing item wording is not always sufficient to remove it.
8.4 Social Desirability as Systematic Score Variance
In Classical Test Theory, an observed score is commonly written as
\[ X = T + E, \]
where \(T\) is the expected score across hypothetical replications of the same measurement procedure and \(E\) is random error.
A key implication is that a stable response distortion produced by a motivating context is not well represented as random error. A more useful conceptual decomposition is
\[ X_{\text{motivated}} = T_{\text{target}} + S_{\text{response}} + E, \]
where \(S_{\text{response}}\) represents one or more systematic influences associated with the response situation. These influences can include dispositional, attitudinal, and situational components (Ziegler et al., 2011).
At the variance level,
\[ \operatorname{Var}(X) = \operatorname{Var}(T_{\text{target}}) + \operatorname{Var}(S_{\text{response}}) + 2\operatorname{Cov}(T_{\text{target}},S_{\text{response}}) + \operatorname{Var}(E). \]
This form makes an important point explicit: the response-related component can be systematic and can covary with substantive trait variance. Consequently, controlling social desirability is not as simple as subtracting a nuisance score that is guaranteed to be independent of the construct.