Variability Indicators in Structural Equation Models Michael Biderman University of Tennessee at Chattanooga

For the past few years I’ve investigated the utility of a structural equation model of faking of personality questionnaires, specifically the Big Five. The model is a CFA containing 1)latent variables representing personality dimensions, and 2)one or more latent variables representing amount of response distortion, i.e., faking. Background: Modeling Faking of Personality Tests

The Basic Faking Model F FE3 FE2 FE1 FE4 FE5 FA3 FA2 FA1 FA4 FA5 FC3 FC2 FC1 FC4 FC5 FS3 FS2 FS1 FS4 FS5 FI3 FI2 FI1 FI4 FI5 E HE3 HE2 HE1 HE4 HE5 A HA3 HA2 HA1 HA4 HA5 C HC3 HC2 HC1 HC4 HC5 S HS3 HS2 HS1 HS4 HS5 I HO3 HO2 HO1 HO4 HO5 E-H A-H C-H S-H O-H E-D A-D C-D S-D O-D E-I A-I C-I S-I I-I E A C S I FP FA Applied to two-condition dataApplied to three-condition data

Beyond the Basic Model: The basic model represents changes in central tendency associated with faking fairly well. Is that all there is? Last year, during manual data entry of Mike Clark’s thesis data (Yes, UTC is a full-service university)... I noticed that some participants seemed to be targeting specific responses, e.g., Since such targeting results in low variability this suggested the possibility that variability of responding might be an indicator of faking. The following describes an attempt to model variability.

5 Other studies of variability Traitedness studies (Britt, 1993; Dwight, Wolf & Golden, 2002; Hershberger, Plomin, & Pedersen, 1995). A person is highly traited on a dimension if the variability of responses to items from the dimension is small. Extreme response style (Greenleaf, 1992). Stability of responses to the same scale over time (Eid & Diener, 1999; Kernis, 2005). No studies of variability of responses in faking situations. None of variability and the Big Five.

1: Biderman & Nguyen, N=203 2: Wrensen & Biderman, N=166 Two-condition data: Honest and Fake Good 50 item IPIP Big Five questionnaire given twice 2-item parcels analyzed 3.Clark & Biderman, N=166 Three-conditions: Honest, Incentive, Instructed faking Three 30-item IPIP Questionnaires given. Whole-scale scores analyzed. Wonderlic Personnel Test (WPT) was given to all participants prior to the first condition. Three used datasets.

7 Measuring Variability To represent variability of responses within dimensions, I computed the standard deviation of responses within each Big Five dimension for each participant for each condition I added the standard deviations as observed variables to the data to which the faking model had previous been applied

8 Datasets 1 and 2 with Standard Deviations added F FE3 FE2 FE1 FE4 FE5 FA3 FA2 FA1 FA4 FA5 FC3 FC2 FC1 FC4 FC5 FS3 FS2 FS1 FS4 FS5 FI3 FI2 FI1 FI4 FI5 E HE3 HE2 HE1 HE4 HE5 A HA3 HA2 HA1 HA4 HA5 C HC3 HC2 HC1 HC4 HC5 S HS3 HS2 HS1 HS4 HS5 I HO3 HO2 HO1 HO4 HO5 SdHE SdHA SdHC SdHS SdHI SdFE SdFA SdFC SdFS SdFI

9 Dataset 3 with standard deviations added E-H A-H C-H S-H O-H E-D A-D C-D S-D O-D E-I A-I C-I S-I I-I E A C S I FP FA SdE-H SdA-H SdC-H SdS-H SdO-H SdE-D SdA-D SdC-D SdS-D SdO-D SdE-I SdA-I SdC-I SdS-I SdO-I

10 Modeling Standard Deviations - 1 Faking leads to elevated central tendency, often resulting in ceiling effects. Ceiling effects lead to lower variability. So the standard deviations were connected to central tendency via regression links. Specifically, standard deviations were regressed onto parcel or scale scores.

11 Modeling Ceiling Effects in Datasets 1 and 2: Standard deviations were regressed onto parcel scores F FE3 FE2 FE1 FE4 FE5 FA3 FA2 FA1 FA4 FA5 FC3 FC2 FC1 FC4 FC5 FS3 FS2 FS1 FS4 FS5 FI3 FI2 FI1 FI4 FI5 E HE3 HE2 HE1 HE4 HE5 A HA3 HA2 HA1 HA4 HA5 C HC3 HC2 HC1 HC4 HC5 S HS3 HS2 HS1 HS4 HS5 I HO3 HO2 HO1 HO4 HO5 SdHE SdHA SdHC SdHS SdHI SdFE SdFA SdFC SdFS SdFI

12 Modeling Ceiling Effects in Dataset 3: Standard Deviations regressed onto scale scores E-H A-H C-H S-H O-H E-D A-D C-D S-D O-D E-I A-I C-I S-I I-I E A C S I FP FA SdE-H SdA-H SdC-H SdS-H SdO-H SdE-D SdA-D SdC-D SdS-D SdO-D SdE-I SdA-I SdC-I SdS-I SdO-I

13 Modeling Standard Deviations - 2 The assumption/hope? was that there are individual differences in variability of responding within dimensions So a latent variable representing such individual differences was added to the model.

14 Modeling variability in Dataset 1 & 2: Adding a “Variability” latent variable F FE3 FE2 FE1 FE4 FE5 FA3 FA2 FA1 FA4 FA5 FC3 FC2 FC1 FC4 FC5 FS3 FS2 FS1 FS4 FS5 FI3 FI2 FI1 FI4 FI E HE3 HE2 HE1 HE4 HE5 A HA3 HA2 HA1 HA4 HA5 C HC3 HC2 HC1 HC4 HC5 S HS3 HS2 HS1 HS4 HS5 I HO3 HO2 HO1 HO4 HO SdHE SdH A SdHC SdHS SdH I SdFE SdFA SdFC SdFS SdFI V

15 Modeling variability in Dataset 3: Adding a “Variability” latent variable E-H A-H C-H S-H O-H E-D A-D C-D S-D O-D E-I A-I C-I S-I I-I E A C S I FP FA SdE-H SdA-H SdC-H SdS-H SdO-H SdE- D SdA- D SdC- D SdS -D SdO-D SdE-I SdA-I SdC-I SdS-I SdI-I V

16 Results Did the regression links significantly improve goodness of fit? Are there individual differences in variability of responding that are captured by the V latent variable?

17 Results of application of Variability model to Dataset 1 Model Χ 2 (1539)= ; p<.001; CFI=.871; RMSEA=.055 Both the regression links and the V latent variable improved model fit. F FE3 FE2 FE1 FE4 FE5 FA3 FA2 FA1 FA4 FA5 FC3 FC2 FC1 FC4 FC5 FS3 FS2 FS1 FS4 FS5 FI3 FI2 FI1 FI4 FI E HE3 HE2 HE1 HE4 HE5 A HA3 HA2 HA1 HA4 HA5 C HC3 HC2 HC1 HC4 HC5 S HS3 HS2 HS1 HS4 HS5 I HI3 HI2 HI1 HI4 HI SHE SHA -.14 SHC -.14 SHS -.07 SHI -.11 SFE -.07 SFA -.16 SFC -.16 SFS -.09 SFI -.09 V ΔΧ 2 (31)= p<.001 Chi-square difference test of regression links: Χ 2 (50)= p<.001 Chi-square difference test of V latent variable: Χ 2 (16)= p<.001

18 Results of application of Variability model to Dataset 2 Model Χ 2 (1539)= ; p<.001; CFI=.816; RMSEA=.066 Chi-square difference test of V latent variable: ΔΧ 2 (16)= p<.001 ΔΧ 2 (31)= p<.001 F FE3 FE2 FE1 FE4 FE5 FA3 FA2 FA1 FA4 FA5 FC3 FC2 FC1 FC4 FC5 FS3 FS2 FS1 FS4 FS5 FO3 FO2 FO1 FO4 FO E HE3 HE2 HE1 HE4 HE5 A HA3 HA2 HA1 HA4 HA5 C HC3 HC2 HC1 HC4 HC5 S HS3 HS2 HS1 HS4 HS5 I HO3 HO2 HO1 HO4 HO SHE SHA -.16 SHC -.14 SHS -.06 SHO -.14 SFE -.13 SFA -.20 SFC -.19 SFS -.09 SFO -.18 V Chi-square difference test of regression links: ΔΧ 2 (50)= p<.001 Again, both the regression links and the V latent variable improve model fit.

19 Results of application of Variability model to Dataset 3 Model Χ 2 (352)= ; p<.001; CFI=.883; RMSEA=.056

20 Tentative Conclusions regarding Variability Model 1) Ceiling effects seem to be successfully modeled by the regressions of standard deviations onto parcels or scale scores. 2) Individual differences in variability of responding to items within dimensions seem to be captured by the V latent variable. Some persons consistently exhibited little variability in responding to items within questionnaire scales. Others exhibit greater variability in responding.

21 What about V and faking? 1) Loadings on V might be larger in faking conditions – magnifying individual differences in variability because some people are targeting while others are not. Mean Standardized Loadings of Standard Deviation indicators on V Dataset 1 HonestIncentive to fakeInstructed to fake Mean loading Dataset 2 HonestIncentive to fakeInstructed to fake Mean loading Dataset 3 HonestIncentive to fakeInstructed to fake Mean loading Tentative Conclusion: Loadings on V are approximately equal in honest and faking conditions.

22 2) V might be related to the faking latent variables. Dataset 1:Correlation of V with F:.04NS Dataset 2:Correlation of V with F:.02NS Dataset 3:Correlation of V with FP:.16 NS Correlation of V with FA:.10NS It appears from these preliminary analyses that variability of responding is not related to faking However, other scenarios and models should be explored What about V and faking?

23 What is V? 1) Perhaps V is related to personality characteristics It appears that V has discriminant validity with respect to the Big Five.

24 What is V? 2) Perhaps V is related to cognitive ability These results suggest that persons with higher CA exhibit less variability of responding.

25 Uses of V How about using it to extract cognitive ability from the Big Five? FA V I WPT e Dataset 1: Multiple R =.57 Dataset 2: Multiple R =.35 Dataset 3: Multiple R =.50 The structural model suggests that there is information on cognitive ability embedded in “noncognitive” personality tests.

26 Conclusions 1) V appears to be an individual difference variable that cuts across personality dimensions. 2)V appears to be unrelated to faking 3) V appears to be independent of the Big Five dimensions. 4) V appears to be related to cognitive ability – persons higher in cognitive ability have lower variability of responding

27 Implications Don’t throw away old datasets. You never know what constructs may be hidden in them.