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ABSTRACT
Extreme response style or, more generally, individual differences in response spacing have been shown to be an influential bias when analyzing questionnaire data. Recently a promising model adjusting for this bias — the differential discrimination model — has been proposed. An advantage to other related approaches is that the model can be fitted using standard structural equation modeling software. However, the model is designed for analyzing continuous item responses, whereas graded response formats are certainly more prominent in behavioral sciences. To resolve this limitation, the present article extends the differential discrimination model to analyzing graded responses. Empirical examples as well as a small simulation study are presented.
Article information
Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.
Ethical Principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.
Funding: This work was not supported by a grant.
Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.
Acknowledgments: The authors thank Dr. Fritz Ostendorf for providing the data used for the second empirical example. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institution is not intended and should not be inferred.
Supplementary Materials
The following MPLUS-code fits the GRDDM to five four-categorical items, such as those analyzed in Example 1.
[01] DATA: FILE IS "Y.txt";
[02]
[03] VARIABLE:
[04] NAMES ARE y1-y5;
[05] CATEGORICAL ARE y1-y5; !define variables as categorical
[06]
[07] MODEL:
[08] tau by y1-y5*; !define factors
[09] alpha by y1-y5*;
[10] tau WITH alpha@0; !fix factor covariance to 0.0
[11] [alpha@1]; !fix alpha intercept to 1.0
[12] tau (sig2t); !label factor variance
[13] alpha (sig2a);
[14] [y1$2-y5$2@0]; !fix middle threshold to 0.0
[15]
[16] MODEL CONSTRAINT:
[17] sig2t = 1 + sig2a; !constrain variance of tau
The following R-code fits the GRDDM to five four-categorical items of Example 1, using the lavaan package. The R-code is split into three parts. The first part [01-14] defines the model. The second part [16] adds the variance constraint on τ. The third part [18-24] calls the lavaan-function, fitting the predefined model. Within the last part, further specifications, such as the estimation algorithm, can be made.
[01] mod <- '
[02] tau =~ y1 + y2 + y3 + y4 + y5 #define factors
[03] alpha =~ y1 + y2 + y3 + y4 + y5
[04] tau ~~ 0*alpha #fix factor covariance to 0.0
[05] tau ~ 0*1 #fix factor intercepts
[06] alpha ~ 1*1
[07] tau ~~ sig2_t*f2 #label factor variance
[08] alpha ~~ sig2_a*f1
[09] y1|t1+0*t2+t3 #fix middle threshold to 0.0
[10] y2|t1+0*t2+t3
[11] y3|t1+0*t2+t3
[12] y4|t1+0*t2+t3
[13] y5|t1+0*t2+t3
[14]'
[15]
[16] cons <- 'sig2_t == 1+sig2_a' #constrain variance of tau
[17]
[18] lavaan(
[19] model = mod,
[20] data = Y,
[21] constraints = cons,
[22] std.ov = TRUE, #fix variance of y* to 1.0
[23] ordered = c('y1','y2','y3','y4',' y5') #define variables as
[24]) #categorical
