Analyzing Dyadic Data with IRT Models

Abstract

Dyadic data frequently occur in social sciences and numerous techniques have been developed for their analysis. The most prominent methods involve using regression, path, and structural equation models. The present contribution extends these approaches by considering Item Response Theory (IRT) Models. Two pivotal dyadic data analysis models, the Actor-Partner Interdependence Model (APIM) and the Common Fate Model (CFM), are built using the Multidimensional Random Coefficients Multinomial Logit Model (MRCMLM). This approach combines the advantages of dyadic data analysis with a model for discrete data, thus allowing for categorical items while drawing inferences based on the estimated true scores on an interval scale.

Appendices

Technical Appendix: APIM Commands

Table 1 Description of ConQuest commands regarding the APIM

APIM Item Fit Indices

Listing 8 Item Fit Indices for the APIM

item::

Item number and label; as no label has been provided, the item number is repeated.

ESTIMATE::

Item parameter estimate; in the dichotomous case, this is the item difficulty parameter [δ _i according to Eq. (1)]. To identify a latent scale, one item per latent dimension is fixed (indicated by an asterisk). By default, ConQuest sets the sum of the item parameters per latent dimension to zero (e.g.: − 1. 452 + 0. 688 + 0. 137 + 0. 147 + (−0. 124) +0.604 = 0). This could be overridden with the command set constraint=cases, causing the mean of the latent variable to be fixed at zero.

ERROR::

Standard error of item difficulty parameter.

MNSQ::

Outfit (UNWEIGHTED FIT) and Infit (WEIGHTED FIT) Index.

CI::

The 95 % confidence interval for the expected value (i.e., 1) of Infit and Outfit.

T::

The t-statistic for the null hypothesis that the Outfit and Infit Index is 1. Values larger than 2 may be considered significant at the 95 % level (corresponds to MNSQ outside the CI).

Extracting the Individual Level Correlation Coefficients

To obtain the individual level correlation coefficients, we use the residuals stored in resid.txt. This file contains 600 lines and 25 columns. The first column is a numerical dyad identifier, followed by four groups of six columns each, comprising the residuals to the respective six items of student/self, student w.r.t parent, parent/self and parent w.r.t student. Any multi-purpose statistics software can be used to obtain the individual level correlation coefficients. We will resort to the R software (R Core Team 2014) for it is freely available (open source) and easy to use. The following script will perform the required steps:

Listing 9 R Script for Computing the CFM Individual Level Correlation Coefficients

The ten statements of Listing 9 perform the following operations:

• In line 1 of the script, we read the content of the file resid.txt and store it in a data.frame named d0.

• Then (line 2) we transform the missing values (ConQuest codes them with -99 by default) to the R missing indicator NA.

• In lines 3–6, the columns obtain more informative variable names (the output file contains no header, therefore, R uses the generic names V1 to V25 by default). This step is merely cosmetic and may as well be omitted.

• Next (line 8), we compute the 25 × 25 correlation matrix of all residuals (omitting the id variable stored in column 1). A schematic view of this matrix is given in Fig. 8.

  Fig. 8

  

  The final CFM

• In line 10, we cut out blocks of correlation coefficients of the residuals of the students’ self-description items with the columns covering the residuals of the students’ assessments of the respective parents (rows 1–6/columns 7–12; grey shaded area termed rA in Fig. 8).

• Analoguously, in line 11, we cut out the correlation coefficients of the residuals of the parents’ self-assessment items with the residuals of the items covering the parents’ assessments of the respective students (rows 13–18/columns 19–24; grey shaded area termed rB in Fig. 8).

• In lines 13 and 14 we prepare two functions, transforming a correlation coefficient to a Fisher’s Z-value (r2z) and backtransforming the latter into a correlation coefficient again (z2r). These functions could easily be enhanced to detect invalid input and issue a corresponding message.

• Finally (lines 16 and 17), we apply the Z-transformation to the two matrix parts, compute the mean and backtransform it to a valid correlation coefficient.

With these steps, we dispose of all required information to draw the complete CFM, depicted in Fig. 7.

CFM Item Fit Indices

Listing 10 Item Fit Indices for the CFM

For an explanation of the column headings see Appendix “APIM Item Fit Indices”.