I would like to model my problem using something similar to Item Response Theory, but my responses are not binary. They are continuous in $[0, 1]$.
What are these models/the research field called?
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2$\begingroup$ Does "linear response" mean that you're dealing with a continuous response (or manifest) variable in [0;1]? $\endgroup$– chlNov 27, 2012 at 15:36
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$\begingroup$ Yes, I changed the title;-) $\endgroup$– user1141785Nov 28, 2012 at 9:01
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1$\begingroup$ Thanks. Could you indicate what is the response variable, precisely? We also need additional information: Do you assume discrete or continuous latent variable(s)? (Some authors, like Bartholomew & Knott, Skrondal & Rabe-Hesketh, or De Boeck, have emphasized the importance of such distinction between latent and manifest variables in the past.) $\endgroup$– chlNov 28, 2012 at 13:20
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$\begingroup$ I'll look at the book. I assume continuous latent variables. I want to model Triathlon finishing time (continuous response variable). There's an athlete who has abilities (swimming, endurance, ...) and item's which have difficulties (route difficulty, climate ...) $\endgroup$– user1141785Nov 28, 2012 at 19:22
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$\begingroup$ B&K book: Table 1.3, p. 11. I would say this has more to do with Factor Analysis, then, but I wonder why response times are bounded in [0;1]. $\endgroup$– chlNov 28, 2012 at 20:27
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3 Answers
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If you have a continuous indicator, then you would use factor analysis. Think of FA as linear regression and IRT it's logistic regression brother.
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$\begingroup$ I realize this question is old, but there are IRT models specifically for this case--bounded response intervals (see my answer below). The second article I list specifically compares these with vanilla confirmatory factor analysis. $\endgroup$– machowApr 2, 2016 at 18:21
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The accepted answer does not give models where the response is bounded between [0,1]. There are IRT models for exactly the case where the response variable is continuous, but bounded in this way.
For example, Samejima [1] describes exactly this case. These models are sometimes called Continuous Response Models (CRM), and the case they're addressing is sometimes called Continuous Response Format (CRF). They are the standard factor analysis setup, but add a transformation to bound the responses within an interval (in the same sense that the standard IRT takes factor analysis, and adds a link so responses will be, say, 0 or 1).
Ferrando [2] has a good paper on this, and the R library estCRM looks like it can implement it (though I haven't used it!).
[1]: Normal ogive model on the continuous response level in the multidimensional latent space
[2]: Theoretical and Empirical Comparisons between Two Models for Continuous Item Response
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$\begingroup$ (+1) See also A Beta Unfolding Model for Continuous Bounded Responses (Psychometrika 79(4):647-674) and A beta unfolding model for continuous bounded responses (Psychometrika 79:647–674). $\endgroup$– chlNov 2, 2020 at 19:51
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I agree that the accepted answer does not point towards specifically dedicated models. See for instance:
- Noel, Y. & Dauvier, B. (2007). A beta item response model for continuous bounded responses, Applied Psychological Measurement, 31, 47-73.
- Noel, Y. (2014). A beta unfolding model for continuous bounded responses, Psychometrika, 79(4), 647-674.
- Verhelst N.D. (2019). Exponential Family Models for Continuous Responses. In: Veldkamp B., Sluijter C. (eds) Theoretical and Practical Advances in Computer-based Educational Measurement. Methodology of Educational Measurement and Assessment. Springer, Cham.
Best,
G.
