Testing fit patterns with polynomial regression models Felix Schönbrodt Ludwig-Maximilians-Universität München, Germany Fit hypotheses, also labeled ‘congruence’, ‘discrepancy’, or ‘congruity’ hypotheses, contain the notion that an outcome is optimal when two predictor variables match well, while incongruent/ discrepant combinations of the predictors lead to a suboptimal outcome. Previous statistical frameworks for analyzing fit hypotheses emphasized the necessity of commensurable scales, which means that both predictors must be measured on the same content dimension and on the same numerical scale. In some research areas, however, it is impossible to achieve scale equivalence, because the predictors have to be measured with di↵erent methods, such as ex- plicit attitudes (e.g., questionnaires) and implicit attitudes (e.g., reaction time task). In this paper, I di↵erentiate numerical congruence from fit patterns, a concept that does not depend on the notion of commensurability, and hence can be applied to fit hypotheses with incom- mensurable scales. Polynomial regression can be used to test for the presence of a fit pattern in empirical data. I propose several new regression models for testing fit patterns which are statistically simpler and conceptually more meaningful than a full polynomial model. An R package is introduced which provides user-friendly functions for the computation, visualiza- tion, and model comparison of several fit patterns. An empirical example on implicit/explicit motive fit demonstrates the usage of the new methods. Keywords: squared di↵erence, congruence, fit, discrepancy, polynomial regression Many psychological theories maintain the notion that the crepancies between explicit and implicit self-esteem pre- congruence (also called fit, match, correspondence, similar- dict well-being (Schröder-Abé, Rudolph, & Schütz, 2007), ity, or discrepancy1) between two constructs has an e↵ect on and discrepancies between implicit and explicit motives pre- some outcome variable. One example is flow theory (Csik- dict well-being (e.g., Hofer & Chasiotis, 2003), psychoso- szentmihalyi, 1975), which maintains that one condition for matic symptoms (Baumann, Kaschel, & Kuhl, 2005), or re- flow experiences is an optimal challenge, which is defined lationship satisfaction and stability of couples (Hagemeyer, as a situation where the perceived ability of the person fits Neberich, Asendorpf, & Neyer, 2013). Furthermore, whole the task difficulty. The optimal challenge can be opera- theories are built on the notion of optimal fit, like person- tionalized with a discrepancy score between task difficulty environment fit theory (Kristof-Brown & Billsberry, 2013) and a person’s ability (e.g., Abuhamdeh & Csikszentmihalyi, or regulatory fit theory (Higgins, 2000). 2009), where deviations into both directions should hinder These examples document the widespread use of fit hy- flow experiences. Similar examples can be drawn from di- potheses. Several studies operationalize these hypotheses verse fields of psychological research. For example, discrep- with an absolute or squared di↵erence of the predictor vari- ancies between explicit and implicit attitudes predict disso- ables. This approach, however, is challenged by theoretical nance reduction behavior (Briñol, Petty, & Wheeler, 2006), considerations which state that for the operationalization of discrepancies between implicit and explicit self-concept pre- fit hypotheses it is necessary that both measures are com- dict scores in intelligence tests (Dislich et al., 2012), dis- mensurable (Edwards, 2002). The principle of commensu- rability, also known as ‘dimensional homogeneity’, states that if two quantities should be compared, added, or sub- tracted, they must be on the same content dimension (‘nom- Felix D. Schönbrodt, Department of Psychology, Ludwig- inal equivalence’, Edwards & Shipp, 2007). For example, Maximilians-Universität München,DRAFT Germany. Acknowledgements. I want to thank Jens Asendorpf, Giulio Costantini, Birk Hagemeyer, it does not make sense to subtract 2.1 meters from 0.6 kilo- Gregor Kappler, Markus Maier, Marco Perugini, John Rauthmann, grams. Furthermore, both measures must be assessed on the and Michael Zehetleitner for valuable comments on a previous ver- 1For clarification, the term ‘discrepancy’ here is used for undi- sion of the manuscript. Correspondence concerning this article rected (aka. non-directional) di↵erence measures (i.e., absolute dif- should be addressed to Felix Schönbrodt, Leopoldstr. 13, 80802 ferences X Y or squared di↵erences (X Y)2). This paper is only München, Germany. Email: [email protected]. Phone: +49 89 | − | − concerned with squared di↵erences as predictors, not with squared 2180 5217. Fax: +49 89 2180 99 5214. di↵erences as outcomes (cf., Edwards, 1995). 2 SCHÖNBRODT same metric (‘scale equivalence’). Two weights (which have both the problems and the solutions can be localized. The the same content dimension) cannot be directly compared third part introduces new statistical models based on polyno- when they are assessed on di↵erent scales like pounds and mial regression, which overcome the identified problems and kilograms. Nominal equivalence is a logical precondition allow to test fit patterns with incommensurable measures. for scale equivalence: When scales measure di↵erent con- Fourth, an empirical example demonstrates how to employ structs, scale equivalence cannot be meaningfully defined. and interpret the new models. Finally, the limitations and These theoretical considerations led to the conclusion that implications of the proposed techniques are discussed. ‘[c]ommensurate dimensions are required for the conceptu- alization and measurement of P-E [person-environment] fit Numerical Congruence vs. Fit Patterns [...]. Without commensurate dimensions, it is impossible to determine the proximity of the person and environment to Science is communicated through verbal definitions, and one another, and the notion of P-E fit becomes meaningless.’ problems arise when di↵erent scientific communities use the (Edwards, Caplan, & Harrison, 1998, p. 31). same label for di↵erent underlying constructs (e.g., Hmel & Commensurable scales typically are achieved by using the Pincus, 2002). Such a situation is present in the psychologi- same response scale with di↵erent item stems, for example cal literature concerning the term congruence (resp. fit, cor- ‘How much money do you actually earn?’ and ‘How much respondence, discrepancy, congruity, or contingency), which money would you like to earn?’ (Edwards & Shipp, 2007). refers to at least two di↵erent constructs. Henceforward, I In practice, however, the actual degree of commensurability will di↵erentiate these two constructs as numerical congru- can be hard to determine as the di↵erent item stems might in- ence and fit patterns. duce response biases, or di↵erential item functioning might bias the meaning of the item when, for example, wifes and Numerical Congruence husbands are compared on the same item (see also Kristof, 1996). Moreover, and central to this paper, in some research It has been argued, probably most pronounced in the re- areas it is impossible to achieve scale equivalence, because search tradition on person-environment fit, that the predic- the predictors have to be measured with di↵erent methods. In tor variables must be commensurable in order to derive an fact, all of the studies described in the first paragraph com- meaningful quantification of the congruence between them pared incommensurable scales, such as reaction time tasks (e.g. Bauer & Hussong, 2009; Caplan, 1987; Edwards, 2002; with Likert scales. Edwards et al., 1998). For example, one could ask employ- But if we cannot subtract kilograms from kilometers, how ees about the desired and the actual travel times in their job: can we subtract milliseconds in a reaction time task from ‘How many days per months you want to travel?’ (desired) points on a Likert scale? These constructs are on inher- and ‘How many days per month do you actually travel?’ (ac- ently di↵erent measurement scales.2 At first view, it seems tual; cf. Edwards, 2002). A numerical comparison of these that in the case of implicit/explicit attitude discrepancies and two quantities is directly possible: Employees with a perfect comparable situations the precondition of commensurability match (desired equals actual) are congruent, and increasing poses an insurmountable challenge. But is it really a com- deviations mean increasing incongruence. As this notion of pelling consequence that in these cases the notion of congru- congruence is closely tied to the actual measurement scale ence ‘becomes meaningless’? on which the variables are located, henceforward I will call it numerical congruence. Numerical congruence requires both In this paper, I propose an appropach that allows to test nominal and scale equivalence. a certain type of fit hypotheses with incommensurable mea- sures by di↵erentiating numerical congruence from fit pat- terns. New statistical models based on polynomial regression Fit Patterns provide a tool for describing and testing these fit patterns in Other strands of psychology, however, conceive of con- empirical data. As will be shown, this approach does not rely gruence in a more abstract and conceptual way. For exam- on the commensurability of predictor variables. The relax- ple, in client-centered therapy (Rogers, 2004) the term con- ation of the commensurability precondition, however, comes gruence is used for a good fit between the real, the perceived, at the