^{*}

Edited by: Neil Conway, Royal Holloway, University of London, United Kingdom

Reviewed by: René Schalk, Tilburg University, Netherlands; M. Teresa Anguera, University of Barcelona, Spain

*Correspondence: Joeri Hofmans

This article was submitted to Organizational Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

A good understanding of the dynamics of psychological contract violation requires theories, research methods and statistical models that explicitly recognize that violation feelings follow from an event that violates one's acceptance limits, after which interpretative processes are set into motion, determining the intensity of these violation feelings. Whereas theories—in the form of the dynamic model of the psychological contract—and research methods—in the form of daily diary research and experience sampling research—are available by now, the statistical tools to model such a two-stage process are still lacking. The aim of the present paper is to fill this gap in the literature by introducing two statistical models—the Zero-Inflated model and the Hurdle model—that closely mimic the theoretical process underlying the elicitation violation feelings via two model components: a binary distribution that models whether violation has occurred or not, and a count distribution that models how severe the negative impact is. Moreover, covariates can be included for both model components separately, which yields insight into their unique and shared antecedents. By doing this, the present paper offers a methodological-substantive synergy, showing how sophisticated methodology can be used to examine an important substantive issue.

Psychological contracts (PCs)—or the individual's perceptions of the mutual obligations of the employee and employer (Rousseau,

These negative consequences can be explained by the fact that, when the employee notices that his/her organization fails to meet its obligations, (s)he is likely to develop an intense negative emotional reaction (i.e., violation feelings), which in turn has several negative attitudinal and behavioral consequences for both the employee and the organization (Morrison and Robinson,

Despite the general awareness that violation feelings result from a two-stage decision-making process in which the employee first assesses whether anything has violated his/her psychological contract and then evaluates the negative emotional impact of these potential violation(s), few studies have explicitly examined violation as a two-stage decision making process (for exceptions, see Griep et al.,

This is an important limitation because a good understanding of the dynamics of psychological contract violation requires theories, research methods and statistical models that explicitly recognize that violation feelings follow from an event that violates one's acceptance limits, after which interpretative processes determine the intensity of these violation feelings. Whereas such dynamic theories—in the form of the dynamic model of the psychological contract (Schalk and Roe,

In what follows, we first discuss the event-based conceptualization of psychological contract violation. Next, we introduce two models that closely align with this event-based conceptualization: the Hurdle regression model and the Zero-Inflated regression model. Apart from a theoretical introduction to these models, we show how these models can be tested using Mplus. Finally, we conclude by comparing our approach to the dominant approaches in the field, discussing how they differ from one another and under which conditions one or the other approach should be used.

Although the psychological contract itself—referring to perceived mutual obligations of the employee and employer (Rousseau,

Importantly, the dynamic model of the psychological contract maintains that not all events lead to violation feelings. In fact, without any major events happening, the psychological contract is in a state of homeostasis, and it is only when the behavior of the organization or the behavior of the employee changes, that the employee will try to accommodate these changes or events within his/her mental model. Crucial for this accommodation process is the idea of acceptance limits, which describe what is considered acceptable for the individual (Schalk and Roe,

Thus, the dynamic model of the psychological contract is not build on the assumption that violation feelings result from a “constant method of accounting” in which people systematically and constantly compare perceived promises to perceived obligations (Schalk and Roe,

Of particular importance is that the sense-making process following the crossing of the acceptance limits can happen subconsciously, which implies that employees might experience violation feelings without being consciously aware of the preceding judgments (Morrison and Robinson,

In the present paper, we offer a way to circumvent this thorny issue by presenting dual regime models, a family of statistical models that allow capturing both processes based on one's violation feelings scores only. In what follows, we first give a short theoretical introduction to dual regime models, after which we show how these models can be tested in Mplus using an illustrative application with fabricated data.

Assume that we follow different employees in their day-to-day job and that we repeatedly (e.g., each working day) ask them to report on their violation feelings using the following question: “Indicate to what extent you experienced feelings of disappointment, frustration and distress toward your organization today.” People can respond to this question by answering “0 = not at all,” “1 = to a small extent,” “2 = to some extent,” “3 = to a moderate extent,” “4 = to a great extent,” and “5 = to a very great extent.” Because psychological contract breaches do not happen very frequently (i.e., Bal et al.,

Dual regime models are statistical models specifically developed to model data being characterized by clumping at zero. To account for the excess of zeros, these models assume that the data are generated according to two different stages (Zorn,

Several dual regime models have been proposed and discussed in the literature, and they can all be classified according to two core model features: (1) the probability distribution that is assumed for the transition stage, and (2) whether or not the events-stage distribution allows for the occurrence of zeros (Zorn,

Note that the appropriateness of one or the other model is both a theoretical and an empirical issue. Regarding the former, most theoretical accounts indicate that a crossing of the acceptance limits does not necessarily lead to the experiencing of violation feelings. For example, the dynamic model of the psychological contract (Schalk and Roe,

In what follows, we will discuss two dual regime models that differ regarding the allowance of zeros in the events-stage distribution: the Hurdle Poisson Regression Model (no zeros are modeled in the events-stage) and the Zero-Inflated Poisson Model (allowing for the occurrence of zeros in the events stage)^{1}

The idea of the Hurdle model is that in the transition stage a “hurdle” needs to be crossed before one moves on to the events stage, which in the context of the dynamic model of the psychological contract maps directly on the crossing of the acceptance limits (Schalk and Roe,

In this model, ϕ_{ij} represents the probability of a zero for person _{ij} represents a predictor variable, γ_{00} represents the intercept and _{0j} the random effect.

The second part of the Hurdle model pertains to the events stage and describes what happens once the hurdle is taken. At this stage, no more zeros are generated in the Hurdle model. Applied to psychological contract research, this implies that the Hurdle model assumes that, once a person experiences a crossing of the acceptance limits of the psychological contract, the individual will per definition experience violation feelings. Stated differently, in a Hurdle model, violation scores of zero do only result from not crossing the acceptance limits of the psychological contract. Therefore, it is said that, in the Hurdle model, all zeros originate from a “structural” source (Hu et al.,

Because once in the events stage, the individual always experiences violation feelings, the events stage is modeled by a truncated-at-zero count model. Such truncated-at-zero count models are typically used to model count data for processes for which zero is not a possible value. Truncated-at-zero count models can take many forms, with examples being the truncated-at-zero Poisson distribution (Mullahy,

In formula 3, ϕ_{ij} again represents the probability of a zero for person _{ij} represents a covariate, and λ_{ij} represents the truncated Poisson mean for counts greater than zero. Finally, ν_{0j} captures the random effect.

Relevant to research on psychological contracts is that the Hurdle model allows for the inclusion of common and unique predictors in both stages of the model. That is, one can include predictors that predict violation of the acceptance limits and predictors that predict violation intensity, with the possibility that any of these predictors can be shared and/or unique. Moreover, in the multilevel Hurdle model, the random effect of the binary part and the random effect of the count part can be allowed to correlate. This might make sense from a theoretical point of view, as the presence of a psychological contract violation at one point in time might be related to the intensity of one's violation feelings at that and other points in time. This phenomenon might for example happen when there are (unmeasured and thus unmodeled) person-related factors that influence both the threshold to perceive a violation and the severity of these violation feelings once violation is experienced. One such a person-related factor might be one's level of Neuroticism, because research on this personality traits shows that people scoring high on Neuroticism both experience more negative situations (i.e., more breaches) and also react more strongly to these negative situations (i.e., stronger violation feelings) (Hampson,

In summary, the Hurdle model is specifically developed to model data generated in two different stages. In the context of psychological contract research, the Hurdle model allows distinguishing between the occurrence of violation feelings and the intensity of the feelings of violation. Moreover, predictors can be included for both violation of the acceptance limits and violation intensity, without requiring that these predictors are the same in both parts of the model. An important characteristic of the Hurdle model is that it assumes that all zeros are generated by failure to cross the hurdle, which means that it assumes that all zero violation feelings scores result from instances where the acceptance limits of the psychological contract were not crossed.

Being a dual regime model, the Zero-Inflated regression model assumes that the data are generated by a two-stage process. However, unlike the Hurdle model, this model assumes that the zero responses are generated by two sources, rather than one. That is, in the Zero-Inflated regression model zeros can arise both in the first stage (i.e., the transition stage) as well as in the second stage (i.e., the events stage). In other words, Zero-Inflated models assume that the zero scores originate from both a “structural” source and a “sampling” source (Hu et al.,

To accommodate the idea that zero violation scores are generated by two different mechanisms or sources, the events stage of the Zero-Inflated models is no longer modeled using a truncated-at-zero count model but using a regular count model. The consequence thereof is that in the event stage, zeros can occur because these zeros are part of the usual count distribution. Very similar to the Hurdle model, Zero-Inflated models can assume different count distributions, such as the Poisson distribution (Lambert,

Because in the ZIP model, zero scores do not only result from failing to pass the transition stage (i.e., not crossing the acceptance limits of the psychological contract), but also from zeros that are generated during the events stage (i.e., not experiencing violation feelings once the acceptance limits of the psychological contract are crossed), the probability of a zero score is given by the following equation:

In equation 5, ϕ_{ij} represents the probability of a zero for person _{ij}) represents the probability of moving to the events stage, or the probability of exceeding the acceptance limits. λ_{ij}, in turn, governs the intensity of the violation feelings when the acceptance limits are exceeded. Because the ZIP model mixes a binary logit model with a Poisson model, the ZIP distribution can be regarded as a mixture of a Poisson distribution and a degenerate component that places all its mass at zero (Lee et al., ^{2}

The formula for the events stage of the multilevel ZIP can be written as follows:

As with the Hurdle model, predictors can be added to both the transition equation (_{ij}_{ij}

In summary, the Zero-Inflated Regression model is a dual regime model specifically developed to model data that are generated in two different stages, which, in the context of psychological contract research, allows distinguishing between the occurrence of crossings of the acceptance limits and the feelings of violation that might follow from it. Moreover, predictors can be included for both the crossing of the acceptance limits part and the violation feelings part, without requiring that these predictors are identical in both parts of the model. Unlike the Hurdle model, the Zero-Inflated Regression model assumes that zeros are generated in both the transition stage and the events stage, which means that it assumes that a zero violation feelings scores results either from instances where the individual did not experience a violation of his/her acceptance limits, or from instances where the individual did experience a such a violation but did not experience violation feelings.

In what follows, we will demonstrate how the Hurdle model and the Zero-Inflated Regression model can be tested using Mplus version 7.31 (Muthén,

Mplus does not have an option to directly test the Poisson Hurdle Regression model. However, it can test a Negative Binomial Hurdle Regression model, which is a Poisson model that is extended with a dispersion parameter. This dispersion parameter allows capturing overdispersion in the Poisson model, which means that it allows the variance of the model to be greater than the mean. The Poisson model can thus be approximated by fixing the dispersion parameter of the Negative Binomial Hurdle Regression model to a very small value.

To instruct Mplus to test a Negative Binomial Hurdle Regression model, one needs to specify “COUNT IS violation (nbh);” in the VARIABLE section of the Mplus syntax. Next, one needs to specify that the dispersion parameter of the Negative Binomial Hurdle Regression model should be fixed to a small value by typing “violation@0.001;” under the %WITHIN% header in the MODEL section. This line instructs Mplus to test a Negative Binomial Hurdle Regression model with a very small dispersion parameter, thereby approximating the Poisson Hurdle Regression model. Further, we test the Multilevel Poisson Hurdle model at the within-person level and at the between-person level by in the MODEL section specifying that the zero-inflated part (referred to as violation#1), as well as the count part (referred to as violation) are predicted by week at the within-person level and by Neuroticism at the between-person level. Because of the nested nature of our data, we allow for random effects for both the zero-inflated part and the count part in the model, which is done by typing “violation#1 violation;” at the between-level. Finally, these random effects can be allowed to correlate, which can be done why specifying “violation#1 WITH violation;”. The full Mplus code for testing a Multilevel Poisson Hurdle model can be seen in Figure

Mplus output for a Multilevel Poisson Hurdle model.

Below, we show the Mplus output for the Multilevel Poisson Hurdle Model (see Figure

Mplus output for a Multilevel Poisson Hurdle model.

Below the model fit information, the model results are printed. The results at the within-person level show that the chances of

Telling Mplus to test a Zero-Inflated Poisson Regression model can be done by specifying that violation is a zero-inflated Poisson variable using “COUNT IS violation (i);” in the VARIABLE section of the syntax (note that “COUNT IS violation (nb);” can be used to test a Zero-Inflated Negative Binomial Regression model). Next, one can test the ZIP model at the within-person level and at the between-person level by in the MODEL section specifying that the zero-inflated part (referred to as violation#1), as well as the count part (referred to as violation) are predicted by week at the within-person level and by Neuroticism at the between-person level. Because of the nested nature of our data, we allow for random effects for both the zero-inflated part and the count part in the model, which is done by putting violation#1 and violation at the between-level. Finally, these random effects can be allowed to correlate, which can be done why specifying “violation#1 WITH violation”. The Mplus code for testing a Multilevel Zero-inflated Poisson model can be seen in Figure

Mplus code for testing a Multilevel Zero-Inflated Poisson model

Below, the Mplus output for the Multilevel Zero-Inflated Poisson Regression Model is shown (see Figure

Mplus output for a Multilevel Zero-Inflated Poisson model.

Next, the model results are printed. The results at the within-person level show that the chances of remaining in the zero state (i.e., the chances of not experiencing breach) increase as weeks go by (

In the present paper, we argue that dual regime models in general, and Hurdle and Zero-Inflated models in particular, deserve to be added to the toolkit of the psychological contract researcher because these models closely mimic the theoretical processes underlying the elicitation of violation feelings via two model components: a binary distribution that models whether an event in one's work environment leads to a crossing of the acceptance limits of the psychological contract, and a count distribution that models how severe the negative impact of this crossing is. Moreover, covariates can be included for both model parts separately, which might yield insight in their unique and shared antecedents. Hence, the adoption of these models by psychological contract researchers might further our understanding of the factors triggering violation feelings in people's day-to-day working lives.

The treatment of violation feelings by these models strongly draws on a dynamic, event-based conceptualization of the psychological contract, according to which violation feelings follow from discrete events that exceed the acceptance limits of one's psychological contract (Schalk and Roe,

Does this mean that one or the other approach is better than the other? To answer this question it is essential to understand that these different approaches in fact address different questions. Asking people at one point in time to consciously reflect on what was promised to them and what they receive is probably a good way to capture stable, decontextualized inter-individual differences in psychological contract fulfillment and breach. However, because of the one-shot, conscious reflection on promises and deliveries, these studies do probably not reflect the dynamic processes governing psychological breach and violation feelings in people's day-to-day life. The strength of our events-based approach is that it mimics this everyday treatment of violation. At the same time, its weakness is that it fails to tap into the processes leading to inter-individual differences in violation (because it does not measure which obligations are (un)fulfilled). Thus, whether the one or the other approach should be used strongly depends on the questions that are being asked. If one wants to learn about how the different components of the psychological contract contribute to inter-individual differences in violation feelings, the traditional way of conceptualizing and measuring violation is probably appropriate. If, in turn, one is interested in capturing how the psychological contract dynamically operates in people's day-to-day working lives, and if one is interested in studying factors that affect the likelihood to experience violation feelings, an event-based approach is better suited.

It is important to stress that our event-based approach, even though it models dynamic repeated measures data, predicts the occurrence and intensity of violation feelings at one point in time. Another way to look at psychological contact dynamics it to study patterns of violation feelings over time. Recent research by de Jong et al. (

When introducing dual regime models, we discussed two of them: the Hurdle Regression model and the Zero-Inflated Regression model. Although we argued that both models can be used to model data generated through a two-stage process, it is important to emphasize that they differ with regard to one crucial aspect. The Hurdle model assumes that all zero observations are structural, which means that this model assumes that, once the acceptance limits of the psychological contract are crossed, the individual per definition experiences violation feelings. The Zero-Inflated model, instead, allows for both structural and sampling zeros, which means that, even when the acceptance limits of the psychological contract are crossed, the crossing might not lead to feelings of violation. Although there are indications in the literature that broken promises do not always lead to violation feelings (Conway and Briner,

In conclusion, when one is interested in studying the ebb and flow of violation feelings in an everyday life context and particularly when the goal is to study predictors of violation feelings, Hurdle Regression model and Zero-Inflated Regression models might be worth looking at. Adopting these methods in research on psychological contracts has the potential to teach us a lot about the features in the situation and the characteristics of the person that trigger dynamic fluctuations in the occurrence and intensity of violation feelings.

The author confirms being the sole contributor of this work and approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{1}Note that the Hurdle Poisson Regression Model and the Zero-Inflated Poisson Model are specific cases of the Hurdle Regression Model and the Zero-Inflated Model in the sense that the events stage is modeled using a truncated-at-zero Poisson and a regular Poisson distribution, respectively. For both the Hurdle and Zero-inflated model, other distributions can be used for modeling the events stage, with one example being the negative binomial distribution. However, because of reasons of simplicity, we primarily focus on the Poisson variants in the present paper.

^{2}Note that this is different from the Hurdle model. In the Hurdle model all zero scores are modeled using a binary logit model, while all nonzero scores are modeled using a truncated-at-zero Poisson mode. Hence, one can test the Hurdle model by testing both models separately (i.e., a two-stage analysis). This is not possible with the Zero-Inflated Poisson model because only part of the zero scores are explained by the binary logit model, while the other zero scores are accounted for by the Poisson distribution. Therefore, both model equations need to be estimated simultaneously.