Expanding the methodological toolbox of HRM researchers: The added value of latent bathtub models and optimal matching analysis

Researchers frequently rely on general linear models (GLMs) to investigate the impact of human resource management (HRM) decisions. However, the structure of organizations and recent technological advancements in the measurement of HRM processes cause contemporary HR data to be hierarchical and/or longitudinal. At the same time, the growing interest in effects at different levels of analysis and over prolonged periods of time further drives the need for HRM researchers to differentiate from traditional methodology. While multilevel techniques have become more common, this article proposes two additional methods that may complement the current methodological toolbox of HRM researchers. Latent bathtub models can accurately describe the multilevel mechanisms occurring in organizations, even if the outcome resides at the higher level of analysis. Optimal matching analysis can be useful to unveil longitudinal patterns in HR data, particularly in contexts where HRM processes are measured on a continuous basis. Illustrating the methods’ applicability to research on employee engagement, this paper demonstrates that the HRM community—both research and practice—can benefit from a more diversified methodological toolbox, drawing on techniques from within and outside the direct field to improve the decision-making process.


| INTRODUCTION
Human resource management (HRM) emerged as a function in the early 20th century to effectively manage and rationalize the employment relationship (Ulrich & Dulebohn, 2015). Nowadays, HRM is increasingly becoming a "science" that aims to enhance the decisions organizations make regarding their human capital (Boudreau & Ramstad, 2005Rasmussen & Ulrich, 2015;Ulrich & Dulebohn, 2015). In creating a basis of evidence for such decisions, HRM scholars have primarily relied on general linear modeling (GLM) such as linear regression. However, the data gathered and compiled in the contemporary HR function is increasingly of hierarchical and longitudinal nature, causing the current methodological toolbox of HRM researchers to fall short (Angrave, Charlwood, Kirkpatrick, Lawrence, & Stuart, 2016;Bersin, 2015).
Methods other than GLM may better account for the complex effects in these new forms of HR data. On the one hand, organizational entities are hierarchical structures, which causes the effects of HRM to occur at and across different levels of analysis simultaneously (Hitt, Beamish, Jackson, & Mathieu, 2007;Wright & Nishii, 2007). On the other hand, as measurement happens on a more continuous basis, HR data structures often consist of many observations nested within subjects over prolonged periods of time (Angrave et al., 2016;Bersin, 2015). Acknowledging the above, scholars have been increasingly moving from GLM applied at a single level of analysis toward multilevel techniques (Boselie, Dietz, & Boon, 2005;Sanders, Cogin, & Bainbridge, 2014;Snape & Redman, 2010). However, the most commonly applied multilevel methods do not work well when examining bottom-up effects, linking individual phenomena to organizational-level outcomes, and they can become overly complex when examining multiple, potentially categorical, variables simultaneously and over prolonged periods of time.
This article proposes two statistical methods that are rarely applied to HRM research questions, despite having added value over and above more traditional methodology. First, bathtub models are proposed as a way to account for multilevel models where the outcome resides at the higher level of analysis. Outperforming traditional aggregation and disaggregation approaches (Bennink, 2014), bathtub modeling can add value to HRM research on, among others, group composition or bottom-up effects. Second, optimal matching analysis (OMA) is advocated for its ability to detect longitudinal patterns. It can reduce large volumes of both categorical and ordinal data into a smaller set of underlying trajectories. Although relatively unknown in the general HRM field, it has been a valuable tool for career pattern analysis (Dlouhy & Biemann, 2015). This article aims to demonstrate the added value of each method to the HRM methodological toolbox by discussing their applicability to research on employee engagement.
After discussing each method's strengths and weaknesses separately, the article concludes with an overview of potential future applications and synergies.

| BATHTUB MODELING AND ENGAGEMENT
Over the past two decades, the influence of HRM on organizational performance has received much scholarly attention (e.g., Becker & Gerhart, 1996;Paauwe, Guest, & Wright, 2013) and a major part of the impact of HRM policies and practices has been demonstrated to be indirect via the behavior of employees (Christian, Garza, & Slaughter, 2011;Harter, Schmidt, & Hayes, 2002;Jiang, Lepak, & Baer, 2012;Kehoe & Wright, 2013;Subramony, 2009). However, an investigation of this mediational process is complex, as it involves measurements at various levels of analysis. The design of HRM policies as well as the implementation of HRM practices commonly occur at either an organizational, functional, departmental, or group level, whereas the behaviors and cognitions they seek to influence are located at the level of the individual employee (Bowen & Ostroff, 2004;Snape & Redman, 2010;Wright & Boswell, 2002;Wright & Nishii, 2007). Although multilevel techniques provide an avenue to test models with such hierarchical structures (e.g., Snijders & Bosker, 1999), they are developed for models where the outcome variable lies at the lower level of analysis. Problems arise when the outcome occurs at a macro-level (e.g., organizational performance) but the predictors reside at a micro-level (e.g., employee behaviors).
Scholars have creatively circumvented modeling such micromacro processes. For example, studies relating HRM implementation and employee engagement to organizational performance have used three different approaches. Nevertheless, all three have their downsides. First, the micro-level scores can be aggregated to the macrolevel. As such, studies have investigated HRM, engagement, and performance at the level of the organization or the work group (e.g., Harter et al., 2002;Jiang et al., 2012;Subramony, 2009;Whitman, Van Rooy, & Viswesvaran, 2010). However, after aggregation to the macro-level, the data loses all information on individual variations (Bennink, Croon, & Vermunt, 2013). To illustrate, an aggregated team score of average engagement can be interpreted either as all employees in the team being averagely engaged, or as the team being a mix of highly engaged and highly disengaged employees. Moreover, aggregation has considerable consequences because the reduction of the sample size-to the size of the macro-level sample-significantly decreases the power of the statistical test (Bennink et al., 2013;Krull & MacKinnon, 1999). As a second approach, scholars have restricted the analysis to micro-level variables. For example, studies have examined employees' individual perceptions of HRM practices and its influence on employees' engagement and individual performance scores (e.g., Christian et al., 2011;Halbesleben, 2010). However, this approach introduces perspective bias as it looks at the impact of the perceptions employees have of the HRM strategies, policies, and practices. Moreover, no conclusions can be drawn regarding the impact on the actual organizational performance, as the models have to rely on micro-level outcomes, such as individual performance evaluations. Finally, disaggregation has been a third approach, in which the macro-level scores (i.e., HRM and organizational performance) are assigned to each micro-level case (i.e., employee). Luckily, this method is not frequently found in published academic studies, as disaggregating scores violates the assumption of independent error terms (Keith, 2005), causing biased standard error estimates, overly liberal tests (Krull & MacKinnon, 1999), and artificially high power (Bennink et al., 2013).
Bathtub models provide a solution that overcomes the problems specific to the aforementioned approaches by correctly modeling the multilevel processes at hand. They offer an opportunity to investigate the relationship between macro-level variables through micro-level mechanisms. Using a latent variable model, a bathtub model raises micro-level responses to the macro-level while taking into account both within-group variance and sampling variability. Subsequently, the latent variable can be used as a regular predictor in the macrolevel model. Applied to employee engagement, bathtub modeling provides an opportunity to examine how HRM policies and practices influence organizational performance through the behaviors of employees. This analysis can be conducted without ignoring the possibility that employee engagement is personal and individually determined (Macey & Schneider, 2008), and without inflating statistical power (Bennink et al., 2013, Krull & MacKinnon, 1999. The following illustrates the modeling process step by step. First, the data requirements are described, followed by the two parts of the model and their respective interpretation. Afterwards, several limitations to bathtub modeling are presented.

| Data requirements
Before conducting a bathtub model, the sample size and the data format need to be considered. The power of a bathtub model largely depends on the sample size at all levels of analysis. Hence, researchers should not only gather data of multiple groups, but they should also ensure high response rates within the groups. Bennink (2014) demonstrates that a sample of as small as 40 groups with 10 respondents per group can already result in nearly unbiased parameter estimates in case of a bottom-up, micro-macro effect (e.g., engagement on organizational performance). In order to detect more complex effects like interactions or indirect effects, larger sample sizes at the macro-level are required. Simulations demonstrate that 200 groups with 10 respondents per group should be enough to detect most effects (Bennink, 2014), suggesting that increasing the macro-level sample size should be the focus for researchers seeking to examine small and/or complex effects.
After data collection, the bathtub modeling requires the data to be structured according to its multilevel nature. Depending on the statistical software used, the model can be estimated on a data set with either a long or a wide format. For long data sets, each row would represent a micro-level case (e.g., employee), and one of the columns would identify the macro-level unit this micro-level case belongs to (e.g., team or work group). Subsequently, a bathtub model can be applied using a multilevel regression approach. For wide data sets, the format is more peculiar: each row needs to represent a macro-level unit, whereas for each micro-level case, each measurement should be stored in a separate column. This format is commonly referred to as the persons-as-variable approach, and it does not work in all software packages (e.g., Mplus). to the group level, it is often referred to as the within-group part.

| Bathtub model
Next, the structural model describes the relationship between the variables at the macro-level. By relating the latent variable, derived from the measurement part, to the other macro-level variables of interest, the bathtub model takes the individual variation that occurs at the micro-level into account. The following elaborates on both parts in more detail.
The measurement part of a bathtub model uses a multilevel latent variable model to raise the observed individual data to the level of the group. Here, the format of the individual, micro-level data is quite important. The latent variable approach was initially proposed by Croon and van Veldhoven (2007), who demonstrated how treating individual scores as exchangeable indicators for a continuous latent variable makes it possible to predict a group-level outcome with lower-level independent variables. It works by estimating an unobserved continuous score at the group-level based on the observed individual data. Because the resulting latent group score reflects the underlying individual data and its variance, it takes the measurement and sampling errors that occur at the micro-level into account. However, this approach was developed specifically for raising continuous variables to the group level using a normally distributed latent variable and, while very useful, the approach has the limitation that it cannot treat categorical variables adequately. Bennink, Croon, and Vermunt (2015) therefore propose an extension of this latent variable approach, which makes it possible to raise categorical variables to the group level using a generalized latent variable modeling framework (Skrondal & Rabe-Hasketh, 2004). This extended model allows micro-level discrete variables to be raised to the macro-level using a categorical latent variable, called a latent class variable. Using this approach, the unobserved heterogeneity at the group level can be estimated using a latent class variable that clusters together groups that are more similar to each other. In this way, individual data can be raised to the group level by creating clusters of macro-level groups (i.e., latent classes) based on the similarity of their micro-level scores. Although it is common practice to have as many latent classes at the macro-level as there are discrete categories at the micro-level (Bennink, 2014), the optimal number of classes may also be estimated based on fit measures like the Bayesian information criterion (BIC), Akaike information criterion (AIC), or chi-square (χ 2 ).
The strength of the measurement model can be assessed using the entropy (R 2 ), with values above .70 reflecting a strong model where classes are adequately distinguished (Vermunt, 2010). This latent class approach is especially useful in the social sciences. To illustrate the above, imagine a study examining the effect of a specific HRM practice, hours of leadership training received by managers, on the performance of teams. The researchers might want to examine whether a part of this effect is indirect, for instance, via the engagement of employees. The bathtub model that corresponds with such a study is presented in Figure 1. Both the leadership training and the team performance occur at a macro-level, and therefore their relationship can be estimated directly using a linear regression model at the group level. In contrast, the engagement data is located at the level of the individual employee and thus requires elevation to the group level using a bathtub model before its involvement can be examined. For this elevation, researchers can choose either a continuous latent variable model or a latent class model. A continuous latent variable would imply that group-level engagement scores run from highly disengaged up to highly engaged groups, following a normal distribution. A linear regression model could then be used to predict team performance, based on the leadership training and the continuous latent engagement scores. Alternatively, a latent class model could be applied. This would result in various macro-level classes, and subsequently, for each team, the probability that it belongs to one of the classes is estimated. For instance, one class could consist of teams where most individuals display an average level of engagement, whereas another class could be composed of teams with a few highly engaged and several disengaged team members. In contrast to an aggregation approach, the latent class model is able to differentiate between these two groups based on their different team dynamics. Afterwards, an analysis of covariance (ANCOVA) model can be run at the macro-level, where team performance is explained by the leadership training and the discrete latent group engagement scores.
Finally, based on the output test statistics, the study's hypotheses can be confirmed or rejected.

| Limitations of bathtub modeling
The micro-macro analysis in general and the bathtub modeling in particular are innovative and well-performing approaches to examine how micro-level mechanisms influence the relationship between macro-level variables. However, there are several limitations.
For example, it is unclear how sample size influences the performance of bathtub models. Although Bennink (2014) demonstrates that the method generally outperforms the traditional aggregation and disaggregation approaches, the more complex bathtub models with indirect or interaction effects can provide equally inaccurate estimates when applied to small macro-level samples. While a sample of 200 groups should be sufficient to provide accurate results for complex models (Bennink, 2014), this number is quite large. The minimum sample size to establish accurate results is unknown and seems to differ from one model to the next. Furthermore, to date, no research has been conducted regarding power in bathtub models with continuous latent variables.
Additionally, it is unknown how missing values impact the performance of bathtub models. Although the standard approach to handling missing values in multilevel research is listwise deletion, multiple imputation is currently undergoing heavy development for the classical top-down effects (Van Buuren, 2011). The impact of missing values and the best way of handling them in multilevel research with bottom-up effects, such as bathtub models, has yet to receive empirical attention.
A further consideration lies in the person-as-variable approach, which considers the employees within a group as interchangeable.
This approach implies that each individual within a group contributes equally to the estimation of the latent score for that group. For constructs like engagement, this seems theoretically sound as each employee's engagement can be considered equally important to the team's latent score. However, there may be HRM research questions in which the data of certain employees can be considered more important to or representative of the group's latent score. For example, when accounting for differences in employment type (e.g., full-/ part-time or temporary/fixed contracts) or when scores follow specific distributions within subgroups of employees (e.g., forced distributed performance evaluations).
Although bathtub modeling clearly outperforms traditional aggregation and disaggregation approaches (Bennink, 2014;Bennink et al., 2013Bennink et al., , 2015Croon & van Veldhoven, 2007), multilevel structural equation modeling (SEM) may function as an alternative. When the outcome variable resides at the micro-level, multi-level SEM can be preferable to bathtub models (Lüdtke et al., 2008). However, similar to the traditional micro-macro analysis (Croon & van Veldhoven, 2007), multilevel SEM only works with continuous data and is unable to handle categorical predictors. Additionally, multilevel SEM on continuous data frequently provides more biased estimates of the bottom-up effects than the latent bathtub model advocated by this paper (Onrust, 2015).
On a final note, bathtub models are very flexible and they run in most statistical software developed for latent variable modeling, including Mplus (Muthén & Muthén, 1998/2016 and Latent GOLD (Vermunt & Magidson, 2013). They can be conducted in R (R Core Team, 2016) as well, using the lavaan package (Rosseel, 2012), but this requires some technical expertise. Similar to SEM, latent bathtub models can be extended to include multiple variables-both continuous and discrete latent variables, with single or with multiple response variables at the micro-level (Bennink et al., 2015). More detailed statistical descriptions of the model can be found in Croon and van Veldhoven (2007) and Bennink et al. (2013Bennink et al. ( , 2014, whereas a syntax for bathtub implementation in Mplus is provided by Bennink (2014).

| OPTIMAL MATCHING ANALYSIS AND EMPLOYEE ENGAGEMENT
The rapid development of HR technology has initiated a trend toward the continuous measurement of personnel behaviors and cognitions.
Mobile applications, social networks, sociometric badges, wearables, and continuous employee feedback systems are rendering more complex and longitudinal HR data (Angrave et al., 2016;Bersin, 2015).
Employee engagement is one of the constructs that organizations increasingly measure on such an ongoing basis, potentially because research demonstrates it is less stable than previously assumed.
In line with the above, researchers call for prolonged periods of HRM researchers have been using methodology other than GLM to examine longitudinal patterns for quite some time (Sanders et al., 2014), but optimal matching analysis (OMA) remains relatively unknown in the field. OMA is a quantitative method originating from the natural sciences, where it has been especially useful in detecting temporal patterns. The method works by assessing the similarity among longitudinal sequences, after which an unsupervised learning algorithm can be used to group the sequences based on their similarity. The result is a categorical variable representing the patterns hidden in the longitudinal data, which can be an insightful classifier on its own or can function as a predictor or outcome variable in further analysis.
OMA has several unique characteristics that make it an addition to the current methodological toolbox of HRM researchers. Similar to other longitudinal methods, OMA only requires a small sample and simulations demonstrate that results are still 95% accurate for samples as small as 50 employees (Dlouhy & Biemann, 2015). In contrast to other methods, however, OMA is a person-oriented method, meaning that the object under observation-the employee or teamis the focus of the analysis, rather than the variance in a specific independent variable (Abbot, 1988). This allows OMA to examine patterns on multiple variables simultaneously, which is a more complex assignment for other longitudinal methods, such as multilevel and latent growth models (Curran, Obeidat, & Losardo, 2010). Hence, OMA can be used to examine employees' patterns on multiple dimensions of engagement (e.g., vigor, dedication, and absorption; Schaufeli et al., 2002) or on engagement data coupled with other constructs. Additionally, OMA functions particularly well with the new forms of HR data. OMA's classification results only increase with the number of nested observations (Dlouhy & Biemann, 2015), while other longitudinal methods soon require higher order polynomials to model observations over prolonged periods of time (Curran et al., 2010). Finally, despite the many observations and dimensions potentially included in OMA, it remains a relatively easy method to implement and interpret. To illustrate these advantages, the following section elaborates on a hypothetical study of weekly employee engagement data, like those gathered by a mobile application. The data requirements are described, and afterwards each of the model steps is explained. Finally, several limitations and alternatives are discussed.

| Data requirements
OMA works by comparing cases based on their temporal sequences.
These sequences consist of a string of elements, each reflecting the temporal state of the case at a specific moment in time. OMA handles these elements as categorical labels at this stage of the analysis, not recognizing any ordinal nature among them. Although this seems a disadvantage, it allows OMA to discover patterns in data that do not necessarily have an underlying order. For example, employees' trajectories across functions, locations, or organizational units (i.e., nominal or categorical variables) can be examined simultaneously with their engagement or performance levels (i.e., ordinal variables).
While an order among elements can be assigned at a later stage of the analysis, at this time, it suffices to label each state with a unique element (e.g., working in finance, with high engagement, and high performance = "A"). Once the elements are determined and labeled, OMA requires the input data set to be transformed into a wide format, where each row represents a case and each column a measurement occasion, so that for each case a sequence of elements arises.
Missing values are common in longitudinal research, but, compared to other methods, OMA handles them relatively easy. As long as no more than 30% of the elements within a sequence are missing, replacing missing values by an additional element (e.g., "X" or "?") will result in a model performance that is nearly equal to that of a complete data set (3% decreased classification accuracy in Dlouhy & Biemann, 2015). Although late joiners, attrition, and other factors may cause sequences to have different lengths, OMA can handle these as long as the length of the shortest included sequence is at least 70% of that of the longest sequence. Nonetheless, specific sequences may contain more than 30% missing values. These sequences can either be deleted or, if missing values gather around the start or end of the sequences, the time frame of observation can be shortened. However, both of these approaches may introduce bias to the results and, as an alternative, researchers can decide to impute the missing values. Despite potential collinearity between sequential and missing elements, gap closure by recursive imputation has been demonstrated to provide accurate estimates of the missing data points (Halpin, 2012).

| Penalty costs
OMA determines the similarity between sequences based on the operations needed to align them-to make them similar. There are two types of operations that can be used to align sequences: indel and substitution. Indel is short for the insertion or the deletion of an element somewhere in the sequence whereas substitution refers to the replacement of an element by another element in that same exact place in the sequence. Both operations need to be assigned a penalty cost to reflect the dissimilarity the operation corrected, and especially the ratio between these respective costs is important (Aisenbrey & Fasang, 2010;Biemann & Datta, 2014). By default, OMA uses a standard indel-substitution cost ratio of 1:2, meaning that deleting an element and inserting another comes at the same penalty cost of a substitution.
This standard ratio implies that OMA views all underlying states as equally dissimilar. However, as discussed earlier, HRM research often uses ordinal variables, which imply that certain states are more similar to each other based on their place in the underlying order. To reflect this similarity, the penalty cost of substitutions between those states can be decreased. Such a decrease can be based solely on theoretical assumptions but, alternatively, the observed transitions in the actual data function as a basis. Here, the rationale is that the observed transition frequency between states provides information about the similarity between these states (Biemann & Datta, 2014).
To illustrate the above, assume three 5-week sequences: E-E-E-E-E, representing a consistently engaged employee; N-N-E-E-E, representing an employee who went from neutral to engaged; and D-D-E-E-E, representing an employee who was disengaged for two weeks. Using the standard cost ratio, either sequence can be changed into the other at a penalty cost of 4: either by two element deletions and two element insertions, or by two substitutions. Alternatively, researchers could set theory-based substitution costs, penalizing transitions between states adjacent in the underlying order at a lower rate. For example, 1.75 for the adjacent levels of engagement and the standard 2 for more "distant" levels. Subsequently, the penalty costs between the first and the last sequence in the above example would still equal 4, whereas those between the first two sequences and the last two sequences would now amount to 3.5. Data-based substitu- for instance, cause transitions toward the extremes of the engagement continuum to be more frequent than the other way around, invalidating this assumption of directional independence. Second, data-based substitution costs will reflect the within-sequence variability of elements and, therefore, the chosen time span of elements has a strong influence. This becomes evident once applied to our engagement example. Assuming engagement fluctuates over time, element transitions would occur relatively frequently when longer spaced time spans, like yearly measurements, are used. In contrast, element repetition would occur frequently in the case of hourly observations. Both time spans have consequences for the data-based substitution costs, and while neither necessarily deteriorates results, researchers should consider that an interdependency exists.

| Clustering the sequences
Once the penalty costs are determined, the optimal matching algorithm can assess the (dis)similarity of each dyad of sequences by aligning them. Although there may be several ways to align two sequences, the algorithm seeks the one with the least penalty costs.
As illustrated earlier, assigning custom substitution costs would thus make the algorithm more prone to use substitution. The process of sequence alignment is repeated for all dyads in the data set and the resulting penalty costs are stored in a Euclidian distance matrix known as the dissimilarity matrix.
Next, a classification algorithm can be applied to the dissimilarity matrix. Any unsupervised learning algorithm that handles Euclidian distance matrices can be used. However, Dlouhy and Biemann (2015) "do not recommend using k-means, median, centroid and single linkage clustering for OMA at all" (p. 171). Out of the eight techniques they tested, Ward's minimum variance method consistently performed best. Irrespective of the chosen algorithm, the result is a categorical variable where cases are assigned to a category based on the underlying patterns in their sequence. In the engagement example, employees will be grouped based on the patterns that have occurred in their weekly engagement levels. One can expect to find, for example, clusters of consistently disengaged, neutral, and engaged employees. Similarly, other clusters may include employees whose engagement has been steadily rising or falling, whose engagement demonstrates certain cyclical patterns, or whose engagement fluctuates randomly. Depending on the assigned penalty costs and the chosen number of clusters, employees with missing values will be added either to the regular clusters or to clusters with specific, recurring patterns of missing values.
OMA's output can be valuable to researchers and practice in at least four ways. First, OMA's relatively simple implementation and interpretation makes it an effective tool to get descriptive insights in longitudinal data patterns. Second, OMA can be useful for identification purposes. For example, based on the cluster output, researchers can effectively identify which employees display certain (dis)engagement patterns and reach out with follow-up interviews or (supportive) interventions. Third, the output can be used as an independent variable in subsequent analyses to study the consequences of following a specific pattern. For example, the displayed engagement patterns could function as a predictor in leadership, performance, or attrition models. Fourth and final, the cluster output can function as a dependent variable for subsequent analysis regarding pattern occurrence. For example, individual differences or HRM practices may explain why certain employees are more likely to display particular engagement patterns.

| Limitations of OMA
A general shortcoming of OMA is that it aims to summarize a database filled with potentially very complex sequential patterns into a (handful of) categorical variable(s). While this has proven useful in certain research fields-including research on DNA, life courses, and careers-it has yet to be tested whether longitudinal measures of employee behaviors and cognitions can be similarly reduced to a set of patterns.
A second challenge relates to setting the substitution costs right.
Several studies illustrate how this ratio should be matched to the specific requirements of the research question and the analysis (Hollister, 2009;Lesnard & Kan, 2011). Other scholars have argued that standard and custom cost ratios lead to similar conclusions (Biemann & Datta, 2014). No study provides insights regarding the optimal settings for research on employee experiences, which may be affected by a wide variety of personal and institutional factors. Moreover, there are rightful concerns regarding the symmetrical nature of the substitution costs (Aisenbrey & Fasang, 2010), which causes transitions between states to be regarded as similar, irrespective of their direction. There seems to be no simple solution to the aforementioned issues, apart from some general best practices regarding the cost-setting procedure (e.g., Gauthier, Widmer, Bucher, & Notredame, 2009).
A third limitation lies in the descriptive nature of OMA. The method can reduce large information volumes into smaller, workable sets of underlying patterns, and complementary analysis may provide insights into why these patterns occur and what they result in. However, researchers seeking to test why, how, and when patterns occur and trajectories develop may turn to other methods.
Here, multilevel and latent growth models can be used to examine the rate of pattern development as well as its causes. Additionally, hazard and Markov models may uncover why and when transitions between states happen. Moreover, time series analysis could be applied to investigate reoccurring patterns and forecast the future state of employees.
Finally, the only software that currently provides a means for automated implementation of OMA is R (R Core Team, 2016). The TraMineR package (Gabadinho, Ritschard, Müller, & Studer, 2011) contains functions that automate the process to a large extent, and only minor specification and customized programming is required.
Additionally, the package includes several visualization functions that facilitate the interpretation of the model's output. However, getting accustomed to the R language and syntax can be effortful.

| DISCUSSION
Previous research has heavily relied on GLM to investigate HRM processes and their potential impact on performance. This article proposes two statistical modeling techniques that, despite their novelty to the field, can be valuable additions to the methodological toolbox of HRM researchers and practitioners. Such additions are necessary seen the growing need particularly in light of the growing need to justify, prioritize, and improve HRM decision making (Boudreau & Ramstad, 2007;Rasmussen & Ulrich, 2015;Ulrich & Dulebohn, 2015) and the new forms of HR data that arise due to technological developments (Angrave et al., 2016;Bersin, 2015).
Using latent variables, bathtub models are put forward as the solution to examine multilevel mechanisms with outcomes at the team or organizational level without decreasing the sample size or neglecting the variation inherent in employees' responses to HRM activities.
Optimal matching analysis is proposed as particularly useful to examine the longitudinal patterns that occur in repeated observations over a prolonged time frame. Research on employee engagement was used to illustrate how each method functions and how they add value over and above the current methods used in HRM research.
Although both bathtub modeling and OMA elevate micro-level data to a macro-level, the two methods strongly vary in their purpose, their complexity, and the expertise required to implement them.
The application of OMA does not require deep statistical or conceptual knowledge and the pattern visualizations facilitate an easy interpretation. However, this simplicity is also reflected in the primarily descriptive insights the method provides. In contrast, the underlying equations as well as the output of bathtub models may be harder to explain to laymen such as business and HR professionals (see Bennink et al., 2013Bennink et al., , 2014Croon & Van Veldhoven, 2007). This increases the difficulty that scholars and HR analytics professionals may experience in translating the latent variable model's results into actionable insights for decision makers.
Nevertheless, both techniques can add value to HRM research on a variety of themes, either applied separately or in synergy with each other and other methods. Recruitment and selection is one field of potential future application. OMA has been frequently applied on career patterns (e.g., Blair-Loy, 1999) and, similarly, by clustering applicants based on their prior work experiences, the method could be valuable for selection purposes. For example, applicants' historic job positions can be coded into unique states based on the associated management responsibilities or the required level of technical expertise. The required 25 months of input data for OMA (Dhouly & Biemann, 2015) could be extracted directly from applicants' résumés, but the digital job market becomes an ever richer data source as well (e.g., LinkedIn, Xing, ResearchGate). The resulting clusters may facilitate decision making in the selection process, similar to applicants' assessment center scores, interviewer ratings, and other recruitment data. Additionally, a latent bathtub model could use such data to examine the effectiveness of recruitment, selection, and/or socialization practices. Irrespective of these practices, bathtub models could also investigate whether certain (combinations of) applicant profiles improve the effectiveness of teams.
The methods may additionally be valuable with regard to workforce planning, facility management, and flexible working arrangements. Recent work by Lesnard and Kan (2011) demonstrates how a two-stage OMA can be used to first cluster the daily work schedules of employees, and subsequently use these clusters to unveil the patterns employees display in their weekly schedules. Using the data collected by sociometric badges and "smart" workplaces, HRM scholars and practitioners could use OMA to uncover patterns in the use of office spaces over time or across locations. This may have direct practical value in terms of the cost reductions related to facility management, but may also be insightful for the design of flexible work arrangements. Moreover, certain configurations of work schedules within teams may have a detrimental impact on their effectiveness. This could be examined by using the cluster output of OMA in subsequent bathtub models, raising the work schedules to a team level and relating them to team performance. Finally, the flexibility of latent variable models makes it a valuable tool to examine a wide variety of contemporary HRM themes. For example, latent approaches can also be used to model unobserved heterogeneity between respondents. This can be valuable to HRM research on team diversity, for example, in terms of location, tenure, age, gender, or cultural background. A more abstract example lies in the investigation of team heterogeneity in terms of individual psychological contracts (e.g., Bakk, Tekle, & Vermunt, 2013). Similarly, the data of e-mail traffic or sociometric badges can be used to examine heterogeneity in terms of the personal networks employees have.
Subsequently, the impact of such heterogeneity on the development, retention, or performance of individuals, teams, and organizations could be assessed. Combined with OMA, this latent heterogeneity of teams could be monitored over longer periods of time to see whether improvement occurs, potentially as a result of changes in HRM policies. Moreover, latent variable models can be used to grasp constructs that are otherwise hard to measure. For instance, a latent performance score could be estimated using multiple indicators of employees' behavior or job output. This latent score has the potential to be a more accurately reflection of employees' actual performance than the separate indicators or their combined average (Murphy, 2008).
In conclusion, bathtub models and OMA can add value on many HRM themes, allowing an investigation of research questions that were previously hard to examine. The methods can stimulate the quality of decision making either by offering a different analytical approach to the issue at hand or by working in synergy with each other and more traditional HRM methodology. Bathtub modeling and OMA are just two examples of methods that are common in other research fields and can benefit the HRM community. We hope that, by demonstrating how these specific methods add value in analyzing the new forms of HR data, HRM researchers and practitioners become more open to methodological developments within and outside the field of HRM. In the end, alternative methodology can offer a different perspective and facilitate improved, more objective decision making regarding human capital.