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Accounting, Organizations and Society 33 (2008) 995–1009 www.elsevier.com/locate/aos

The appropriateness of statistical methods for testing contingency hypotheses in management accounting research Jonas Gerdin *, Jan Greve ¨ rebro University, SE-701 82 O ¨ rebro, Sweden Department of Business Administration, O

Abstract In recent years, the contingency-based management accounting literature has been criticized for being fragmentary and contradictory as a result of methodological limitations. This study adds to this picture by showing that the theoretical meaning of some commonly used statistical techniques is unclear, i.e. the functional forms are not precise enough to be able to discriminate between several sometimes even conflicting theories of contingency fit. The study also shows that the techniques differ significantly in terms of how interaction effects between context and management accounting are modeled. This implies that some methods are only appropriate when theory predicts interaction effects in general while others are only appropriate in cases where theory specifies a more precise functional form of interaction such as symmetrical or crossover interactions. Based on these observations, several recommendations for future research are proposed. Ó 2007 Elsevier Ltd. All rights reserved.

Introduction In recent years, several literature reviews have highlighted that many different ways of conceptualizing ‘contingency fit’ between context and Management Accounting System (MAS) have been used in the literature (Chenhall, 2003; Luft & Shields, 2003) and that few researchers fully acknowledge the difficulties of relating these forms * Corresponding author. Tel.: +46 19 30 30 00; fax: +46 19 33 25 46. E-mail addresses: [email protected] (J. Gerdin), jan. [email protected] (J. Greve).

to each other (Gerdin & Greve, 2004). There has also been a growing interest in (and debate about) how individual statistical techniques have been applied in contingency-oriented MAS research (Dunk, 2003; Gerdin, 2005a, 2005b; Hartmann, 2005; Hartmann & Moers, 1999, 2003). The purpose of this paper is to combine these two streams of research by providing a systematic analysis of the appropriateness of commonly used statistical techniques for testing the different forms of fit found in the literature. In so doing, we propose a conceptual framework which identifies a number of possible perspectives of contingency fit. Unlike most of the

0361-3682/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.aos.2007.07.003

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existing MAS literature (e.g. Chenhall, 2003; Gerdin & Greve, 2004; Luft & Shields, 2003), the framework explicitly elaborates on the distinction between a matching and a multiplicative model of fit (Schoonhoven, 1981). The framework also contributes to the more general discussion about the use of statistical techniques in contingency research (Donaldson, 2001; Drazin & Van de Ven, 1985; Meilich, 2006; Venkatraman, 1989) by highlighting that the paradigm seems to accommodate at least three levels of theory specification. The paper proceeds as follows. Drawing upon seminal contingency work, three levels of precision in the functional form of context/MAS interactions and four principal and conflicting approaches to contingency fit are identified. Next, it is discussed to what extent statistical methods frequently applied in contingency-based MAS research can be used to test the different levels of interaction and to distinguish between the four approaches. This results in several conclusions and recommendations for future research which finalize the paper.

and business strategy if the organization is to perform well (Burns & Stalker, 1961; Donaldson, 2001; Drazin & Van de Ven, 1985; Lawrence & Lorsch, 1967; Pennings, 1992; Woodward, 1965). Galbraith (1973, p. 2) formulated this core idea of contingency theory in the following way: 1. There is no one best way to organize. 2. Any way of organizing is not equally effective. These statements imply that the effectiveness of organization structures is contingent on context—i.e. there is no universally best way to organize—and that, in a particular context, certain structure(s) will outperform other structures. Schoonhoven (1981, p. 351) referred to this particular form of relations between variables as interactions. Generally, an interaction effect exists whenever the effect of an independent variable (structure) on the dependent variable (performance) varies due to the values of a third variable (context) (Jaccard & Turrisi, 2003). As illustrated in Table A, Fig. 1, an interaction effect thus implies that a change in structure has a more positive (or negative) effect on performance in different contexts (Luft & Shields, 2003). Given the broad format of this type of interaction, it will henceforth be referred to in terms of ‘general interaction’ and represents the first level of theory specification.

Levels of theory specification in the contingency paradigm The essence of contingency theory is that organizations must adapt their structure to contingencies such as the environment, organizational size

Table A

Table B Performance

Performance

Performance

CH

CH

CL

Low

High Structure

a

Table C

CH

CL

CL

Low

High Structure

Low

High Structure

CH and CL denote high and low levels of the contingency factor, respectively.

Fig. 1. Illustration of an interaction (monotonic) function (Table A), a symmetrical interaction (non-monotonic) function (Table B) and a crossover interaction function (i.e. both non-monotonic and disordinal function) (Table C)a.

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However, much of the early contingency work, e.g. Burns and Stalker (1961) and Galbraith (1973), seems to involve a higher level of theory specification (henceforth referred to in terms of the second level) where interaction effects take a symmetrical form (Schoonhoven, 1981). As illustrated in Table B in Fig. 1, a symmetrical interaction implies that a certain structure is high-performing when contingency is low (CL), while another structure is high-performing when contingency is high (CH). Some contingency theorists have brought the idea of symmetrical interaction one step further as they have also specified the relation between contingency fit and performance (henceforth referred to as the third level of theory specification). For example, Woodward (1965, p. 69), in her pioneering study of the fit of span of control to technology, noted that all firms in a state of fit had equal performance ratings even though there were three different fits, one for each technology category. Hence, organizations in states of fit outperformed those in misfit—both within and between contextual subgroups—which suggests that it is the degree of fit between structure and context that is the principal explanation of observed variance in organizational performance, not structure or context alone.1 Pfeffer (1982, p. 148) referred to this relation in terms of ‘the consonance hypothesis’ implying that ‘‘those organizations that have structures that more closely match the requirements of the context are more effective than those that do not’’. However, when we claim that there are no generally more effective contexts, we need to demonstrate that the symmetrical interaction effect also is disordinal, i.e. the ranking order of structure related to performance changes within the observable range of data (Cohen, Cohen, West, & 1

In fact, some scholars even propose that structural contingency theory, at least in its original form, postulated so-called iso-performance, i.e. that organizations with structures that match contexts perform about equally albeit they operate in different contexts, while organizations in states of misfit perform below that level (Donaldson, 2001; Drazin & Van de Ven, 1985; see also Schoonhoven’s interpretation of Galbraith’s (1973) theory at p. 353 and also her operationalization of contingency fit at p. 352).

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Aiken, 2003) (see Table C, Fig. 1). Such effect is sometimes referred to in terms of a ‘crossover interaction’ (Cohen et al., 2003).

Conceptualizations of contingency fit Over the years, there has been considerable debate in the contingency theory literature (for overviews see Donaldson, 2001; Drazin & Van de Ven, 1985; Pennings, 1992; Venkatraman, 1989). The following three issues of controversy are directly linked to the conceptualization of contingency fit:2 – Congruence vs. contingency. Is fit postulated, or must it be explicitly shown that deviations from optimal context/structure combinations lower organizational performance? – Cartesianism vs. configurationalism. Is fit a continuum between pairs of contingency and structure dimensions that allows frequent and small movements by organizations from one state of fit to another, or is it the internal consistency of multiple contingency and structural elements with organizations having to make ‘quantum jumps’? – Matching vs. multiplicative relationships. Is fit a line with many optimal combinations of context and structure where any deviations affect performance equally, or is it assumed that there are only two optima and that the effect of deviations differs across different levels of context? While the meanings and implications of the former two controversies on management accounting research have been discussed in several previous 2

Note that the terminology in the literature is somewhat confusing. For example, when Drazin and Van de Ven (1985, p. 514) use the term ‘congruence’, it refers to an unconditional association between context and structure, while Donaldson (2001, pp. 186–189) use the same term to denote a specific type of conditional association between context, structure and performance—an association which yet other researchers have referred to in terms of matching models (Schoonhoven, 1981; Venkatraman, 1989). In the present study, we decided to handle this diversity by using an adapted version of the terminology proposed by Gerdin and Greve (2004).

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studies (see e.g. Chenhall, 2003; Gerdin & Greve, 2004; Luft & Shields, 2003), the third controversy has received much less attention (but see Hartmann & Moers, 1999, p. 299). This is unfortunate since also the distinction between a matching and a multiplicative model of contingency fit has important consequences for the choice of appropriate statistical methods. In the matching form of fit, it is assumed that for every value of context, there is a unique value of structure at which performance is maximized. As illustrated by Tables A and C in Fig. 2, a particular structure optimizes performance at a low value of contingency (e.g. low structure at C1) while another structure maximizes performance at a high value of contingency (e.g. high structure at C5). Judging from the mathematical operationalizations of this form of fit in

previous studies (see e.g. Schoonhoven, 1981, p. 352), so-called iso-performance is often assumed, i.e. all states of fit produce the same level of performance (see also Woodward, 1965, and the discussions in Donaldson, 2001, & Van de Ven & Drazin, 1985). However, iso-performance is not an inevitable consequence of the symmetry assumption (Donaldson, 2001). Furthermore, Table A illustrates that any deviations in either direction from the unique optimal value of structure reduce the level of performance. Put in another way, each point on the fit line represents the optimum on a non-linear function where performance is the dependent variable and structure is the independent variable (see Table C, Fig. 2). Thus, the contingency factor acts as a moderator because it ‘‘determines which charac-

Table Aa High

Structure

Table Ba

P2

P4

P6

P8

P10

P4

P6

P8

P10

P8

P6

P8

P10 P8

P8

P10 P8

P6

P4

P10

P8

P6

P4

P2

High

Structure

P6

Low

P2

P4

P6

P8

P10

P4

P5

P6

P7

P8

P6

P6

P6

P6

P6

P8

P7

P6

P5

P4

P10 P8

P6

P4

P2

Low Low

High

Low

Contingency a

Contingency

The indices denote the level of organizational performance.

Table Cb

Table Db

10

C5

8

C4

6 C3

4 C2

2

Performance

Performance

High

10

C5

8

C4

6

C3

4

C2

2

C1

C1 Low

High Structure

b The

Low

High Structure

indices denote the level of contingency.

Fig. 2. Illustration of a matching form of interaction (Tables A and C) and a multiplicative form of interaction (Tables B and D).

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teristic produces high levels of effectiveness of the organization’’ (Donaldson, 2001, p. 6) and it ‘‘effectively shifts the curvilinear relationship between the structural and outcome variables’’ (Meilich, 2006, p. 165). Finally, matching models seem to imply that a deviation of one unit from the optimal structure/ contingency combination has the same effect on performance across all levels of the contingency (see the indices in Table A, Fig. 2). When Schoonhoven (1981, p. 355) examined some of Galbraith’s (1973) statements, she concluded that he implicitly understands contingency relationships as multiplicative forms of fit. That is, the effect of a structural variable on performance increases, or decreases, as a result of changes of the contingency level (see Fig. 2, Tables B and D). Contrary to the matching form, multiplicative interaction thus assumes that performance is a linear function of structure at each value of contingency. The assumption about linearity in conjunction with the assumption about non-monotonic relationships implies that the contingency factor (moderator) determines both the direction and amplitude of structure’s effect on performance (cf. Table D in Fig. 2 where the structure-performance relationships at C2 and C5 not only have different slopes, but also different signs). The conflicting assumptions about linearity in the matching and the multiplicative form have important consequences. Note especially that the multiplicative form implies that only two structures (the extreme values) are assumed to be optimal (cf. the typical ‘saddle form’ in Table B (Southwood, 1978)). Implicitly, the multiplicative interaction form thus postulates so-called heteroperformance, i.e. that different contingency/structure combinations produce different maximum performances. Also note that in contexts around the inflexion point C3 in Fig. 2, Table D, all structures produce about the same level of performance. None of these features are compatible with a matching view, where fit results from numerous combinations of contingency and structure, yet for each value of contingency, only one structure can be optimal. It can be concluded that within the contingency literature, there has been a number of controver-

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sial issues in terms of what constitutes fit between context and structure and how fit is attained. In addition to Cartesian models of fit (such as the matching and multiplicative model), there are theories predicting that fit is the degree of adherence to configurations consisting of multiple context and structure variables (Drazin & Van de Ven, 1985; Venkatraman, 1989). As mentioned above, there is also a set of so-called congruence models that do not aim to explain variations in performance in terms of (mis)fit as they typically postulate that only high performers exist to be observed. In the previous section, we also pointed out that there are at least three broad variants of contingency theory that differ in terms of their level of specificity. Some theories seem to predict interaction effects per se between context and structure (i.e. general interaction), while others prescribe non-monotonic relationships (i.e. symmetrical interaction) and, sometimes, also disordinal relationships (i.e. crossover interaction). When we combine the different models of fit with the different levels of theory specification, we get a number of possible variants of contingency theory (see Table 1). Arguably, as these theoretical variants make very different assumptions about how fit is obtained, they typically require different statistical test methods. More precisely, the method(s) used should (at least ideally) have the capacity to test for the assumptions made at the chosen level of theory specification—no more, no less—and, at the same time, have a functional form that corresponds with the model of fit predicted. The functional form should preferably also be precise enough to be able to discriminate between the different models of fit. As will be demonstrated in the next section, this is not always the case.

Commonly used statistical techniques in management accounting studies to conceptualize contingency fit According to several recent literature reviews, many different statistical techniques have been used in the literature (see e.g. Chenhall, 2003; Gerdin & Greve, 2004; Hartmann & Moers, 1999; Luft

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Table 1 Variants of contingency theory (the head of the table is an adapted version of that in Gerdin and Greve (2004, p. 318))a

Forms of fit Levels 3. Crossover interaction 2. Symmetrical interaction 1. General interaction

Contingency Cartesian Matching Multiplicative

Congruence Configuration

a

As all congruence models seem to predict at least symmetrical relationships (i.e. that different contexts are associated with different structures), the lower right cell is considered as theoretically not relevant.

Table 2 Commonly used statistical methods in MAS studies to conceptualize and test the existence of contingency fit Contingency fit as:

Selected references:

Difference in means

Abernethy and Brownell (1999), Abernethy and Lillis (1995), Chenhall and Langfield-Smith (1998) Duncan and Moores (1989), Gerdin (2005b), Govindarajan (1988), Haka (1987), Merchant (1981, 1984), Selto et al. (1995) Abernethy and Brownell (1999), Abernethy and Lillis (1995), Khandwalla (1972), Merchant (1981, 1984), Simons (1987) Macintosh and Daft (1987)

Bivariate correlation Difference in correlation coefficients (strength) Difference in correlation coefficients (form) Linear regression coefficients Difference in regression coefficients An indirect path coefficient

& Shields, 2003). In Table 2, frequently occurring techniques identified in these literature reviews are outlined. In the following sections, these statistical techniques will be briefly described and classified in terms of their level of theory specification and it will be discussed to what extent they can discriminate between the four forms of contingency fit outlined above.3 3 Note that these techniques will be dealt with separately below. In many of the empirical studies cited, however, several ways of testing contingency fit have been used. This implies that some of the drawbacks and limitations related to individual statistical techniques discussed may have been addressed by the researchers cited insofar as they deliberately have used several complementary statistical analyses (see also the concluding discussion).

Brownell (1982), Duncan and Moores (1989), Frucot and Shearon (1991), Kaplan and Mackey (1992), Simons (1987) Abernethy and Brownell (1997), Bisbe and Otley (2004), Brownell and Merchant (1990), Chong (1996), Perera et al. (1997) Baines and Langfield-Smith (2003), Bouwens and Abernethy (2000), Chenhall and Morris (1986), Chong and Chong (1997)

Fit as difference in means In congruence-type of studies, this way of conceptualizing contingency fit implies that a sample of organizations, subunits or the like is divided into subgroups and the MAS characteristics of the subgroups are compared (see e.g. Abernethy & Lillis, 1995). Since all congruence models seem to predict symmetrical relationships, but it is unclear whether crossover interactions are postulated, it will henceforth be argued that significant results imply (but do not formally test) either of the two types of interactions. In contingency-type of MAS research, we have found two approaches. One is that of Chenhall and Langfield-Smith (1998) who used cluster analysis to form organizational gestalts and then exam-

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ined whether the differences in average performance scores between clusters were statistically significant. As the authors expected that different strategies would be associated with different sets of management control techniques and practices, this way of testing contingency fit implies a configuration approach predicting symmetrical interactions. In the other approach, the typical procedure is first to create subgroups consisting of fits and misfits in two contexts, respectively, and then to show that the performance score of the fits is superior to that of the misfits (see e.g. Abernethy & Brownell, 1999). In our view, this way of conceptualizing fit implies that a crossover interaction is predicted (i.e. relationships are symmetrical and, furthermore, no context is generally more effective). However, this test provides very little information about the form of the association. For example, is a difference in MAS the result of an incremental adaptation to context, or are quantum jumps required (cf. the Cartesian and the Configuration approach, respectively)? Furthermore, is a significant result a crude expression of a fit line with numerous optimal solutions, or does it indicate that MAS should either be maximized or minimized in order to maximize performance (cf. the matching and the multiplicative form of interaction, respectively)? So to conclude, the different variants of analyzing means found in the literature may imply both symmetrical and crossover interactions. Furthermore, for some uses it is clear which form of fit is being tested, for others this is not the case. Fit as bivariate correlation In some congruence-type of studies, fit is operationalized as a statistically significant correlation between context and MAS (see e.g. Merchant, 1981, 1984). Given that this application of correlation analysis is only used to test predictable correlations between pairs of context and MAS variables, it can be argued that it focuses on the form of the relationship. As a result, bivariate correlation analysis seems to correspond well with a Cartesian variant of congruence-type of contingency fit. In contingency-type of studies, bivariate correlation analysis has been used to test the effect of fit on performance in so-called deviation-score or

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residual analyses (see e.g. Duncan & Moores, 1989; Gerdin, 2005b). The typical procedure is to first compute deviations between a theoretically or empirically derived fit line and actual positions, and then to correlate the deviation-score with performance. Arguably, this way of using bivariate correlation analysis is founded on a matching theory insofar as the fit line encompasses an infinite number of unique combinations between context and MAS, each of them assumed to produce maximum performance, and each unit of deviation from the fit line affects performance equally (see Tables A and C in Fig. 2). Furthermore, since any deviation from the optimal line is explained by misfit alone (context has no direct effect), it implies that a crossover interaction is predicted. Notably, however, bivariate correlation analysis has also been used to test symmetrical and general interactions. An example of the former type is the study of Govindarajan (1988) which predicted a negative correlation between the degree of distance from ideal configurations in two contexts and organizational performance, but no assumptions about ‘context effects’ were made (see also Selto, Renner, & Young, 1995). An example of the latter type of interaction is the study of Haka (1987) which expected that firms using sophisticated capital budgeting models outperformed matched non-users in high levels of context but not in low levels. As Haka used a non-parametric correlation analysis based on rank order, however, it is unclear whether the underlying relationship is expected to be linear (cf. the matching model) or non-linear (cf. the multiplicative model). To conclude, bivariate correlation analysis has been employed for testing congruence theories as well as matching and configuration theories at the crossover and symmetrical interaction levels, respectively. The technique has also been used to test the existence of general interactions. Fit as difference in correlation coefficients: strength To our knowledge, subgroup correlation analysis focusing on differences in strength has above all been applied in contingency-type of MAS studies (but see e.g. Khandwalla, 1972). For example,

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Abernethy and Brownell (1999, p. 198) computed correlation between strategic change and performance for two MAS sub-samples (see also Simons, 1987; Merchant, 1981, 1984; Abernethy & Lillis, 1995). In our view, an analysis focusing on differences in strength between subgroups does neither correspond with a matching, nor a configuration perspective. The premise is that these theories give little reason to expect that the predictive ability of MAS on performance should differ across different levels of the contingency variable (see also the discussion in Drazin & Van de Ven, 1985). In contrast, this is what a multiplicative form of fit implies. That is, for contingency levels around in the inflexion point in Table D, Fig. 2, the predictive ability of MAS on performance is very low, which implies that the correlation is expected to be low, while the opposite holds for the extreme values of the contingency variable (see C1 and C5). Note also that the method represents the first level of theory specification (i.e. predicts general interactions), since a significant difference in correlation per se neither shows that a high-correlation subgroup generally outperforms a low-correlation group, nor that a certain MAS is appropriate in certain contexts and another MAS in others.

crossover interaction effect is present (see Table B and C in Fig. 1). However, it is unclear as to which form of fit is supported. To illustrate, again recall the relationships depicted in Fig. 2 above. From both a matching and a multiplicative perspective, we would expect a negative correlation between MAS and performance for low levels of the context (see C1 and C2 in Tables C and D) and a positive correlation for high levels (C4 and C5). However, while both models of fit are supported by different signs of the subgroup correlation coefficients, we should not conclude that the higher the correlations, the better the support of these types of relationships. In fact, it could be argued that this use of correlation analysis is better suited to test a specific form of configuration theory, namely, when it is expected that firms benefit from the highest possible values on a set of management control dimensions in one context, and from the lowest possible values in the opposite context (see e.g. Govindarajan, 1988). To conclude, subgroup correlation analysis focusing on the form of relationships can generally be used to test the existence of symmetrical interactions, but it cannot differentiate between a matching, a multiplicative and a configuration model. Fit as linear regression coefficients

Fit as difference in correlation coefficients: form Congruence-type of studies focusing on the different forms of correlation coefficients typically predict that a particular MAS is positively correlated with the extent of one context and negatively correlated with the extent of another one (e.g. Macintosh & Daft, 1987). In our view, such a use of subgroup correlation analysis seems to imply a Cartesian variant of contingency fit insofar as a continuous (and linear) relationship between context and MAS is predicted. In Contingency-type of studies, fit is supported by demonstrating that there are statistically significant differences between subgroups, produced by the correlation between MAS and performance being positive for one contextual subgroup and negative for the other subgroup. Accordingly, some information about whether relationships are symmetrical is provided, but none about whether a

This approach to conceptualizing and testing contingency fit typically implies that regression equations of the following form are fitted to the data: Y ¼ b0 þ b1 X 1 þ b2 X 2 þ b3 X 3 þ e

ð1Þ

In congruence-type of MAS studies, main effect regression analysis has primarily been used to test whether MAS design/use (Y) is associated with one or several contingency factors (Xi) (see e.g. Kaplan & Mackey, 1992). However, regressions in the ‘opposite direction’ can also be found in the literature. For instance, Simons (1987) used a logit regression model in which competitive strategy was modeled as the dependent variable (Y) and a number of management control and environmental factors were independent variables (Xi). Importantly, however, the theoretical interpretation of main effect regressions can be unclear.

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For example, the Kaplan and Mackey study cited above seems to imply a Cartesian variant since each independent contextual variable is expected to have an effect on the degree of MAS use. The analysis performed by Simons (1987), in contrast, seems to imply a configuration variant as the logit regression examines whether the overall pattern of control system attributes differs between strategic types. In contingency-type of studies, a ‘fit term’ is often regressed on performance (Brownell, 1982; Duncan & Moores, 1989; Frucot & Shearon, 1991). Such regression typically has the following format (Hartmann & Moers, 1999; Venkatraman, 1989): Y ¼ b0 þ b1 X þ b2 Z þ b3 jX  Zj þ e

ð2Þ

where X is a MAS attribute, Z is context and Y is performance. The ‘fit term’ jX  Zj is a deviationscore, indicating the lack of fit between X and Z. Arguably, this way of using linear regression analysis resembles the above discussed use of bivariate correlation analysis to establish a relationship between a deviation-score and performance. By implication, it means that this method is not suitable for detecting multiplicative or configurative relationships between (mis)fit and performance. Unlike the use of bivariate correlation analysis discussed above, however, linear regression coefficients disclose the nature of the relation (Duncan & Moores, 1989), i.e. the potential contextual influence on performance is separated from the interaction effect as such in the analysis. This means that interaction effects will be detected also in situations where performance is strongly influenced by context, i.e. when the interaction is symmetric (but not disordinal). Yet, if b2 in Eq. (2) is not significant, or if the variables X and Z are not included in the equation, and b3 is still significant, this indicates a crossover interaction. Accordingly, depending on how the regression is written and the results of the test, this statistical technique can be used to explore the existence of both symmetrical and crossover interactions. Fit as difference in regression coefficients One frequently used technique for testing the existence of a significant difference in regression

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coefficients is the moderated regression analysis (MRA) (Hartmann & Moers, 1999). An MRA typically has the following format: Y ¼ b0 þ b1 X þ b2 Z þ b3 X  Z þ e

ð3Þ

where Y is a dependent variable, X is an independent variable, Z is a moderator, X * Z is the moderating effect that Z has on the relationship between X and Y, and e is the error variable (Cohen et al., 2003; Jaccard & Turrisi, 2003). Although MRA has been used in congruencetype of studies (e.g. Perera, Harrison, & Poole, 1997), a significant interaction term per se is not required to conclude that a congruence relationship exists. Rather, the inclusion of such term (in addition to main effect coefficients) merely implies that the functional form of the context/MAS relationship is ‘better’ specified. In contingency-type of MAS studies (Bisbe & Otley, 2004; Brownell & Merchant, 1990; Chong, 1996), however, the interaction term is crucial as it tests the existence of general interaction effects. That is, a significant term indicates that the effect of MAS on performance differs across different levels of context, but no information is provided about whether the effect is non-monotonic (symmetrical interaction) or disordinal (crossover interaction). Furthermore, since this form of MRA is based on ‘linear by linear’ interactions (Cohen et al., 2003; Jaccard & Turrisi, 2003), it is unsuitable to test the existence of curve linearity predicted by the matching model (cf. Table C in Fig. 2). In the MAS literature, there are also examples of where regression coefficients in different subgroups have been compared in order to test for interaction effects. For example, Abernethy and Brownell (1997) first divided their sample into four contextual groups and then regressed simultaneously three types of control on managerial performance in each of the four groups. Like MRA, this way of using main effect regression analysis is inconsistent with matching theory insofar as matching theory gives little reason to expect that MAS’s effect on performance should differ across different levels of context. So to conclude, both MRA and comparisons of main effect coefficients between subgroups seem appropriate for testing a multiplicative model of

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fit. Furthermore, both test for the existence of general interactions. Fit as an indirect path coefficient In MAS research, contingency fit is often analyzed by the introduction of a mediating variable. That is, the effect of the independent variable on the dependent variable operates completely or partially through the mediating variable (Gerdin & Greve, 2004; Luft & Shields, 2003; Venkatraman, 1989). In congruence-type of studies, different properties of MAS are typically modeled as the dependent variable (e.g. Bouwens & Abernethy, 2000; Chenhall & Morris, 1986). Arguably, while a path analysis may reveal associations between variables, only direct relations can express a congruence relationship. Hence, a congruence relationship is supported whenever an indirect path is prevalent, but only in the sense that the mediator is significantly associated with MAS. Also contingency-type of studies using pathanalytical techniques are common in the MAS literature (e.g. Baines & Langfield-Smith, 2003; Chong & Chong, 1997). However, although these methods contribute to our understanding about what determines design and use of MAS, they do not test for the existence of interaction effects between context and MAS on performance. To illustrate, consider a theory predicting that highperforming firms in dynamic environments rely on externally oriented MASs (Proposition A), and that high-performing firms in stable environments rely on internally oriented MASs (Proposition B). While a path analysis in a sense could confirm Proposition A by showing that dynamism is positively correlated with externally oriented MASs which, in turn, is positively correlated with performance, such a result would contradict Proposition B (because it implies that stability is positively correlated with an internally oriented MAS which, in turn, is negatively correlated with performance). Accordingly, path analyses cannot explore the existence of moderated causal relationship(s) predicted by contingency theory (Donaldson, 2001; Schoonhoven, 1981), i.e. where the relationship between MAS and performance is moderated by context.

Results of the analysis of statistical methods In Table 3, the different uses of statistical methods found in the contingency-style MAS literature have been sorted into the classificatory framework developed above. A method’s row position thus denotes the level of specificity of the theory being tested and the column denotes the particular model of fit being tested. A line that crosses several columns represents the method’s inability to discriminate between the models of fit in question. Finally, a dotted line marks that the interaction effect is postulated rather than formally tested. Several observations can be made based on Table 3. A first observation is that studies have been made in most theoretically relevant subcategories of contingency theory. This is noteworthy in its own right as it reinforces recent claims suggesting that contingency-based MAS research is diverse (Chenhall, 2003; Gerdin & Greve, 2004; Luft & Shields, 2003) and, furthermore, suggests that future research not only should consider the different forms of fit present in the literature, but also the different levels of theory specification. Given the purpose of this study, however, a more important conclusion related to this observation is that, generally, different statistical methods have been used to test these sub-theories (an exception is congruence-type of studies in which all methods have been employed). Overall, this multiplicity is a necessity as these different variants of contingency theory make very different assumptions about the nature of relationships. A second observation is that some methods have difficulties in distinguishing between different forms of fit. For example, when mean values of performance between subgroups are compared and a significant crossover interaction in the predicted direction is identified, this indicates that MAS is contingent on context. However, a crossover interaction effect may be predicted in all forms. Consequently, such comparison of mean values helps to identify contingency variables; however, it does not specify the functional form of the fit relationship. A third observation is that the number and quality of methods is unequally distributed amongst sub-theories. For some subcategories, several func-

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Table 3 Results of the analysis of statistical methods used in management accounting researcha Forms of fit

Contingency

Congruence

Cartesian Levels

Matching

Configuration Multiplicative

Regression coefficients

3. Crossover interaction

Bivariate correlation All methods

Difference in means Bivariate correlation

2. Symmetrical interaction

Regression coefficients

Difference in means Difference in correlation (form)

All methods

Difference in regression coefficients

1. General interaction

Difference in correlation (strength) Bivariate correlation

a A double-headed line signifies that the statistical method has been used to test a particular sub-theory. A line that crosses several columns denotes that the method cannot discriminate between the forms of fit in question. Finally, an unbroken line means that the interaction effect is formally tested while a broken line signifies that the interaction effect is postulated.

tionally precise methods have been used (e.g. matching theories predicting crossover interactions) while, for others, the opposite is the case (e.g. for multiplicative type of studies predicting symmetrical or crossover interactions). As will be developed in more detail below, these different prerequisites imply that researchers sometimes need to prioritize between several conceivable alternatives and, sometimes, need to come up with complementary or ‘new’ techniques in order to compensate for an existing lack of appropriate ones. A fourth observation is that one of the methods previously discussed, i.e. path analysis, can only be used to test congruence-type of theories. However, this is not to say that path analysis generally should be avoided when a contingency-form of fit is predicted. After all, there might be situations where theory suggests a combination of the two forms of causality (Chenhall, 2003). A fifth observation is that all methods can be used to test for symmetrical/crossover interactions in congruence-type of studies although, again, the interaction effects are postulated rather than empirically examined. Interestingly, however, a closer look suggests that this stream of research is dominated by two different objectives. One is

to identify the contingency factors as such (see e.g. Abernethy & Lillis, 1995; Kaplan & Mackey, 1992; Merchant, 1984). The other objective is to refine a congruence model by, for example, specifying under what conditions congruence will occur (see e.g. Khandwalla, 1972; Perera et al., 1997), or by further specifying the causal relations (see e.g. Bouwens & Abernethy, 2000).

Suggestions for future research Based on these observations, a procedure for the analysis of the appropriateness of statistical methods may be sketched. However, before so doing, the study should be positioned theoretically by specifying the form of fit and the level of interaction (see Table 3). Specify the form of fit. This is an important step as earlier studies have suggested that the theoretical assumptions underpinning congruence- and contingency-type of models, respectively, and Cartesian- and Configuration-type of models, respectively, are so different that they should be considered as incompatible (see e.g. Chenhall, 2003; Gerdin & Greve, 2004). As mentioned

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above, however, the conflicting assumptions underpinning matching and multiplicative models of fit have been less thoroughly discussed, in particular in the management accounting literature. We argue that this poses a threat to theory development as it is often unclear what the underlying theory actually looks like (for an excellent exception, see Bisbe & Otley, 2004). That is, while it is often clearly stated what type of MAS is expected at the extreme values of context (i.e. in the contexts where the two models typically make identical predictions), little is said about the pattern between them. There is also a risk that the two models become mixed within the scope of individual studies. For example, it may first be proposed that ‘the higher the level of context, the greater the use of MAS information if the organization is to perform well’ and then stated in the formal hypothesis that ‘the higher the level of context, the greater the effect of MAS information on performance’. Again, however, the first statement has a matching form as it predicts that the appropriate level of MAS vis-a`-vis contexts maximizes performance (cf. the multiplicative form which predicts that performance can be maximized only by maximizing or minimizing MAS). The latter statement, in contrast, represents a multiplicative form as it is assumed that the effect of MAS on performance increases/decreases at a changing rate across different levels of context (cf. the matching model which assumes that performance changes at a fixed rate). Specify the level of interaction. As mentioned above, this dimension of theory specification has, to our knowledge, not been fully recognized in the contingency literature. Indeed, Schoonhoven (1981) argued that contingency theory implies symmetrical interactions. However, our analysis of the MAS literature suggests that there are at least two alternatives which are so different that they should be explicitly recognized in future studies, namely, theories predicting general interactions and theories predicting crossover interactions. When the form of fit and the interaction level has been specified, the next step is to choose or develop appropriate method(s) for testing hypotheses. Based on the above analysis of Table 3, we propose the following guidelines for future research. First, use methods whose functional form

is precise enough to be able to discriminate between the different forms of fit. For example, if a matching form of fit at the crossover interaction level is predicted, bivariate correlation or regression analysis should be preferred to an analysis of mean values since the latter method provides no information about the form of the underlying context/MAS relationship. Second, more informative methods should be preferred to less informative. This implies, for example, that the sole use of subgroup analysis is recommended only when groups represent true categories because of the loss of information that follows when ‘artificial’ categories are created on the basis of continuous variables (Cohen et al., 2003; Jaccard & Turrisi, 2003). Importantly, however, it also implies that we should preferably use those methods that provide the strongest link between the verbal statements and their mathematical formulation. And, as we see it, this may imply that some statistical techniques in Table 3 should not be used. For instance, although it was argued above that a subgroup correlation analysis focusing on differences in strength seems to be consistent with a multiplicative form of fit and can be used to test the existence of general interactions, it would be difficult to argue that such method is more appropriate than MRA for testing the sub-theory in question. Indeed, it can be argued that such recommendations should go without saying. However, our literature review shows that the choice of statistical test method is rarely based on explicit and theoretically grounded arguments explaining why the particular method(s) used should be preferred to other conceivable alternatives. Third, and finally, in situations where no appropriate method can be found in the literature, we are forced to combine existing methods or to come up with new ones. Irrespective which alternative is chosen, however, the general recommendation is to ensure that the method—or combination of methods—can both discriminate between different models of fit and test for the particular level of interaction assumed. Accordingly, if a multiplicative model at the crossover interaction level is predicted, a method with a precise functional form such as MRA may well be complemented with (but not substituted for) an ‘unprecise’ method like

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ANOVA since only the latter tests for the existence of a crossover interaction effect (cf. Table 3).4 Hopefully, these guidelines can serve as a means of selecting/developing appropriate statistical tests in empirical studies. However, at least three remarks should be made. First, although the classificatory framework highlights that there are more variants of contingency theory than has typically been noted in the MAS literature, it should be pointed out that also each sub-theory may include several, possibly competing variants. For example, matching models may predict both isoperformance (Schoonhoven, 1981) and hetero-performance (Donaldson, 2001). Importantly, however, such theoretical variants imply that the commonly used statistical methods described above may have to be tailored to the particular assumptions made. For example, if a matching model predicts iso-performance, the regression model depicted in Eq. (2) should preferably be changed so that this assumption is explicitly recognized and tested (for an example, see e.g. Schoonhoven, 1981, p. 352). Again, such adaptations are not only necessary to ensure a high correspondence between theory and statistical tests, but will also reduce the risk of erroneous conclusions. As Jaccard and Turrisi (2003, p. 21) pointed out; ‘‘failure to obtain a statistically significant interaction [. . .] may reflect the presence of an alternative functional form rather than the absence of a moderated relationship.’’ Second, while we have tried to cover the most commonly applied ways of using the statistical techniques, we as researchers should be openminded to other ways of using them. For example, subgroup correlation analysis focusing on differences in strength criticized above may be used to show that a lower performing subgroup has significantly lower correlation between context and MAS than has a higher performing subgroup because misfits are expected to be more scattered around the fit line (cf. the matching form of fit).

4

An alternative is to perform additional analyses of the MRA regression to examine whether the interaction effect is non-monotonic (Schoonhoven, 1981, 376–377) and disordinal (Cohen et al., 2003, p. 288).

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Third, and importantly, the recommendations above start out from the assumption that existing theory can be used to develop strong and precise predictions. While the differences between congruence- and contingency-type of models and between Cartesian- and Configuration-type of models are rather obvious, the differences between a matching and a multiplicative model are more subtle. Indeed, the frequent use of MRA and the scarce use of matching methods, indicate a preference for the multiplicative model as the explanation of observed performance differences. However, since the theoretical arguments for choosing this model are not always clearly stated and the verbal arguments sometimes seem to correspond better with a matching model, the specific form of fit seems to be an open question in many studies. And if this is the case, we should consider using techniques such polynomial regression which concurrently tests for the existence of several models of fit (Edwards, 2001; Meilich, 2006).

Conclusion Our objective has been to examine the appropriateness of commonly used statistical methods in the contingency-based MAS literature. In so doing, we developed a framework (Table 1) which highlights that we need to make more specific choices of theory than has typically been noted. In particular, we should pay more attention to the theoretical differences between matching- and multiplicative-type of models, and between interaction levels. Given these specifications, the analysis summarized in Table 3 can hopefully serve as a means of selecting/developing appropriate statistical tests consistent with the particular sub-theory in question. However, it is important to ensure that the technique (or set of techniques) has a precise functional format and cover all the assumptions made.

Acknowledgements The authors gratefully acknowledge the useful comments made by the anonymous reviewers,

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