A Mixture Regression Model
6.4 Limitations of Mixture Models for the Selection of Comparison Partners
capacities compared to company no. 2235,but there are other caveats combined with them. Hotel no. 2995 disqualifies itself by having poor profit figures which are even outperformed by the company under evaluation; hotel no. 2914 has the lowest efficiency score of all four comparative companies.
Obstacles to using mixture models fall into different classes. Benchmark- ing studies based on mixture regression models depend on the limitations of the standard mixture regression models when distinguishing between controllable and uncontrollable variables,in other words differentiating between constraints and conditional variables. Another obstacle is that serious problems arise when managers operate without guidance in aggregating the
‘best practice partners’ in order to achieve a set of target values for future hotel management. One option to address aggregation problems is to weight the respective companies with their efficiency values,or to use their final set of posterior probabilities which decided on the group membership. However, both strategies are more intuitive than theory driven,and thus they will not be further investigated.
The need for a multiobjective approach in efficiency and performance studies is well documented by Table 6.16,which shows the results from a correlation analysis between the efficiency scores of the three models and a composite score which was generated by simply summinge(Y1),e(Y2) and e(Y3) for each individual company in the database. The efficiency scores of all three models show a high (and positive),thus plausible,influence on the observed profit of the companies in the database. An indication of the need for multiobjective methods for the selection of comparison partners can be derived from the even higher correlation between the composite efficiency scores,denoted as e(SUM),and the gross profit generated by the companies.
The relationship between these composite efficiency scores and the gross profit variable is plotted in Fig. 6.3.
6.4 Limitations of Mixture Models for the Selection of
with the EM algorithm,such as its sensitivity to local optima and the identification problems due to multi-collinearity in the predictor variables.
Other problems include choosing the optimal parameter specifications and specifying the appropriate number of segments.
In the following,specific problems are discussed that only occur in the application of mixture regression models to discover comparison partners in panel databases.
6.4.1 General criticism of the central tendency method
One principal criticism is related to this property of central tendency. Mixture regression analysis involves statistical estimation of the parameters of a fitted line. The criterion for fit is that the line reflects the general structure of the data, so it tends to go through the centre of the data points. When measuring efficiency as a company’s distance from the regression line,this ‘best practice’
line will always result in similar numbers of efficient and inefficient companies.
In this respect,using mixture regression models does not differ substantially from using ordinary calculations of mean key ratios as usually done in traditional benchmarking studies. Their advantage,compared to the latter technique,lies in the capability of the analyst to consider scale differences which are determined by the slope of the regression line,hence,yielding individual efficiency scores that are more realistic than those obtained from just subtracting a company’s key ratio from one single mean ratio generated by all companies in a specific group.
Fig. 6.3. Efficiency scores versus gross profit (composite model).
Under mixture regression analysis the company efficiency is assumed to follow a probability distribution which belongs to the exponential family. The universal property of mixture models simplifies the specification phase prior to analysis and is a real advantage of this technique compared to ordinary regression models.
Successful measurement of efficiency also depends on fitting the right functional form of the curve to the data. Fitting a wrong shape yields incorrect efficiency measures. As in ordinary regression analysis,mixture regression analysis requires prior specification of the shape of the curve to be fitted since it assumes a linear relationship between input and output variables. However, this may be too restrictive and carries the risk of fitting a curve to the data that has the wrong shape. Testing for the appropriateness of the functional form of the curve and choosing alternative models until a better shape is achieved addresses this problem but can result in a computationally very cumbersome procedure.
6.4.2 Disregarding different characteristics of input variables As mentioned previously,the true efficiency of a company is not directly observable,so it has to be inferred from variables that are observable,such as input and output variables. Assume that the quality of the approximation is sufficient and the analysis shows inefficiencies for a given company. It follows that analysis based on the model should give indications of unprofitable management by comparing the input resources of the given company with the efficient companies in the same mixture regression group. This provides the basic process of target setting. The implication that targets can be set is inherent in comparative studies.
The obstacles that arise in target setting with mixture regression analysis as the underlying methodology arise from severe problems related to distinct treatment of discretionary and non-discretionary variables. In mixture regres- sion models all independent variables are supposed to be discretionary. This limitation leads to situations where objectives set by the best-practice companies obviously cannot be targeted by a specific (inefficient) company due to environmental constraints. This problem is true for both types of non-discretionary variables: for the physical characteristics of a property (e.g.
number and mix of rooms) as well as (and especially) for variables determined by a company’s market area (e.g. location of a hotel). As proven by the case study of hotel no. 2235 in the present text,the segmentation proposed by the mixture regression analysis did not sufficiently reflect the distinct characteristics of the non-discretionary variables.
However,there is an alternative approach that can distinguish between controllable and uncontrollable variables in mixture regression models. One may define the dependent variable as a key ratio derived from significant output and controllable input factors and include only non-discretionary input
factors as dependent variables in the mixture regression model. This model formulation will have the advantage of separating controllable and uncontrol- lable effects,but at the same time will reduce the benchmarking procedure to a simple clustering of single key ratios. Additionally it will not support the managers’ target setting activities,as artificial composite companies made up by best practising companies,which are identified by the mixture regression model, will still vary for a given firm.
6.4.3 Problems with real-time benchmarking systems
In comparison with direct numerical optimization of the likelihood,the convergence of the EM algorithm is slow depending on the data distribution and the initial estimates for the parameters. In the present survey,sometimes more than 150 iterations have been necessary to reach convergence which required between 4 and 6 minutes computation time for each individual run on a Pentium 500 MHz computer system.6
This,in combination with other requirements (e.g. restarting the program in order to avoid local optima),makes for a slow analysis process. Applications in real-time systems,like the introduced benchmarking system in Chapter 5, will suffer from severe performance problems. Possibilities to avoid the occur- rence of local optima convergence,for instance,the use of simple clustering procedures ora prioricluster solutions to obtain an initial partition of the data, may speed up the system significantly.
6.4.4 Inadequacy of multiobjective problems
The analyses of the three mixture regression models have shown that, although the output factors were not contradictory in nature,the results for each of the introduced models presented a very different picture. Although a few hotels had consistently high rankings and a few had consistently low rankings for all three efficiency measures,the majority exhibit considerable variation depending on the indicator chosen. As a result,it is more unclear how to present an overall picture to the manager. The weakness of mixture regression modelling in multiobjective situations is even more clear when someone plans to include more contradictory target variables,for instance, customer satisfaction indicators and expenditures for service operations.
The inadequacy of the mixture regression approach for multiobjective problems is because,using regression analysis,the structure of a system of equations showing the interactions between the many types of output and input variables has to be specified. The calculation of a composite efficiency score by simple summation of all output factors considered,as introduced in
6 Windows NT (Version 4.0).
the present case study,assumes equal importance of all output factors under evaluation and lacks any substantial theory. A more sophisticated solution could,for instance,foresee a hierarchical system of company objectives. How- ever,despite it sometimes being computationally cumbersome,a complicated system of equations may not be followed easily,making it difficult to apply mixture regression analysis.