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30 T. Heimonen and M. Virtanen 5.2 Discriminant analysis If we wish to predict membership of a particular group we can use DA. In this type of analysis we choose independent variables that we think might predict the membership of a certain group. The assumptions underlying DA are that the population is multinormal and the variances of groups are equal (Appendix 1). Linear combinations are compared to a cross-matrix of the data so that total variance of cross-products (Stotal) is partitioned into variances associated with differences between groups (S bg ) and within groups (Swg) expressed in equation form as follows: S total S bg Swg (3) After determining the overall relationship between groups and independent variables (predictors), the next step is the analysis of discriminant functions that make up the relationship (Tabachnik and Fidell, 2007). The best discriminant functions are those for which the gap between centroids of different groups is the largest. Z i = i1X 1 + i2X i2 + + inX n (4) where Xin denotes explanatory variables. In the DA we used the same explanatory variables as in the regression analysis, but only those variables that were statistically significant in the best model are reported. Age, the number of auxiliary business names, the branch of industry dummy, and the solvency factor were discriminating variables in the standardised canonical discriminant function in Model 1 (Table 3). Model 1 classifies correctly 78.2% of the total sample and 97.6 percent of NHG and NHS firms. However, the correct classification without cross-validation was only 15.4%. In a matched pair comparison with Model 2, the classification rate was 64.1% for the overall sample as well as for the other groups (Table 3, Model 2). The cross validation rate for unselected cases was 67.2%. In summary we can conclude that ignorance of a priori probabilities results in poorer performance of the discriminant model. The obvious reason for this is that DA is used for classification purposes, as is LRA. In DA the discriminant functions are defined and the results should be interpreted as expressing the straightforward likelihood of belonging to a specific group. In our analysis, classification of cases was quite good compared, for example, to Moreno and Casillas (2007). Cross validation of the data improves the robustness of the model. Compared to LRA the results are similar, but in DA we cannot tell anything about the direction of impact of a single independent variable. 5.3 Logistic regression analysis In the logistic regression model, the dependent variable is dichotomic and can have one of two alternative values. If we wish to test whether a firm belongs to the group of HG businesses, the dependent variable receives a value of 0 if it is a NHG company and 1 if it is a HG. Similarly successful firms could be classified into two separate groups [non-highly successful firm (NHS) = 0; HS firm = 1]. When we wish to test both HG and high success we combine these two variables by multiplying them. Then HG and
Characteristics of successful gazelles 31 HS are each allocated a value 1, whilst 0 denotes the absence of HG, high success or both. where y i = + (X) + , (5) y 0 for low growth or success 1, for HG and high success X matrix of the predictor variables. The probability that yi = 1 will be i i i i P y =1 x = exp x 1 + x (6) The list of predictor variables is presented in Appendix 3. In the logistic regression the impact of changes to the coefficients on the probability of an event occurring depends on the initial probability of the event. In Table 4, LRA is presented first without taking into account a priori probabilities (Model 1) and thereafter with matched selection of NHS and NHG businesses (Table 4, Model 2). In addition to the number of auxiliary business names, age, business volume factor and services as a branch of industry, which were found to be significant in the DA, trade and manufacturing as well as the liquidity factor were also found to be statistically significant in the LRA. In model 2, which included a matched pair comparison and cross validation, age and services as a branch of industry were no longer statistically significant. The number of auxiliary business names was the only variable that decreased the odds ratio with respect to classification as a HG and HS firm. These results are partly in the line with the conclusion of Ettlinger and Tufford (1996) that high performing firms seem to invest in labour; however, in the current study it is not possible to tell whether labour investments exceeded capital investments made by the firms. In Model 1 the overall classification rate of the model was almost 80%, but this was due to the correct classification of NHS and NHG firms; in contrast, the correct classification of HS and HG firms was very poor (26.7%). It should be noted that in the logistic regression model it is always possible to achieve at least 50% accuracy by simply setting the prediction for each observation to correspond to the most frequent outcome (Hoetker 2007). Thus the model is not appropriate since it does not reach the level of a priori probability (0.5) for the HS and HG group. When a priori probabilities are taken into account and similar numbers of cases are selected for the analysis, the results change so that age is no longer a statistically significant variable (Table 3, Model 2). Moreover different dummies for branch of industry are statistically significant in the forced model. The overall classification rate of the model was 63.8%. The correct classification rate of selected cases was 62.3% for NHG and NHS firms and 65.3% for HG and HS firms. Unselected cases all belonged to NHS and NHG and their classification rate was 66.3%.
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Aalto University publication series
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Author Tomi Heimonen Name of the do
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Acknowledgements This dissertation
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CONTENTS PART 1: Overview of the di
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List of Articles This thesis is bas
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1 Introduction Some scholars claim
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1 Introduction In addition, the dur
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1 Introduction Figure 1: Framework
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1 Introduction factors that affect
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1 Introduction 1.4 Key concepts and
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2 Literature review and conceptual
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3 Research methodology Constructivi
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3 Research methodology and the meth
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3 Research methodology The data use
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3 Research methodology 3.4 Models a
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3 Research methodology reliable. Se
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3 Research methodology concluded th
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4 Results, evaluation of the studie
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References Abetti, P. A. (2001). Ac
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References Bhide, A. (2000). The or
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References Creswell, J. W. (2003).
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References Golafshani, N. (2003). U
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References Leitch, C., Hill, F., an
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References Pellikka, J. and Laurone
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References Storper, M. (1992). The
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Appendixes Overview: Appendix 1: Ke
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Appendixes High growth When examini
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Appendixes profitability, liquidity
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Appendixes Appendix 1 Continued (Ke
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Appendixes Appendix 2 Construction
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Appendixes Appendix 2 Continued (Co
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