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32 T. Heimonen and M. Virtanen Table 4 LRA of combined growth and success Model 1 (no matched pair comparison) Model 2 (matched pair comparison and cross validation) Dependent variable (0,1): (0 = non-high growth and high success firms 1 = high growth and high success firms) Coeff. S.E. Wald Df Exp. (b) Coeff. S.E. Wald Df Exp. (b) Independent variables Constant –0.243 0.340 0.513 1 0.784 –1.953 0.689 8.047 1 0.142 Number of auxiliary business names –0.191* 0.109 3.046 1 0.826 –0.265** 0.127 4.367 1 0.767 Age of company –0.057*** 0.022 6.990 1 0.944 Growth of the number of staff 0.006** 0.003 6.206 1 1.006 0.010** 0.004 6.245 1 1.010 Branch of industry: Services including Kibs firms –0.889** 0.304 8.545 1 0.411 Trade 0.857* 0.454 3.565 1 2.357 Manufacturing 1.146** 0.553 4.293 1 3.147 Liquidity factor 0.606*** 0.176 11.877 1 1.833 0.660** 0.238 7.721 1 1.936 Business volume factor –1.117* 0.598 3.483 1 0.327 Model summary Number of firms 321 321 (27 missing cases) (27 missing cases) A priori probability firms with 0.766 (n = 246) 0.500 (n = 75) other combinations A priori probability firms with high growth 0.234 (n = 75) 0.500 (n = 75) and success Log likelihood 293.996 178.523 Hosmer and Lemeshow test sig. 0.568 0.934 Cox and Snell R 2 0.158 0.191 Nagelkerke R 2 0.238 0.254 Classification rates (%) Overall classification 79.8 63.8 Firms with other combinations 95.9 62.3 Firms with high growth and high success 26.7 65.3 Cross validation 66.3 Note: Level of significance *p .1, **p .05, ***p .01
Characteristics of successful gazelles 33 As a summary of LRA and in answer to the question about the impact of ignorance of a priori probabilities on the results of the analysis, we can state that ignorance of a priori probabilities has a significant impact on classification statistics. This means that the model does not predict the correct classification better than random selection. Even if the overall classification rate is almost 80%, the correct classification of the ‘winners’ is only 27%. When a priori probabilities are taken into account the classification rate of HG and HS businesses is 65%. We conclude that, in order to be a robust method for analysing skewed data, cross validation should be used to take into account a priori probabilities of dichotomic groups. The results of LRA models should be interpreted differently from linear regression. LRA results should be interpreted as the function of the estimated model. In other words, the value of the likelihood function expresses the probability or odds of belonging to some predefined group but the interpretation presupposes continuous explanatory variables. In LRA it is recommended that the number of observations exceeds 500 in order to avoid systematic overestimation of coefficients. The variance of the error term in the logistic regression was not constant for all the observations. When small samples are used this is associated with heteroscedasticity, which reduces the effectiveness of the model. However, according to Pindyck and Rubinfeld (1976), the presence of heteroscedasticity does not in and of itself result in either biased or inconsistent parameter estimates. 6 Conclusions and comparison of different analyses The growth of the number of staff and the profitability factor were the only variables that were statistically significant in regression models of both growth and success. Thus, we can conclude that growth and success are not the same phenomenon and, indeed, may be inversely related. Regression analysis is not an appropriate way to analyse HG and HS businesses when time series analysis is not possible. The reason is that in the entrepreneurial market, growth and success are dynamic and presuppose in most cases discontinuity, vague environment and a holistic approach. Moreover, the explanatory power of success models will probably be quite modest if the model specification is based on all the statistical requirements. In DA, the number of auxiliary business names, the branch of industry (service sector) and the solvency factor were statistically significant variables in the discriminant function and allowed the correct classification of almost two thirds of the observations (64%). Without cross validation the model also included age as a statistically significant variable in the function. The total classification rate of this model was more than three quarters (78%), demonstrating that skewness of the distribution distorts the results quite substantially. Compared to LRA, in addition to the number of auxiliary business names, the growth of the number of staff, both trade and manufacturing industries, as well as the liquidity factor were discovered to be statistically significant in the LRA. Both logistic regression and DA produce unsatisfactory results if a priori probabilities are not taken into account. If we ignore a priori probabilities and skewness of the data we may obtain invalid results even if the statistical indicators of the fit of the model and its characteristics are satisfactory. There is no sense in using sophisticated statistical methods to analyse the data if it does not produce better results than the toss of a coin.
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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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not increased compared to Eastern F
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