Although the negative intercept (a) is in itself meaningless, Model I is 

 correct because there is no basis for constraining a. 



II. A PROBLEM WITH THE CUSTOMARY APPROACH 



There are many cases where the logic of the application dictates the 

 response at a particular value of X. For example, if the response is some 

 change that is regressed against time then the response must be when X = 

 (Fig. 3). If there is no elapse time, there can be no change. If the linear 

 assumption is valid, the appropriate conceptual mode is 



Y = 3 X + e (Model II) 



and the customary predictive equation (based on Model I) is inappropriate and 

 may give poor estimates of 3 (see Fig. 4). Yet the vast majority of regres- 

 sion programs (e.g., SPSS, IMSL, IBM's 5110 package, and TI-59) do not allow 

 specification of a zero intercept or any constraint through a known point. 

 Statistical texts usually do not cover this topic either. However, formulas 

 for the zero-intercept case are given by Brownlee (1965) and Krumbein (1965) . 





• 



•^ 



y^ 



a> 





^< 



% 



o> 



^ 



^ 





c 



J**^ 







o 



^^ 









^^ 







o 



y^^ 







= 



Time 







Figure 3. Application of Model II forces a zero-intercept solution. 



Model 11-^ )3 =0.63 

 Model I -^ ? =0.34 



Figure 4. Model II estimates an increase in Y per unit increase in X 

 that is nearly twice that predicted using Model I. The phy- 

 sical relationship between X and Y dictates which model 

 should be adopted. If Model II is appropriate the solution can 

 be obtained using a simple artifice described in this report 

 to modify results of standard computer programs intended for 

 Model I. 



