Algebra and Multiple Linear Regression Part 4 - Concluding Remarks

Starting from this expression, one can execute the derivation just as in the case of the full equation (i.e., the equation including the constant term), and arrive at a set of equations that result in the least square expression for an equation that passes through the origin. We will not dwell on this point since it is not common in practice. We will use this concept to fit the data presented, just to illustrate its use, and for the sake of comparison, ignoring the fact that without the constant term these data are overdetermined, while they are not overdetermined if the constant term is included - if we had more data (even only one more relationship) they would be overdetermined in both cases. 

If we take our original set of data, as expressed in equations 7-5a-7.5c [1], and add one more relationship to them, we come up with the following situation  

We now have the situation we discussed earlier we have four relationships among a set of data, and only three possible variables (even including the b0 term) that we can use to fit these data. We can solve any subset of three of these relationships, simply by leaving one of the four equations out of the solution. If we do that we come up with the 

Eliminating equation 7-1 Eliminating equation 7-2 Eliminating equation 7-3 Eliminating equation 7-4 

The last entry in this table, the results obtained from eliminating equation 7-4, of course represents the results obtained from the original set of three equations, since eliminating equation 7-4 from the set leaves us with exactly that same original set. However, even though there does not seem to be much difference between the various equations represented by equations 7-2a -7-2d, clearly the fitting equation depends very strongly upon which subset of these equations we choose to keep in our calculations. 

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Matrix Algebra and Multiple Linear Regression Part 4 - Concluding Remarks... 

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