## 3 Methodology
### Testing and Relaxing the Classical Assumptions
#### 3.3.2 Diagnostics of the Linearity Assumption
In the first step of MSRA, we selected a quadratic model as a second order Maclaurin series expansion of a scalar function (\(y\)) of five real variables [46]. This is a typical choice, because it provides enough flexibility to approximate a wide variety of relationships. However, this selection is arbitrary. Researchers can choose such set of regressors that suits their specific case. For example, the square roots of the independent variables can also be added. MSRA can work with any set of regressors that has \(x_{1}=1\), as long as the count of the regressors does not exceed roughly half of the count of data points. The first four instructions of MSRA are initialization of the algorithm, whereas the main body of the procedure consists of instructions 5 to 13. In the main body, we calculate the \(p\)-values of the \(t\)-tests for significance for all current regressors. It will be executed on every step of the backward elimination procedure. Those steps are divided into three phases. In the first phase (the main body and instructions 14-17), we drop one insignificant regressor per step (selecting the one with maximal \(p\)-value), but always keep the regression constant \(b_{1}\) in front of the unit regressor. In the second phase (instructions 18-19), we deal with the unit regressor. If the regression constant \(b_{1}\) is significant, then MSRA stops otherwise we drop the unit regressor. In the third phase (the main body and instructions 20-23) we deal with a model without a regression constant. There, we drop one insignificant regressor per step (selecting the one with maximal \(p\)-value). Typically, the third phase will consist of one step, while more steps would be rarely observed. We believe that MSRA at least to some extent handles the four objections to the stepwise regression formulated before the algorithm. The multiple testing problem from objection (a) is not that relevant because the decision to drop a regressors is driven by \(t\)-tests that have failed to reject the hypothesis. In that context, we observe \"false discovery at a step\" when we falsely reject \(H_{0}\) and, therefore, falsely keep the regressor in the model. Therefore, the multiple testing makes the dropping decision harder instead of easier as the objection (a) implied (note that the term \"false discovery at a step\" means keeping the regressor in the model at that step). MSRA does not use \(R^{2}_{adj}\) as a selection criterion, but instead uses series of \(t\)-statistics. The influence of the count of parameters, \(p\), on the \(t\)-test results is much smaller than that on the quantitative performance measure adjusted coefficient of determination. All that allows us to avoid the worst effects formulated in objection (b). The MSRA algorithm uses SVD decomposition of the design matrix instead of inverting the information matrix. Therefore, the problem of perfect multicollinearity is solved. The objection (c) argument for partial multicollinearity is also not applicable, because in any step the procedure starts again and regressors can change their significance from step to step. The difficult problem for the treatment of the regression constant \(b_{1}\) formulated in objection (d) is the best part of MSRA. In the first phase, the unit regressor is singled out never to be dropped. In the second phase, the unit regressor is treated exactly as the other regressors (it just \"happened\" that all other regressors are significant). In the third phase, there is no unit regressor, so the question of its treatment is irrelevant. MSRA assures that the unit regressor will never be dropped at an earlier step and at the same time the resulting model will contain only significant parameters. Some authors rightly advocate that any form of automatic stepwise regression should be used in conjunction with the statistics practitioner judgement to increase the quality of the regressors set using expert knowledge [37] (p. 878).