In regression model \(\boldsymbol {y} = \boldsymbol {X} \boldsymbol {\beta } + \boldsymbol {u}\) with collinearity, we study the effectiveness of variable selection procedure via t-tests (VSP-T) within the traditional ordinary least squares (OLS) framework. First, the structural effect of collinearity on the power of t-test and VSP-T is clarified, and the performance measures of VSP-T are proposed, through which the two-sided UMPI test is shown to have a more difficulty to distinguish the nonnull hypothesis from the null hypothesis than the one-side one. Second, in a new comprehensive analytical framework, each individual OLS estimator is expressed as a simple regression model to characterize the linearly algebraic relations of each OLSE, “power deflation factor (PDF)” (collinearity factor), and \(\boldsymbol t\) -statistic. In particular, while t-statistic is linearly related to the PDF, each OLSE in the t-statistic is shown to be not directly affected by the PDF. Third, t-statistics are shown to be correlated ( \(\boldsymbol {X}\) , a.e.), and so the model selection process (MSP) via VSP-T that deletes a variable with minimum \(|t|\) -value is not legitimate unless the order statistics of corelated \(|t|\) s are treated together with the identifiability of collinear structure in \(|t|\) s. In other words, besides PDFs, t-values are not directly comparable nor linearly ordered due to the nonzero correlations among t-statistics. To directly show the ineffectiveness, a uniformly most powerful invariant (UMPI) test for testing the equality of the means of two t-statistics is derived to know how the t-values are clustered. Finally, it is shown that in the framework of the Neyman-Pearson testing theory, MSP via VSP-T is hardly effective at least theoretically so long as the single error term \(\boldsymbol {u}\) is shared with submodels in a repeated MSP via VSP-T, besides collinearity and correlation problems.