Numerical Analysis in Least Squares Regression with an Application to the Abortion-Crime Debate

Authors

  • William Anderson,

    Corresponding author
    1. Department of Statistical Science, Cornell University, Ithaca, NY
      *Martin T. Wells, Charles A. Alexander Professor of Statistical Sciences, Department of Statistical Science, Cornell University, Ithaca, NY 14853; email: mtw1@cornell.edu. Anderson is a Visiting Assistant Professor in the School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853; email: wla6@cornell.edu.
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  • Martin T. Wells

    Corresponding author
    1. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY
      *Martin T. Wells, Charles A. Alexander Professor of Statistical Sciences, Department of Statistical Science, Cornell University, Ithaca, NY 14853; email: mtw1@cornell.edu. Anderson is a Visiting Assistant Professor in the School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853; email: wla6@cornell.edu.
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*Martin T. Wells, Charles A. Alexander Professor of Statistical Sciences, Department of Statistical Science, Cornell University, Ithaca, NY 14853; email: mtw1@cornell.edu. Anderson is a Visiting Assistant Professor in the School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853; email: wla6@cornell.edu.

Abstract

Concepts of numerical analysis with applications to least squares problems are introduced in a manner the practitioner can readily apply to his or her own research problems, especially in the social sciences. Numerical analysis is mainly concerned with the accuracy and stability of numerical algorithms. We frame these concerns in terms of forward and backward error, two important concepts in helping understand the quality of the computed answers. The goal of numerical computing is to obtain correct, approximate answers to the true solution. We extended this forward and backward error framework to issues in least squares problems and check the condition of the regression problem via condition numbers. The more ill-conditioned the data are, the more sensitive the computed solution is to perturbations in the data, and the more unstable the computed solutions become. Condition numbers can also be used to signal the presence of solution degrading collinearity in regression problems. We apply the various numerical analysis tools outlined with some model diagnostics to the abortion-crime debate, and show the regression analysis used in various papers addressing the abortion-crime debate cannot be trusted.

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