Statistical Analysis Course by Penn state University

  1. Welcome to STAT 505! | STAT 505 (psu.edu) Applied Multivariate Statistical Analysis
  2. Welcome to STAT 501! | STAT 501 (psu.edu) Regression Methods

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Visualizing Ellipse rotation and associated parametric equations

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  Diagonal matrix Ellipse axes are parallel to coordinate axes (no rotation). Zeros in off-diagonals means zero correlation. Same values in diagonals Ellipse is rotated 45 degrees if correlation (off-diagonal) is positive regardless of its magnitude.  Similarly -45 degrees if correlation is negative. Correlation matrix Correlation matrix has 1 in diagonals and values between -1 and 1 inclusive in off-diagonals. Ellipse size remains the same (always touches square of side 2 units).  Whether the correlation is positive or negative can be observed by the orientation of the ellipse. The amount of correlation can be interpreted by how thin the ellipse is. How It Works Below is a list of parametric equations starting from that of a general ellipse and modifying it step by step into a prediction ellipse, showing how different parts contribute at each step. Reference How to Draw Ellipse of Covariance Matrix (cookierobotics.com)
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Robust covariance matrix estimation in SPSS

   IN SPSS,  Produces the parameter estimates along with robust or heteroskedasticity-consistent (HC) standard errors. When  Parameter estimates with robust standard errors  is selected, the following methods are available for the robust covariance matrix estimation. HC0 Based on the original asymptotic or large sample robust, empirical, or "sandwich" estimator of the covariance matrix of the parameter estimates. The middle part of the sandwich contains squared OLS (ordinary least squares) or squared weighted WLS (weighted least squares) residuals. HC1 A finite-sample modification of HC0, multiplying it by N/(N-p), where N is the sample size and p is the number of non-redundant parameters in the model. HC2 A modification of HC0 that involves dividing the squared residual by 1-h, where h is the leverage for the case. HC3 A modification of HC0 that approximates a jackknife estimator. Squared residuals are divided by the square of 1-h. HC4 A modification of HC0 tha...
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Vignettes for Matrix concepts, related operations

  Geometric Transformations, compute the matrix of a rotation transformation and visualize it. Wolfram|Alpha Examples: Geometric Transformations (wolframalpha.com) Reference Change of Basis – GeoGebra Eigenvalues of a Matrix Calculator - Online Eigen Values λ Finder (dcode.fr) Eigenvectors of a Matrix Calculator (with Eigenvalues) - Online (dcode.fr) Matrix Visualization | Sarah Greer (sygreer.com)   Matrix visualizer (utexas.edu)  gives rotation by angle. Eigenvalue Calculator: Wolfram|Alpha (wolframalpha.com) Determinant Calculator: Wolfram|Alpha (wolframalpha.com)
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