Coefficient Of Determination/proportionate Reduction In Error
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shows the predicted value of Y for New York. The authors ask us to consider the following situation. Suppose we didn't know the actual Y, the percentage of residents of New York who have a bachelor's degree. Suppose proportionate reduction in error definition further that we did not have knowledge of X, New York's median household income
Proportionate Reduction In Error Is Conceptually
(Frankfort-Nachmias and Leon-Guerrero 2011:419). What would we do in this situation? Do you have any ideas? You would use the mean value proportionate reduction in error formula as your estimate. That would be 28.56%. When we know median household income, we can use the regression equation to estimate New York's percentage of residents with a bachelor's degree. Using the equation on page 441, proportionate reduction in error symbol we would predict 29.26% of New York residents have bachelor's degrees. We have improved the prediction by 0.70. This is obviously an improvement, but we really want to see how much we improve the predictions for all cases. Table 13.5 on page 416 uses median household income and percentage of residents with a bachelor's degree to illustrate how to calculate the error sum of squares (SSE). Because we already have the total sum
Proportionate Reduction In Error Can Be Symbolized By
of squares, we can create the regression sum of squares, or SSR, as indicated in the following slide. Now we can put all this together and consider the coefficient of determination (r²), which calculates the two measures of error for all cases in a particular research problem and gives us an overall idea of how much we reduce error by using the linear model. It is a PRE measure. Coefficient of Determination (r²) A PRE measure reflecting the proportional reduction of error that results from using the linear regression model. The total sum of squares (SST) measures the prediction error when the independent variable is ignored. The error sum of squares (SSE) measures the prediction errors when using the independent variable and the linear regression equation. The coefficient of determination in this case is 0.79. This indicates that in this example we reduce our prediction error by 79 percent. It also shows that the independent variable, median household income, explains 79 percent of the variation in the dependent variable, percentage with a bachelor's degree. The pie chart, Figure 13.9 on page 423 in the textbook, illustrates one way to think about r². Because the independent variable explains about 79 percent of the total variation in the dependent variable, that leaves about 21 percent of the variat
of making observations which are possibly subject to errors of all types. Such measures quantify how much having the observations available has reduced the loss (cost) the proportionate reduction in error is a measure of the of the uncertainty about the intended quantity compared with not having those the proportionate reduction in error is a measure of the quizlet observations. Proportional reduction in error is a more restrictive framework widely used in statistics, in which the
Proportional Reduction In Error Lambda
general loss function is replaced by a more direct measure of error such as the mean square error. Examples are the coefficient of determination and Goodman and Kruskal's lambda.[1] The https://learn.bu.edu/bbcswebdav/pid-826908-dt-content-rid-2073693_1/courses/13sprgmetcj702_ol/week06/metcj702_W06S01T06_assessing.html concept of proportional reduction in loss was proposed by Bruce Cooil and Roland T. Rust in their 1994 paper. Many commonly used reliability measures for quantitative data (such as continuous data in an experimental design) are PRL measures, including Cronbach's alpha and measures proposed by Ben J. Winer (1971). It also provides a general way of developing measures for https://en.wikipedia.org/wiki/Proportional_reduction_in_loss the reliability of qualitative data. For example, this framework provides several possible measures that are applicable when a researcher wants to assess the consensus between judges who are asked to code a number of items into mutually exclusive qualitative categories (Cooil and Rust, 1995). Measures of this latter type have been proposed by several researchers, including Perrault and Leigh (1989). References[edit] ^ Upton G., Cook, I. (2006) Oxford Dictionary of Statistics, OUP. ISBN 978-0-19-954145-4 Cooil, B., and Rust, R. T. (1994), "Reliability and Expected Loss: A Unifying Principle," Psychometrika, 59, 203-216. (available here) Cooil, B., and Rust, R. T. (1995), "General Estimators for the Reliability of Qualitative Data," Psychometrika, 60, 199-220. (available here) Rust, R. T., and Cooil, B. (1994), "Reliability Measures for Qualitative Data: Theory and Implications," Journal of Marketing Research, 31(1), 1-14. (available here) Winer, B.J. (1971), Statistical Principles in Experimental Design. New York: McGraw-Hill. Perreault, W.D. and Leigh, L.E. (1989), “Reliability of Nominal Data Based on Qualitative Judgments,” Journal of Marketing Research, 26, 135-148 Retrieved from "https://en.wikipedia.org/w/index.php?title=Proportional_reduction_in_loss&oldid=735653331" Categories: Comparison of assessments Navigation
interval data X-axis = i.v., Y-axis = d.v. direction: positive (up to the right), negative (down to the right) using X to predict Y line summarizes relationship simple linear regression equation: Y = a + bX a = Y-intercept value of Y http://faculty.washington.edu/ddbrewer/s231/s231regr.htm when line crosses Y-axis (i.e., when X = 0) b = slope change in Y with https://www.coursehero.com/file/p3roic7/as-the-proportionate-reduction-in-error-PRE-that-is-the-proportion-of-the/ a unit change in X (rise/run) regression line minimizes squared errors (least squares line) SSE = sum of squared errors MSE = mean squared error errors in prediction = residuals Issues to consider: influence of outliers - outliers can suppress or drive relationships nonlinear relationships - linear regression based on assumption of linearity extrapolation - predicting y based on regression equation and reduction in x outside of observed range of x values - be careful! heteroschedasticity - errors in prediction vary over range of values Correlation strength of association - degree of clustering about line r2 as a PRE measure - strength of association Proportional Reduction in Error (PRE) PRE = (E1 - E2) / E1 E1 = errors in predicting d.v. based on distribution of d.v. E2 = errors in predicting d.v. when prediction reduction in error is based on i.v. ranges between 0 and 1 how much can we reduce errors in predicting d.v. by considering i.v.? 100% reduction (1.0) - perfect prediction 0% reduction (0.0) - i.v. does not help in predicting r2: E1 = predict values of Y based on Y bar (mean) sum of (Y - Y bar)2 E2: predict Y based on X and regression line sum of (Y - Y hat)2 (deviation of observed Y from regression line, squared) Correlation: r Pearson correlation coefficient or product-moment coefficient indicates how closely observed values fall around regression line/clustering about line & direction of association r = square root of r2 and takes sign of slope ranges between -1 and 1 negative r = negative relationship positive r = positive relationship 0 = no relationship strength of r which is stronger: -.2 or +.1? -.5 or +.75? absolute value of r indicates strength general guide for interpreting strength of r (absolute value) 0 - .2 = weak, slight .2 - .4 = mild/modest .4 - .6 = moderate .6 - .8 = moderately strong .8 - 1.0 = strong r standardizes the degree of association, regardless of units of measurement one approach to computing r: standardize each case's value on x and y: (X - X bar) / s.d. on X (Y - Y bar) / s.d. on Y for each case, multiply standardized X value by standa
Upload Documents Write Course Advice Refer your Friends Earn Money Upload Documents Apply for Scholarship Create Q&A pairs Become a Tutor Find Study Resources by School by Literature Guides by Subject Get Instant Tutoring Help Ask a Tutor a Question Use Flashcards View Flashcards Create Flashcards Earn by Contributing Earn Free AccessLearn More > Upload Documents Write Course Advice Refer your Friends Earn MoneyLearn More > Upload Documents Apply for Scholarship Create Q&A pairs Become a Tutor Are you an educator? Log in Sign up Home Simon Fraser STAT STAT 285 Stat 285 Regression Analysis Notes As the proportionate reduction in error pre that is SCHOOL Simon Fraser COURSE TITLE STAT 285 TYPE Notes UPLOADED BY harrycheew PAGES 8 Click to edit the document details This preview shows pages 6–8. Sign up to view the full content. View Full Document as the proportionate reduction in error ( PRE ), that is, the proportion of the prediction error that can be reduced by knowing the independent variable. The proportionate reduction in error ( PRE ) due to X is PRE = (SS total – SS error ) / SS total = SS regression / SS total = 73.476 / 74.8 ≈ 0.99 Thus, 99% of the error in predicting John’s height is reduced by taking his age into account. Regression and Pearson’s Correlation r 2 = (SS total – SS error ) / SS total = 73.476 / 74.8 ≈ 0.99 The squared correlation ( r 2 ) is called the coefficient of determination. That is the proportion of variance in Y determined or explained by X. This preview has intentionally blurred sections. Sign up to view the full version. View Full Document The complementary quantity 1- r 2 is called the coefficient of non-determination. That is, the proportion of variance in Y that is not explained by X is 1- r 2 . 1- r 2 = SS error / SS total In our example, 1- r 2 = 1- 0.99 = 0.01 = 1% Thus, only about 1% of the variance in John’s height is not explained by his age. Regression and Analysis of Variance (1- r 2 ) * SS t