14_corr_regre1

14_corr_regre1 - Regression and correlation Involve...

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Correlation & Regression, I 9.07 4/1/2004 Regression and correlation Y, Z, …) Involve bivariate, paired data, X & Y – Height & weight measured for the same individual – IQ & exam scores for each individual – Height of mother paired with height of daughter Sometimes more than two variables (W, X, Regression & correlation Regression vs. correlation rule Concerned with the questions: Does a statistical relationship exist between X & Y, which allows some predictability of one of the variables from the other? How strong is the apparent relationship, in the sense of predictive ability? Can a simple linear rule be used to predict one variable from the other, and if so how good is this rule? E.G. Y = 5X + 6 Regression: – Predicting Y from X (or X from Y) by a linear Correlation: – How good is this relationship? 1
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pair. Scatter plot: height vs. weight 40 45 50 55 60 65 70 75 80 140 170 190 200 ( i (i Regression line First tool: scatter plot For each pair of points, plot one member of a pair against the corresponding other member of that In an experimental study, convention is to plot the independent variable on the x-axis, the dependent on the y-axis. Often we are describing the results of observational or “correlational” studies, in which case it doesn’t matter which variable is on which axis. 150 160 180 210 Weight lbs) He ght nches) 2 nd tool: find the regression line technique isn’t appropriate) 40 45 50 55 60 65 70 75 80 140 170 190 200 ( ( We attempt to predict the values of y from the values of x, by fitting a straight line to the data The data probably doesn’t fit on a straight line Scatter The relationship between x & y may not quite be linear (or it could be far from linear, in which case this The regression line is like a perfect version of what the linear relationship in the data would look like 150 160 180 210 Weight lbs) Height inches) 2
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How do we find the regression line that best fits the data? • We don’t just sketch in something that looks good fit,” find the equation of the best fit line Straight Line x=0) x y a b First, recall the equation for a line. Next, what do we mean by “best fit”? Finally, based upon that definition of “best General formula for any line is y=bx+a b is the slope of the line a is the intercept (i.e., the value of y when “best fit” mean? i i y i i i (y i –y i ’) 2 i i Minimizing sum of squared errors X Y y i y i y i –y i Least-squares regression: What does •I f y is the true value of y paired with x , let ’ = our prediction of y from x We want to minimize the error in our
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14_corr_regre1 - Regression and correlation Involve...

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