Linear Regression

# Linear Regression - Now this brings us to linear regression...

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Now this brings us to linear regression . This is a way of 1. Determining if there is a linear relationship between 2 variables 2. What this relationship is Correlation only tells us that there is a relation. Regression also tells us what that relation is. It does not tell us whether there is causality however. Liner regression is a way of drawing a straight line through the data and finding the “best fit”. By itself it cannot determine causality! So for 2 variables X and Y, we estimate: Y = a + bX + e Where a is a constant b is the coefficient on X or the slope of the linear relationship e is the residuals or the amount by which the values vary from the predicted values Diagrammatically we draw a straight line through the scatter gram and compare the predicted values with the actual values where the difference is the error term. We want to minimize these deviations from the line so that: i n i c i i e Y Y = - = ) ( 1 is minimized where Y c is the predicted or calculated value of Y for the value of X. This is usually done by squaring the error terms and minimizing the sum of the squares – this is usually called the least squares method. (Why do we use squared values to minimize? What this does is put more weight on the larger errors. We will not prove it, but it can be proven that the least-squares method gives the best, linear, unbiased estimate of the relationship. It also lets us invoke the Pythagorean rule for right-angled triangles)

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So we have our computed line for the relationship between Y and X. If Yc
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Linear Regression - Now this brings us to linear regression...

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