You can see that b and r will be equal whenever the ratio of the two standard deviations is 1.0. One more time: the correlation is the slope when both variables are measured as z scores (that is, when both X and Y are measured as z scores, r = b. Others find it more helpful to memorize a simple formula or two. Some people find that describing the various relations between r and b is helpful in understanding them. This is another way of saying that r and b are related r is the harmonic mean of the two regression slopes and r-square is the product of the two slopes. Therefore, it follows that the product of the two slopes equals r-squared, and that the harmonic mean of the two slopes is r. We can do the same for the regression of X on Y and if we do, we find that It is an example of the linear model where we have the grand mean of Y and an effect for X). (You have already seen this equation once before. If we work with z y for a moment, we have The predicted value for Y measured as a z score is. Of course b equals r when X and Y are measured in z scores. The means of both variables are zero, so a is zero and drops out of the equation. This says that the predicted value of X or Y in z-scores is just the predictor times r. If we make predictions in z scores instead of observed scores, we have So the geometric interpretation of r is the size of the angle between the regression of y on x and of x on y. The size of the interior angle is related to r. For values of r between 0 and 1, there will be two regression lines that form an interior angle less than 90 degrees, as illustrated in the figure below. If r = 0, then the two lines will be the respective means of Y and X, and the regression lines will be the same as the major axes of the figure. Otherwise, there will be two distinct lines. If we do so, the regression lines will only be the same if there is no error of prediction, that is, if and only if r = 1. We can also regress X on Y to predict values of X from values of Y. When we compute a regression equation, we regress Y on X to make our best predictions. Which tells us the average of the z-score cross products. We have discussed the algebraic interpretation of r, that is, (4) talk about when you might prefer r to b or vice versa. (3) show how you can have different vales of r but the same slope (and vice versa), and (1) review the connection between correlation and regression, presenting a geometric interpretation of correlation based on regression, In this section, I want to accomplish four things: You are now in a better position to appreciate how they are similar and different. You have been introduced to both correlation and regression. What implication does this distribution have for the accuracy of predicted values of Y? What are the main things that influence the shape and variance of this distribution?ĭescribe the sampling distribution of the regression line. Describe each term and say what the regression equation means.ĭescribe a concrete example (provide both words and numbers) in which two different groups could have the same correlation but different raw score b weights.ĭescribe a concrete example (provide both words and numbers) in which two different groups could have the same raw score b weights but different correlations.ĭescribe the sampling distribution of r. Write the regression equation for predicting Y from X in z-scores.
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