![]() There are several points outside the ellipse at the right side of the scatter plot. From the density ellipse for the Displacement by Horsepower scatter plot, the reason for the possible outliers appear in the histogram for Displacement. In the Displacement by Horsepower plot, this point is highlighted in the middle of the density ellipse.īy deselecting the point, all points will appear with the same brightness, as shown in Figure 17. This point is also an outlier in some of the other scatter plots but not all of them. In Figure 16, the single blue circle that is an outlier in the Weight by Turning Circle scatter plot has been selected. It's possible to explore the points outside the circles to see if they are multivariate outliers. The red circles contain about 95% of the data. That’s the topic of the next section.The scatter plot matrix in Figure 16 shows density ellipses in each individual scatter plot. It is a surprising mathematical fact that no matter what the shape of the scatter plot, the same equation gives the “best” among all straight lines. Endnote #Įven though we won’t establish the mathematical basis for the regression equation, we can see that it gives pretty good predictions when the scatter plot is football shaped. Note that if the slope is negative, then for every unit increase in \(x\), the average of \(y\) decreases. In general, the slope of the regression line can be interpreted as the average increase in \(y\) per unit increase in \(x\). This is true for all values of \(w\) in the sample. The second group is estimated to be 0.2 inches taller, on average. Specifically, consider a group of dogs whose weight is \(w\) pounds, and the group whose weight is \(w+1\) pounds. The slope reflects the difference in the average heights of two groups of dogs that are 1 pound apart in weight. The slope is positive, and it is important to note that this does not mean that we think basset hounds get taller if they put on weight. The slope of the line is measures the increase in the estimated height per unit increase in weight. To calculate the equation of the regression line, we need the slope and the intercept. Suppose the observed correlation \(r\) is 0.5, and that the summary statistics for the two variables are as in the table below: Suppose that our goal is to use regression to estimate the height of a basset hound based on its weight, using a sample that looks consistent with the regression model. The slope of 3.57 pounds per inch means that the average pregnancy weight of the taller group is about 3.57 pounds more than that of the shorter group. Another way to think about the slope is to take any two consecutive strips (which are necessarily 1 inch apart), corresponding to two groups of women who are separated by 1 inch in height. Notice that the successive vertical strips in the scatter plot are one inch apart, because the heights have been rounded to the nearest inch. Pounds more than our prediction for the shorter woman. Thus the equation of the regression line can be written as: When the variables \(x\) and \(y\) are measured in standard units, the regression line for predicting \(y\) based on \(x\) has slope \(r\) and passes through the origin. In regression, we use the value of one variable (which we will call \(x\)) to predict the value of another (which we will call \(y\)). The average of these heights will be less than 1.5 standard units. Some will be taller, and some will be shorter. It doesn’t say that all of these children will be somewhat less than 1.5 standard units in height. For example, it says that if you take all children whose midparent height is 1.5 standard units, then the average height of these children is somewhat less than 1.5 standard units. Keep in mind that the regression effect is a statement about averages. In general, individuals who are away from average on one variable are expected to be not quite as far away from average on the other. Children whose midparent heights were below average turned out to be somewhat taller relative to their generation, on average. Regression to the mean also works when the midparent height is below average. This is called “regression to the mean” and it is how the name regression arises. In other words, we predict that the child will be somewhat closer to average than the parents were. If the midparent height is 2 standard units, we predict that the child’s height will be somewhat less than 2 standard units. In terms of prediction, this means that for a parents whose midparent height is at 1.5 standard units, our prediction of the child’s height is somewhat less than 1.5 standard units. But for more moderate values of \(r\), the regression line is noticeably flatter. When \(r\) is close to 1, the scatter plot, the 45 degree line, and the regression line are all very close to each other. ![]()
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