第12章線形回帰と相関

Data rarely fit a straight line exactly. Usually, you must be satisfied with rough predictions. Typically, you have a set of data whose scatter plot appears to "fit" a straight line. This is called a Line of Best Fit or Least-Squares Line.

The third exam score, x, is the independent variable and the final exam score, y, is the dependent variable. We will plot a regression line that best "fits" the data. If each of you were to fit a line "by eye," you would draw different lines. We can use what is called a least-squares regression line to obtain the best fit line.

Consider the following diagram. Each point of data is of the the form (x, y) and each point of the line of best fit using least-squares linear regression has the form (x, ŷ).

The ŷ is read "y hat" and is the estimated value of y. It is the value of y obtained using the regression line. It is not generally equal to y from data.

The term y₀ – ŷ₀ = ε₀ is called the "error" or residual. It is not an error in the sense of a mistake. The absolute value of a residual measures the vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line.

If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates that actual data value for y.

In the diagram in Figure 12.16, y₀ – ŷ₀ = ε₀ is the residual for the point shown. Here the point lies above the line and the residual is positive.

ε = the Greek letter epsilon

For each data point, you can calculate the residuals or errors, y_i - ŷ_i = ε_i for i = 1, 2, 3, ..., 11.

Each |ε| is a vertical distance.

For the example about the third exam scores and the final exam scores for the 11 statistics students, there are 11 data points. Therefore, there are 11 ε values. If you square each ε and add, you get

This is called the Sum of Squared Errors (SSE).

Using calculus, you can determine the values of a and b that make the SSE a minimum. When you make the SSE a minimum, you have determined the points that are on the line of best fit. It turns out that the line of best fit has the equation:

where $a=\overline{y}-b\overline{x}$ and $b=\frac{\Sigma (x-\overline{x})(y-\overline{y})}{\Sigma {(x-\overline{x})}^2}$.

The sample means of the x values and the y values are $\overline{x}$ and $\overline{y}$, respectively. The best fit line always passes through the point $(\overline{x},\overline{y})$.

The slope b can be written as $b=r\left(\frac{s_y}{s_x}\right)$ where s_y = the standard deviation of the y values and s_x = the standard deviation of the x values. r is the correlation coefficient, which is discussed in the next section.