Geometry of Regression
Since graphical portrayal brings an added dimension to the understanding of the regression line we will go through the regression line for various values of 'a' and 'b'. As you know the regression line is Y = a + bx. Here 'a' is the Y-intercept and 'b' is the slope.
Since the regression line intersects the 'Y' axis at the point 'a', 'a' is called the 'Y' intercept. That is, the value of 'a' determines at what point the line intersects the 'Y' axis.
Let us understand the concept better, by taking the value of 'b' as zero, and calculating the regression line for various values of 'a'.
What happens when a = 0, b = 0?
The regression line is Y = 0.
Now let us take the value of a as 1, and b as 0.
The regression line will be Y = 1.
So, one can see that when b is 0, the regression line is a straight line parallel to the X-axis, and the line intercepts the Y-axis at point 1 which is the value of a.
What happens when a = 2, b = 0?
The regression line Y = 2.
The regression line is a straight line parallel to the X-axis, and intercepts the Y-axis at point 2, which is equal to 'a'.
What happens when a is negative. Let us see when a = -1 and b = 0.
The regression line will be equal to Y = -1.
Thus, we see that the regression line intercepts the y-axis at -1, which is equal to 'a'.
Let us see what happens when a = 0 and b = 1. The regression equation becomes Y = X.
The line can be found in the I and III quadrants. This line goes through the origin. This is because a = 0. Had 'a' been equal to 1, the line would have intercepted the Y-axis at 1 as shown in the figure.
When 'a' changes from 0 to 1, the entire line has shifted upward by 1 unit.
Let us see what happens when a = 0 and b = 2
The regression line now becomes Y = 2X. Let us now plot the regression line, when the values of X are 1, 2, 3, 4, then the corresponding values of Y = 2, 4, 6, 8.
Now compare this graph with the previous one. What can you notice. Yes, the line has become steeper. Why? Because, previously the value of b was 1, now the value of b = 2. Thus, the value of 'b' determines how steep the line is. In other words 'b' determines the slope of the line.
Can the slope be only positive? No, it can be negative too.
What happens when a = 0 and b = -1? The regression line becomes Y = -X.
What do we observe? Now, the line slopes downward from the left to the right. What does this imply? It shows that for a unit change in 'X', the value of 'Y' declines by a unit 'b'.
We know that any straight line, with the exception of the vertical line, can be characterized by its slope. So, what do we mean by 'slope'. By 'slope' we mean basically the inclination of a line - whether it rises or falls as you move from left to right along the X-axis and the rate at which the line rises or falls (in other words, how steep the line is).
We have seen that the slope of a line may be positive, negative, zero or undefined. A line with a positive slope rises from left to right. For such a line the value of 'Y' increases as 'X' increases (or Y decreases as X decreases). A line having a negative slope falls from left to right. For such a line the value of Y decreases as X increases (or Y increases as X decreases). This means that X and Y behave in an inverse manner. A line having a zero slope is horizontal. As X increases or decreases, Y stays constant. Vertical lines have a slope which is undefined.
The slope of a line is quantified by a number. The sign of the slope indicates whether the line is rising or falling. The magnitude (absolute value) of the slope indicates the relative steepness of the line. The slope tells us the rate at which the value of Y changes relatively to changes in the value of X. The larger the absolute value of the slope, the steeper the angle at which the line rises or falls.