O'Reilly logo

Growth Curve Modeling: Theory and Applications by Michael J. Panik

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

4

ESTIMATION OF TREND

4.1 LINEAR TREND EQUATION

Let us consider what we shall call a purely mathematical (linear) model involving Y as a function of time t or

images

where β0 is the (constant) vertical intercept and β1 is the (constant) slope. Here β0 is the value of Y when t = 0 while β1 = ΔYt (=rise/run). Specifically, β1 is the rate of change in Y per unit change in t. So if β1 = 5, then when t increases by one unit, Y increases by five units; and if β1 = −3, then as t increases by one unit, Y decreases by three units.

To obtain Equation 4.1, we need only two specific points in the (t, Y)-plane (Fig. 4.1). That is, β1 = ΔYt = (Y2Y1)/(t2t1); and at, say, (t2,Y2), β0 = Yβ1t = Y2β1t2. For instance, suppose (t1, Y1) = (1, 2) and (t2, Y2) = (4, 7). What is the equation of the line passing through these two points?

Clearly β1 = ΔYt = (7 − 2)/(4 − 1) = 5/3 while, at (t1, Y1), β0 = 2 − (5/3)(1) = 1/3. Hence our particularization of Equation 4.1 is Y = (1/3) + (5/3)t. So when, t = 0, Y = 1/3; and, for β1 = 5/3, when t increases by one unit, Y increases by 5/3 units.

What if we have more than two points in the (t, Y)-plane? Specifically, we may have a scatter of points such as the one depicted in Figure 4.2.

images

FIGURE 4.1 The line through the points (t1,Y1) and (t2,Y2).

FIGURE ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required