CHAPTER 06.04: NONLINEAR REGRESSION: Exponential Model: Transformed Data: Part 1 of 2
In this segment, we're going to take an example of the exponential model of regression. So we have an exponential model which we want to be able to use for regressing to a particular data, and what we're going to do is we're going to use transformed data. We're going to use transformed data to so do, and we'll talk about, a little bit later, about what the transformed data means. In the previous segment, I already talked the theory about using the transformed data to develop the exponential model. So let's talk about the example. The example is about an isotope of technetium, which is used for gallbladder scanning, and an isotope, technetium 99m isotope is used to be able to do a CT scan of a gallbladder. Now, as we know that this is a radioactive material, so we want to be able to figure out that how long does it take for a person to lose this isotope from the body. The half-life of technetium 99m is about 6 hours, so it doesn't take too long for one to go back to the normal levels of radiation which we get from the sun, for example. But let's go ahead and see, figure out for ourselves that if we take the intensity of the isotope versus time data, will we be able to figure out that how much of the isotope is still in the body? So what I'm going to do is I'm going to give you time, which is in hours, versus relative intensity data. So that's relative intensity. What I mean by relative intensity is that this is the intensity of the isotope when related to what is happening at 0 hours. So what that means is that at 0, the relative intensity would be, of course, 1, because you are dividing by the intensity at time, t equal to 0. At other points, the data is given to us, at 1, it is 0.891, at 3, it is 0.708, at 5, it is 0.562, at 7, it is 0.447, at 9, it is 0.355. So you're already seeing that down around 6 hours or so, you're losing about half of it, so the half-life is, of technetium 99m actually is 6 hours. But let's go ahead and figure that what happens, what is the relative intensity at 24 hours. Does it go down very low so that the patient is not worried about having radioactive material in their bodies? So what we're going to do is we are going to use the exponential model to be able to figure out what the relative intensity of the technetium isotope is at 24 hours. So we're going to use the exponential model to do the regression. So, but we're going to use the transformed data for this. So what that means is that if I have gamma is equal to a e to the power b t, I'm going to transform the data corresponding to this model so that I can use the linear regression formulas. So what does that mean? It's that I'm going to take the log of both sides, for example, in this case, that's how I'll be able to do the transformation of the data so that I can use the linear regression model. So I get log of gamma is equal to log of a, plus b t, and what I'm going to do is I'm going to take this to be my z, I'm going to assume this to be my a0, this to be my a1, that's what I'm going to take, do those substitutions. So if I do those substitutions, I'll get z is equal to a0, plus a1 times t, that's what I'm getting. Now this is a linear . . . this is a linear relationship between z and t, as you can see that it's a straight line, or a first-order polynomial, relationship between z and t. Now, what that means is that if I am able to find z versus t data, then I'll be able to find a0 and a1. So if I am able to find a0 and a1, then I can find my a and b, which b is same as a1, while log of a is equal to a0, so a will be equal to e to the power a0. So that means that, let's suppose, for example, once I find a0, log of a is a0, so a will be e to the power a0, so that's how I'll be able to find my a, and b is nothing but the value of a1, so that's how I will be able to find the constants of the exponential model. Keep in again mind that the linear relationship is between z and t. So in no way should you consider that we are linearizing the model. We're not linearizing the model. All we are doing is we're transforming the data itself so that there's a linear relationship between z and t, not between gamma and t, not between the relative intensity and time, it's an exponential model. We are developing, trying to see if there's a linear relationship between z and t, and from that we can backtrack the constants of the exponential model, which are a and b. So as you can see that z is equal to the log of gamma, which means that what we'll have to do is we'll have to take all the individual gamma values, and calculate our zi values to be able to do that. Now, how will I find a0 and a1 of this? a1 will be equal to n times summation, zi ti, minus summation zi, summation, ti, divided by n times summation, ti squared, minus summation, ti, whole squared. So that's the . . . coming from the linear regression formulas, that's what I'm going to use. All these summations are going from 1 to n, so that's why I have not put the limits of the summation there, I should have, but all the limits of the summation are 1 to n, where n is the number of data points which I have. And a0 is nothing but z-bar, minus a1 t-bar, where z-bar is the average value of the z values, and t-bar is the average value of the t values. So that's how I will be able to find a1 and a0 from the z versus t . . . z versus t data, and then I'll be able to find a and b, which are the constants of the exponential model from the values of a0 and a1 which I find from the linear regression formulas here. So I'm going to draw a table so that I can calculate these zi value, which are the values which I need from the original data. So I'm going to calculate the zi values, and then also I'm going to develop a table so that I can show the summations which I need. I need the summation of zi ti, summation of zi, summation of ti, and summation of ti squared, that's what I need.