Example

X-ray

Data are available that represent the number of surviving marine bacteria $N(t)$ following exposure to 200 kilo-volt x-rays for periods of time $t$ ranging from 6 to 90 minutes. In each case the initial number of bacteria was $N(0) = 40,000$. Biological theory suggests that $N(t) = N(0) \exp^{\beta t}$ for some underlying constant $\beta$.

How would you describe the relationship between Time and $N(t)$? You are probably less sure that it’s linear (i.e., you likely would not use a straight line to illustrate this relationship).

Biological theory suggests that $N(t) = N(0) \exp^{\beta t}$ for some underlying constant $\beta$.

You may agree that this plot suggest that an exponential relationship looks plausible.

This is where we could just look directly to the biological theory

$$\begin{eqnarray*} N(t) &=& N(0)\exp^{\beta t}\\ \end{eqnarray*}$$

But you notice that we can re-arrange this equation such that

$$\begin{eqnarray*} N(t) &=& N(0)\exp^{\beta t}\\ \frac{N(t)}{N(0)} &=& \exp^{\beta t}\\ \log\bigg(\frac{N(t)}{N(0)}\bigg) &=& \beta t. \end{eqnarray*}$$

Notice that this is of a similar setup to the straight line form $y=\alpha + \beta x$ with $y=\log \bigg(\frac{N(t)}{N(0)} \bigg)$, $x=t$ and $\alpha=0$.

$$\begin{eqnarray*} y &=& \alpha + \beta x\\ \log \bigg(\frac{N(t)}{N(0)} \bigg) &=& 0 + \beta t\\ \end{eqnarray*}$$

Let’s now plot $y=\log \bigg(\frac{N(t)}{N(0)} \bigg)$ against $x=t$ (Time).

Now describe the relationship between time $t$ and $\log\bigg(\frac{N(t)}{N(0)} \bigg)$. It appears linear.

$\newline$

Data: $(y_t,t), \quad i=1, \dots, n; \quad n=15.$

$y_t$ = transformed number of marine bacteria at time $t$ (vertical axis)

$t$ = time (horizontal axis)

Possible model: $$y_t= \beta t + \epsilon_i \quad \quad \mathrm{for} \, t=1, \dots, n.$$

where $y_t=\log\bigg(\frac{N(t)}{N(0)}\bigg).$

In this example, we have met the idea of a transformation which is something we will make further use of. As a result of the transformation, we are able to draw a plot such that the anticipated relationship is linear and we were able to write down a linear model to model the transformed data. This will come in handy later in the course.