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.