January 27, 2014

\[ \newcommand{\esp}[1]{\mathbb{E}\left(#1\right)} \newcommand{\var}[1]{\mbox{Var}\left(#1\right)} \]

A model for continuous data for a single individual can be represented mathematically as follows: \[ y_{j} = f(t_j) + e_j, \quad \quad 1\leq j \leq n, \] where

\((t_1,t_2,\ldots , t_n)\) is the vector of sampling times.

\(f\) is the

*structural model*. It is a function of time: \(f(t_j)\) is the value of \(f\) evaluated at time \(t_j\).\((e_1, e_2, \ldots, e_n)\) are called the

*residual errors*. It is common to assume that the residual errors are normally distributed but we can also use any centered probability distribution, i.e., such that \(\esp{e_j} =0\).

Then, \(f(t_j)\) is the predicted value of \(y_j\):

\[ \tilde{y}_j = f(t_j) \]

We usually state continuous data models in a slightly more flexible way: \[ y_{j} = f(t_j) + g(t_j ;\xi)\varepsilon_j , \quad \quad 1\leq j \leq n, \] where now

\(g\) is called the

*residual error model*. It depends on some parameters \(\xi\). It may also be a function of the time \(t\).\((\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n)\) are the

*standardized residual errors*. For identifiability reasons, we suppose that these come from a probability distribution which is centered and has unit variance: \(\esp{\varepsilon_j} = 0\) and \(\var{\varepsilon_j} =1\), such as the standard normal distribution \({\cal N}(0,1)\).

The choice of a residual error model \(g\) is very flexible and allows us to account for many different hypotheses we may have on the error’s distribution.

\(g=a\). That is, \(y_j=f(t_j)+a\varepsilon_j\) and \(\xi=a\).

In the following example, \[ f(t) = 25\left( e^{-0.6 t} - e^{-t}\right) \]

You can select the standard deviation \(a\) of the residual errors and the number \(n\) of sample times.

Then, you can plot

the observations \((y_j)\) versus the sampling times \((t_j)\),

the observations \((y_j)\) versus the predictions \((\tilde{y}_j)\), where \(\tilde{y}_j = f(t_j)\),

the residual errors \((e_j)\) versus the sampling times \((t_j)\), where \(e_j=y_j - \tilde{y}_j = a\varepsilon_j\),

the residual errors \((e_j)\) versus the predictions \((\tilde{y}_j)\).

The median is obtained with \(\varepsilon_j=0\) while the first and third quartiles are obtained, respectively, with \(\varepsilon_j=-0.6745\) and \(\varepsilon_j=0.6745\).

\(g=bf\). That is, \(y_j=f(t_j)+b f(t_j) \varepsilon_j\) and \(\xi=b\).

\(g=a+bf\). That is, \(y_j=f(t_j)+(a+b f(t_j)) \varepsilon_j\) and \(\xi=(a,b)\).

An alternative combined error model assumes that \[y_j=f(t_j)+ a \varepsilon_j^{(1)}+b f(t_j)) \varepsilon_j^{(2)}\] where \(\varepsilon_j^{(1)}\) and \(\varepsilon_j^{(2)}\) are independent. If \(\varepsilon_j^{(1)}\) and \(\varepsilon_j^{(2)}\) are normally distributed, then there exists a sequence of standard normal random variables \((\varepsilon_j)\) such that \[y_j=f(t_j)+ \sqrt{a +b f(t_j)} \varepsilon_j\] Here, \(g=\sqrt{a^2+b^2f^2}\) and \(\xi=(a,b)\).

**Remark:** In this example, these two combined error models are almost indistinguishable.

The assumption that the distribution of any observation \(y_{j}\) is symmetrical around its predicted value is a very strong one. If this assumption does not hold, we may want to transform the data to make it more symmetric around its (transformed) predicted value. In other cases, constraints on the values that observations can take may also lead us to transform the data.

Basic model can be extended to include a transformation of the data: \[ u(y_{j})=u(f(t_{j}))+ g(t_{j},\xi_i)\varepsilon_{j} , \] where \(u\) is a monotonic transformation (a strictly increasing or decreasing function). As we can see, both the data \((y_{j})\) and the structural model \(f\) are transformed by the function \(u\) so that \(f(t_{j})\) remains the prediction of \(y_{j}\).

Let us see now some examples of basic transformations, assuming a constant error model (in the domain of the transform data), i.e. \(g=a\).

- If \(y\) takes nonnegative values, a log transformation can be used: \(u(y) = \log(y)\).

We can then write the model with one of two equivalent representations: \[
\begin{aligned}
\log(y_{j})&=\log(f(t_{j}))+ a\varepsilon_{j} \\
y_{j}&=f(t_{j})\, e^{ a\varepsilon_{j} }.
\end{aligned}
\] This model is sometimes called *exponential error model*. We remark that when \(a\) is small, the exponential error model is similar to the proportional error one since \(e^{a\varepsilon_j} \approx 1 + a\varepsilon_j\).

- If \(y\) takes its values between 0 and 100, a logit-like transformation can be used: \(u(y) = \log(y/(100-y))\).

We can then write the model with one of two equivalent representations: \[ \begin{aligned} \log\left(\frac{y_{j}}{100-y_{j}}\right)&=\log\left(\frac{f(t_{j})}{100-f(t_{j})}\right)+ a\varepsilon_{j} \\ y_{j}&=\frac{ 100 f(t_{j})}{f(t_{j}) + (100-f(t_{j}))\, e^{ -a\varepsilon_{j}} }. \end{aligned} \]

In the following example, \[ f(t) = \frac{100}{1+e^{-(t-5)}}\]