Bootstrapping the Blue-Green Algae.

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The biologists have suggested two different treatments, A and B say, with four and two levels, respectively. The levels within treatment A are:

1. No shaking and no CO₂ aeration,
2. CO₂ bubbled through the culture,
3. Continuous shaking, but no CO₂ aeration, and
4. Continuous shaking and CO₂ bubbled trough the culture.

Treatment B indicates the addition of two specific additives, that is, the levels within treatment B are

1. Additive X added and
2. Additive Y added.

Hereby the biologists have gathered eight samples. The samples have been monitored daily for two weeks and the algal density was recorded. The growth curves are shown in Figure 1.

The data illustrated in Figure 1 are evidently growth data. When dealing with growth data non-linear modelling is the path to follow, and we consider the non-linear model named the Mitscherlich equation

• i is the i'th day (i=1,…,14),
• j is the j'th level of treatment A, and
• k indicates additive X or Y.

The eight different models were fitted in S-Plus using the non-linear least squares function nls, see Table 1 below for results. For all eight sets of parameter estimates we find a relatively low standard error.

To examine the effects of the different treatments, we propose a bootstrapping method, which permutes the residuals obtained when fitting the models. Hence the null hypothesis becomes

The assumption of no day effect implies that observations are independent, which is certainly not the case (cf. Figure 1).  With this in mind we carry on with the following bootstrap algorithm:

Hereby we permute the obtained residuals within each treatment allowing us to create the 95% bootstrap confidence intervals. The results are shown in Figure 2.

The serial dependence present in the time series have been ignored so far. Modelling the depencies which are obviously present in Figure 1 are done by an autoregressive process or order 1, i.e. an AR(1) process. This model can be fitted by using the mixed random effects model implemented in S-Plus. Hence the model becomes

Hence the assumptions can be stated as in the following null-hypothesis

We now present a bootstrap method to perform a parametric test on the above hypothesis. The method is based on the nlme-function in S-Plus, which fits any non-linear mixed effects model and returns the estimated parameters. Moreover, it is also possible to obtain the fitted values, the residual values for any given model, and what is more important the correlation structure of the residuals. For each curve we generate a set of residuals with the same correlation structure as the ones estimated with the nlme-function. These sets of residuals are added to the corresponding curves whereby we obtain new sets of observed values representing growth curves, which we again fit with the non-linear Mitscherlich equation in S-Plus. The obtained estimated parameters are recorded and we again simulate new residual values and so forth. In this way bootstrapped confidence intervals are obtained by ordering the recorded parameters and extracting the percentiles corresponding to a prescribed confidence level. The algorithm is sketched below:

 The obtained estimates and bootstrap confidence intervals based on 200 simulations are shown in Table 2 below. Also the standard confidence intervals based on an approximatively normal theory from the first fit with the nlme-function is shown.

We observe that more of standard intervals are narrower than the bootstrap intervals. Since estimation of non-linear mixed effects models is highly computer intensive we only managed to bootstrap a sample of 200 sets of parameters. This number is far from enough to ensure bootstrap confidence intervals comparable with standard confidence intervals.