MINI CASE 10.1

Here are three examples of people using the binomial in very different contexts. In the first case the Water Authority is using it to test if standards have been breached, the second sees its use in total quality management and the third applies it to a clinical trial.

(a) The 2004 Water Quality Control Policy for California specifically states that a Binomial Model Statistical Evaluation is to be used:

Binomial Model Statistical Evaluation

Once data have been summarized, RWQCBs shall determine if standards are exceeded. The RWQCBs shall determine for each averaging period which data points exceed water quality standards. The number of measurements that exceed standards shall be reported in the water body fact sheet.

When numerical data are evaluated, all of the following steps shall be completed:

A. For each data point representing the averaging period, the RWQCB shall answer the question: Are water quality standards met?

B. If the measurement is greater than the water quality standard, objective, criterion, or evaluation guideline, then the standard is exceeded.

C. Sum the number of samples exceeding the standard, objective, criterion, or evaluation guideline.

D. Sum the total number of measurements (sample population).

E. Compare the result to the appropriate table (i.e., Tables 3.1, 3.2, 4.1 or 4.2).

F. Report the result of this comparison in the water body fact sheet.

Source: California's Clean Water Act, Section 303(d) List, September, 2004

(b) Many industrial processes use Acceptance Sampling as a method of monitoring quality. A random sample is taken at given time intervals and the proportion defective noted. This proportion is then plotted on a chart which has Upper and Lower Control Limits marked. If the plot is outside of these limits, the process is deemed out of control. You can see that this is a binomial process since the item is either defective, or not defective. The Control Limits are calculated from the binomial distribution. (It would be fair to say that views of quality control have moved on from just using these samples to ideas of TQM (Total Quality Management) and Taguchi methods.

(c) Statistical Process Control (based on the binomial distribution) has also been applied to clinical trials

As a clinical trial is a process, too, and as there are quality characteristics that can be measured, statistical quality control methods can also be applied to clinical trials. Benefits of the usage of statistical quality control in clinical trials: If it is relatively cheap, it is not personnel intensive, it can centrally be performed from the monitoring centre it is not work intensive and it provides a fast and direct access to information.

However there are some special characteristics of clinical trials that have to be taken into account when applying statistical quality control methods. Differently from the production industry there are no production runs, which would form a natural sample basis. There is no sampling of observation. The sample size varies from time interval to time interval. Patient values will change over time, due to therapeutic effects. Repeated measurements are possible, as patients may have more than one visit in a given time interval. The process 'clinical trial' cannot be directly corrected.

The statistical methods presented in this context have been selected for the situation of clinical trials. Generally two types of data are considered:

(a) Measurement data which are modelled using the normal distribution;

(b) Event data, where the Binomial and Poisson distribution is used.

Basically all quality characteristics are monitored over calendar time, but only values, which were taken at the same individual patient time in the trial, are used in analysis because of potential therapeutic effects. Event data and measurement data for Shewhart charts are aggregated over calendar time intervals. Here all observations in a time interval have the same time value, the interval midpoint time.

All methods have been adopted for the characteristics of clinical trials.

Source: http://www.meduniwien.ac.at/medstat/misc/QC/teil1.html; (accessed 15 June 2011)

Reproduced by permission of Gerhard Svolba