Calculating necessary sample size

In day-to-day practical experimentation, sample sizes probably owe more to tradition than logic. PhD students use five rats not on the basis of any rational calculation but simply because their supervisor always used five (and they in turn used five because their supervisors used five and so on ad infinitum.) There is now increasing recognition that hereditary malpractice ought to be replaced by soundly based optimization. 

It is a sin to perform an experiment that is either too small or too large. Experiments that are excessively large are obviously a waste of resources. An answer could have been arrived at with less expenditure of time, effort and cash. If animals or humans were the subjects of the experiment, then there is also the ethical issue of unnecessary inconvenience or suffering. Experiments that are too small are perhaps even worse. If numbers are inadequate then the statistical 

3 Size of experimental effect we want to be able to detect 

The necessary sample size will be governed by the smallest difference we want to be able to detect. The latter is not based on statistical considerations. It can only be defined by somebody with expertise in the particular field. They should be able to say what sort of size of effect needs to be detectable. 

There is always a temptation to be a super-perfectionist and insist that, if there is even the slightest difference, you want to know about it. However, if you set the detection limit to an excessively low value, the necessary sample size will probably escalate to something unmanageable We have to accept that we cannot detect infinitesimally small effects and we must draw the limit at some reasonable point. 

---

The requirements and means of calculating necessary sample size depends on the desired (or practical) comparative sizes of test and control groups. 

Figure 8.4 implies that, if we want to be able to calculate necessary sample size, we will need to supply values for each of the factors on the left of the diagram. At this point it would be useful to make a few additional comments about these three factors... 

Using a statistical package to calculate necessary sample size... 

For those commendably simple experiments that result in 2 x 2 contingency tables, some statistical packages include simple routines to calculate necessary sample size. For anything more complex, you are on your own - quite right too The routine will require you to provide values for the following ... 

Many statistical packages provide a routine that will generate a 95 per cent Cl for the extent of any change in a proportion within a 2 x 2 table and this can be used to check for practical significance. For experiments that can be described in a 2 x 2 contingency table, it is simple to calculate necessary sample sizes. 

This means that the necessary sample size will be very large if the systems 0 and 1 differ too much. For example, for AS/kB = —15, N > 3 x 108 is needed to reach a 95% accuracy level. In this case, it is better to introduce intermediates and perform multistage calculations instead. 

With the analysis above, we can answer some important practical questions related to MFEP calculations. Should an MFEP calculation be used at all How many stages are needed How should the intermediates be formulated What is the necessary sample size for each stage ... 

Show how to calculate the necessary sample sizes for an experiment that will be analysed by a two-sample t-test... 

Table 8.2 Generic output for calculation of necessary sample size for the rifampicin/theophylline experiment...

The design of all experiments should involve a rational calculation of how large the samples need to be. For an experiment that will be analysed by a two-sample f-test, the main influences upon necessary sample size are ... 

Figure 16.3 Calculation of necessary sample size for a contingency chi-square test...

Calculating the necessary sample size for our IUD expulsion trial... 

As with any form of sample-based research, it is important to know in advance how much data (completed questionnaires) will be required. We can calculate a necessary sample size using the approaches described earlier in this book. We just need to identify the primary question, determine what statistical analysis will be used to answer it and then perform a sample size calculation in the normal way. 

A meta-analysis for continuous data cannot be calculated unless the pertinent standard deviations are known. Unfortunately, clinical reports often give the sample size and mean ratings for the various groups but do not report the standard deviations (or standard error of the mean), which are necessary for effect size calculations. Thus, investigators should always report the indices of variability (e.g., confidence intervals, SDs) for the critical variables related to their primary hypothesis. 

Before - identify the primary question, primary end-point and primary statistical analysis establish equivalence limits where necessary perform power and sample size calculations. 

Polymer Neutralization Number. A sample of polymer was weighed to the nearest 0.1 mg. (sample sizes of 0.5-1.0 gram). The polymer was dissolved in tetrahydrofuran and if necessary warmed on a steam bath. The solution was titrated with standard potassium hydroxide in methanol using phenolphthalein as indicator. Calculation ... 

For a sample size of 10, and with a 5% level of significance, the critical value of Q, from tables, is 0.464. The calculated 2-value exceeds this critical value, and therefore this point may be rejected from subsequent analysis. If necessary, the remaining data can be examined for further suspected outliers. 

Two or more successive gas-phase equilibrations give all the necessary data to calculate the concentrations in the aqueous phase. The mathematics of multiple-phase equilibration are presented in Reference 10. A plot of the log of the compound s concentration in each gas phase equilibration vs. the number of equilibrations produces a straight line (Figure 1). The negative slope of this line is the log of the distribution coefficient plus 1. The intercept is the product of the initial concentration of the unknown compound, its distribution coefficient, and a constant related to sample size, the instrument, and its sensitivity. 

Software for determining the observations required for work sampling study is readily available today. These programs perform aU the statistical calculations required to determine sample sizes and confidence intervals. For example, calculations for 90%, 95%, and 99% confidence intervals for a sample can be calculated. They can also provide the number of samples necessary to achieve the desired confidence for a specified degree of accuracy. 

In practice, the sample size calculation is not done by hand, but by computer programs (such as the free G Power ), which lets you choose a statistical test, asks for the necessary inputs (i.e., a, P, variance, and effect size), and gives you the minimum required sample size. They can also be used to determine the power of your test given the sample size, a, S, and effect size. Because of all these unknowns, it is a good idea to consult a biostatistician on these matters if possible. 

---