Statistical analysis from repeat runs

The errors of the entire rig were determined by repeat runs at a constant overall feed saturation of 70%. This value was more appropriate for on-line monitoring and control of the second cut of fractional protein precipitation. The process was run off-line as well. The relationship between the errors for individual stages of preparation and measurements and the overall error is discussed below.

shown in Figure 5.12 (p. 176) against measurement cycle. The mean is 0.0154 /s with a coefficient of variation of 7.30% for the redissolved samples. The supernatant mean is 1.36x10'^ /s and the coefficient of variation 14.4%. The ADH fraction remaining soluble is 0.0879.

A detailed statistical analysis is now considered. Every measurement is subject to random errors that contribute to the final error. Where they are minor, a similar order of magnitude and equally likely to be positive or negative the frequency distribution is described as normal or Gaussian having a symmetrical bell-shape. Biased observations lead to skewness. If a key parameter is uncontrolled there may be incoherent scatter. The standardised normal curve has zero mean and unit standard deviation. The total enclosed area is one, the ordinate being relative frequency.

- i f .

y =■ ■ e ^ [ 5. 2]

y27T

where z is the multiples of the standard deviation. 6 8% of the observations occur within ±1 standard deviation from the mean and 95% within ± 2 standard deviations from the mean. The worst possible case may be defined as 3 standard deviations above or below the mean, which includes 99.7% of all measurements.

Consequently the distributions are examined using a normal probability axis for the dependent variable. The transformation of the scale enables the standard normal distribution to be represented as a straight line. The data are arranged in order of increasing magnitude and the percentage cumulative distribution plotted. The highest point may not be shown since the ordinate does not reach 1 0 0%. (Alternatively the percentage cumulative frequency is found not with the number of observations, n, as the denominator but n+ 1.) However the lower and upper ends are of less interest; the measurements are nearest the tails of the distribution. The arithmetic mean may be obtained at the estimated intercept of 50% cumulative

frequency.

The results roughly fit a straight line, see Figure 5.13 (p. 177). There are several more precipitated feed measurements between 0 and + 1 standard deviations from the mean than are expected. Fewer supernatant measurements lie in this range, being numerous between -1 and 0 standard deviations from the mean. Despite limited sets of data, Gaussian distributions are followed.

It is important to consistently reject any wild measurements which occur from experimental failures but not the ones that represent actual phenomena. There otherwise would be a significant effect on the analysis. Various techniques are available to remove the outliers completely, replace them with a designated value such as the next greatest extreme or assign a weighting factor (Barnett and Lewis, 1984). Hence Chauvenet’s criterion for normal distributions is used (Holman, 1978). A measurement is eliminated if the probability of the deviation from the mean is less than l/2n. A new mean and standard deviation are calculated, but the test must only be applied throughout each set once. For instance where there are twenty readings those beyond 2.24 times the standard deviation from the mean are rejected. This is obtained using integral tables of the standardised normal function.

The limits by the criterion for the precipitated feed samples are 0.0129 and 0.0180 /s. Point 17 (Figure 5.12, p. 176) is rejected and the mean is recalculated as 0.0156 /s. The coefficient of variation is reduced to 5.12%. The limits are from 0.919x10'^ to 1.79x10^ /s for the supernatant samples, which include all measurements. Thus the ADH fraction remaining soluble becomes 0.0868. It is seen that elimination of the suspicious measurement leads to a decrease in the coefficient of variation, whilst the mean rate and ADH fraction remaining soluble are relatively unchanged.

Any pattern by the sequence itself is also significant. The Shewhart chart is used in process control to maintain stability about a target value. Corrective action may be taken if one measurement is outside three standard deviations from the mean.

two successive measurements fail to be within warning lines located at two standard deviations or seven consecutive results are on the same side of the mean. The principle is amended here for elimination from the entire data set based on retrospective evaluation of the sample mean and standard deviation. A measurement that fails the first condition would be rejected by the Chauvenet criterion in the above case. One further infringement occurs at point 11 for the redissolved run which is the seventh of eight samples in series greater than the mean, though this would be satisfactory if point 17 was removed.

These statistical techniques may be performed for the ADH fraction remaining soluble, instead of the precipitated feed and supernatant groups of measurements separately. The range is from 0.0667 at point 9 to 0.122 at point 17 having a coefficient of variation of 19.0% (Figures 5.13-4, pp. 177-8). The distribution is again approximately Gaussian. All are acceptable by the Chauvenet criterion and the control chart.

The experiment was repeated off-line but with continuous precipitation at 70% overall feed saturation. The mean rate for eighteen precipitated feed samples is 0.0160 /s (Figure 5.15, p. 179) and the coefficient of variation is 3.79%. There was no ADH activity in the manually centrifuged samples. Fourteen measurements are between -1 and 0 standard deviations from the mean (Figure 5.16, p. 180). The remainder are scattered above. None are rejected by the Chauvenet criterion or exhibit a trend with time.

Data from preliminary on-line runs at 50% overall feed saturation are briefly presented in Figure 5.17 (p. 181). The mean for the redissolved samples is 0.0261 /s, the coefficient of variation being 7.64%. The mean is 0.0123 /s for the supernatant samples. The coefficient of variation is 17.1% and the ADH fraction remaining soluble 0.471.

The overall errors are attributed in Figure 5.18 (p. 182) to the individual equipment. The total variance for any number of independent random variables is

equal to the sum of the variances. By subtracting the precipitated feed samples coefficient of variation squared from that of the supernatant samples and taking the square root the coefficient of variation owing to the microcentrifuge unit alone may be predicted. At 70% overall feed saturation this is 12.4% or 13.4% if the outlier test is applied, and at 50% overall feed saturation it is 15.3%. Since the propagation of the independent errors depends on their squares the largest values predominate. The continuous precipitation rig error (section 5.2) is negligible here and the coefficients of variation for the precipitated feed samples are assumed to represent the stopped-flow analyser.

The difference between the coefficients of variation for the precipitated feed samples and the tracer studies indicates the additional error of on-line ADH assays of the diluted sample stream. The value is lower where measured off-line suggesting the continuous precipitation achieved is reproducible. The microcentrifuge appears to be the major source of error in comparing the resolubilised feed and supernatant data, for instance by resuspension or re-entrainment from the tubing. Whereas it may be difficult to further eliminate these effects it has been shown that those of the stopped-flow analyser were decreased such as by the tracer studies. The errors of large magnitude are important because of the square relationship. Not much is gained by trying to reduce the minor ones. The coefficients of variation at 70% overall feed saturation are lower than at 50%, possibly as the ADH fraction remaining soluble profile is less steep. The level of error is generally high throughout and this will determine the approach for on-line control.