Replication errors assumptions

Various calibration schemes similar to those given in Section 2.2.8 were simulated. The major differences were (1) the assumption of an additional 100% calibration sample after every fifth determination (including replications) to detect instrument drift, and (2) the cost structure outlined in Table 4.6, which is sununarized in Eq. (4.2) below. The results are depicted graphically in Figure 4.5, where the total cost per batch is plotted against the estimated confidence interval CI(X). This allows a compromise involving acceptable costs and error levels to be found. 

With unequal variances we cannot speak of an "overall standard error". In that case s2 computed by (3.11) yields an unbiased estimate of the constant a2 in the weighting coefficients. Therefore, s 2 = s2/wi is an unbiased estimate of the error variance. If we have a different independent estimate of the same variance, for example computed from the replicates at the value x of the independent variable, then our assumptions can be checked by an F-test, involving the ratio of the two estimates, see e.g. Himmelblau (ref. 5). Though this is the best way to measure the goodness-of-fit, it requires additional information (i.e., replicates), not always available. 

Accuracy is the closeness of the mean of a set of replicate analyses to the true value of the sample. Often, it is only possible to assess the accuracy of one method relative to another by comparing the means of replicate analyses by the two methods using the t test. The basic assumption, or null hypothesis, made is that there is no significant difference between the mean value of the two sets of data. This is assessed as the number of times the difference between the two means is greater than the standard error of the difference (t value). 

One of the main motivations of using synthetic DNA for cellular engineering seems to be at odds with the random nature of directed evolution. Traditionally, PCR-based methods have been used to create sequence diversity, inspired by the fact that mutations in nature commonly arise from errors in DNA replication. PCR-based methods are preferred when there is no prior knowledge about where mutations are likely to influence the traits of interest, but are limited in that the sequence diversity that results is restricted and biased. With single base mutations per codon - a common assumption with most protocols - only 5.7 amino acids are accessible per position on average, and in most cases, the resulting set of amino acids does not accurately represent the spectrum of physicochemical properties of naturally-occurring residues [76]. 

An important feature of the replication-mutation kinetics of Eq. (2) is its straightforward accessibility to justifiable model assumptions. As an example we discuss the uniform error model [18,19] This refers to a molecule which is reproduced sequentially, i.e. digit by digit from one end of the (linear) polymer to the other. The basic assumption is that the accuracy of replication is independent of the particular site and the nature of the monomer at this position. Then, the frequency of mutation depends exclusively on the number of monomers that have to be exchanged in order to mutate from 4 to Ij, which are counted by the Hamming distance of the two strings, d(Ij,Ik) ... 

This conservation is a consequence of assumption (ii), namely, that mutants originate exclusively through erroneous replication and not through external interferences such as radiation or chemical attack. (If this assumption is relaxed, destruction terms must be subtracted in the conservation law to balance the additional first-order off-diagonal mutation terms.) The nondiagonal elements of the value matrix depend strongly on the Hamming distance d(i,k) between template i and erroneous replica k. For the uniform error rate model the expression reads... 

Any experimental procedure will be afflicted by a random error. The error variance, a , is not known to the experimenter, but it is possible to obtain an unbiassed estimate, s, of the variance by replication of an experiment. For synthetic chemical systems, we have seen that it is reasonable to assume that the experimental errors are normally and independently distributed. It is also reasonable to assume that the variance is constant in a limited experimental domain. These assumptions cannot be taken for granted. They must always be checked, e.g. by plotting the residuals in different ways. Such diagnostic tests will be discussed later on. 

Two points merit emphasis in the above exercise a), The statistical confidence Interval for the outcome s based on S and its SE (using a 2-sided Student s-t) SE but not S is used also for the estimation of Lp. b) The confidence Interval, and Lj, and Lp (and its upper limit) are correct for normally distributed random errors. Faired T, B comparisons and a moderate number of replicates tend to make these assumptions reasonably good this is an important precaution, given the widely varying blank distributions of such difficult measurements. Perhaps the most important consequence of the paired comparison InjJuced, symmetry, is that the expected value for the null signal [B - B ] will be zero -- ie, unbiased. Systematic error bounds, some deeper implications of paired... 

From the replicate observations made for a certain level combination we can obtain an estimate of experimental error. For example, the yields observed for the two replicates of run no. 1 were 57% and 61%. Since they are genuine replicates and were performed in random order, we can take the variance of this pair of values, which is 8, as an estimate of the typical variance of our experimental procedure. Strictly, it is an estimate relative to the combination of levels from which the two results were obtained - at 40 °C temperature and catalyst A. However, if we assume that the variance is the same for the entire region investigated, we can combine the information from aU of the runs and obtain a variance estimate with more degrees of freedom. In practice, this assumption customarily works quite weU. If any doubt arises, we can always perform an F test to verify its vahdity. 

One of the assumptions of one-way (and other) ANOVA calculations is that the uncontrolled variation is truly random. However, in measurements made over a period of time, variation in an uncontrolled factor such as pressure, temperature, deterioration of apparatus, etc., may produce a trend in the results. As a result the errors due to uncontrolled variation are no longer random since the errors in successive measurements are correlated. This can lead to a systematic error in the results. Fortunately this problem is simply overcome by using the technique of randomization. Suppose we wish to compare the effect of a single factor, the concentration of perchloric acid in aqueous solution, at three different levels or treatments (0.1 M, 0.5 M, and 1.0 M) on the fluorescence intensity of quinine (which is widely used as a primary standard in fluorescence spectrometry). Let us suppose that four replicate intensity measurements are made for each treatment, i.e. in each perchloric acid solution. Instead of making the four measurements in 0.1 M acid, followed by the four in 0.5 M acid, then the four in 1 M acid, we make the 12 measurements in a random order, decided by using a table of random numbers. Each treatment is assigned a number for each replication as follows ... 

In Table XLII are shown regression coefficients for raw vs. cooked carotene contents for four vegetables, spinach, green beans, Fordhook chard, and rhubarb chard. Details of the experiment are contained in the table. Two regression coefficients were calculated for each set of replications. Calculation of the regression for fresh value on cooked is based on the assumption that the cooked values were measured without error, and the regression of cooked values on fresh is based on the assumption that the fresh values are measured without error. One or the other of these assumptions is often convenient to make, even though neither is entirely correct. 

In the statistical treatment of data, it is assumed that the handful of replicate experimental results obtained in the laboratory is a minute fraction of the infinite number of results that could, in principle, be obtained gi en infinite time and an infinite amount of sample. Statisticians call the handful of data a sample and view it as a subset of an infinite population, or universe, of data that exists in principle. The laws of statistics apply strictly to populations only when applying t hese laws to a sample of laboratory data, we must assume that the sample is truly representative of the population. Because there is no assurance that this assumption is valid, statements about random errors are necessarily uncertain and must be couched in terms of probabilities. 

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