Experimental variance

Example 12 Suppose Analyst A made five observations and obtained a standard deviation of 0.06, where Analyst B with six observations obtained 5-3 = 0.03. The experimental variance ratio is ... 

The experimental sample on which the frequency histogram is based has an experimental mean m and an experimental variance s, which are... 

Usually there is no opportunity to repeat the measurements to determine the experimental variance or standard deviation. This is the most common situation encountered in field measurements. Each measurement is carried out only once due to restricted resources, and because field-measured quantities are often unstable, repetition to determine the spread is not justified. In such cases prior knowledge gained in a laboratory with the same or a similar meter and measurement approach could be used. The second alternative is to rely on the specifications given by the instrument manufacturer, although instrumenr manufacturers do not normally specify the risk level related to the confidence limits they are giving. 

Experimental variance Permission granted by OSHA for the use of an alternative method of worker protection during an approved experiment to demonstrate or validate new safety and health techniques. The variance terminates upon study com pletion unless another type of variance is issued by OSHA. 

Uncertainties relating to the determination of accurate quantitative results are not relevant in these experiments. The observed experimental variance of the INAA results is a summation of the variances of homogeneity and the relevant analytical components as shown in Equation (4.8) ... 

A randomized crossover design has theoretical appeal because it eliminates the largest source of experimental variance interindividual variability. This could significantly enhance statistical power and permit much smaller samples sizes to detect a treatment effect. Unfortunately, a crossover design is appropriate only in rare cases in psychopharmacology, namely in studies ... 

Statistical tests on the significance of the above coefficients are possible if we have estimates of the experimental variance from the past or if we can repeat some experiments or even the total design. Alternatively, variance can be computed if the so-called central point of a design is (sampled and) measured repeatedly. This is a point between all factor levels. In Fig. 3-1 this is the point between location 1 and location 2 and between depth 1 and depth 2. The meaning of another term which is used, zero level, will be clear after we have learned how to construct general designs. 

There is only one way to circumvent additional experiments, namely if the experimental variance is known a priori as a2 and the number of experiments k exceeds the number of estimated coefficients na in the model. 

The application of the F test with = 10-1 = 9 degrees of freedom and q, = 10(5-1) = 40 degrees of freedom, indicated the existence of an apparent between-instrument effect at higher turbidity values of 50 FNU and 200 FNU. Under these circumstances the best estimate of the uncertainty of the turbidity measurement was considered to be the pooled estimate of variance obtained from ten individual values of experimental variance of the observations made on each instrument. Consequently, relative uncertainties of measurement as high as 0.017 FNU at a nominal value of 0.5 FNU, and 1.87 FNU at a nominal value of 100 FNU were esti-... 

AT, maximum lipase activity T, temperature M, moisture OO, % olive oil a0, alr a2, a3, fl4, and a5, model parameters S2experim, experimental variance. 

After the parameter estimation, the discarding of the mixing rules begins using the F-Test, that points out with 99% confidence whether or not the tested model is equivalent to the experimental variance. If there is an equivalence the discrimination goes to the next step on the other hand the model is discarded and the discrimination procedure continues for the remaining models. If no models is discarded three consecutive times the confidence is decreased down to 95% until one or more models are eliminated. 

When the experimental variance, o-fxp( i), is larger than the variance in the property, green body. For this case, only an effective homogeneity can be determined. The experimental variance is established by repeated measurements of the property for a single sample. 

To estimate the experimental variance, a 30-parameter polynomial in the four independent variables was constructed and fitted to the data in the same manner. Reduced versions of this polynomial were then tested until a minimum value of the residual mean square, S/ n — pj), was found... 

Normalization is a very important step, as it aims to reduce experimental variance. Normalization is most often performed by dividing each spectrum by a normalization factor (Figure 2G). The most popular normalization factor is calculated as the total ion count (TIC), which is the sum of all ion intensities in a spectrum. Several studies discovered that in MSI the assumptions for TIC applicability hold true only for very homogeneous tissues. In heterogeneous samples, more robust normalization factors based on the median or the TIC with exclusion of very localized mass signals have been proposed (35-37). 

The standard deviation of log is 0.171, which is quite high. For example, for 1-pentanol in SDS the predicted is 603 250. Nevertheless, it is a good fit taking into account the number of solubilizates covered by a single equation and the fact that the differ considerably for the various investigations. If, for example, we compare this result with the experimental results given in Table 6.1, it is well within the experimental variance. 

Low Avoidance (RLA). The fifth strain was an unselected strain of the original Roman stock, the Roman Control Avoidance (RCA) strain. As can be seen from Fig. 2, by making use of discrete behavioural strains and thus increasing the experimental variance while holding the error variance constant, a clear correlation is apparent between the two characters. This study has since been replicated on 8 strains in which the correlation between GABA production and ChE activity in the sensori-motor cortex was +0.88 (P < 0.001). 

Having carried out as many experiments as there are coefficients in the model equation and not having any independent measurements or estimation of the experimental variance we cannot do any of the standard statistical tests for testing the significance of the coefficients. However all factorial matrices, complete or fractional, have the fundamental property that all coefficients are estimated with equal precision and, like the screening matrices, all the coefficients have the same unit, that of the response variable. This is because they are calculated as contrasts of the experimental response data, and they are coefficients of the dimensionless coded variables X. They can therefore be compared directly with one another. 

When the number of experiments exceeds the number of effects calculated, or when the experimental variance can be estimated independently (for example by repeating certain experiments) the significance of the coefficients may be estimated and the analysis of variance of the regression may be carried out. The methods that are shown below are for saturated or near saturated designs, where the experiments are too costly or too long to carry out and the main criterion in choosing a design is its R-efficiency. 

The residual mean square MS sm can be considered as an estimate of the variance and its square root as an estimate of the standard deviation of the experimental technique. (A certain degree of caution is in order here - the residuals will, as we have seen, include random variation, but may also be partly a result of shortcomings in the model. For the moment we have no way of telling. When some of the experiments are replicated the dispersion of the results about the mean value allows another estimation of the experimental variance to be obtained, this time without any bias.)... 

This is an unbiased estimate of the experimental variance cP. As the number of degrees of freedom increases this becomes a more reliable estimate of the "true" standard deviation o. 

The two estimations, s and s, may be considered as significantly different. The mathematical model is thus rejected and it is therefore s, derived from the error sum of squares which is retained as an estimate of a. Fisher s test may be carried out a second time to compare s with our estimate of the experimental variance s- F = 161.8. With... 

As we continue with the example we will use the following values for the estimated experimental variance and standard deviation ... 

When we have an estimate of the experimental variance (either from repeating experiments, as here, or by having carried out more distinct experiments than there are coefficients in the model, or from having done some preliminary experiments), the statistical significance of the individual coefficients may be calculated. 

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