Interaction confidence limits

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The model contains eleven parameters (constant term, four linear coefficients, six cross-product coefficients) and the design contains 16 experimental runs. With the assumption that interaction effects involving three or more factors have negligible influence on the yield, the residual sum of squares, RSS = I ef, would then give an estimate of the experimental error variance, s2 = RSS/(16 — 11), with five degrees of freedom, the estimate of s2 obtained in this way was used to compute 95% confidence limits of the estimated parameters. 

This means that three- and four-factor interaction effects were assumed to be small compared to the experimental error. The model contains 11 parameters and the experimental design contains 16 runs. The excluded higher order interaction effects allow an estimate of the residual sum of squares, SSE, which will give an estimate of the experimental error variance, s with 16 - 11 = 5 degrees of freedom. This estimate can then be used to compute confidence limits for the estimated model parameters so that their significance can be evaluated. 

Taking these confidence limits into account, it is seen that the variations in response are largely described by the linear terms and the interaction term for x-. The coefficients of the other terms are not significant. 

In practice, confidence limits for random-effect models will be wider than for fixed-effect models and this is a more realistic representation of the true uncertainties if we are interested in prediction. This may accord with our intuition in the sense that in borderline cases where the result was otherwise significant, a large centre-by-treatment interaction might cause us to doubt that it was genuine. 

This result lends support to our conclusion about the five effects that do not fit the straight line in the normal plot. The values of two others, the 12 and 123 interaction effects, lie practically at the confidence limit. The conclusions do not change much in relation to the analysis of the full design, but it is important to recognize that to obtain an error estimate we are combining variances that differ by up to four orders of magnitude. This is a violation of the normal error hypothesis that is the basis for a... 

In the kinetics shown in Fig. 5.9, the standard error (to 95 % confidence limits) for H2O2, P and R2 was calculated to be 0.15 3.8 x 10 , 0.027 1.4 x 10 and 0.17 3.3 X 10 respectively (using 5 x 10" realisations). Within the limits of the error, all three algorithms predict the correct recombination yield, allowing Slice to be used with reasonable confidence. Although for the recombination kinetics ten slices were found to be sufficient, this may not be sufficient to model the exchange interaction, which is responsible for creating electron polarisation. An analysis of this is now presented in the next section. 

The relevant measurements were performed during in total 6 month distributed over 4 years during which 5.7 1010 muonium atoms were in the interaction region. Out of those, one event fell within a 99% confidence interval of all relevant distributions (Fig. 12). The expected background due to accidental coincidences was 1.7(2) events. Thus an upper limit on the conversion probability of... 

Method (a), the use of the position of the coulombic attraction theory minimum with the Od = 0 value for g, leads to the same mathematical formula for s as that expressing the Donnan equilibrium. However, we cannot say that this constitutes a derivation of the Donnan equilibrium from the coulombic attraction theory because it does not correspond to a physical limit. If Od = 0 really were the case, there would be no reason for the macroions to remain at the minimum position of the interaction potential. Nevertheless, the identity of the two expressions is an interesting result. Because Equation 4.20 is derived in the case in which there is no double layer overlap and Equation 4.1 (the Donnan equilibrium) is likewise derived without reference to the overlap of the double layers, it is precisely in this limit that the calculation should reproduce the Donnan equilibrium. The fact that it does gives us some confidence that our approximations are not too drastic and should lead to physically significant results when applied to overlapping double layers. 

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