Unweighted least-squares analysis

Show that if the relative error ajk is constant, then an unweighted linear Arrhenius least-squares analysis is correct. 

The unweighted least squares analysis is based on the assumption that the best value of the rate constant k is the one t,hat minimizes the sum of the squares of the residuals. In the general case one should regard the zero time point as an adjustable constant in order to avoid undue weighting of the initial point. An analysis of this type gives the following expressions for first-and second-order rate constants... 

These plots can also provide information about the assumption of constant error variance (Section III) made in the unweighted linear or nonlinear least-squares analyses. If the residuals continually increase or continually decrease in such plots, a nonconstant error variance would be evident. Here, either a weighted least-squares analysis should be conducted (Section III,A,2) or a transformation should be found to stabilize the error variance (Section VI). 

In the above analysis, y was considered to be a reaction rate. Clearly, any dependent variable can be used. Note, however, that if the dependent variable, y, is distributed with constant error variance, then the function z will also have constant error variance and the unweighted linear least-squares analysis is rigorous. If, in addition, y has error that is normal and independent, the least-squares analysis would provide a maximum likelihood estimate of A. On the other hand, if any transformation of the reaction rate is felt to fulfill more nearly these characteristics, the transformation may be made on y, ru r2 and the same analysis may be applied. One common transformation will be logarithmic. 

If this procedure is followed, then a reaction order will be obtained which is not masked by the effects of the error distribution of the dependent variables If the transformation achieves the four qualities (a-d) listed at the first of this section, an unweighted linear least-squares analysis may be used rigorously. The reaction order, a = X + 1, and the transformed forward rate constant, B, possess all of the desirable properties of maximum likelihood estimates. Finally, the equivalent of the likelihood function can be represented b the plot of the transformed sum of squares versus the reaction order. This provides not only a reliable confidence interval on the reaction order, but also the entire sum-of-squares curve as a function of the reaction order. Then, for example, one could readily determine whether any previously postulated reaction order can be reconciled with the available data. 

Lagtime, 75 Laplace transform, 82 Larmor precessional frequency, 155, 165 Laser pulse absorption, 144 Lattice energy, 403 Law of mass action, 60, 125 Least-squares analysis linear, 41 nonlinear, 49 univariate, 44 unweighted, 44, 51 weighted, 46, 51, 247 Leaving group, 9, 340, 349, 357 Lennard-Jones potential, 393 Lewis acid-base adduct, 425 Lewis acid catalysis, 265 Lewis acidity, 426... 

The average component numbers from these unweighted fits are 1n close agreement with the results from the weighted least squares analysis as expected. The standard deviation, and hence the unreal lability of the results from a single series, 1s generally higher In the estimation of component number from the slope than from the Intercept. This observation Is attributed to the variances In slope and Intercept which. In turn, are functions of the peak capacities and the number of counted peaks (6). The variance In the estimation from the Intercept Increases with the value m, and a reversal of the trend Is observed In Set E. 

Table II contains the optimal empirical resolutions calculated from the weighted least squares analysis of counted peak maxima. The empirical r from each set was used to calculate, from an unweighted east squares fit, a component number from each series in that set. The results for each set were averaged, and the mean and standard deviation for each set are reported In Table II.

So here we are with an unweighted non-linear least-squares program we encounter precisely the same problems as with unweighted linear least squares. The problem lies in defining which experimental parameter carries the dominant uncertainty, and selecting that parameter as the dependent one. In the unweighted least-squares analysis we used SI v in the... 

An unweighted least-squares analysis such as illustrated here usually suffices for chemical analysis, but for the most precise values one might want to use weighted least squares as developed by G. Nowogrocki et al., Anal. Chim. Acta 122 (1979) 185, G. Kateman et al., Anal. Chim. Acta 152 (1983) 61, and H. C.Smit etal., Anal. Chim. Acta 153 (1983) 121. 

Inhibition Constants In inhibition experiments, dataware collected for 4 to 5 different substrate concentrations at each of 4 to 6 different inhibitor concentrations (ranging from zero added inhibitor, to sufficient inhibitor to reduce the observed activity by approximately 80%). Lineweaver-Burk plots of the resulting data were analyzed by unweighted linear least squares analysis, yielding a slope and ordinate intercept at each inhibitor concentration. (Due to its prevalence in the literature, Lineweaver-Burk analysis was the analytical method of choice. Because of the magnitude of error in our data was independent of velocity, and... 

Replots of slopes and/or ordinate intercepts versus inhibitor concentration were analyzed by unweighted linear least squares analysis. All replots were found to be linear, with correlation coefficients greater than 0.99+. The inhibitor constants derived from slope replots are designated k, and those from ordinate intercept replots are designated Kint, The values so obtained were inserted into the velocity equation describing the observed mechanism of inhibition and were used to generate the lines shown in the final figures. 

Obtained from a unweighted least-squares analysis (Atkinson et al., 2006). 

While the unweighted least squares method of data analysis is commonly used for the determination of reaction rate constants, it does not yield the best possible value for k. There are two principal reasons for this failure. 

Structural Solution and Refin ent. The structure was solved by analysis of similar structuies and expanded using Fourier maps. All atoms were refined anisotropically. The final cycle of full-matrix least-squares refinement was based on 555 observed reflections (7 > 3.00o(7)) and 70 variable parameters and converged (largest parameter shift was 0.00 times its esd) with unweighted and weighted agreement foctors of... 

In the context of the analysis of enzyme kinetics it is sometimes stated that one should always use a non-linear least-squares method for such data, because the usual, unweighted least-squares fits depend on the particular analysis method (Lineweaver-Burk, Hanes, etc.) used. We have seen in section 3.5 that the latter part of this statement is correct. But how about the former ... 

The weighted least-squares routine shown below provides the adjustable parameters and their standard deviations. If that is all you need, you may want to use it also as your general least-squares routine, especially after you have incorporated it in a menu or given it a toolbar icon (in which case it is easier to use and more readily accessible than the Regression routine in the Analysis ToolPak). When using it for unweighted least squares, merely leave the second column empty. Alternatively, if you desire the routine to provide more statistical information, you can modify it to do so. Remember, you are at the controls here. 

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