Non-constant variance

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

Mitchell, and Hills ( ) They use weighted least squares to resolve the non-constant variance of the response signal for different concentrations, whereas we transform the response to achieve constant variance. 

The solution to the problem of non-constant variance (or heteroscedasticity) rests in several suggestions. The simplest is to limit the range of the graph ( 1 ). The range, however, would be so small that it would be ineffective to use it practically. 

Another solution to the problem of non-constant variance is to transform the response data. A common way of transforming data has been by taking the logarithms of both the response and amount variables ( 8-10 ). However, for all the data we looked at, the log transformation has been too strong. See Tables I and V. 

However, with improper transformation the calculation of confidence bands and amount interval estimates is erroneous because of the non-constant variance ... 

When both X and Y are measured with error, this is called error-in-variables (EIV) regression, which will be dealt with in a later section. When x is not fixed, but random and X and Y have a joint random distribution, this is referred to as conditional regression, and will also be dealt with later in the chapter. When the residual s have non-constant variance, this is referred to as hetero-... 

III. TREATMENT OF DATA WITH NON-CONSTANT VARIANCE A. Weighted Multilinear Regression... 

Other possibilities are described in the literature, which deal with cases where the variance depends on the response, or where the errors are correlated, or where the errors depend on the order in which the experiments are carried out. For the most part these are outside the scope of this book. However the use of transformations of response values to correct for a non-constant variance, as well as non-normality, is widespread, ever since it was proposed by Box and Cox (2), and this is the subject of the following two sections. 

Jacquez, J. A. Mather, F. J. and Crawford, C. R. in "Linear Regression with Non-Constant, Unknown Error Variances Biometrics 1968, 2 , 607. 

Transformation Power of Selected Data Sets. Hartley statistic values are shown in Tables I-III for fenvalerate, chlordecone, and chlorothalonil. In each case a power transformation was found of sufficient size at a 93% probability which satisfied the H criterion. For fenvalerate the power of 0.15 was satisfactory for constant variance. For chlordecone the whole range of powers from 0.30 to 0.10 satisfied the critical H value (listed in order of increasing transformation power). Despite apparent non-constancy of data for chlorothalonil shown in Table III, the critical H was satisfied for the range in power transformation from 0.23 to 0.10. 

To model the relationship between PLA and PLR, we used each of these in ordinary least squares (OLS) multiple regression to explore the relationship between the dependent variables Mean PLR or Mean PLA and the independent variables (Berry and Feldman, 1985).OLS regression was used because data satisfied OLS assumptions for the model as the best linear unbiased estimator (BLUE). Distribution of errors (residuals) is normal, they are uncorrelated with each other, and homoscedastic (constant variance among residuals), with the mean of 0. We also analyzed predicted values plotted against residuals, as they are a better indicator of non-normality in aggregated data, and found them also to be homoscedastic and independent of one other. 

A quarter of a century ago Behnken [224] as well as Tidwell and Mortimer [225] pointed out that the linearization transforms the error structure in the observed copolymer composition with the result that such errors after transformation have no longer zero mean and constant variances. It means that such transformed variables do not meet the requirements for the least-squares procedure. The only statistically accurate means of estimation of the reactivity ratios from the experimental data is based on the non-linear least-squares procedure. An effective computing program for this purpose has been published by Tidwell and Mortimer (TM) [225]. Their method is considered to be such a modification of the curve-fitting procedure where the sum of the squares of the difference between the observed and computed polymer compositions is minimized. 

For most immunoassays, the mean response is a non-linear function of the anal3de concentration (the dose-response curve) and the variance of replicate measurements is a non-constant function of the mean response, as seen in Fig. 9.3. 

Case 2. In this case, scaling may be used to account for non-constant noise level (error variance), and the applied weight corresponds to the inverse of residual variance of each variable (it seems similar to unit variance scaling but instead of the standard deviation of the variable, the residuals standard deviation, as assessed by replicates, is used), down-weighting variables whose uncertainty is higher. 

Where f(x) is tlie probability of x successes in n performances. One can show that the expected value of the random variable X is np and its variance is npq. As a simple example of tlie binomial distribution, consider tlie probability distribution of tlie number of defectives in a sample of 5 items drawn with replacement from a lot of 1000 items, 50 of which are defective. Associate success with drawing a defective item from tlie lot. Tlien the result of each drawing can be classified success (defective item) or failure (non-defective item). The sample of items is drawn witli replacement (i.e., each item in tlie sample is relumed before tlie next is drawn from tlie lot tlierefore the probability of success remains constant at 0.05. Substituting in Eq. (20.5.2) tlie values n = 5, p = 0.05, and q = 0.95 yields... 

We note that for the harmonic Hamiltonian in (5.25) the variance of the work approaches zero in the limit of an infinitely slow transformation, v — 0, r — oo, vt = const. However, as shown by Oberhofer et al. [13], this is not the case in general. As a consequence of adiabatic invariants of Hamiltonian dynamics, even infinitely slow transformations can result in a non-delta-like distribution of the work. Analytically solvable examples for that unexpected behavior are, for instance, harmonic Hamiltonians with time-dependent spring constants k = k t). 

Note, however, that in the absence of micromixing, )n is constant so that this term will be null. Nevertheless, when micromixing is present, the spurious scalar dissipation term will be non-zero, and thus decrease the scalar variance for inhomogeneous flows. 

Note that the effect of the spurious dissipation term can be non-trivial. For example, consider the case where at t — 0 the system is initialized with p = 1, but with ( )i varying as a function of x. The micromixing term in (5.387) will initially be null, but ys will be non-zero. Thus, p2 will be formed in order to generate the correct distribution for the mixture-fraction variance. By construction, the composition vector in environment 2 will be constant and equal to )2. 

A simpler desaiption of the system is desired and it can be obtained using equilibrium constants that describe aU eqnilibria in the systan obviously only a thermodynamic equilibrium constant must be made use of since, as discussed above, at variance with stoichiometric constants, they account for all the molecular level interactions in a very simple way, and notably, their chromatographic and non-chromatographic estimates can be compared to validate the retention mechanism they describe. 

The value of of the BaTiO, ceramics is lower than that reported for BaTiOj single crystal [9] (along [100] =4 000). This may be due to the structural and compositional variances. Meanwhile, the size of crystalline particle may affect the dielectric constant , that is, when the particle size is lower than certain value, the constant will decrease with the decrease of the size. In addition to those mentioned above, porosity and the existence of low dielectric constant affect the non-ferroelectric layers at the metal-ferroelectric interface and the grain boundaries. 

The resistivity measurements along the transverse c direction are also interesting. Actually one can infer that is directly related to the physics of the a — b planes and therefore could probe whether transport proceeds via collective modes or independent quasi-particles. Jacobsen et al. [80], were the first to report a non-monotonic temperature dependence of p,. in (TMTSF)2PFj passing through a well characterized maximum at r , = 80 K (under ambient pressure) at variance with the T-dependence in the a direction [80]. A recent pressure study of this effect has shown that T evolves under pressure and reaches about 300 K at 10 kbar [81]. The constant volume data for p,(7) in the metallic regime below... 

We consider one of these designs, 2 full-factorial, fractional-factorial or Plackett-Burman design, of N experiments. The postulated model of p coefficients consists of a constant term, main effects and possibly first-order interactions. All diagonal coefficients (coefficients of variance) in the dispersion matrix are equal to l/N and all non-diagonal terms are zero. 

---