Response-error relationship

A commonly accepted fitting approach is the four-parameter logistic equation (4PL) used to fit the mean concentration-response relationship, and the power-of-the-mean equation to fit the response-error relationship [15,18]. In some applications a fifth parameter describing the asymmetry of the curve might be incorporated, ensuring a higher versatility of the model [18]. 

Y is the expected response and X is the corresponding concentration. A, B,C and D are the four parameters of the equation, where A gives an estimate of expected response at zero dose, B is the slope (e.g., response/concentration) in the middle of the calibration curve, C is the IC50 and D is the expected response at infinite dose. This curve satisfies all the conditions specified for the response-concentration relationship and also closely approximates the mass-action equations [20]. Weighting of the results is recommended for fitting dose-response data from immunoassay, in order to compensate the heterogeneity of response variances in the response-error relationship [17,18]. 

In the method of Ekins, a response-error relationship (RER) is established by plotting the error of the response variable against the response (e.g., 10 replicates for each dose plotting the SD against the response Fig. 15.13). 

Fig. 15.13. Response error relationship (A) and precision profiles (B) to optimize (particularly AM-)EIA according to Ekins (1979). The immunoreactant concentrations which will give minimum variance, will give also minimum variance when small amounts of sample are added and allows the highest detectability. First, the R is measured at different responses (reliability is not high with a few replicates. Section 15.1). The error in dose (AC) or the relative error (AC/C coefficient of variation) can then be determined by the AR (corresponding to the R of that dose) and the sensitivity of the dose-response curve as shown in Fig. 15.1. A change in, e.g., the antibody concentration may change the precision profile from (a) to (b), which has a greater detectability, but less precision at high antigen concentrations.

The topic of nonlinear calibration for LBAs, such as immunoassays, has been reviewed in detail in a number of publications [4,8,9,15 17]. Typically, immunoassay calibration curves are inherently nonlinear [9]. Because the response error relationship is a nonconstant function of the mean response, weighting is needed to account for the heterogeneity in response variances. The four- or five-parameter logistic models are accepted widely as the standard models for fitting nonlinear sigmoidal calibration data [3 5,8,9,16,17], This model can be described... 

Of all the requirements that have to be fulfilled by a manufacturer, starting with responsibilities and reporting relationships, warehousing practices, service contract policies, airhandUng equipment, etc., only a few of those will be touched upon here that directly relate to the analytical laboratory. Key phrases are underlined or are in italics Acceptance Criteria, Accuracy, Baseline, Calibration, Concentration range. Control samples. Data Clean-Up, Deviation, Error propagation. Error recovery. Interference, Linearity, Noise, Numerical artifact. Precision, Recovery, Reliability, Repeatability, Reproducibility, Ruggedness, Selectivity, Specifications, System Suitability, Validation. 

CPU time. In response to these slow and rigorous calculations, many fast heuristic approaches have been developed that are based on intuitive concepts such as docking [10], matching pharmacophores [19], or linear free energy relationships [20], A disadvantage of many simple heuristic approaches is their susceptibility to generalization error [17], where accuracy of the predictions is limited to the training data. 

Figure 3 shows the PCA score plot of the same data of figure 2 after the application of equation 4. The application of linear normalization to an array of linear sensors should produce, on the PCA score plot, one point for each compound, independent of its concentration, and achieve the highest possible recognition. Deviations from ideal behaviour, as shown in figure 3, are due to the presence of measurement errors, and to the non-linear relationship between sensor response and concentration. 

Biochemical oxygen demand (BOD) is one of the most widely determined parameters in managing organic pollution. The conventional BOD test includes a 5-day incubation period, so a more expeditious and reproducible method for assessment of this parameter is required. Trichosporon cutaneum, a microorganism formerly used in waste water treatment, has also been employed to construct a BOD biosensor. The dynamic system where the sensor was implemented consisted of a 0.1 M phosphate buffer at pH 7 saturated with dissolved oxygen which was transferred to a flow-cell at a rate of 1 mL/min. When the current reached a steady-state value, a sample was injected into the flow-cell at 0.2 mL/min. The steady-state current was found to be dependent on the BOD of the sample solution. After the sample was flushed from the flow-cell, the current of the microbial sensor gradually returned to its initial level. The response time of microbial sensors depends on the nature of the sample solution concerned. A linear relationship was foimd between the current difference (i.e. that between the initial and final steady-state currents) and the 5-day BOD assay of the standard solution up to 60 mg/L. The minimum measurable BOD was 3 mg/L. The current was reproducible within 6% of the relative error when a BOD of 40 mg/L was used over 10 experiments [128]. 

Regression a statistical method for investigating the relationship between a dependent variable or response, and one or more independent or predicated variables. If a response y can be expressed as a systematic function of x plus an error term ( ) ... 

The steady-state current depended on the concentration of ammonia. A linear relationship was observed between the current decrease and the ammonia concentration below 42 mg l-1 (current decrease 4.7 iA). The minimum concentration for the determination of ammonia was 0.1 mg 1" (signal to noise, 20 reproducibility, 5 %). The current decrease was reproducible within 4 % of relative error when a sample solution containing 21 mg 1 of ammonium hydroxide was employed. The standard deviation was 0.7 mg 1 in 20 exper-ments. The response time of the sensor was within 4 min. 

---