Serial correlation coefficient

With this transformation, the linear regression model, using the ordinary least-squares method of determination, is valid. However, to employ it, we need to know the population serial correlation coefficient, P. We estimate it by r. The population Equation 3.9 through Equation 3.11 will be changed to population estimates ... 

Step 1 Estimate the population serial correlation coefficient, P, with the sample correlation coefficient, r. It requires a regression through the origin or the (0, 0) points, using the residuals, instead of y and x, to find the slope. The equation has the form... 

The analysis of residuals (y — y ), in the form of the serial correlation coefficient (SCC), provides a useful measure of how much the model deviates from the experimental data. Serial correlation is an indication of whether residuals tend to mn in groups of positive or negative values or tend to be scattered randomly about zero. A large positive value of the SCC is indicative of a systematic deviation of the model from the data. 

The limit of detection, limit of quantitation, and linear dynamic range are to be determined by serial dilution of a sample. Three replicate measurements at each level are recommended, and the acceptance criterion for calibration linearity should be a prespecilied correlation coefficient (say, an r2 value of 0.995 or greater). 

Petersson (1988b) developed an efficient urea analyzer for undiluted blood samples by using a urease-covered ammonium ion selective electrode in an FIA system. Forty samples per hour could be determined with a useful measuring range up to 40 mmol/l and a serial CV of 1%. The sensor was stable for 25 days. The correlation coefficient with a routinely used method was 0.99. Variations in the hematocrit level had only a small effect on the measurement. 

The standards and samples (n = 6) were analysed by using a Perkin Elmer 100 FAAS system with an air-acetylene flame. The metals were determined at the following wavelengths Cu, 324.8 nm Fe, 248.3 nm Pb, 217.0 nm Mn, 279.5 nm Ni, 232.0 nm Zn, 213.9 nm. Calibrations were produced by serial dilution of 1000 igml-1 stock solutions in the range 0-10 xgml-1. All calibration graphs exhibited linear relationships for each metal, except zinc, i.e. y = mx + c. For zinc, a curved relationship was obtained, i.e. y = ax2 +bx + c. All correlation coefficients were >0.96. 

Cognizance of experimental error is critical for understanding the meaning of the coefficients and conducting the experiments themselves. As a first principle, factorial experimental designs should be run in fully randomized order. This is necessary in order to break any serial correlations that may exist. For example, suppose the design of Table 3.1 were run in the order presented. Further suppose that ambient temperature is an important but unrealized factor affecting the responses. 

Some statisticians prefer an easier method than the Cochrane-Orcutt procedure for removing serial correlation— the first difference procedure. As previously discussed, when serial correlation is present, P, the population correlation coefficient, tends to be large P > 0), so a number of statisticians recommend just setting P= and applying the transforming Equation 3.25 (Kutner et al., 2005)... 

A set of static friction data which contain a trend are shown in Figure 4. These data are said to be non-stationary. The existence of these trends may be demonstrated using either serial correlation analysis or a periodic data averaging technique known as the run-test. Serial correlation involves calculation of the autocorrelation coefficient r at a lag k, which is a measure of the correlation between events at a distance k apart, and is given by ... 

It should be noted, however, that the use of a serial dilutions approach will depend on the range that has been established for the method during the validation phase. This is because serial dilutions often result in very broad ranges of concentration. The disadvantage of this is that the correlation coefficient (R ) can remain relatively unaltered, with quite substantial errors in measurement—particularly at the low concentration end of the plot. In Figure 3.11, we can see that a 40% error in the measurement at 7.5 pg/mL will still give an acceptable (correlation coefficient B) when the limit has been set at R > 0.999. 

Several commercially available mixes of pesticide compounds were used for calibration of the system by serial dilution and external standard techniques. A least squares regression was applied to curve fit the calibration data. Calibration curves for organochlorine pesticides (OCPs) were linear ranging from below 0.01 to lOng/pL with correlation coefficients ranging from 0.994 to 0.999 (Tables 4.12 and 4.13). 

Because of the inductive effect of the second nitrogen atom, neither the high stability constant nor the high antibacterial action of oxine is quite reached (the latter falls short by only one twofold serial dilution). However, the last seven substances in Table 11.9 all have stability constants of the same order. For these seven substances, the antibacterial action runs parallel to the partition coefficient, which establishes the correlation. 

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