Serial correlation models

Draper and Smith [1] discuss the application of DW to the analysis of residuals from a calibration their discussion is based on the fundamental work of Durbin, et al in the references listed at the beginning of this chapter. While we cannot reproduce their entire discussion here, at the heart of it is the fact that there are many kinds of serial correlation, including linear, quadratic and higher order. As Draper and Smith show (on p. 64), the linear correlation between the residuals from the calibration data and the predicted values from that calibration model is zero. Therefore if the sample data is ordered according to the analyte values predicted from the calibration model, a statistically significant value of the Durbin-Watson statistic for the residuals in indicative of high-order serial correlation, that is nonlinearity. 

Stage 1. The MeOH/H20/NaCl data are subjected to the correlation procedure described previously which gives values of the Wilson energy constants (Zi and Z2) and a new set of data for temperature and vapor composition that are internally consistent (see Table I). The small values of the standard deviation and the bias indicate good quality data in the salt effect field. For the analysis of serial correlation among the residuals we use the Durbin-Watson test (9). A run of positive or negative signs in the series of residuals is some indication that the model... 

At this stage, time-varying random effects should be examined by plotting the residuals obtained using a simple OLS estimate of (3, ignoring any serial correlation or random effects in the model, against time. If this plot shows no trends and is of constant variance across time, no other random effects need to be included in the model, save perhaps a random intercept term. If, however, a systematic trend still exists in the plot then further random effects need to be included in the model to account for the trend. 

Any decent model used to analyse a series of n-of-1 trials will allow for at least three sources of variation pure between-patient variability, within-patient variability and a random effect for patient-by-treatment interaction. Since a series of measurements are being obtained, however, it may be inappropriate to assume that within-patient errors are independent (or more formally that the correlations between measures are equal). If patients are subject to spells of illness for example, then two measurements taken during two administrations of the same drug are more likely to be similar if the administrations are close together rather than far apart. This phenomenon can be referred to as serial correlation and is potentially a problem for the analysis of n-of-1 trials. (It would also be a problem for multiperiod cross-overs, the standard analysis of which, however, ignores the even more serious problem of patient-by-treatment interaction.)... 

Whenever there is a time element in the regression analysis, there is a real danger of the dependent variable correlating with itself. In the literature of statistics, this phenomenon is termed autocorrelation or serial correlation in this text, we use the latter as descriptive of a situation in which the value, y is dependent on y, i, which, in turn, is dependent on y, 2. From a statistical perspective, this is problematic because the error term, e,-, is not independent—a requirement of the linear regression model. This interferes with least-squares calculation. 

Let us do an actual problem, Example 3.1, the data for which are from an actual D value computation for steam sterilization. Biological indicators (strips of paper containing approximately 1 x 10 bacterial spores per strip) were affixed to stainless steel hip joints. In order to calculate a D value, or the time required to reduce the initial population by 1 logio, the adequacy of the regression model must be evaluated. Because bo and bi are unbiased estimators, even when serial correlation is present, the model y = bo + biX + et may still be useful. However, recall that e, is now composed of e,- i + di, where the 4s are N 0, 1). 

Note that the residuals plotted over the time exposures do not appear to be randomly centered around 0, which suggests that the linear model may be inadequate and that positive serial correlation may be present. Table 3.4 provides the actual x, y, data values, the predicted values y, and the residuals e,. 

The easiest way to correct this is to remove the data where x = 0 and x = 5. We kept them in this model to evaluate serial correlation because if we... 

As previously stated, most serial correlatirai problems point to the need for another, or several values for the x, variable. For instance, in Example 3.1, the sterilization hip joint study, looking at the data, the researcher noted that the temperature fluctuation in the steam vessel was 2.0°C. In this type of study, a range of 4.0°C can be very influential. For example, as x,, x,y and x,j are measured, the bier vessel cycles throughout the +2.0°C range. The serial correlation would tend to appear positive due to the very closely related temperature fluctuations. A way to correct this situation partially would be to add another regression variable, X2, representing temperature. The model would then be... 

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 ... 

Given that the serial correlation is eliminated, the model can be retransformed to the original scale ... 

Note that we have not tested the model for fit at this time (linearity of model, serial correlation, etc.), as we combine everything in the modelbuilding chapter. 

In Chapter 3, it was noted that residual analysis is very useful for exploring the effects of outliers and nonnormal distributions of data, for how these relate to adequacy of the regression model, and for identifying and correcting for serially correlated data. At the end of the chapter, formulas for the process of... 

However, before selection procedures can be used effectively, the researcher should assure that the errors are normally distributed, and the model has no significant outhers, multicoUinearity, or serial correlation. If any of these are problems, they must be addressed first. 

All the tests and conditions we have discussed earlier should also be used, such as multicollinearity testing, serial correlation, and so on. The final model selected should be in terms of application to the population, not just one sample. This caution, all too often, goes unheeded, so a new model must be developed for each new set of data. Therefore, when a final model is selected, it should be tested for its robustness. 

The book presents a well-defined procedure for adding or subtracting independent variables to the model variable and covers how to apply statistical forecasting methods to the serially correlated data characteristically found in clinical and pharmaceutical settings. The standalone chapters allow you to pick and choose which chapter to read first and hone in on the information that fits your immediate needs. Each example is presented in computer software format. The author uses MiniTab in the book but supplies instructions that are easily adapted for SAS and SPSSX, making the book applicable to individual situations. 

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. 

Before any reliability analysis is started, tests for trends and serial correlations must be performed to evaluate the type of the available TBF data and to select the best analysis method. If there is a trend in the dataset, this means that the data are not identically distributed. The presence of serial correlation shows that the data are not independent. In the case of no trend and no serial correlation, data are called iid (independent and identically distributed) and the RP method is the best method for reliability analysis, while in the case of having trend and serial correlation, the NHPP method is used for modelling. 

The advantage of the mixing tank model approach is its relative simplicity, intuitive accessibility, and easy correlation with pharmacokinetic models. However, the physical basis for considering a segment of the small intestine as one or more serial mixing tanks is limited, although such an assumption has been commonly and successfully utilized in the physical and biological sciences. 

Ensures diat die model for the population response is correctly specified—reasonable for population pharmacokinetics Serial concentrations measured from an individual are likely to be correlated Constant intraindividual variance is frequently violated and typically accounted for widi error models that specify the G vs. concentration relationship die distribution of G over (time) is defined by die underlying structural model Historical requirement for inference unrealistic for nonlinear models particularly with biologic data... 

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