Outlying Response Variable Observations

Sometimes, a set of normal-looking Xi values may be associated with an extreme y, value. The residual value, e,- = y, — y, often is useful for evaluating y, values. It is important with multiple linear regressirai to know where the influential values of the regression model are. Generally, they are at the extreme ends, but not always. 

We have discussed residual e, analysis in other chapters, so we will not spend a lot of time revisiting this. Two forms of residual analyses are particularly valuable for use in multiple regression semi-Studentized and Studentized residuals. A semi-Studentized residual, ej, is the ith residual value divided by the square root of the mean square error. 

The hat matrix can be of use in another aspect of residual analysis, the Studentized residual. Recall that H = X(X X) X is the hat matrix. Y = HY is the predicted vector, Y, the product of the x H matrix times the Y value vector. e = l- H)Y, the error of the residual vector can be determined by subtracting the n x H matrix from ann x n identity matrix, I, and multiplying that result by the Y vector. 

We are interested in the Studentized residual, which is the ratio of c, to S(e,), where 

Large Studentized residual values are suspect, and they follow the Sm-dent s t distribution with n k — I degrees of freedom. Extreme residuals are directly related to the y, values, in that e, = y, — y,. 

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In bioanalysis, extracted samples are usually stored in either autosampler vials or wells in a plate (such as 96-well plate) sealed with pierceable caps or covers. During injection, the autosampler needle has to pierce the caps or covers to load samples. The debris may completely or partially block the autosampler needle, which would result in no sample or variably low sample volumes injected. Accordingly, no or randomly low IS responses are observed. As most autosamplers have a built-in needle flushing mechanism, the debris in the needle might be flushed out later partially or completely. Therefore, the injected volume can be back to normal at a later time without an operator s intervention. Apparently, when a needle will be blocked and when the blocked needle will be cleared by flushing, as well as how it will be blocked (completely or partially) are difficult to predict. Hence, there would be no clear pattern for this type of IS variations. However, the affected injections normally have lowered IS responses (Fig. 9). Despite lowered IS responses, the accuracy of quantitation can usually be maintained except for situations where no or very low amount of samples are injected, resulting in responses outside the limit of linear range or unacceptable S/N. 

The criterion employed for a positive response in this assay is a reproducible statistically significant increase in mutation frequency (weighted mean for duplicate treated cultures) over the concurrent vehicle control value (weighted mean for four independent control cultures). Ideally, the response should show evidence of a dose-response relationship. When a small isolated significant increase in mutation frequency is observed in only one of the two duplicate experiments, then a third test should be carried out. If the third test shows no significant effects, the initial increase is likely to be a chance result. In cases where an apparent treatment-related increase is thought to be a result of unusually low variability or a low control frequency, comparison with the laboratory historical control frequency may be justified. 

The overall response to the reaction variables is very similar in the carbonylation and reductive carbonylation reactions. This may indicate similar catalysts and reaction mechanisms. In the carbonylation reaction Co(CO) " was identified by its characteristic CO stretching frequency ( v(CO) r 1890 cm" as the dominant species present in high pressure infrared experiments carried out at 170 °C and 5000 psig. Similar results were obtained in the reductive carbonylation of methanol. It is known that Co(CO) " rapidly reacts with CH I to yield CH C(0)Co(C0) (J9) however, in the carbonylation and reductive carbonylation reactions acyl-cobalt complexes are not observed by infrared under catalytic conditions. This indicates that once formed, the acyl complex rapidly reacts as outlined by Equations 7 and 8. 

The variable responses observed are probably the main drawback for the practical use of essential oil as miticides. It must be pointed out that the same plant species often produces essential oils with variable composition because of environmental and/or genetic factors many species have varieties, the so called chemotypes for instance at least seven chemotypes are known for Thymus vulgaris [88,89]. Also the extraction process influences the composition of the essential oils. For these reasons, it is advisable that authors report the composition of the essential oils used in the biological investigations. Unfortunately, only one paper reported this important information [64]. 

The response of many instruments is linear as a function of the measured variable, if variations due to experimental conditions or the instrument are taken into account. The objective is to determine the parameters of the linear equation that best represents the observations. The primary hypothesis in using the method of least squares is that one of the two variables should be without error while the second one is subject to random errors. This is the most frequently applied method. The coefficients a and b of the linear equation y = ax + b, as well as the standard deviation on a and on the estimation of y have been obtained in the past using a variety of similar equations. The choice of which formula to use depended on whether calculations were carried out manually, with calculator or using a spreadsheet. However, appropriate computer software is now widely used. 

The experimental matrix corresponding to a given design determines the settings of the variables [(-1) or ( + 1)] for each experiment. Once the series of experiments has been carried out, estimates of the coefficients b0, bt, and by-are calculated from the observed response values Y using the method of least squares for fitting the data. 

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