Confidence intervals multiple regression

Also in multiple regression, confidence intervals for the parameters can be derived. From a practical point of view, it is, however, more important to test if single... 

Fig. 15.9. Determination of the Multiple Of Normal Activity (MONA). It is assumed that all sera to be tested have parallel dose-response curves (I-lII), the slope (m) of which with simple regression analysis is obtained from the log transformation in Fig. 15.8. The number of times a serum should be diluted over that of the reference serum (MONA) can then be calculated and is independent of the original dilution (i.e., a = b = MONAn). However, to obtain the MONA of the various sera, their EIA-absorbance values are determined at the same dilution (e.g., D ). The confidence interval of the mean of MONA-values, important for seroconversion tests, can similarly be established.

The most important statistical parameters r, s, and F and the 95% confidence intervals of the regression coefficients are calculated by Eqs. (20) to (23) (for details on Eqs. (20) to (23), see Refs. 39 to 42). For more details on linear (multiple) regression analysis and the calculation of different statistical parameters, as well as other validation techniques (e.g., the jackknife method and bootstrapping), see Refs. 33,39 12 ... 

The researcher can always take each x point and perform a t-test confidence interval, and this is often the course chosen. Although from a probability perspective, this is not correct from a practical perspective, it is easy, useful, and more readily understood by audiences. We discuss this issue in greater detail using indicator or dummy variables in the multiple linear regression section of this book. 

There are times when a researcher would like to compare multiple regression function lines. One approach is to construct a series of 95% confidence intervals for each of the y values at specific x, values. If the confidence intervals overlap, from regression line A to regression line B, the researcher simply states that no difference exists, and if the confidence intervals do not overlap, the researcher states that the y points are significantly different from each other at a (see Figure 2.37). 

In the equations, n represents the number of datum points used to derive the equation, r is the multiple correlation coefficient, s is the standard deviation from regression, F is the F statistic for variance of each additional variable, the values in parentheses after the equation coefficients are for construction of confidence intervals and the Roman numbers in subscript refer to the substituent position.)... 

Tnieness [6], [7], A systematic error can occur as an additive error (e.g., an undetected blank) or a multiplicative error (e.g., an incorrect titer). Systematic errors are detected by analyzing a short series of m samples with known contents x,- and found contents v, as the results. Evaluation by linear regression (Chap. 3.5) yields y = a + bx. An intercept o =)= 0 is due to an additive systematic error, whereas a slope b =t= 1 indicates a multiplicative error. The significance of a and b is tested by verifying that 0 and 1 do not belong to the 95 % confidence intervals around a and b, see Equation (5.2). 

A point which may need emphasis, stated clearly in Hunter ( 2 ), is the precise interpretation of the confidence band about the predicted amount. This is important since without a clearly understood meaning, the interval will not be useful for assessing the precision of the predicted amounts or concentrations nor for comparing the results from various laboratories. Another reason the user of these methods must understand the interpretation is because increased precision can be achieved in at least two ways -by additional replication of the standards, which reduces the width of the confidence band about the regression line, and by performing multiple determinations on the unknowns, which reduces the width of the interval about the mean instrument response of the unknown. The interval for U is then given by... 

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