Case study 7.2 Multiple regression

Student s t-tests can be used to determine the significance of the parameters. In order to do so, the t-value from the t-distribution is needed (see Appendix C)  

This value should be compared with the calculated tots = bp /se bp), which are shown in the ANOVA table (Table 7.3). For both parameters Ubs 2.02, thus both parameters in the model are significant. 

The confidence interval for the mean response, /r(xo), at xo = 100, can be calculated according to 

A die-casting process was examined and experiments conducted in order to study the effects of the furnace temperature (xi) and the die-closing time (X2) on the temperature difference of the die surface (y). The data are shown in Table 7.5. 

The experimental data are presented below in vector and matrix notation  

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The methodology section of the study must have been informative enough to permit calculation of effect size and p values. That is, the author must have presented differences between groups and standard deviations, or r-values. In those cases where multiple regression was used, the effect size of IQ differences at blood levels of 10 /ig/dl and 20 jig/dl was calculated from the regression equation. The standard deviation of the control group was the denominator. 

Gonzalez, A. G., TWo Level Factorial Experimental Designs Based on Multiple Linear Regression Models A Tutorial Digest Illustrated by Case Studies, Analytica Chimica Acta 360, 1998, 227-241. 

In the case study, dummy variables were used to evaluate seasonality and the trend over a period of three years with a dry summer (1989, 1990, 1991). To evaluate the seasonality, an additional series is assigned for each month the series is equal to one or zero. This means the addition of 11 new series or dummy variables (the twelfth month variable is redundant) for a multiple regression. To evaluate the trend the twelfth dummy variable, dry summer , is equal to one in 1989, equal to two in 1990, and equal to three in 1991. The following new dummy variables were created ... 

These special cases of multiple linear regression analysis have been developed for the determination of the impact of individual molecular substructures (independent variables) on one dependent variable. Both techniques are similar yet, the Free-Wilson method considers the retention of the unsubstituted analyte as base, while Fujita-Ban analysis uses the less substituted molecule as reference. These procedures have not been frequently employed in chromatography only their application in QSRR studies in RP TLC and HPLC have been reported. 

Variable Selection in Multiple Linear Regression Analysis of Selwood et al. Data Set—A Case Study. 

The data were statistically analyzed using the SOLO Statistical System (BMDP Statistical Software, Inc., Los Angeles, CA) on a personal computer. Differences between groups were tested by the Mann-Whitney test or a paired t-test in cases where paired data sets were tested. Possible relationships were studied with (multiple) linear regression using least-square estimates. 

When the studied case concerns obtaining a relationship for the characterization of a process with multiple independent variables and only one dependent variable, we can use a multiple linear regression ... 

Fig. 10 illustrates the results of a multifactor cross-classification analysis from the same study, for white men age 45-64, with five risk factors dichotomized (including plasma glucose 1-hour post-50-gm-oral-load). With exclusion from the analysis of hypertensives on treatment and diabetics on treatment, glucose and rate of major ECG abnormalities were significantly related in two cases (noted by asterisks), but not in two others, and in only one of four comparisons for white women age 45-64 (Fig. 11). Similar inconsistent results were obtained with the more elegant multiple logistic regression technique (Fig. 12) where in only two of four analyses (after exclusion of hypertensives on treatment) were the values greater than 2.00 obtained indicating a significant relationship between post-load plasma glucose and major ECG abnormalities.

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