Normal probability plots, effects

For the subplot analysis it appears that the effects due to A, and the interaction between B and Humidity are real, with some evidence of an interaction between B and Temperature. It is possible to split the two degrees of freedom for Temperature and Humidity into linear and quadratic contrasts and to construct a normal probability plot for the whole plot contrasts. This would reveal important effects due to the linear components of both Temperature and Humidity. 

Normal probability plots or half normal probability plots (Bimbaun plots) [24,29] are graphical methods that help to decide which factors are significant. Effects that are normally distributed around zero are effects... 

Figure 3.3 Normal probability plot of the normalized effects for the resolution between epianhydrotetracycline and tetracycline obtainedfrom the fractional factorial design...

The real power of the use of half-normal probability plots, however, comes with data that are likely to have embedded outliers. These data profoundly distort the half-normal plots, as illustrated with the data for methyl isobutyl ketone shown in Figure 9. The plot shows neither normal random error nor significant effects cleanly. Thus, this... 

Compound recovery data for duplicate runs differed by 2-15, depending on the compound. Half-normal probability plot analysis of the new data for the anomalous compounds indicated none of the distortion encountered earlier. Results for acetone and tetrachloroethylene now indicated only random variation with no significant outliers. Results for 2,4-dichlorophenol and 2,5-dichlorophenol indicated a significant pH effect. A significant interaction effect (AB) was detected between variables pH and primary column type for the dichlorophenols and also for methyl isobutyl ketone. This interaction effect indicates that at approximately low pH (pH 2), compound recoveries for dichlorophenols will be greater when a C18 phase is used as the primary column. The half-normal plot for 2,5-dichlorophenol is shown in Figure 10. In examining data for all the compounds from the 23 replicate factorials, this interaction consistently appears for phenolic compounds. 

Figure 3 shows a half-normal probability plot of the effect estimates obtained from the Design-Expert software package. Three main effects, A (pressure), B (power), and E (gap) are important. Because the main effects are aliased with three-factor interactions, this interpretation is probably correct. There are also two two-factor interaction alias chains that are important, AB = CE and AC = BE. Because AB is the interaction of two strong main effects, those of pressure and... 

Figure 3. Half-normal probability plot of the effect estimates from the plasma etching experiment in Table 3.

Figure 3. Fialf-normal probability plot of the factor effects from the 2s factorial experiment on hue.

Box and Meyer did not present any formal inference procedures for using this statistic to identify dispersion effects. The use seems to be informal for screening factors with large effects from those with no or small effects on dispersion, for example, by making a normal probability plot of the statistics see Montgomery (1990) for an application of this idea. 

Like many classical methods of data analysis, the normal probability plot has limitations. It is only useful if there are several factors, and clearly will not be much use in the case of two or three factors. It also assumes that a large number of the factors are not significant, and will not give good results if there are too many significant effects. However, in certain cases it can provide useful preliminary graphical information, although probably not much used in modern computer based chemometrics. 

Excluding the intercept term, diere are seven coefficients. A normal probability plot can be obtained as follows. First, rank die seven coefficients in order. Then, for each coefficient of rank p calculate a probability (p — 0.5)/7. Convert diese probabilities into expected proportions of die normal distribution for a reading of appropriate rank using an appropriate function in Excel. Plot the values of each of die seven effects (horizontal axis) against die expected proportion of normal distribution for a reading of given rank. 

The effects obtained are estimates of true effects and a statistical analysis can be carried out. A useful graphical method to determine whether effects are significant consists of drawing normal probability plots (see Fig. 6.9) or half-normal plots. Non-significant effects are normally distributed around zero and tend to fall on a straight line in those plots while significant effects deviate from the line. Fig. 6.9 shows that for... 

Fig. 6.9. Normal probability plot lor the effects from Tabic 6.2 A. B and AB are considered significant.

Significant effects, i.e. effects that are significantly larger than could be due to experimental variability, can be identified by means of both graphical and statistical methods. The graphical method that is used most often is the normal probability plot explained in the preceding section (Fig. 6.9). The statistical tests are often based on a /-test, where the test statistic can be written as... 

The are a number of ways of doing this. If the experiments have been replicated, ANOVA will reveal which effects are statistically significant. Otherwise, we rely on the fact that most of the effects are probably small and distributed randomly about zero. Thus, we look for the effects with the largest absolute values that stand out from the others. Making a normal probability plot of the distribution of their values is a widely used method. 

Normal probability plots to discern significant effects... 

If it is assumed that interaction effects between three or more variables can be neglected, it is seen that the estimated effect can be used as estimators of the model parameters. The 16 model parameters have been estimated from 16 runs. There are no degrees of freedom left to give an estimate of the residual variance which might have been used as an estimate of the error variance. Neither was any previous error estimate available. The significance of the estimated parameters must be assessed from a Normal probability plot. 

In addition to the use of normal probability plots previously mentioned, i.e. to assess significant effects from screening experiments, and to check residuals after fitting models, there are some other useful applications which are briefly discussed in this section. 

A normal probability plot of the estimated effects is shown in Fig. 6.13. 

A normal probability plot of the estimated coefficients is shown in Fig.17.2. It is seen that variables Xj and are significant, and that they also show a significant interaction effect. These are the same conclusions as were reached in the original report, where five derived responses were anaylzed one at a time. 

The analysis utilized the approach of normal probability plots, as no reliable historical estimate of variation was available. For each primary response the observed factor effects were plotted on normal probability paper in rank order (Fig. 2). Here, factor effects are estimated as the difference of the average response at the high level (-I-) from the average response of the low level (-). On these plots, values that deviate markedly from the general trend line indicate significant effects. The largest factor effects (in absolute value) are labeled. 

Figure 3 Normal probability plot of effects on burn.

The resulting print-out from MINITAB is given in Figure 2. However, the easiest way to interpret the data is to look at the pareto chart of calculated effects (Fig. 3) and related normal probability plot (Fig. 4). The vertical line on the pareto chart in Figure 3 corresponds to a P-value of 0.10 for each calculated effect. In other words. 

The MINITAB analysis Printout is given in Figure 8. The pareto chart (Fig. 9) and corresponding normal probability plot (Fig. 10) show that there are no statistically significant main effects or interactions present. Note that the bars are all to the left of the verticle line on the pareto chart and none of the points on the probability plot are labeled. In addition, looking at the data... 

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