Half-normal plots

Estimated effects, as well as residuals from eight-run designs, give plots in which the points are rather far apart. It may be difficult to see how a line should be drawn. A plot with more densely marked points can be obtained if the absolute values of the estimated effects, hj or the residuals, Cj, are plotted in the interval P = [50 - 100 %]. 

A word of caution on the use of half-normal plots is found in a book by Daniel.[10] Sometimes, the sign of an estimate can provide extra information, which will be lost if the absolute values are plotted. It is therefore advisable, not to use half-normal plots routinely. 

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Probability plot Q-Q plot P-P Plot Hanging histogram Rootagram Poissonness plot Average versus standard deviation Component-plus-residual plot Partial-residual plot Residual plots Control chart Cusum chart Half-normal plot Ridge trace Youden plot... 

Daniel, C. (1959). Use of half-normal plots in interpreting factorial two-level experiments. Technometrics 1, 311-341. 

Zahn, D. A. (1975). Modifications of and revised critical values for the half-normal plot. Technometrics 17, 189-200. 

Half-normal plots are a modification in which the absolute value of X is used rather than the actual value itself (20). This technique has the benefit of a plot whose X axis starts at 0. 

Use of Half-Normal Plots with Factorial Data. The application of this method to the factorial data is straightforward. If, for any given compound, the data from the factorial experiment occurred simply as the result of random variation about a fixed mean, and the changes in the levels of the variables had no real effect at all on the percent recovery, then the 15 main effects and interactions, representing 15... 

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. 

Analysis of data from the factorials indicates that pH has a consistently significant effect on compound recoveries. A summary of the effect of pH level on compounds used in the study is given in Table VI. There is also an interaction between pH and primary column sorbent type for some compounds. This interaction suggests that at low sample pH, a C18 column will produce the best extraction efficiencies for phenolic compounds. The effect of adding methanol to the sample before extraction clearly produced odd results when the recovery data from the 24 factorial was analyzed by using half-normal plots. This effect will be studied in future work. Additionally, different elution solvents will be examined as well as new sorbent phases as they become available. 

Figure 3 shows a half-normal plot for hue from the full experiment. Two effects clearly stand out and these are the main effects of factors A and F. All the other effects appear consistent with a null hypothesis of no effect. These conclusions are reinforced by other analyses. Fitting a model with all main effects and two-factor interactions results in highly significant effects for factors A and F. No other effects are significant at the 5% level, but the main effect of B and the AC interaction are both quite close, with values less than 0.075. Analysis by Lenth s (1989) method, discussed in Chapter 12, also finds that the only significant effects are those for A and F. 

Subsequently, Zahn (1969,1975ab) considered some variations on the iterative methods of Daniel (1959) and Birnbaum (1959), but his results were primarily empirical. The subjective use of half-normal plots remains a standard methodology for the analysis of orthogonal saturated designs, but the development of objective methods is progressing rapidly. 

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. 1. Half normal plot of effect of design variables.

Normal probability plots or, half-normal plots are recommended instead of histograms for detecting deviations from normality. For a normally distributed random variable X with mean 0 and variance a2, a... 

Figure 1.7 Normal, half-normal, and QQ plots for 100 simulated observations from a normal distribution (left), chi-squared distribution with four degrees of freedom (middle), and student s T-distribution with four degrees of freedom (right). If the data are consistent with a normal distribution, the resulting plots should all show approximate linearity with no curvatures. The normal plot and QQ plot are usually indistinguishable. The half-normal plot is usually more sensitive at detecting departures from normality than the normal or QQ plot.

Figure 3.9 Half-normal plot of coefficients (paracetamol effervescent tablet). 

Several representations of factor effects (effect graphic effects, normal [dot. half-normal plot, etc.) permit us to determine the influence of the factors studied. Wc can see that factors X, Xj, X are the only inHuent factors on the variation of the yield (Fig. 10-12). and the factors X, and Xj have hardly any effect. An increase of the solution B addition temperature (Xi) from 25 C to 45 C entails a diminution of the yield. 

As an alternative, the half normal probability plot [3] can be used to analyse the data. Figure 8.3 shows our half normal plot for the data in Kukovecz et al. (2005). In this analysis, we found that none of the factors is statistically significant at the 95% confidence level. 

Designing experiments for maximum information from cyclic oxidation tests and their statistical analysis using half normal plots (COTEST)... 

An alternative approach, which can be used when there are no or only one or two degrees of freedom, is to use the graphical method of half normal plotting to show which effects are statistically significant. The results for Alloy 800 are analysed by half normal plots instead of analysis of variance or regression as there are only two degrees of freedom for the experimental error variance. Similar to ANOVA and regression, analysis by half normal plots still requires that the data are independent, approximately normal and with constant variance. 

Half normal plots give a visual indication of which factors are statistically significant. The technique is useful when there are few or no degrees of freedom available for a residual mean square in ANOVA or regression. Half normal plots are therefore useful when the experimental design is saturated and all effects are of interest. The half normal plot is a type of probability plot where a numerical value for the factor effects is plotted on the vertical axis against the expected normal order statistics on the horizontal axis (see Grove and Davis, 1997). 

As the standard test matrix has been followed, the effects diagrams in Figs 18.3 to 18.5 above will give the same message as half normal plots. The advantage of half normal plots is that they show which effects are statistically significant. The numerical values of the factor effects for the half normal plot need to be found from orthogonal contrasts. 

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