HORTOFRUTICOLAS
1. EVOLUCION CRECIENTE DE LA COMPETENCIA
1.1. Competencia comunitaria I. En frutas de huesoI.En frutas de hueso
The coefficients of two factors, sulfur anhydride (factor 2) and the elimination of water (factor 4), are much larger than the others. These are therefore the ones that are most influential in achieving our goal. But are there other influential factors? Knowing where to draw the line between influential and non-influential factors is a delicate question that uses the value of the experimental error (Chapter 5). Since the number of trials in a fractional factorial is limited, this value is not always available. Analysis of variance (ANOVA) gives an initial answer, but we should not forget that the residuals contain interactions that have not been taken into account. We assume the model is proper, but in fact it may not be.
According to the chosen model, we can obtain an RMSE (root mean square error) of reasonable size. The analysis of variance is a partial response to the problem of other inflated factors. There are two other tools that allow us to have an idea of the relative importance of the coefficients. These are the Pareto diagram and the Daniel (Normal) chart. These tools help the experimenter to draw a line between influential and non-influential factors. In this situation, we can use several types of reasoning that complement each other, using them in concert to develop a final answer.
One method of reasoning says that all effects lower than a certain value don’t influence the response noticeably. For example, the experimenter may say that a variation of the ratio (factor 6) less than 0.5 is not significant. In this case, there are only two influential factors. If the experimenter decides to put the limit at 0.25, there are four influential factors.
A second method uses the Pareto diagram (Figure 7.2). All coefficient values are placed in decreasing order of the absolute value of their coefficients. This facilitates the choice of the limit between significance and non-significance: save the coefficients that are above a certain value, and throw out those that are below it. There is not a big difference between the two methods, except the second uses a picture, which makes the
comparisons easier.
Figure 7.2 Pareto chart of the sulfonation experiment
A third method supports both previous modes of reasoning. We use a Daniel diagram, implemented in JMP as the Normal Plot. The small-valued coefficients follow a normal distribution, are assimilated into the experimental error, and align along a line called the Henry line. High-value coefficients do not follow a normal distribution, so they don’t follow the Henry line. This partitions the two populations, the experimental error and the effects we need to take into account (Figure 7.3). With the Daniel diagram, the
arbitrariness of the decision is considerably diminished.
If possible, its best to use other statistical techniques that help to make a good decision.
For example, if the experimenter has made repetitions (duplicate runs under the same experimental conditions), a real estimate of experimental error can be calculated to obtain the limit between significant and non-significant factors. A commonly chosen limit is two or three times the standard deviation. The decision of how many standard deviations to use is based on the experimental error and the knowledge of the risks of incorrect conclusions.
Regardless of the reasoning you use, it is always a good idea to examine a Pareto chart and a Daniel diagram. They are good decision-making tools.
Figure 7.3 The Daniel diagram shows the separation of significant effects
The examination of the Pareto and Daniel diagrams shows that we should retain a single influential factor: the sulfuric anhydride concentration in the sulfuric acid (factor 2).
This experiment also shows that the elimination of water (factor 4) is slightly influential.
The decision to keep or eliminate this factor is not based on statistics. Here, reasoning based on chemical best practice must be the guide. Since the drainage of water affects the concentration of the sulfuric acid, it is chemically sound to regard water elimination (factor 4) as slightly influential, and to keep it.
The other factors and interactions are too small and do not contribute much to explaining the response.
In a choice, there is always an arbitrary element, but in the case of designed experiments, the choice is never absolute and can always be modified later in a new analysis.
7.2.4 Study Conclusion
The relative importance of the contrasts is indicated by the bar chart (Figure 7.1), the Pareto Plot (Figure 7.2), and the Daniel plot (Figure 7.3).
The contrasts
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12,A
13,A
14,A
15,A
16 ,A
23 andA
26 are small. Using the second assumption, we see that all the coefficients aliased in these contrasts are negligible. There are therefore no second-order interactions among the factors.Two influential factors remain (Figure 7.4): the quantity of sulfur anhydride present in the sulfuric acid (factor 2) and the elimination of water during the reaction (factor 4).
Figure 7.4 Effects of the two influential factors
The amount of sulfur anhydride (SO3) present in the sulfuric acid has a negative effect.
The response is high if we choose the low level; that is, we shouldn’t add SO3 to the sulfuric acid.
Water elimination during the reaction has a positive effect. Therefore, the response will be higher if we choose its high level—that is, if we eliminate the water. This effect is only slightly elevated and might be ignored if we relied simply on statistics. However, from reasoning based on chemical best practice, the elimination or water raises the acid concentration and helps the reaction form sulfonic acids. The influence of this factor cannot be ignored.
The sulfonation of benzene, in the presence of toluene and of xylenes, is therefore minimized if water is eliminated from concentrated sulfuric acid that does not include sulfur anhydride.