Analytical Methods Committee, AMCTB No 55

Received 1st March 2013, Accepted 1st March 2013

First published on 11th March 2013

Good experimental design is important in many studies of analytical and other chemical processes. Complete factorial designs, which study all the factors (experimental variables) affecting the system response, using at least two levels (values) for each factor, can give rise to an unacceptably large number of trial experiments. This is because even apparently simple processes may be affected by a large number of factors. Moreover these factors may affect the system response interactively, i.e. the effect of one factor may depend on the levels of others. Any interactions must also be distinguished from random measurement errors. So it is more common to use partial factorial designs in which some information, especially about interactions, may be sacrificed in the interests of a manageable number of experiments.

Suppose we wish to study four factors. Four experiments will be then insufficient, so we shall have to use eight experiments in a PB design, and have seven factors. This means that three of the latter will be dummy factors; they will have no chemical meaning at all. However it turns out that the apparent effects of these dummy factors can be used to estimate the random measurement errors (see below). The more dummy factors there are, the better the estimate of such errors, so it is not uncommon for experimenters to use a larger PB design than is strictly necessary, thus getting higher quality information on the significance of each “real” factor.

PB designs utilise two levels for each factor, the higher level being denoted “+” and the lower “−” as usual. A further feature of the PB method is that the + and − signs for the individual trial experiments are assigned in a cyclical manner. If we utilise eight experiments with seven factors labelled A–G, the levels for the first experiment might be:

Such sequences of + and − signs are provided by generating vectors and are widely available in the literature and in software packages. The levels for the second experiment, again with four + and three − signs, are then obtained by moving the last sign for the first experiment to the beginning of the line, giving:

This cyclical process is repeated for the first seven experiments. For the eighth experiment all the factors are set at the low (−) level, giving an overall design in which there are 28 + signs and 28 − signs, each factor having been studied four times at the higher level and four times at the lower. The effect of each factor is then readily determined from the expression:

2[∑(y+) − ∑(y−)]/N |

where N is the total number of experiments, eight in this case. The (y+) terms are the responses when a given factor is at its high level, and the (y−) terms reflect the responses for that factor set to its low level. It can be shown that the effects for the main factors determined in this way are not confounded with each other (see AMCTB 36).

EXPERIMENT | FACTORS | Result y | ||||||
---|---|---|---|---|---|---|---|---|

A | d1 | B | d2 | C | d3 | D | ||

1 | + | − | − | + | − | + | + | 10 |

2 | + | + | − | − | + | − | + | 9 |

3 | + | + | + | − | − | + | − | 10 |

4 | − | + | + | + | − | − | + | 9 |

5 | + | − | + | + | + | − | − | 8 |

6 | − | + | − | + | + | + | − | 7 |

7 | − | − | + | − | + | + | + | 7 |

8 | − | − | − | − | − | − | − | 7 |

Effect | +1.75 | +0.75 | +0.25 | +0.25 | −1.25 | +0.25 | +0.75 | * |

SS | 6.125 | 1.125 | 0.125 | 0.125 | 3.125 | 0.125 | 1.125 | * |

F-value | 13.4 | * | 0.3 | * | 6.8 | * | 2.5 | * |

From these results we can see that, for example, the effect of factor A is 0.25(10 + 9 + 10 + 8 − 9 − 7 − 7 − 7) = +1.75. Similarly it can be shown that the effects of B, C, and D are +0.25, −1.25 and +0.75 respectively. Clearly a negative effect, as obtained here with factor C, means that moving that factor from a high to low value increases the system response (fluorescence intensity in this case) rather than decreasing it. The effects of the dummy factors d1, d2 and d3, are found by the same method to be +0.75, +0.25, and +0.25 respectively.

SS = N × (estimated effect)^{2}/4 |

The sums of squares for A, B, C, and D are thus 6.125, 0.125, 3.125, and 1.125 respectively. Each of these sums of squares has just one degree of freedom, so their mean square values (i.e., variances) are the same as the SS ones. The sums of squares for the dummy factors d1, d2, and d3 are similarly found to be 1.125, 0.125, and 0.125 respectively. The mean sum of squares for these estimates of the random measurement errors is thus 0.458: this has three degrees of freedom as there are three dummy variables. Each of the individual factors A–D can now be compared with this estimated random error using a one-tailed F-test at the p = 0.05 significance level. So for factor A the value of F is 6.125/0.458 = 13.37. The critical value of F_{1,3} at p = 0.05 is 10.13, so we can conclude that the effect of changing the level of factor A is significant. The same approach shows that factors B, C and D seem to have no significant effect. Such calculations are in practice performed using suitable software such as Minitab®, so once the trial experiments are complete the conclusions can be drawn at once.

However, the popularity of PB methods comes with a significant health warning. PB designs are ideal for screening purposes in systems where it is desired to identify a few main factors affecting the outcome, and where interactions are not significant. Theory shows that while the main factors in a PB design are not confounded, there is strong confounding between the main factors and any two-factor interactions that may arise. So if there are significant interactions, PB methods could provide misleading results. In recent years much attention has been given to diagnostic approaches for revealing interactions in PB designs. These are beyond the scope of this paper; but it is worth noting that if dummy factors seem to have unexpectedly high effect values, this might be a sign that interactions are indeed present.

This Technical Brief, drafted by J.N. Miller, was prepared for the Analytical Methods Committee by the Statistical Subcommittee.

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