SEGUNDA PARTE
B. Políticas y programas del Gobierno sobre la atención y educación de los menores
The terminology, Design of Experiments (DoE), is a statistical concept that is widely used in research quality control and product or process improvement. To gain a good understanding of DoE, some basic statistical background of quality control and assurance needs to be introduced first.
Today, the modern definition of quality is more preferable: quality is inversely proportional to variability. Variability particularly refers to unwanted or harmful variability, but there are situations in which variability is actually good. Based on the modern definition of quality, quality improvement is defined as the reduction of variability in processes and products or the reduction of waste.217
Every product possesses a number of elements that jointly describe what the user or consumer thinks of as quality. These parameters are often called quality characteristics, and they may be of several types:217
1. Physical: length, weight, voltage, viscosity 2. Sensory: taste, appearance, colour
3. Time orientation: reliability, durability, serviceability
The different types of quality characteristics can relate directly or indirectly to the dimensions of quality discussed above. Data on quality characteristics is classified as either ‘attributes’ or ‘variables’. Variables data are usually continuous measurements, such as length, viscosity or voltage. On the other hand, attributes data usually describe discrete data, often taking the form of counts. A value of
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measurement that corresponds to the desired value for that quality characteristic is called the nominal or target value for that characteristic. The largest allowable value for a quality characteristic is called the upper specification limit (USL), and the smallest allowable value for a quality characteristic is called the lower specification limit (LSL).217
There are three management aspects of quality improvement: quality planning, quality assurance, and quality control and improvement. Since variability is often a major source of poor quality, statistical techniques are the major tools of quality control and improvement. The major statistical techniques for quality control and improvement are: 217
Statistical process control (SPC): A control chart is one of the primary techniques.
Design of experiments (DoE): This is extremely helpful in discovering the key variables influencing the quality characteristics of interest in the process. One major type of designed experiment is the factorial design, in which factors are varied together in such a way that all possible combinations of factor levels are tested.
Acceptance sampling: This is defined as the inspection and classification of a sample of units selected at random from a larger batch or lot and the ultimate decision about disposition of the lot.
Obviously, the DoE method is the technique that will be focused upon, in this research. In statistics, the quality characteristic of interest is called a response, and the variables influencing the quality characteristic of interest are called factors. Before further review of DoE, the concepts of quality engineering and corresponding statistics must first of all be introduced to the area of materials science. Any properties of a material that we would like to study, such as tensile strength, modulus or toughness are actually quality characteristics, or known as responses in statistical terminology. Any variables influencing the properties of the material such as the concentration of fillers, the processing temperature or the crystallisation conditions etc. are the corresponding factors of the responses. Therefore, DoE approach can be a very powerful statistical tool when investigating materials science, to predict accurate responses from the input factors, whilst minimising experimental time.
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A designed experiment or DoE is a test, or series of tests, in which purposeful changes are made to input variables of a process so that we may observe and identify corresponding changes in the output response. When there are several factors of interest in an experiment, a factorial design should be used, where designs factors are varied together. The effect of a factor is defined as the change in response produced by a change in the level of the factor. This is called a main effect because it refers to the primary factors in the study. In some experiments, the difference in response between the levels of one factor is not the same at all levels of the other factors. When this occurs, there is an interaction between the factors. When an interaction is large, the corresponding main effects have little meaning. Thus, knowledge of the interaction is often more useful than knowledge of the main effect, since a significant interaction can mask the significance of main effects. Therefore, interactions between the factors should be considered seriously.
A factorial design with k factors, each at two levels, is called a 2k factorial design, because each complete replicate of the design has 2k runs. The 22 Design is the simplest 2k design, and the design can be represented geometrically as a square with the 22 = 4 runs forming the corners of the square.
Figure 2.3-1 Test matrix of the 22
factorial design.217
The figure above shows the 4 runs in a tabular format often called the test matrix. Each run of the test matrix is on the corners of the square and the – and + signs in each row show the settings for the variables A and B for that run. The effect of factor A and B, and the interaction AB, can be calculated by the method called the analysis of variance (ANOVA), and the equations are shown below:217
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Effect A: 𝐴 =2𝑛1 [𝑎 + 𝑎𝑏 − 𝑏 − (1)] (2.3-1) Effect B: 𝐵 =2𝑛1 [𝑏 + 𝑎𝑏 − 𝑎 − (1)] (2.3-2) Interaction AB: 𝐴𝐵 = 2𝑛1 [𝑎𝑏 + (1) − 𝑎 − 𝑏] (2.3-3) where a, b, ab and (1) are the totals of all n observations taken at these design points (Figure 2.3-1).
The 22 design is the simplest one, so for the 2k design, regression model and residual analysis are used. The equation below is a typical regression model for the 22 design.
𝑦 = 𝛽0+ 𝛽1𝑥1+ 𝛽2𝑥2+ 𝛽12𝑥1𝑥2+ 𝜖 (2.3-4)
where y is the response and the factors A and B are represented by coded variables 𝑥1and 𝑥2, and the AB interaction is expressed by the cross-product term in the model, 𝑥1𝑥2. The coefficients 𝛽0, 𝛽1, 𝛽2 and 𝛽12 are named regression coefficients, and 𝜖 is a random error term.
If the response is well modelled by a linear function of the independent variables, then the function is known as the first-order model (Equation 2.3-4). On the other hand, if there is curvature in the system, a model with a higher order must be used, such as second-order model. The second-order is most widely used, because it is more accurate than the first-order model, but it is much simpler than the models with higher degree given that it can provide adequate accuracy. Equation 2.3-5 shows a full second-order regression model.
𝑦 = 𝛽 + 𝛽1𝑥1+ 𝛽2𝑥2 + 𝛽12𝑥1𝑥2 + 𝛽11𝑥12+ 𝛽22𝑥22+ 𝜀 (2.3-5)
The significance of the models can be examined by calculating and comparing F- ratios or p-values. The calculation methodology cited by Montgomery 32 is briefly introduced here. Assume a single factor with a different levels or treatments has n observations for each treatment. Total sum of squares (SST), treatment sum of
squares (SSTreatments) and error sum of squares (SSE) can be calculated as follows:
𝑆𝑆𝑇 = ∑ ∑𝑛 (𝑦𝑖𝑗 − 𝑦̅)2
𝑗=1 𝑎
𝑖=1 (2.3-6)
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= 𝑆𝑆𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠+ 𝑆𝑆𝐸
where 𝑦𝑖𝑗 represents the jth observation taken under treatment 𝑖, 𝑦̅𝑖∙ stands for the
average of the observations under the 𝑖 th treatment and 𝑦̅ presents the grand average. The mean square (MS) and F0 can be calculated by the following equations:
𝑀𝑆 = 𝑆𝑆/𝐷𝑓 (2.3-7) 𝐹0 = 𝑆𝑆𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠𝑎−1 𝑆𝑆𝐸 [𝑎(𝑛−1)] =𝑀𝑆𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠 𝑀𝑆𝐸 (2.3-8)
where 𝐷𝑓 is the number of degrees of freedom that corresponds to the sum of
squares in Equation 2.3-6. The ratio F0 has an F distribution with (a-1) and a(n-1)
degrees of freedom. If the F0 value computed is greater than the critical F value
related to a certain significance level (α), that is F0 > Fα, a-1, a(n-1), then the
corresponding effect is considered as statistically significant. A p-value is an alternative approach, in which the p-value is equal to the probability above (F0) in the
Fα, a-1, a(n-1) distribution:
𝑝 − 𝑣𝑎𝑙𝑢𝑒 = 𝑃[𝐹𝑎−1,𝑎(𝑛−1)> 𝐹0] (2.3-9) The significance level (α) is usually set as 0.05. With F0 or p-value, it is possible to
evaluate whether the variables are significant or the terms in Equation 2.3-5 are necessary.
Generally, the regression models can be simplified by removing the negligible terms with p-values greater than 0.10. On the other hand, the terms with p-values lower than 0.05 can be considered as important and kept in the models. Residual analysis needs to be carried out to confirm the adequacy of the model used. ANOVA assumes that the model errors (and as a result, the observations) are normally and independently distributed with the same variance in each factor, so the validity of the assumption needs to be checked. The normality assumption can be verified by constructing a normal probability plot of the residuals, while the independence assumption can be verified by using plots of residuals against run order.
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Figure 2.3-2 An example of 3D response surface generated by Design Expert®
Response surfaces can be plotted according to the regression model generated, which are 3D plots showing the plane of predicted response values generated by the regression model. A 3D response surface has many uses such as predicting the response value at a particular point. Figure 2.4-2 shows an example of 3D surface generated in a DoE experiment.
Within a typical design of experiments approach, a central composite design (CCD) is widely used to analyse a second order response surface. Generally, 2k factorial runs, 2k axial runs, and at least one centre point are required by a CCD, for a set of investigations based upon k factors. In Figure 4.2 is shown the central composite design when k is equal to 2. These show that each numeric factor is varied over 5 levels: ± alpha (axial points), ± 1 (factorial points) and the centre point (0,0).
Figure 2.3-3 Central composite design for k = 2
The sparsity of effects principle states that the direct effects and two-factor interactions usually dominate in a system whilst the higher order interactions are
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negligible. For this research programme, CCD would be employed for fitting second order surface response models.
DoE methodologies have been widely used in experimental research, and some studies of manufacturing process relating to polymer-based products have been reported in literature. In the field of polymers, DoE approach has been used to investigate blow moulding218, electrospinning of nanofibers219,220, mixing and blending processes for polyolefin compounds221,222 and polymer-based composites223–225.
Recently, as part of this research project, a study of single screw extrusion dynamics by the DoE approach has been published1. Other reported application of DoE in process research have included pharmaceutical products226, UV-curable coatings227 and sintered powder metal copper extrudates228.