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Multifactor RSM Tutorial (Part 1 – The Basics)
Response Surface Design and Analysis
This tutorial shows the use of Design-Expert? software for response surface methodology (RSM). This class of designs is aimed at process optimization. A case study provides a real-life feel to the exercise. Due to the specific nature of the case study, a number of features that could be helpful to you for RSM will not be exercised in this tutorial. Many of these features are used in the General One Factor, RSM One Factor or Two-Level Factorial tutorials. If you have not completed all of these tutorials, consider doing so before starting in on this one. We will presume that you can handle the statistical aspects of RSM. For a good primer on the subject, see RSM Simplified (Anderson and Whitcomb, Productivity, Inc., New York). You will find overviews on RSM and how it’s done via Design-Expert in the online Help system. To gain a working knowledge of RSM, we recommend you attend our Response Surface Methods for Process Optimization workshop. Call Stat-Ease or visit our website, https://www.doczj.com/doc/ee7285331.html,, for a schedule. The case study in this tutorial involves production of a chemical. The two most important responses, designated by the letter “y”, are: ? ? y1 - Conversion (% of reactants converted to product) y2 - Activity.
The experimenter chose three process factors to study. Their names and levels can be seen in the following table.
Factor
A – Time B - Temperature C - Catalyst
Units
minutes degrees C percent
Low Level (-1)
40 80 2
High Level (+1)
50 90 3
Factors for response surface study You will study the chemical process with a standard RSM design called a central composite design (CCD). It’s well suited for fitting a quadratic surface, which usually works well for process optimization. The three-factor layout for the CCD is pictured below. It is composed of a core factorial that forms a cube with sides that are two coded units in length (from -1 to +1 as noted in the table above). The stars represent axial points. How far out from the cube these should go is a matter for much discussion between statisticians? They designate this distance “alpha” – measured in terms of coded factor levels. As you will see Design-Expert offers a variety of options for alpha.
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Central Composite Design for three factors Assume that the experiments will be conducted over a two-day period, in two blocks: 1. Twelve runs: composed of eight factorial points, plus four center points. 2. Eight runs: composed of six axial (star) points, plus two more center points.
Design the Experiment
Start the program by finding and double clicking the Design-Expert software icon. Take the quickest route to initiating a new design by clicking the blank-sheet icon on the left of the toolbar. The other route is via File, New Design (or associated Alt keys).
Main menu and tool bar Click on the Response Surface folder tab to show the designs available for RSM.
Response surface design tab The default selection is the Central Composite design, which will be used for this case study. Click on the down arrow in the Numeric Factors entry box and Select 3. Ignore the option of including categoric factors in your designs (leave at default of 0).
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To see alternative RSM designs for three factors, click on the choices for BoxBehnken (17 runs) and Miscellaneous designs, where you find the 3-Level Factorial option (32 runs, including 5 center points). Now go back and re-select the Central Composite design. Before entering the factors and ranges, click the Options at the bottom of the CCD screen. Notice that it defaults to a Rotatable design with the axial (star) points set at 1.68719 coded units from the center – a conventional choice for the CCD. For Center points increase the number to the normal default of 6 and press the Tab key.
Default CCD option for alpha set so design will be rotatable Many of the options are statistical in nature, but one that produces less extreme factor ranges is the “Practical” value for alpha. This is computed by taking the fourth root of the number of factors (in this case 3? or 1.31607). See RSM Simplified Chapter 8 “Everything You Should Know About CCDs (but dare not ask!)” for details on this practical versus other levels suggested for alpha in CCDs – the most popular of which may be the “Face Centered” (alpha equal one). Press OK to accept the rotatable value. Using the information provided in the table on page 1 of this tutorial (or on the screen capture below), type in the details for factor Name (A, B, C), Units and levels for low (-1) and high (+1), by tabbing or clicking to each cell and entering the details given in the introduction to this case study.
Completed factor form
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You’ve now specified the cubical portion of the CCD. As you did this, Design-Expert calculated the coded distance “alpha” for placement on the star points in the central composite design. Alternatively, by clicking an option further down this screen, you could have entered values for alpha levels and let the software figure out the rest. This would be helpful if you wanted to avoid going out of operating constraints. Now go back to the bottom of the central composite design form. Leave the Type at its default value of Full (the other option is a “small” CCD, which we do not recommend unless you must cut the number of runs to the bare minimum). You will need two blocks for this design, one for each day, so click on the Blocks field and select 2.
Selecting the number of blocks Notice that the software displays how this CCD will be laid out in the two blocks. Click on the Continue button to reach the second page of the “wizard” for building a response surface design. You now have the option of identifying Block Names. Enter Day 1 and Day 2 as shown below.
Block names Press Continue to enter Responses. Select 2 from the pull down list. Then enter the response Name and Units for each response as shown below.
Completed response form At any time in the design-building phase, you can return to the previous page by pressing the Back button. Then you can revise your selections. Press the Continue button to get the design layout (your run order may differ due to randomization).
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Design layout (only partially shown, your run order may differ due to randomization) Design-Expert offers many ways to modify the design and how it’s laid out on-screen. Preceding tutorials, especially in Part 2 for the General One Factor, delved into this in detail, so go back and look this over if you haven’t already. Click the Tips button for a refresher.
Save the Data to a File
Now that you’ve invested some time into your design, it would be prudent to save your work. Click on the File menu item and select Save As.
Save As selection You can then specify the File name (we suggest tut-RSM) to Save as type *.dx7” in the Data folder for Design-Expert (or wherever you want to Save in).
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File Save As dialog box
Enter the Response Data – Create Simple Scatter Plots
Assume that the experiment is now completed. Obviously at this stage the responses must be entered into Design-Expert. We see no benefit to making you type all the numbers, particularly with the potential confusion due to differences in randomized run orders. Use the File, Open Design menu and select RSM.dx7 from the Design-Expert program Data directory. Click on Open to load the data. Let’s examine the data, which came in with the file you opened (no need to type it in!). Move your cursor to the top of the Std column and perform a right-click to bring up a menu from which you should select Sort by Standard Order (this could also be done via the View menu).
Sorting by Standard (Std) Order Next go to the Block column and do a right click. Choose Display Point Type.
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Displaying the Point Type (Notice that we widened the column so that you can see how some points are labeled “Fact” for factorial and others “Center” for center point, etc. Do this by placing your mouse cursor over the border and, when it changes to a double-arrow [?], drag it where you want. Some times this does not work the first time, but do not be discouraged: It will probably work the second time. What a drag!) Before doing the analysis, it might be interesting to take a look at some simple plots. Click on the Graph Columns node which branches from the design ‘root’ at the upper left of your screen. You should now see a scatter plot with factor A:Time on the X-axis is set at and the response of Conversion on the Y-axis. It will be much more productive to see the impact of the control factors on response surface graphics to be produced later. For now it would be most useful to produce a plot showing the impact of blocks, because this will be literally blocked out in the analysis. On the floating Graph Columns tool click on the X Axis downlist symbol and select Block.
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Graph Columns feature for design layout The graph shows a slight correlation (0.152) of conversion by block. Change the Y Axis to Activity to see how it’s affected by the day-to-day blocking (not much!).
Changing response (resulting graph not shown) Finally, to see how the responses correlate with each other, change the X Axis to Conversion.
Plotting one response versus the other (resulting graph not shown) Feel free to make other scatter plots. Notice that you can also color the by selected factors, including run (the default). However, do not get carried away with this, because it will be much more productive to do statistical analysis first before drawing any conclusions.
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Analyze the Results
You will now start analyzing the responses numerically. Under the Analysis branch click the node labeled Conversion. A new set of buttons appears at the top of your screen. They are arranged from left to right in the order needed to complete the analysis. What could be simpler?
Begin analysis of Conversion Design-Expert provides a full array of response transformations via the Transform option. Click Tips for details. For now, accept the default transformation selection of None. Click on the Fit Summary button next. At this point Design-Expert fits linear, twofactor interaction (2FI), quadratic and cubic polynomials to the response. To move around the display, use the side and/or bottom scroll bars. You will first see the identification of the response, immediately followed in this case by a warning: “The Cubic Model is Aliased.” Do not be alarmed. By design, the central composite matrix provides too few unique design points to determine all of the terms in the cubic model. It’s set up only for the quadratic model (or some subset). Next you will see several extremely useful summary tables for model selection. Each of these tables will be discussed briefly below. The table of “Sequential Model Sum of Squares” (technically “Type I”) shows how terms of increasing complexity contribute to the total model. The model hierarchy is described below: ? ? “Linear vs Block”: the significance of adding the linear terms to the mean and blocks, “2FI vs Linear”: the significance of adding the two factor interaction terms to the mean, block and linear terms already in the model,
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“Quadratic vs 2FI”: the significance of adding the quadratic (squared) terms to the mean, block, linear and two factor interaction terms already in the model, “Cubic vs Quadratic”: the significance of the cubic terms beyond all other terms.
?
Sequential Model Sum of Squares For each source of terms (linear, etc.), examine the probability (“Prob > F”) to see if it falls below 0.05 (or whatever statistical significance level you choose). So far, the quadratic model looks best – these terms are significant, but adding the cubic order terms will not significantly improve the fit. (Even if they were significant, the cubic terms would be aliased, so they wouldn’t be useful for modeling purposes.) Scroll down to the next table for lack of fit tests on the various model orders.
Summary Table: Lack of Fit Tests The “Lack of Fit Tests” table compares the residual error to the “Pure Error” from replicated design points. If there is significant lack of fit, as shown by a low probability value (“Prob>F”), then be careful about using the model as a response predictor. In this
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case, the linear model definitely can be ruled out, because its Prob > F falls below 0.05. The quadratic model, identified earlier as the likely model, does not show significant lack of fit. Remember that the cubic model is aliased, so it should not be chosen. Scroll down to the last table in the Fit Summary report, which provides “Model Summary Statistics” for the ‘bottom line’ on comparing the options.
Summary Table: Model Summary Statistics The quadratic model comes out best: It exhibits low standard deviation (“Std. Dev.”), high “R-Squared” values and a low “PRESS.” The program automatically underlines at least one “Suggested” model. Always confirm this suggestion by looking at these tables. Check Tips for more information about the procedure for choosing model(s). Design-Expert now allows you to select a model for an in-depth statistical study. Click on the Model button at the top of the screen next to see the terms in the model.
Model results
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The program defaults to the “Suggested” model from the Fit Summary screen. If you want, you can choose an alternate model from the Process Order pull-down list. (Be sure to do this in the rare cases when Design-Expert suggests more than one model.)
The options for process order At this stage you could press the Add Term button and insert higher degree terms with integer powers, such as quartic (4th degree). However, for this case study, we’ll leave the selection at Quadratic. You could now manually reduce the model by clicking off insignificant effects. For example, you will see in a moment that several terms in this case are marginally significant at best. Design-Expert also provides several automatic reduction algorithms as alternatives to the “Manual” method: “Backward,” “Forward” and “Stepwise.” Click the down arrow on the Selection list box to use these. Click on the ANOVA button to produce the analysis of variance for the selected model. The ANOVA table is available in two views. By default it will add text providing brief explanations and guidelines to the reported statistics. To turn this off, choose View, Annotated ANOVA. Notice that this toggles off the check mark (√).
Statistics for selected model: ANOVA table
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The ANOVA in this case confirms the adequacy of the quadratic model (the Model Prob>F is less than 0.05.) You can also see probability values for each individual term in the model. You may want to consider removing terms with probability values greater than 0.10. Use process knowledge to guide your decisions. Next, see that Design-Expert presents various statistics to augment the ANOVA – most notably various R-Squared values. These look very good.
Post-ANOVA statistics Scroll down to bring the following details on model coefficients to your screen. The mean effect shift for each block is listed here too. (Under certain circumstances the display may be adversely affected when scrolled. To rectify this problem, maximize the screen by clicking the icon at upper right of Windows.)
Coefficients for the quadratic model Again scroll down to bring the next section to your screen: the predictive models in terms of coded versus actual factors (shown side-by-side below). Block terms are left out. These terms can be used to re-create the results of this experiment, but they cannot be used for modeling future responses.
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Final equation: coded versus actual You cannot edit any of the ANOVA outputs. However, you can copy and paste the data to your favorite Windows word processor or spreadsheet.
Diagnose the Statistical Properties of the Model
The diagnostic details provided by Design-Expert can best be digested by viewing plots the come with a click on the Diagnostics button. The most important diagnostic, the normal probability plot of the residuals, comes up by default.
Normal probability plot of the residuals
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The data points should be approximately linear. A non-linear pattern (look for an Sshaped curve) indicates non-normality in the error term, which may be corrected by a transformation. There are no signs of any problems in our data. At the left of the screen you see the Diagnostics Tool palette. First of all, notice that residuals will be studentized unless you uncheck the first box on the floating tool palette (not advised). This counteracts varying leverages due to location of design points. For example, the center points carry little weight in the fit and thus exhibit low leverage. Each button on the palette represents a different diagnostics graph. Check out the other graphs if you like. Explanations for most of these graphs were covered in prior Tutorials. In this case, none of the graphs indicate any cause for alarm. Now click the option for Influence. Here’s where you find the find plots for externally studentized residuals (better known as “outlier t”) and other plots that may be helpful for finding problem points in the design. Also, from here you can click Report to bring up a detailed case-by-case diagnostic statistics, many of which have already been shown graphically. (In previous versions of Design-Expert, this report appeared under ANOVA.)
Diagnostics report The note below the table (“Predicted values include block corrections.”) alerts you that any shift from block 1 to block 2 will be included for purposes of residual diagnostics. (Recall that block corrections did not appear in the predictive equations shown in the ANOVA report.) Also note that one value of DFFITS is flagged. As we discussed in the General One-Factor Tutorial (Part 2 – Advanced Features), this statistic stands for difference in fits. It measures the change in each predicted value that occurs when that response is deleted. To see what program Help says about DFFFITs, right-click the number.
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Accessing context-sensitive Help Given that only this one diagnostic is flagged, it probably is not a cause for alarm. Press X to close out the screen tip provided by the program’s Help system.
Examine Model Graphs
The diagnosis of residuals reveals no statistical problems, so you will now generate the response surface plots. Click on the Model Graphs button. The 2D contour plot of factors A versus B comes up by default in graduated color shading.
Response surface contour plot Note that Design-Expert will display any actual point included in the design space shown. In this case you see a plot of conversion as a function of time and temperature at a mid-level slice of catalyst. This slice includes six center points as indicated by the dot at the middle of the contour plot. By replicating center points, you get a very good power of prediction at the middle of your experimental region. The Factors Tool comes along with the default plot. Move this floating tool as needed by clicking on the top blue border and dragging it. The tool controls which factor(s) are
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plotted on the graph. The Gauges view is the default. Each factor listed will either have an axis label, indicating that it is currently shown on the graph, or a red slider bar, which allows you to choose specific settings for the factors that are not currently plotted. The red slider bars will default to the midpoint levels of the factors not currently assigned to axes. You can change a factor level by dragging the red slider bars or by right clicking on a factor name to make it active (it becomes highlighted) and then typing the desired level in the numeric space near the bottom of the tool palette. Click on the C:Catalyst toolbar to see its value. Don’t worry if it shifts a bit – we will instruct you on how to reset it in a moment.
Factors tool with factor C highlighted and value displayed Click down on the red bar with your mouse and push it to the right.
Slide bar for C pushed right to higher value As indicated by the color key on the left, the surface becomes ‘hot’ at higher response levels, yellow in the ’80’s and red above 90 for conversion. To enable a handy tool for reading coordinates off contour plots, go to View, Show Crosshairs Window.
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Showing crosshairs window Now move your mouse over the contour plot and notice that Design-Expert generates the predicted response for specific values of the factors that correspond to that point. If you place the crosshair over an actual point, for example – the one at the upper left corner of the graph now on screen, you also get that observed value (in this case: 66).
Prediction at coordinates of 40 and 90 where an actual run was performed Now press the Default button to put factor C back at its midpoint. Then switch to the Sheet View by clicking on the Sheet button.
Factors tool – Sheet view
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In the columns labeled Axis and Value you can change the axes settings or type in specific values for factors. Return to the default view by clicking on the Gauges button. At the bottom of the Factors Tool is a pull-down list from which you can also select the factors to plot. Only the terms that are in the model are included in this list. If you select a single factor (such as A) the graph will change to a One Factor Plot. From this view, if you now choose a two-factor interaction term (such as AC) the plot will become the interaction graph of that pair. The only way to get back to a contour graph is to use the menu item View, Contour.
Perturbation Plot
Wouldn’t it be handy to see all your factors on one response plot? You can do this with the perturbation plot, which provides silhouette views of the response surface. The real benefit from this plot is for selecting axes and constants in contour and 3D plots. Use the View, Perturbation menu item to select it.
The Perturbation plot with factor A clicked to highlight it For response surface designs the perturbation plot shows how the response changes as each factor moves from the chosen reference point, with all other factors held constant at the reference value. Design-Expert sets the reference point default at the middle of the design space (the coded zero level of each factor). Click on the curve for factor A to see it better. (The software will highlight it with a different color.) In this case, you can see that factor A (time) produces a relatively small effect as it changes from the reference point. Therefore, because you can only plot contours for two factors at a time, it makes sense to choose B and C, and slice on A.
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Contour Plot: Revisited
Let’s look at the contour plot of factors B and C. Return to the contour plots via the View, Contour selection.
Back to Contour view In the Factors Tool right click on the Catalyst bar palette. Then select X1 axis by left clicking on it.
Making factor C the x1-axis You now see a catalyst versus temperature plot of conversion, with time held as a constant at its midpoint. The colors are neat, but what if you must print the graphs in black and white? That can be easily fixed by right-clicking over the graph and selecting Graph Preferences.
Graph preferences Click the Graphs 2 tab and change the Contour graph shading to Std Error shading.
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Design-Expert 软件在响应面优化法中的应用 （王世磊郑州大学450001） 摘要：本文简要介绍了响应面优化法，以及数据处理软件Design-ExpertDesign-Expert的相关知识，最后结合实例，介绍该软件在响应面优化法上的应用实例。 关键词：数据处理，响应面优化法，Design-Expert软件 1.响应面优化法简介 响应面优化法,即响应曲面法( Response Surface Methodology ，RSM)，这是一种实验条件寻优的方法，适宜于解决非线性数据处理的相关问题。它囊括了试验设计、建模、检验模型的合适性、寻求最佳组合条件等众多试验和统计技术；通过对过程的回归拟合和响应曲面、等高线的绘制、可方便地求出相应于各因素水平的响应值[1]。在各因素水平的响应值的基础上，可以找出预测的响应最优值以及相应的实验条件。 响应面优化法，考虑了试验随机误差；同时，响应面法将复杂的未知的函数关系在小区域内用简单的一次或二次多项式模型来拟合，计算比较简便，是降低开发成本、优化加工条件、提高产品质量、解决生产过程中的实际问题的一种有效方法[2]。 响应面优化法，将实验得出的数据结果，进行响应面分析，得到的预测模型，一般是个曲面，即所获得的预测模型是连续的。与正交实验相比，其优势是：在实验条件寻优过程中，可以连续的对实验的各个水平进行分析，而正交实验只能对一个个孤立的实验点进行分析。 当然，响应面优化法自然有其局限性。响应面优化的前提是：设计的实验点应包括最佳的实验条件，如果实验点的选取不当，使用响应面优化法师不能得到很好的优化结果的。因而，在使用响应面优化法之前，应当确立合理的实验的各因素与水平。 结合文献报道，一般实验因素与水平的选取，可以采用多种实验设计的方法，常采用的是下面几个： 1.使用已有文献报道的结果，确定响应面优化法实验的各因素与水平。 2.使用单因素实验[3]，确定合理的响应面优化法实验的各因素与水平。 3.使用爬坡实验[4]，确定合理的响应面优化法实验的各因素与水平。 4.使用两水平因子设计实验[5]，确定合理的响应面优化法实验的各因素与水平。 在确立了实验的因素与水平之后，下一步即是实验设计。可以进行响应面分析的实验设计有多种，但最常用的是下面两种：Central Composite Design-响应面优化分析、Box-Behnken Design-响应面优化分析。 Central Composite Design，简称CCD，即中心组合设计，有时也成为星点设计。其设计表是在两水平析因设计的基础上加上极值点和中心点构成的，通常实验表是以代码的形式编排的,实验时再转化为实际操作值（,一般水平取值为0,±1,±α,其中0为中值,α为极值, α=F*（1/ 4）； F 为析因设计部分实验次数, F = 2k或F = 2 k×（1/ 2 ），其中 k为因素数，F = 2 k×（1/ 2 一般 5 因素以上采用，设计表有下面三个部分组成[6]：(1) 2k或 2 k×（1/ 2 ）析因设计。(2)极值点。由于两水平析因设计只能用作线性考察,需再加上第二部分极值点,才适合于非线性拟合。如果以坐标表示,极值点在相应坐标轴上的位置称为轴点(axial point) 或星点( star point) ,表示为(±α,0,…, 0) , (0,±α,…, 0) ,…, (0, 0,…,±α)星点的组数与因素数相同。(3)一定数量的中心点重复试验。中心点的个数与CCD设计的特殊性质如正交
食品科学研究中实验设计的案例分析 —响应面法优化超声波辅助酶法制备燕麦ACE抑制肽的工艺研究 摘要：选择对ACE 抑制率有显著影响的四个因素：超声波处理时间(X1)、超声波功率(X2)、超声波水浴温度(X3)和酶解时间(X4)，进行四因素三水平的响应面分析试验，经过Design-Expert优化得到最优条件为超声波处理时间28.42min、超声波功率190.04W、超声波水浴温度55.05℃、酶解时间2.24h，在此条件下燕麦ACE 抑制肽的抑制率87.36%。与参考文献SAS软件处理的结果中比较差异很小。 关键字：Design-Expert 响应面分析 1.比较分析 表一响应面试验设计 因素 水平 -1 0 1 超声波处理时间X1(min) 20 30 40 超声波功率X2(W) 132 176 220 超声波水浴温度X3(℃) 50 55 60 酶解时间X4(h) 2.Design-Expert响应面分析 分析试验设计包括：方差分析、拟合二次回归方程、残差图等数据点分布图、二次项的等高线和响应面图。优化四个因素（超声波处理时间、超声波功率、超声波水浴温度、酶解时间）使响应值最大，最终得到最大响应值和相应四个因素的值。 利用Design-Expert软件可以与文献SAS软件比较，结果可以得到最优，通过上述步骤分析可以判断分析结果的可靠性。 2.1 数据的输入
2.2 Box-Behnken响应面试验设计与结果 2.3 选择模型
2.4 方差分析 在本例中，模型显著性检验p<0.05，表明该模型具有统计学意义。由图4知其自变量一次项A，
B，D，二次项AC,A2,B2,C2,D2显著(p<0.05)。失拟项用来表示所用模型与实验拟合的程度，即二者差异的程度。本例P值为0.0861>0.05，对模型是有利的，无失拟因素存在，因此可用该回归方程代替试验真实点对实验结果进行分析。 图 5 由图5可知：校正决定系数R2(adj)(0.9788>0.80)和变异系数（CV）为0.51%，说明该模型只有2.12%的变异，能由该模型解释。进一步说明模型拟合优度较好,可用来对超声波辅助酶法制备燕麦ACE抑制肽的工艺研究进行初步分析和预测。
试验设计与优化方法，都未能给出直观的图形，因而也不能凭直觉观察其最优化点，虽然能找出最优值，但难以直观地判别优化区域．为此响应面分析法（也称响应曲面法）应运而生．响应面分析也是一种最优化方法，它是将体系的响应（如萃取化学中的萃取率）作为一个或多个因素（如萃取剂浓度、酸度等）的函数，运用图形技术将这种函数关系显示出来，以供我们凭借直觉的观察来选择试验设计中的最优化条件． 显然，要构造这样的响应面并进行分析以确定最优条件或寻找最优区域，首先必须通过大量的量测试验数据建立一个合适的数学模型（建模），然后再用此数学模型作图． 建模最常用和最有效的方法之一就是多元线性回归方法．对于非线性体系可作适当处理化为线性形式．设有m个因素影响指标取值，通过次量测试验，得到n组试验数据．假设指标与因素之间的关系可用线性模型表示，则有应用均匀设计一节中的方法将上式写成矩阵式或简记为式中表示第次试验中第个因素的水平值；为建立模型时待估计的第个参数；为第次试验的量测响应（指标）值；为第次量测时的误差．应用最小二乘法即可求出模型参数矩阵B如下将B阵代入原假设的回归方程，就可得到响应关于各因素水平的数学模型，进而可以图形方式绘出响应与因素的关系图． 模型中如果只有一个因素（或自变量），响应（曲）面是二维空间中的一条曲线；当有二个因素时，响应面是三维空间中的曲面．下面简要讨论二因素响应面分析的大致过程． 在化学量测实践中，一般不考虑三因素及三因素以上间的交互作用，有理由设二因素响应（曲）面的数学模型为二次多项式模型，可表示如下：通过n次量测试验（试验次数应大于参数个数，一般认为至少应是它的3倍），以最小二乘法估计模型各参数，从而建立模型；求出模型后，以两因素水平为X坐标和y坐标，以相应的由上式计算的响应为Z坐标作出三维空间的曲面（这就是2因素响应曲面）．应当指出，上述求出的模型只是最小二乘解，不一定与实际体系相符，也即，计算值与试验值之间的差异不一定符合要求．因此，求出系数的最小二乘估计后，应进行检验．一个简单实用的方法就是以响应的计算值与试验值之间的相关系数是否接近于1或观察其相关图是否所有的点都基本接近直线进行判别．如果以表示响应试验值，为计算值，则两者的相关系数R定义为其中对于二因素以上的试验，要在三维以上的抽象空间才能表示，一般先进行主成分分析进行降维后，再在三维或二维空间中加以描述．等等………… 2注意事项 对于构造高阶响应面，主要有以下两个问题： 1，抽样数量将显著增加，此外，普通的实验设计也将更糟。 2，高阶响应面容易产生振动。 响应面法(response surface methodology，记为RSM)最早是由数学家Box和Wilson于1951年提出来的。就是通过一系列确定性的“试验”拟合一个响应面来模拟真实极限状态曲面。其基本思想是假设一个包括一些未知参量的极限状态函数与基本变量之间的解析表达式代替实际的不能明确表达的结构极限状态函数。响应面方法是一项统计学的综合试验技术，用于处理几个变量对一个体系或结构的作用问题，也就是体系或结构的输入(变量值)与输出(响应)的转换关系问题。现用两个变量来说明:结构响应Z与变量x1,x2具有未知的、不能明确表达的函数关系Z=g(x1,x2)。要得到“真实”的函数通常需要大量的模拟，而响应面法则是用有限的试验来回归拟合一个关系Z= g’(x1,x2)，并以此来代替真实曲面Z=g(x1,x2)，将功能函数表示成基本随机变量的显示函数，应用于可靠度分析中。响应面方法实际上源于一种试验设计方法，试验设计方法是用来研究设计参数对模型设计状况影响的一种取样策略，决定了构造近似模型所需样本点的个数和这些点的空间分布情况。目前广泛应用于计算机仿真试验设计的主要方法是拉丁超立方体抽样和均匀设计，这两种试验设计能应用于多种多样的模型，且对模型的变化具有稳健性。 3响应面分析
响应面法试验设计与优化方法，都未能给出直观的图形，因而也不能凭直觉观察其最优化点，虽然能找出最优值，但难以直观地判别优化区域．为此响应面分析法（也称响应 曲面法）应运而生．响应面分析也是一种最优化方法，它是将体系的响应（如萃取化学中的萃取率）作为一个或多个因素（如萃取剂浓度、酸度等）的函数，运用图 形技术将这种函数关系显示出来，以供我们凭借直觉的观察来选择试验设计中的最优化条件． 显然，要构造这样的响应面并进行分析以确定最优条件或寻找最优区域，首先必须通过大量的量测试验数据建立一个合适的数学模型（建模），然后再用此数学模型 作图． 建模最常用和最有效的方法之一就是多元线性回归方法．对于非线性体系可作适当处理化为线性形式．设有ｍ个因素影响指标取值，通过次量测试验，得到ｎ组试验 数据（）．假设指标与因素之间的关系可用线性模型表示，则有应用均匀设计一节中的方法将上式写成矩阵式或简记为式中表示第次试验中第个因素的水平值；为建 立模型时待估计的第个参数；为第次试验的量测响应（指标）值；为第次量测时的误差．应用最小二乘法即可求出模型参数矩阵Ｂ如下将Ｂ阵代入原假设的回归方 程，就可得到响应关于各因素水平的数学模型，进而可以图形方式绘出响应与因素的关系图．模型中如果只有一个因素（或自变量），响应（曲）面是二维空间中的一条曲线；当有二个因素时，响应面是三维空间中的曲面．下面简要讨论二因素响应面分析的 大致过程． 在化学量测实践中，一般不考虑三因素及三因素以上间的交互作用，有理由设二因素响应（曲）面的数学模型为二次多项式模型，可表示如下：通过ｎ次量测试验 （试验次数应大于参数个数，一般认为至少应是它的３倍），以最小二乘法估计模型各参数，从而建立模型；求出模型后，以两因素水平为Ｘ坐标和ｙ坐标，以相应 的由上式计算的响应为Ｚ坐标作出三维空间的曲面（这就是２因素响应曲面）． 应当指出，上述求出的模型只是最小二乘解，不一定与实际体系相符，也即，计算值与试验值之间的差异不一定符合要求．因此，求出系数的最小二乘估计后，应进 行检验．一个简单实用的方法就是以响应的计算值与试验值之间的相关系数是否接近于１或观察其相关图是否所有的点都基本接近直线进行判别．如果以表示响应试 验值，为计算值，则两者的相关系数Ｒ定义为其中 对于二因素以上的试验，要在三维以上的抽象空间才能表示，一般先进行主成分分析进行降维后，再在三维或二维空间中加以描述．
学校 食品科学研究中实验设计的案例分析 —响应面法优化超声波辅助酶法制备燕麦ACE抑制肽的工艺研究 摘要：选择对ACE 抑制率有显著影响的四个因素：超声波处理时间(X1)、超声波功率(X2)、超声波水浴温度(X3)和酶解时间(X4)，进行四因素三水平的响应面分析试验，经过Design-Expert优化得到最优条件为超声波处理时间28.42min、超声波功率190.04W、超声波水浴温度55.05℃、酶解时间2.24h，在此条件下燕麦ACE 抑制肽的抑制率87.36%。与参考文献SAS软件处理的结果中比较差异很小。 关键字：Design-Expert 响应面分析 1.比较分析 表一响应面试验设计 因素 水平 -1 0 1 超声波处理时间X1(min) 20 30 40 超声波功率X2(W) 132 176 220 超声波水浴温度X3(℃) 50 55 60 酶解时间X4(h) 1 2 3 2.Design-Expert响应面分析 分析试验设计包括：方差分析、拟合二次回归方程、残差图等数据点分布图、二次项的等高线和响应面图。优化四个因素（超声波处理时间、超声波功率、超声波水浴温度、酶解时间）使响应值最大，最终得到最大响应值和相应四个因素的值。 利用Design-Expert软件可以与文献SAS软件比较，结果可以得到最优，通过上述步骤分析可以判断分析结果的可靠性。
2.1 数据的输入 图 1 2.2 Box-Behnken响应面试验设计与结果 图 2 2.3 选择模型
2.4 方差分析 在本例中，模型显著性检验p<0.05，表明该模型具有统计学意义。由图4知其自变量一次项A，
响应面法： 响应面法的基本思想是通过近似构造一个具有明确表达形式的多项式来表达隐式功能函数.本质上来说,响应面法是一套统计方法,用这种方法来寻找考虑了输入变量值的变异或不确定性之后的最佳响应值[1]。 响应面法与正交法的区别 正交试验设计则注重如何科学合理地安排试验，可同时考虑几种因素，寻找最佳因素水平组合；但它不能在给出的整个区域上找到因素和响应值之间的一个明确的函数表达式即回归方程，从而无法找到整个区域上因素的最佳组合和响应值的最优值．因此，人们期望找到一种试验次数少、周期短，求得的同归方程精度高、能研究几种因素间交互作用的回归分析方法，响应面分析方法的很大程度上满足了这些要求． 响应面法与正交法的应用 1.童心等在正交实验联用响应面法优化脱皮马勃总生物碱提取的研究中得到了响应面法比正交法能得到更精确的因素水平量，从而有更好的实验结果。 传统的正交设计方法是一种用线性数学模型进行设计的设计方法，可以找出多个因素水平的最佳组合。但是正交设计只能分析离散型数据，具有精度不高，预测性不佳的缺点。响应面法采用非线性模型，能求得高精度的回归方程，进行合理预测来找出最优工艺条件[2]。 张崟等在响应面法和正交试验对骨素酶解工艺优化的比较实验中发现响应面法得出的最优工艺所得的水解度比正交试验高出15.4 %[3]。 总的来说，正交实验比响应面设计实验次数更少，但是响应面设计可以得到一个回归方程，并能得到精确的因素水平值。 参考文献 [1] 王永菲,响应面法的理论与应用 [2] 童心, 正交实验联用响应面法优化脱皮马勃总生物碱提取的研究 [3] 张崟, 响应面法和正交试验对骨素酶解工艺优化的比较
响应面分析法优化反应条件的中心组合设计分组 组数pH 时间温度摩尔比 一. 加热温度为130℃（17组） （1）（9组） 1 3 4.50 10.00 130.00 1.25 3 17 4.50 150.00 130.00 1.25 25 6 7.00 80.00 130.00 1.25 26 11 7.00 80.00 130.00 1.25 27 14 7.00 80.00 130.00 1.25 28 8 7.00 80.00 130.00 1.25 29 5 7.00 80.00 130.00 1.25 4 24 9.50 150.00 130.00 1.25 2 1 9.50 10.00 130.00 1.25 （2）（4组） 9 18 4.50 80.00 130.00 0.50 21 16 7.00 10.00 130.00 0.50 22 20 7.00 150.00 130.00 0.50 10 13 9.50 80.00 130.00 0.50 （3）（4组） 11 7 4.50 80.00 130.00 2.00 23 27 7.00 10.00 130.00 2.00 24 22 7.00 150.00 130.00 2.00 12 21 9.50 80.00 130.00 2.00 组数pH 时间温度摩尔比 二. 加热温度为110℃（6组） 5 19 7.00 80.00 110.00 0.50 7 10 7.00 80.00 110.00 2.00 13 28 7.00 10.00 110.00 1.25 14 26 7.00 150.00 110.00 1.25 17 12 4.50 80.00 110.00 1.25 18 23 9.50 80.00 110.00 1.25 三．加热温度为150℃（6组） 6 9 7.00 80.00 150.00 0.50 8 4 7.00 80.00 150.00 2.00 15 15 7.00 10.00 150.00 1.25 16 29 7.00 150.00 150.00 1.25 19 2 4.50 80.00 150.00 1.25 20 25 9.50 80.00 150.00 1.25
Response Surface Methods (RSM) 响应面设计方法通过更多的水平实验方案，拟合二阶以上的模型，帮助我们找到设计的最优点。 BLOCK 例如：本实验需要分两天完成，那么两天中因为其他不可控制因素的变化可能会对试验造成影响，那么就可以设置2个BLOCK，软件会在两个BLOCK中设置对应的几个中点试验重复，检查中点试验的重复性是否良好，以观察这些不可控制因素对试验造成多大影响，从而最大限度的降低试验中不可控制因素对试验的干扰。再例如，本实验其中一部分在甲实验室完成，另一部分要在乙实验室完成，那么就可以设置2个BLOCK，原因同上。 回归分析（Analysis ） Transform: 对模型做一些数学变换，比如对数变换、倒数变换，目的是让因子和响应之间的关系变得简单，比如线性化 Fit Summary: 对模型做不同种类的拟合，比如线性拟合、二次拟合、三次拟合等等，目的是帮助我们看看哪种拟合效果最好F(x) Model: 在选定数学变化，以及决定采用哪种拟合方式以后就可以在这里对模的细节进行设置了，比如要保留那些因子项和交互项。
ANOVA: 方差分析，软件会自动对模型进行拟合，然后根据残差对各种因素的贡献做方差分析，让我们知道那些项是关键的，必须在模型中保留。方程的显著性检验，系数显著性检验回归方程。 Diagnostics: 在做完拟合之后，用图示的方式给出分析结果，比如残差的正态性、分布的随机性等等 Model Graph: 用图形的方式告诉用户模型是什么样子的，比如用等高线来描述响应和因子之间的函数关系。 预测优化（Optimization） Criteria: 得到模型之后我们就可以用它来预测最佳的设计参数是多少了。这里Criteria是给出优化的条件，比如各个因子的取值范围、优化的目标是什么等等 Solution: 在Criteria中设定了优化的约束和目标之后，这里就会给出优化的结果，一般是用列表的形式给出一些详细设计参数供我们参考 Graphs: 用图形的方式给出解空间的形状，或者是解在设计空间的位置，如右图。 Diagnostics Tools Diagnostics Normal Plot of Residuals 残差的正态图，越接近一条直线，说明回归模型拟合得越好Residuals vs. Predicted
响应面分析 响应面实验考察的范围比较窄，如果不先确定存在最大响应值的区域的话，很有可能在响应面实验时无法得到最值。在B&B上有一篇文章就通过具体的实例证明了这一点:第一次响应面没有得到最值，经过分析发现考察区域本身不存在最值点。经过进一步搜索后确定了一个存在最值的区域，再进行响应面实验就成功了。最陡爬坡法就是一个经典的搜索考察区域、逼近最值空间的方法。 最陡爬坡法在运用中存在两个问题，一是爬坡的方向，二是爬坡的步长。前者根据效应的正负就可以确定:如果某个因素是正效应，那么爬坡时就增加因素的水平;反之，即减少因素水平。而对应爬坡步长，则要稍微复杂些。 以下是自己对软件使用的一些想法，挺凌乱的，怕日后忘了，先写下来: 应用design expert应注意的问题:在析因实验设计中，如果至少有一个是数量因子，则在分析中得到的fit summary是不可靠的，不能应用其中suggest的方程(线性/二次/三次等，一般来说suggest都是一次方程)，如何选择方程要尽量考虑以下几点: 1.尽量考虑较高次的方程 2.满足所选方程不会aliased(在方差分析里看) 3.model要显著(在方差分析里看) https://www.doczj.com/doc/ee7285331.html,ck of fit要不显著(在方差分析里看)。 5. 诊断项里的残差要近似符合正态分布。 特别是第四条，如果发现lack of fit显著了，那么很可能是漏掉了某项交互作用，对于A B两因素的二次方程而言，如果出现 lack of fit ,考虑下是否漏掉A2B AB2 A2B2 等. 只有当试验中有重复的点时，才能计算拟合不足。
对于响应面设计而言:由于一般的响应面设计就那几种，如2因素，得到的方程就绝对不会含有A2B AB2 A2B2 这些项，这是因为响应面设计的实验点数太少，这些项就如同A3 B3一样会被aliased的。 总之两句话:对于响应面设计，在f(x)里的model比较简单，都是二次的，一般默认的那几个A, B , AB, A2 ,B2就OK了。 对于含有数量因子的析因设计，如发现因子间存在二次关系，这个时候就要小心了，除了响应面里面的那几个外，是否还存在 A2B, AB2, A2B2等(判断标准就是上面5条) 要注意的是，析因实验与响应面设计的一个区别是:析因必须对每个因素的每个水平交叉重复N次(N>=2)，对于析因实验来说，不重复就无法分离交互作用和纯误差对响应变量的贡献。而响应面只需对中心点重复N次(由响应面的方法而定)，其余的点做一次就够了。 lack of fit，失拟检验，评估模型的拟合度。如果 p 值小于您选择的 a 水平，则证明模型未能与数据准确拟合。您可能需要添加项，或者变换数据，以便更准确地为数据建模。插值和拟合是统计中经常用到的两种方法，是解决离散点近似符合某函数的问题实验或者实际测量得到一系列的点A1,A2,A3,… 想要知道这些点近似符合哪个函数 插值就是经过这些点，做出多项式函数或者其他函数，来作为这些点的近似函数而拟合就是尽量靠近这些点，做出多项式函数或者其他函数，来作为这些点的近似函数 这两个的区别是:插值出来的函数肯定要过所有的点，也就是说所有的点都在这个函数上。而拟合出来的函数不一定过所有的点，但所有的点到这个函数的距离的某个运算式是最小的，或者说拟合出来的函数是所有近似函数中误差最小的那个。
DX7-04C-MultifactorRSM-P1.doc Rev. 4/12/06
Multifactor RSM Tutorial (Part 1 – The Basics)
Response Surface Design and Analysis
This tutorial shows the use of Design-Expert? software for response surface methodology (RSM). This class of designs is aimed at process optimization. A case study provides a real-life feel to the exercise. Due to the specific nature of the case study, a number of features that could be helpful to you for RSM will not be exercised in this tutorial. Many of these features are used in the General One Factor, RSM One Factor or Two-Level Factorial tutorials. If you have not completed all of these tutorials, consider doing so before starting in on this one. We will presume that you can handle the statistical aspects of RSM. For a good primer on the subject, see RSM Simplified (Anderson and Whitcomb, Productivity, Inc., New York). You will find overviews on RSM and how it’s done via Design-Expert in the online Help system. To gain a working knowledge of RSM, we recommend you attend our Response Surface Methods for Process Optimization workshop. Call Stat-Ease or visit our website, https://www.doczj.com/doc/ee7285331.html,, for a schedule. The case study in this tutorial involves production of a chemical. The two most important responses, designated by the letter “y”, are: ? ? y1 - Conversion (% of reactants converted to product) y2 - Activity.
The experimenter chose three process factors to study. Their names and levels can be seen in the following table.
Factor
A – Time B - Temperature C - Catalyst
Units
minutes degrees C percent
Low Level (-1)
40 80 2
High Level (+1)
50 90 3
Factors for response surface study You will study the chemical process with a standard RSM design called a central composite design (CCD). It’s well suited for fitting a quadratic surface, which usually works well for process optimization. The three-factor layout for the CCD is pictured below. It is composed of a core factorial that forms a cube with sides that are two coded units in length (from -1 to +1 as noted in the table above). The stars represent axial points. How far out from the cube these should go is a matter for much discussion between statisticians? They designate this distance “alpha” – measured in terms of coded factor levels. As you will see Design-Expert offers a variety of options for alpha.
Design-Expert 7 User’s Guide
Multifactor RSM Tutorial – Part 1 ? 1