Response Surface¶

Part 1/3 – The Basics¶

Response Surface Design and Analysis¶

This tutorial, the first of three in this series, shows how to use Stat-Ease 360^® 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.

If you are in a rush to get the gist of design and analysis of RSM, hop past all the “Note” sections. However, if/when you find the time, it will be well worth the effort to explore these “by-the-ways.”

We will presume that you are knowledgeable about the statistical aspects of RSM. For a good primer on the subject, see RSM Simplified (Anderson and Whitcomb, Productivity, Inc., New York, 2016). To gain a working knowledge of RSM, we recommend you attend our Modern DOE for Process Optimization workshop. Visit www.statease.com and follow the “Learn DOE” link for more details on this and other educational resources from Stat-Ease.

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 are shown in the following table.

Factors for response surface study

You will study the chemical process using 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.

Note

The three-factor layout for this 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, the program offers a variety of options for alpha.

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 and click the blank-sheet icon on the left of the toolbar and then click Response Surface from the list of designs on the left to show the designs available for RSM.

Response surface menu¶

The default selection is the Central Composite design, which is used in this case study.

If not already entered, click the up arrow in the Numeric Factors entry box and Select 3 as shown below.

Selecting three numeric factors¶

Note

Before entering factors and ranges, click Options. Notice that it defaults to a Rotatable design with the axial (star) points set at 1.68179 coded units from the center – a conventional choice for the CCD.

Default CCD option for alpha set so design is rotatable¶

Many 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 3^¼ 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 equals one). Press OK to accept the rotatable value. (Note: you won’t get the “center points in each axial block” option until you change to 2 blocks in this design, as below).

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 Low and High levels.

Completed factor form¶

You’ve now specified the cubical portion of the CCD. As you did this, the program calculated the coded distance “alpha” for placement on the star points in the central composite design.

Now return to the bottom of the central composite design form. Leave Type at its default value of Full (the other option is a “small” CCD, which we do not recommend unless you must reduce the number of runs to the bare minimum). You will need two blocks for this design, one for each day, so click the Blocks field and select 2.

Selecting the number of blocks¶

Notice the software displays how this CCD will be laid out in the two blocks – for example, 4 center points will go in one and 2 in the other. Click Next 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 Next to enter Responses. Select 2 from the pull down list. Now 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 Finish to view the design layout (your run order may differ due to randomization).

Design layout (your run order may differ due to randomization)¶

Enter the Response Data – Create Simple Scatter Plots¶

Assume that the experiment is now completed. At this stage, the responses must be entered into the program. We see no benefit to making you type all the numbers, particularly with the potential confusion due to differences in randomized run orders. Therefore, use the Help, Tutorial Data menu and select Chemical Conversion from the list.

Let’s examine the data! Click on the Design node on the left to view the design spreadsheet. Move your cursor to Std column header and right-click to bring up a menu from which to select Sort Ascending (this can also be done via a double-click on the header).

Sorting by Standard (Std) Order¶

Now right-mouse click the Select column header (top left cell) and choose Space Point Type.

Displaying the Point Type¶

Notice the new column identifying points as “Factorial,” “Center” (for center point), and so on. Notice how the factorial points align only to the Day 1 block. Then in Day 2 the axial points are run. Center points are divided between the two blocks.

Unless you change the default setting for the Select option, do not expect the Type column to appear the next time you run the program. It is only on temporarily at this stage for your information.

Before focusing on modeling the response as a function of the factors varied in this RSM experiment, it will be good to assess the impact of the blocking via a simple scatter plot. Click the Graph Columns node branching from the design ‘root’ at the upper left of your screen. You should see a scatter plot with factor A:Time on the X-axis and the Conversion response on the Y-axis.

For now, it is most useful to produce a plot showing the impact of blocks because this will be literally blocked out in the analysis. Therefore, on the Graph Columns tool click the button where Conversion intersects with Block as shown below. Then change Color By to Space Type.

Plotting the effect of Block on Conversion¶

The graph visually shows there is not much of a difference between the center point results for block 1 and 2. Bear in mind that this will be filtered out mathematically so as not to bias the estimation of factor effects.

Graph Columns feature for design layout¶

Change the Y Axis to Activity (by clicking down the column one box) to see how it’s affected by the day-to-day blocking (even less).

Changing response (resulting graph not shown)¶

Next, to see how the responses correlate, change the X Axis to Conversion. Now that we have 2 numeric factors along the axes, we can see the correlation between them. In the upper left of the legend you will see the correlation number is 0.224, showing slight correlation.

Plotting one response versus the other (resulting graph not shown)¶

You may also note there is a faded pink color in the box in the grid for this graph, denoting the slight upward correlation.

Now for a really awesome scatterplot in 3D, change the X Axis to A:Time, the Y Axis to C:Catalyst and the Z-Axis to Conversion. This provides a dramatic view of conditions leading to maximizing the response. Grab it with your mouse and rotate it around. This looks quite promising!

3D scatterplot enabled by plotting the response on the Z Axis as a function of two factors¶

Continue exploring relationships with the graph columns tools. However, do not get carried away with this, because it will be much more productive to do statistical analysis first – before drawing any conclusions.

Analyze the Results¶

Now let’s start analyzing the responses numerically. Under the Analysis branch click the node labeled Conversion and press the Start Analysis button. A new set of tabs 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¶

Stat-Ease 360 provides a full array of response transformations via the Transform option. Click Tips for details. For now, accept the default transformation selection of None.

Now click the Fit Summary tab. At this point the program fits linear, two-factor interaction (2FI), quadratic, and cubic polynomials to the response. At the top is the response identification, immediately followed below, 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 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 tables for model selection. Each table is discussed briefly via sidebars in this tutorial on RSM.

The table of “Sequential Model Sum of Squares” (technically “Type I”) shows how terms of increasing complexity contribute to the total model.

Fit Summary tab¶

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 program is indicating (via bold highlighting) 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.) Move down to the Lack of Fit Tests pane for Lack of Fit tests on the various model orders.

The “Lack of Fit Tests” pane compares residual error with “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 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.

Look over the last pane in the Fit Summary report, which provides “Model Summary Statistics” for the ‘bottom line’ on comparing the options

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 viewing these tables.

The program allows you to select a model for in-depth statistical study. Click the Model tab at the top of the screen to see the terms in the model.

Model results¶

The program defaults to the “Suggested” model shown in the earlier Fit Summary table.

Click the ANOVA tab to produce the analysis of variance for the selected model.

Statistics for selected model: ANOVA table¶

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, move over to the Fit Statistics pane to see various statistics to augment the ANOVA. The R-Squared statistics are very good — near to 1.

Post-ANOVA statistics¶

Next, move down to the Coefficients pane to bring the following details to your screen, including the mean effect-shift for each block, that is; the difference from Day 1 to Day 2 in the response.

Coefficients for the quadratic model¶

Press Coded Equation to bring the next section to your screen — the predictive models in terms of coded factors. Click Actual Equation for the predictive models in terms of actual factors. 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.

Final equation: coded¶

Final equation: actual¶

You cannot edit any ANOVA outputs. However, you can copy and paste the data to your favorite word processor or spreadsheet. Also, as detailed in the One-Factor RSM tutorial, th eprogram provides a tool to export equations directly to Excel in a handy format that allows you to enter whatever inputs you like to generate predicted response. This might be handy for clients who are phobic about statistics. 😉

Diagnose the Statistical Properties of the Model¶

The diagnostic details provided by the program can best be grasped by viewing plots available via the Diagnostics tab. The most important diagnostic — normal probability plot of the residuals — appears in the first pane.

Normal probability plot of the residuals¶

Data points should be approximately linear. A non-linear pattern (such as an S-shaped curve) indicates non-normality in the error term, which may be corrected by a transformation. The only sign of any problems in this data may be the point at the far right. Click this on your screen to highlight it as shown above.

Now click the Resid. vs Run tab.

Residuals versus run¶

Now you can see that, although the highlighted run does differ more from its predicted value than any other, there is really no cause for alarm due to it being within the red control limits.

Next move to the Cook’s Distance tab.

Cook’s Distance — the first of the Influence diagnostics¶

Nothing stands out here.

Move on to the Leverage tab. This is best explained by the previous tutorial on One-Factor RSM so go back to that if you did not already go through it. Then skip ahead to DFBETAS, which breaks down the changes in the model to each coefficient, which statisticians symbolize with the Greek letter β, hence the acronym DFBETAS — the difference in betas. For the Term click the down-list arrow and select A as shown in the following screen shot.

DFBETAS for term A¶

You can evaluate ten model terms (including the intercept) for this quadratic predictive model (see sidebar below for help).

Now move on to the Report tab in the bottom-right pane to bring up detailed case-by-case diagnostic statistics, many which have already been shown graphically.

Diagnostics report¶

Examine Model Graphs¶

The residuals diagnosis reveals no statistical problems, so now let’s generate response surface plots. Click the Model Graphs tab. The 2D contour plot of factors A versus B comes up by default in graduated color shading.

Response surface contour plot¶

Note

The program displays 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 appears on the right with the default plot. Move this around as needed by clicking and dragging the top blue border (drag it back to the right side of the screen to “pin” it back in place. The tool controls which factor(s) are plotted on the graph.

Note

Each factor listed in the Factors Tool has either an axis label, indicating that it is currently shown on the graph, or a slider bar, which allows you to choose specific settings for the factors that are not currently plotted. All slider bars default to midpoint levels of those factors not currently assigned to axes. You can change factor levels by dragging their slider bars or by left-clicking factor names to make them active (they become highlighted) and then typing desired levels into the numeric space near the bottom of the tool. Give this a try.

Click the C: Catalyst toolbar to see its value. Don’t worry if the slider bar shifts a bit — we will instruct you how to re-set it in a moment.

Factors tool showing factor C highlighted and value displayed¶

Left-Click the bar with your mouse and drag 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.

Note

To enable a handy tool for reading coordinates off contour plots, go to View, Show Crosshairs Window (click and drag the titlebar if you’d like to unpin it from the left of your screen). Now move your mouse over the contour plot and notice that the program generates the predicted response for specific factor values corresponding to that point. If you place the crosshair over an actual point, for example – the one at the far upper left corner of the graph now on screen, you also see that observed value (in this case: 66).

Prediction at coordinates of 40 and 90 where an actual run was performed¶

P.S. See what happens when you press the Full option for crosshairs.

Now press the Default button on the Factors Tool to place factor C back at its midpoint.

Note

Open the Factors Sheet by clicking the Sheet… button on the Factors Tool.

Factors sheet¶

In the columns labeled Axis and Value you can change the axes settings by right-clicking, or type in specific values for factors. Give this a try. Then close the window and press the Default button.

P.S. The Terms list on the Factors Tool is a drop-down menu from which you can also select the factors to plot. Only the terms that are in the model are included in this list. At this point in the tutorial this should be set at AB. If you select a single factor (such as A) the graph changes to a One-Factor Plot. Try this if you like, but notice how the program warns if you plot a main effect that’s involved in an interaction.

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 of this plot is when selecting axes and constants in contour and 3D plots. See it by mousing to the Graphs Toolbar and pressing Perturbation or pull it up from the View menu via New Graph.

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. The program sets the reference point default at the middle of the design space (the coded zero level of each factor).

Click the curve for factor A to see it better. The software highlights it in a different color as shown above. It also highlights the legend. In this case, at the center point, you 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.

Contour Plot: Revisited¶

Let’s look at the plot of factors B and C. Start by clicking Contour on the Graphs toolbar. Then in the Factors Tool right-click the Catalyst bar and select X1 axis by left clicking 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.

Contour plot of B:temperature versus C:catalyst¶

The contour plots are highly interactive. For example, right-click up in the hot spot at the upper middle and select Add Flag.

Adding a flag¶

That’s enough on the contour plot for now — hold off until Part 3 of this tutorial to learn other tips and tricks on making this graph and others more presentable. Right-click and Delete flag to clean the slate.

3D Surface Plot¶

Now to really get a feel for how the response varies as a function of the two factors chosen for display, select 3D Surface from the Graphs Toolbar. You then will see three-dimensional display of the response surface. If the coordinates encompass actual design points, these will be displayed. On the Factors Tool move the slide bar for A:time to the right. This presents a very compelling picture of how the response can be maximized. Right-click at the peak to set a flag.

3D response surface plot with A:time at high level¶

You can see points below the surface by rotating the plot. Move your mouse over the graph. Click and hold the left mouse button and then drag.

Seeing a point beneath the surface¶

Seeing an actual result predicted so closely lends credence to the model. Things are really looking up at this point!

Remember that you’re only looking at a ‘slice’ of factor A (time). Normally, you’d want to make additional plots with slices of A at the minus and plus one levels, but let’s keep moving — still lots to be done for making the most of this RSM experiment.

Analyze the Data for the Second Response¶

This step is a BIG one. Analyze the data for the second response, activity. Be sure you find the appropriate polynomial to fit the data, examine the residuals and plot the response surface. Hint: The correct model is linear.

Before you quit, do a File, Save to preserve your analysis.