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All we know is that this flat surface is one side of the 'hill'. If we start at 0, in coded units, then we can do a series of single experiments on this path up the 'hill' of the steepest descent. If we take steps of 1 in coded units, this would be five minutes in terms of the time units. And for each step along that path, we would go up 0. The response is plotted and shows an increase that drops off towards the end. This is a pretty smooth curve and in reality, you probably should go a little bit more beyond the peak to make sure you are at the peak.

But all you are trying to do is to find out approximately where the top of the 'hill' is. If your first experiment is not exactly right you might have gone off in the wrong direction!

So you might want to do another first-order experiment just to be sure. Or, you might wish to do a second order experiment, assuming you are near the top. This is what we will discuss in the next section. The second order experiment will help find a more exact location of the peak. The point is, this is a fairly cheap way to 'scout around the mountain' to try to find where the optimum conditions are.

Remember, this example is being shown in two dimensions but you may be working in three or four-dimensional space! You can use the same method, fitting a first-order model and then moving up the response surface in k dimensional space until you think you are close to where the optimal conditions are. If you are in more than 2 dimensions, you will not be able to get a nice plot. But that is OK. The method of steepest ascent tells you where to take new measurements, and you will know the response at those points.

You might move a few steps and you may see that the response continued to move up or perhaps not - then you might do another first order experiment and redirect your efforts. The point is, when we do the experiment for the second order model, we hope that the optimum will be in the range of the experiment - if it is not, we are extrapolating to find the optimum. In this case, the safest thing to do is to do another experiment around this estimated optimum.

Since the experiment for the second order model requires more runs than experiments for the first order model, we want to move into the right region before we start fitting second-order models. This second order model includes linear terms, cross product terms and a second order term for each of the x's. The linear terms just have one subscript.

The quadratic terms have two subscripts. To fit this model, we are going to need a response surface design that has more runs than the first order designs used to move close to the optimum. This second order model is the basis for response surface designs under the assumption that although the hill is not a perfect quadratic polynomial in k dimensions, it provides a good approximation to the surface near the maximum or a minimum.

Assuming that we have 'marched up this hill' and if we re-specified the region of interest in our example, we are now between 80 - 90 in terms of time and - in terms of temperature. We would now translate these natural units into our coded units and if we fit the first order model again, hopefully we can detect that the middle is higher than the corner points so we would have curvature in our model, and could now fit a quadratic polynomial. After using the Steepest Ascent method to find the optimum location in terms of our factors, we can now go directly to the second order response surface design.

A favorite design that we consider is sometimes referred to as a central composite design. The central compositive design is shown on Figure Pick a radius along this line and place a new point at that radius. The effect is that each factor is now measured at 5 levels - center, 2 corners and the 2 star points.

This gives us plenty of unique treatments to fit the 2nd order model with treatment degrees of freedom left over to test the goodness of fit. Replication is still usually done only at the center point. RSM dates from the 's. Early applications were found in the chemical industry. We have already talked about Box.

Box and Draper have some wonderful references about building RSMs and analyzing them which are very useful. In many experiments more than one response is of interest for the experimenter. Furthermore, we sometimes want to find a solution for controllable factors which result in the best possible value for each response. This is the context of multiple response optimization, where we seek a compromise between the responses; however, it is not always possible to find a solution for controllable factors which optimize all of the responses simultaneously.

Multiple response optimization has an extensive literature in the context of multiple objective optimization which is beyond the scope of this course. Here, we will discuss the basic steps in this area.

As expected, multiple response analysis starts with building a regression model for each response separately. For instance, in Example One of the traditional methods way to analyze and find the desired operating condition one is overlaid contour plots.

This method is mainly useful when we have two or maybe three controllable factors but in higher dimensions it loses its efficiency. This method simply consists of overlaying contour plot for each of the responses one over another in the controllable factors space and finding the area which makes the best possible value for each of the responses. Figure This area might be of special interest for the experimenter because they satisfy given conditions on the responses.

Another dominant approach for dealing with multiple response optimization is to form a constrained optimization problem. The Design-Expert software package solves this approach using a direct search method. Another important procedure that we will discuss here, also implemented in Minitab, is the desirability function approach.

In this approach the value of each response for a given combination of controllable factors is first translated to a number between zero and one known as individual desirability. Individual desirability functions are different for different objective types which might be Maximization, Minimization or Target. If the objective type is maximum value, the desirability function is defined as. Individual desirability is then used to calculate the overall desirability using the following formula:.

Now, the design variables should be chosen so that the overall desirability will be maximized. A favorite design that we consider for a second order model is referred to as a central composite design. Here is an example in two dimensions: Example The value of these points is something greater than 1. Why is it something greater than 1? If you think about the region of experimentation, we have up to now always defined a box, but if you think of a circle the star points are somewhere on the circumference of that circle, or in three dimensions on the ball enclosing the box.

All of these are design points around the region where you expect the optimum outcome to be located. Typically the only replication, in order to get some measure of pure error, is done at the center of the design.

The data set for the Example The analysis using the Response Surface Design analysis module is shown in the Ex In the last section we looked at the Example In this section we examine a more general central composite design. This is a common design. Much of this detail is given in Table As the number of factors increases, you can see the efficiencies that are brought to bear. The spherical designs are rotatable in the sense that the points are all equidistant from the center.

Rotatable refers to the variance of the response function. A rotatable design exists when there is an equal prediction variance for all points a fixed distance from the center, 0.

This is a nice property. If you pick the center of your design space and run your experiments, all points that are equal distance from the center in any direction, have an equal variance of prediction. You can see in the table above that the difference in the variation between the spherical and rotatable designs are slight, and don't seem to make much difference.

But both ideas provide justification for selecting how far away the star points should be from the center. Why do we take about five or six center points in the design? The reason is also related to the variance of a predicted value.

When fitting a response surface you want to estimate the response function in this design region where we are trying to find the optimum. We want the prediction to be reliable throughout the region, and especially near the center since we hope the optimum is in the central region. By picking five to six center points, the variance in the middle is approximately the same as the variance at the edge.

If you only had one or two center points, then you would have less precision in the middle than you would have at the edge. As you go farther out beyond a distance of 1 in coded units, you get more variance and less precision.

What we are trying to do is to balance the precision at the edge of the design relative to the middle. How do you select the region where you want to run the experiment?

Remember, for each factor X we said we need to choose the lower level is and the upper level for the region of experimentation. We usually picked the -1 and 1 as the boundary.

If the lower natural unit is really the lowest number that you can test, because the experiment won't work lower than this, or the lower level is zero and you can't put in a negative amount of something, then, the star point is not possible because it is outside the range of experimentation. Generally, if you are not up against a boundary then this is not an issue and the star points are a way to reach beyond the region that you think the experiment should be run in.

The issue isn't selecting the coding of the design relative to the natural units. You might lose some of these exact properties, but as long as you have the points nicely spread out in space you can fit a regression function. The penalty for not specifying the points exactly would be seen in the variance, and it would be actually very slight.

We can create central composite designs using a full factorial, central composite designs with fractional factorials, half fraction and a quarter fraction, and they can be arranged in blocks. Later, we will look at the Box-Behnken designs. Here is the design that results:. Block 2 is the second half fraction of the factorial part with two center points.

The third block consists of six star points, plus to center points. Each of the three blocks contains two center points and the first two blocks have half of the corner points each. The third block contains the star points and is of size 8.

Rollover the words 'Block 1', 'Block 2', and 'Block 3' in the graphic above. Do you see how they use center points strategically to tie the blocks together? They are represented in each block and they keep the design connected. They are either up or down, in or out, right or left. The center points have zero on all three axes, truly the center of this region.

We have designed this to cover the space in just the right way so that we can estimate a quadratic equation. Using a Central Composite Design, we can't estimate cubic terms, and we can't estimate higher order interactions.

However, we would have wasted a lot of resources to do it. The CCD allows us to estimate just linear and quadratic terms and first order interactions. This example is from the Box and Draper book and the data from Tables 9. This example has three variables and they are creating a polymer, a kind of plastic that has a quality of elasticity. The measurement in this experiment is the level of elasticity. We created the design in Minitab for this experiment, however the data only has two center points:.

Variables A and B are the concentration of two ingredients that make up the polymer, and C is the temperature, and the response is elasticity. There are 8 corner points, a complete factorial, 6 star points and 2 center points. Find optimal factor settings. Source Estimate Std. However, many statisticians do not think an interaction term should be included in a model unless both main effects are also included. Interaction plots confirm the need for an interaction term in the model.

We perform a residuals analysis to validate the model assumptions. We generate a normal plot, a box plot, a histogram and a run-order plot of the residuals. The residual plots do not indicate problems with the underlying assumptions. From the above output, we make the following conclusions.

The R 2 is reasonable for fitting Uniformity well known to be a difficult response to model. The residual plots do not reveal any major violations of the underlying assumptions. The interaction plot shows why an interaction term is needed parallel lines would suggest no interaction. A comparison of the full model and the model containing just the main effects and squared pressure terms indicates that there is no significant difference between the two models.

Thus, we will proceed with the model containing main effects and the squared pressure term. The fact that the stepwise procedure selected a model for Stress containing a term that was not significant indicates that all output generated by statistical software should be carefully examined.

In this case, the stepwise procedure identified the model with the lowest AIC Akaike information criterion , but did not take into account contributions by individual terms. Other software using a different criteria may identify a different model, so it is important to understand the algorithms being used.

We perform a residuals analysis to validate the model by generating a run-order plot, box plot, histogram, and normal probability plot of the residuals. The residual plots do not indicate any major violations of the underlying assumptions. Now enter the response Name and Units for each response as shown below. 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. Click the Tips button for a refresher. Design layout your run order may differ due to randomization. Click the File menu item and select Save As. Assume that the experiment is now completed. 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. 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. Now right-mouse click the Select column header top left cell and choose Space Point Type.

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 Design-Expert. 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. You should see a scatter plot with factor A:Time on the X-axis and the Conversion response on the Y-axis.

The correlation grid that pops up with the Graph Columns can be very interesting. Block versus run or, conversely, run vs block is also highly correlated due to this restriction in randomization runs having to be done for day 1 before day 2. It is good to see so many white squares because these indicate little or no correlation between factors, thus they can be estimated independently.

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 floating Graph Columns tool click the button where Conversion intersects with Block as shown below. The graph visually shows there is not much of a difference between block 1 and 2. The points on Day 1 and Day 2 are both spread around about the same average value. Bear in mind that whatever the difference may be between blocks, it will be filtered out mathematically so as not to bias the estimation of factor effects.

Finally, 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. Plotting one response versus the other resulting graph not shown. You may also note there is a faded pink color in the box you clicked in the grid to get this graph, denoting a slight upward correlation as you go from left to right on the graph.

Feel free to make other scatter plots, but the ones done thus far will be most valuable so it will be best to move on from here. Notice you can also color selected factors, including run default. For example, choose Color by Block to see which points were run in block 1 black and block 2 red.

However, do not get carried away with this, because it will be much more productive to do statistical analysis first — before drawing any conclusions. Under the Analysis branch click the node labeled Conversion.

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? 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. Now click the Fit Summary tab. At this point Design-Expert fits linear, two-factor interaction 2FI , quadratic, and cubic polynomials to the response. By design, the central composite matrix provides too few unique design points to determine all the terms in the cubic model.

Next you will see several extremely useful tables for model selection. Each table is discussed briefly via sidebars in this tutorial on RSM. Use the blue layout buttons to choose how many panes are visible on your screen at once.

For each source of terms linear, etc. So far, Design-Expert 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. 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. Always confirm this suggestion by viewing these tables. From the main menu select Help, Screen Tips or simply press the lightbulb icon for more information about the procedure for choosing model s.

Design-Expert 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. If you want, you can choose an alternative model from the Process Order pull-down list. Be sure to try this in the rare cases when Design-Expert suggests more than one model. Also, 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. 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. Use process knowledge to guide your decisions.

The R-Squared statistics are very good — near to 1. 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. Press Coded Equation to bring the next section to your screen — the predictive models in terms of coded factors. Click Actual Equation for the 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. However, you can copy and paste the data to your favorite word processor or spreadsheet. This might be handy for clients who are phobic about statistics. The diagnostic details provided by Design-Expert can best be grasped by viewing plots available via the Diagnostics tab.