Stefan H. Steiner
University of Waterloo
Los Alamos National Laboratory
Bethany J. Giddings White
University of Waterloo
University of Waterloo
Journal of Statistics Education Volume 15, Number 1 (2007), www.amstat.org/publications/jse/v15n1/steiner.html
Copyright © 2007 by Stefan H. Steiner, Michael Hamada, Bethany J. Giddings White, Vadim Kutsyy, Sofia Mosesova, and Geoffrey Salloum all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.
Key Words:Constrained experimental region; Generalized linear model; Optimal design; Poisson regression; Robust parameter design.
The experiment involved the simple task of blowing soap bubbles. The objective was to determine an optimal bubble solution consisting of a mixture of water, dishwashing soap and glycerin. Minimal prior knowledge of the science behind bubbles is necessary to complete this project. However, in addition to the educational value associated with planning, conducting and analyzing an experiment, students also have the opportunity to learn some science; bubbles are soap films which people have devoted their entire careers to studying (Boys 1959; Isenberg 1992).
In our teaching experience three separate groups of students have conducted versions of this project. The experimental design and data come from one of these groups. For the illustrative purposes of this article, we present a possible analysis of these data. The instructor has the flexibility to vary what is required and the amount of guidance offered depending on the level of the students' knowledge. Throughout the article, we discuss how the bubble project can be adapted in a variety of ways for use in teaching.
We present the issues and choices that the students need to confront in completing this project by following the structured empirical problem-solving framework PPDAC (Oldford and MacKay 2001). PPDAC is an acronym for a five-stage process:
We find PPDAC useful to structure the design and analysis of any experiment and have used it in our teaching and consulting. We present the bubble mixture experiment project by discussing each of the five stages of PPDAC. At each stage, we describe the choices made in our example project along with their rationale. As well, we discuss options for making the project either less or more complicated.
However, we start by giving some background on mixture experiments. To complete this project the students must recognize that the bubble experiment is a mixture experiment and apply the appropriate methods. This may mean independent research for the students.
Figure 1: A Two Dimensional View of the Three Component Mixture Experimental Region. All mixtures must lie on or inside the triangle whose equation is S+W+G=1.
In the bubble project, the experimental region is further constrained because we know, before running the experiment, that some combinations will not produce any bubbles (e.g. 100% water) or be very poor. We discuss this issue further in the Plan section.
For a mixture experiment, the standard linear first-order response model for a normally distributed response Y is
Note that there is no intercept in this model. If the intercept were included, the design matrix would be singular because the proportions of water, glycerin and soap must add to one (i.e. S+W+G=1). Additional models commonly used for mixture experiments include the quadratic model, special cubic model, and the rarely used full-cubic model (Cornell 2002):
The choices of response and population attribute are tied to the definition of best or optimal bubble mixture, and have a big impact on an appropriate analysis of the subsequent experimental data. In the bubble project, the students decided an optimal solution produced, on average, the most bubbles. Therefore, the response of interest was the number of bubbles and the attribute was the average number of bubbles across all conditions in the population. Other possible choices for response include: the size of the biggest bubble, the time the bubbles survive, whether or not any bubbles where made, etc. In each case, it is critical that the student think about both whether the response reflects their definition of optimal, and whether the proposed response can be precisely and accurately measured. Students can be quite inventive; for instance, to measure time, they thought of blowing bubbles onto a dish so that the bubble can be observed until it pops. A possible extension to the basic project is to use more than one response. This will lead to multiple analyses and possibly the need to compromise in some way when making recommendations.
In the choice of a response, the instructor needs to be careful the project does not become too complex for the students to handle. To assess this, they need to think ahead to the analysis stage. For instance, with a response defined as the number of bubbles, a standard regression model with normally distributed errors may need to be replaced by a Poisson regression model. Poisson regression is an advanced topic that students would learn about in a course on generalized linear models. Alternatively, a standard analysis with the usual normal error assumption that ignores the discrete nature of the count response may be acceptable. An appropriate choice depends on the data. Another possibility is an analysis on a transformed scale. For instance, in another application of this experiment, time to a bubble popping was used as the response and the log time was well modeled using a standard normal regression model. The instructor should be prepared to discuss the pros and cons of each modeling choice. The choice of response can be a great lead-in to a class discussion of model assumptions and more advanced types of models.
Next, the students needed to define their target population by answering the question “average over what?” For example, they may decide they want to find the best bubble solution for all different conditions, including different bubble making devices, environmental conditions, etc. Alternatively, they may decide to look for the best bubble solution only for typical summer conditions (e.g. hot and humid) in their part of the world. In a class project, the students have control over how ambitious they want to be. In other applications, the goals of an experiment are usually driven by outside considerations, and we need to try to achieve them through a good plan that makes appropriate use of the experimental principles of blocking, replication and random assignment. The students typically come back to the definition of the population after they have given more thought in the Plan stage to what is realistic, given the time and cost constraints inherent in a class project.
Figure 2: Bubble Experiment Cause-and-Effect Diagram.
While the students should not spend a large amount of effort developing a cause and effect diagram, they should try to think broadly about all the possible inputs that could influence their chosen response. There are many explanatory variates that could (with our limited knowledge) influence the number of bubbles produced, of which many are not mixture components. Consideration of all the possibly important explanatory variates can have a large impact on the experimental plan.
In the example, the students decided to deliberately vary the water type and soap brand in the experiment as well as the three mixture components: soap, water and glycerin. By choosing two different water types (spring and tap) and two different dishwashing soap brands (Joy® and Ivory®), they planned to also consider whether the optimal bubble formulation depended on the levels of these two factors. As well, they decided to look for a robust bubble solution that works well no matter what sort of water or soap was used. See Taguchi (1987) for more on robust design experiments. Using the robust design terminology we refer to soap brand and water type as noise factors.
What should be done with all the other possibly important explanatory variates? One option is to (try to) hold them fixed during the experiment. This makes the experiment easier to run (assuming it is not difficult to hold the explanatory variates fixed), but restricts the generalizability of the conclusions. Remember the choice of target population in the Problem stage. Another option is to create blocks within which some explanatory variates are held fixed, and then replicate the design over a number of blocks. For instance, if we thought there might be a large day effect, we could block by day and repeat the design over two (or more) different days. This helps achieve the goal of assessing the effect of changing the mixture components while at the same time allowing us to check that the results obtained under one set of conditions (one day) are also valid for other conditions (i.e. the second day).
In the example, the students decided to hold all other explanatory variates fixed. This choice was made mostly because it was easy. Blocking was rejected because the planned experiment was already complex with three mixture factors and two noise factors. Also, while they felt that temperature and humidity were probably important explanatory variates, running an experiment where they changed the temperature and humidity was not practical given the available resources.
To accurately compare the bubble formulations, they used a battery-operated device to blow the bubbles. This avoided possible problems associated with a person blowing bubbles such as speed of a person's breath and temperature. An assortment of bubble-blowing toys were considered and tested with a standard store-bought bubble solution. One of the most consistent battery-operated devices was chosen to run the experiment. To ensure reliable results, the batteries were replaced prior to running the experiment. Also for consistency, each student had one task in the running the experiment. One student made all the bubble solutions; a second operated the bubble making device, while the third student did all the bubble counting and recording. This was a good idea as it reduced the chance that student-to-student differences could influence the results.
These considerations led the students to the following constraints on the experimental region: G 0.15, S 0.35, and W 0.98; the first two constraints imply that 0.50 W. Then, based on some trial and error, the students decided they also needed a lower limit on the amount of soap. As a result, they further restricted the design region with S 0.04. Finally, because glycerin is much more expensive than dish soap which is in turn more expensive than water, they decided to restrict water to greater than 60% (i.e. W 0.6). Thus the final experimental region was defined by the constraints: 0.04 S 0.35, 0.60 W 0.98 and G 0.15,. This selected constrained experimental region is displayed in the right panel of Figure 3. These decisions are somewhat arbitrary and should be reviewed for future projects using the results presented later in this article. Alternatively, the cost constraints could be imposed more formally through a cost function.
|Figure 3a||Figure 3b|
Figure 3: Constrained Experimental Region G 0.15,
0.5 W 0.98
Left panel shows constrained region relative to complete simplex. Right panel shows the constrained region in close-up.
From the restricted experimental region the students must select a design. In the example, the students selected 12 component mixtures spread out over the constrained experimental region based on their judgment. The 12 chosen mixtures are shown in Figure 4. They used 12 formulations because they felt that was a manageable number and would allow them to fit the special cubic model (3) while still retaining some degrees of freedom for assessing model fit.
Figure 4: Proposed Mixture Design Showing the 12 Mixture Proportions.
Clearly other choices of design points and the number of mixture combinations are possible and are probably better. Students can extend the project by looking for an improved design. However, finding an optimal design within a constrained mixture region is nontrivial. The best design, defined in terms of prediction error, depends on the proposed model. One possibility for finding near optimal designs is to use a genetic algorithm as in Goldfarb, Borror, Montgomery and Anderson-Cook (2005). Finding an optimal design in the context of a generalized linear model makes the task harder still. Another alternative is to consider distance-based (space filling) designs where the selection criterion attempts to spread the design points uniformly over the feasible region (see Johnson et al. 1990). Distance-based designs have the advantage that they are not model dependent.
To explore how varying the type of water and soap brand would affect the number of bubbles produced, the students used a crossed control-by-noise array. In other words, they ran 48 trials using all possible combinations of the twelve bubble formulations, two types of water and two brands of soap. As mentioned earlier all other explanatory variates in the cause-and-effect diagram were to be held fixed during the experiment. Another option is to add more noise factors and conduct a screening experiment to determine the set of factors that contribute to the effectiveness of the bubble solution. With many noise factors, a fractional design in the noise factors could have been used.
To reduce the effect of potentially unknown confounding variables, the order in which the students created and ran the treatments was randomized. The design and run order (as well as the results) are given in Appendix A.
To partially counteract any variability that could be attributed to the bubble device, the students also planned to pass each solution through the bubble blowing process five times (i.e. they conducted five repeats for each run). The difference between replicates, where each of the 48 bubble formulations would be made again from scratch, and repeats, where the same bubble formulation is used to generate additional responses, is very important for the students to understand. Clearly replication captures additional sources of error, such as mixing variation. We discuss the issue of repeats versus replicates further in the Analysis section.
In some applications of this project censoring of the response could become an issue. If there were a large number of bubbles, it would not be possible to accurately count the actual number of bubbles. Instead, we could then record a right-censored value (e.g. there were at least 25 bubbles). Similarly, with time to a bubble popping as the response, right censoring may arise if the times are too long and the bubbles have not popped after, say, half an hour.
There were a number of practical issues that arose while conducting the experiment. One of the primary difficulties was measuring the exact proportions of glycerin and soap. Glycerin, in particular, is very viscous and is difficult to pour into solutions. To aid in the process, the students measured the required amount of water first and then tried to remove any remaining glycerin and soap by dipping the measuring spoons into the water. This was a very effective technique, as they seemed to get most of the glycerin and soap off the measuring spoons.
The experiment was time consuming to run, with mixing the 48 solutions taking the bulk of the time. This is clearly an important consideration that should be considered in the Plan stage. Also, it can be very helpful to make a few trial runs before conducting the whole experiment. This allows everyone to get a better idea of the process and time involved as well as learn a lot about the logistics of running the experiment. If necessary, the experimental plan could be altered and/or scaled back if time constraints are a concern.
Figure 5: Plot of Number of Bubbles in Each Repeat by Run
Solid Line Showing the Average Count at Each Run.
Figure 5 shows there are no wild outliers. Also, while it is difficult to assess within run (i.e. repeat to repeat) variation from only five repeats, there is no evidence that a Poisson assumption (i.e. that the standard deviation should increase as the square root of the mean) is unreasonable.
In our example there are five repeats per run. There is no replication because each of the 48 bubble solutions was made only once (this was enough work!). In modeling, using the repeats as replicates does not result in an appropriate analysis as the experimental error would likely be underestimated and there would be too many significant effects. The standard approach to handle repeats is to calculate a performance measure, such as an average or sum, across the repeats for each run and to do the analysis using the performance measure. In the example, the students decided to use the total number of bubbles produced across the five repeats as the performance measure. This is appropriate in this case as they planned to fit a Poisson regression and the sum of independent Poisson counts is still Poisson with mean equal to the sum of individual means.
Next, the students plotted the data to informally examine the effect of the noise factors. Figure 6 shows the total number of bubbles across the five repeats by mixture formulation with different plotting symbols for the two levels of each of the two noise factors. Clearly the number of bubbles produced does not seem to depend much on the water type, but depends strongly on the type of soap, with Joy producing many more bubbles, on average, than Ivory.
|Water Noise Factor||Soap Noise Factor|
|Figure 6a||Figure 6b|
Figure 6: Total Bubble Count (across the five repeats) Versus Mixture Combination
Left plot: Solid circle = Spring, Open circle = Tap for Water noise factor. Right plot: Solid circle = Joy, Open circle = Ivory for Soap noise factor.
Now they were ready to consider formal models. Because the chosen performance measure was the total number of bubbles produced across the five repeats for each run, the students fit a generalized linear model (McCullagh and Nelder 1989) with a Poisson response and a log link to the data.
In general, the Poisson regression for a response Y is written:
where is the mean number of bubbles, X is a vector of covariates and is a vector with the corresponding coefficients. Due to the inclusion of additional covariates (noise factors) that are not a part of the mixture the students needed to consider extensions to models (1) – (4) to model . For example, coding the two levels of the noise factors: water type (NW) and soap brand (NW), as –1 and +1, the necessary extension to the quadratic model (2) is given by (5). Note that as we add noise factors, the number of model parameters multiply by the number of noise combinations.
The students fit both the extended quadratic (5) and special-cubic forms of the mixture model. The special-cubic model crossed with separate terms for each of the four combinations of the noise factors had 28 parameters while the corresponding quadratic model had 24 parameters. Fitting the quadratic and special cubic models gave residual deviances of 100.85 and 93.825 respectively. To compare the models they tested the hypothesis . Comparing the likelihood ratio test statistic (7.025) to a chisquare distribution with 4 degrees of freedom resulted in a p-value of 0.135. They also determined the Akaike Information Criterion (AIC) for the special-cubic model and the AIC for the quadratic model and got 390.91 and 389.94 respectively. Because they could not reject the null hypothesis and the quadratic model had a smaller AIC, the students concluded that it was preferred; i.e. the additional special-cubic terms did not help the model fit.
In the quadratic model fit, the students then noticed that none of the terms involving the water type noise factor were significant. As a result, all such terms were removed from the model leading to what they referred to as the reduced quadratic model. See Appendix B for the details of the reduced quadratic model fit. The significant terms included: WS, GNS, WGNS and SGNS. The model fit is summarized as
As there are still non significant parameters in (6) a further reduction of the model is possible. However, for simplicity in the presentation we proceed with model (6).
The deviance residuals from the reduced quadratic model (6) are plotted in Figure 7. There are no obvious problems with the model fit, and there is no pattern over time.
Figure 7: Deviance Residuals From Reduced Quadratic Model Versus Run Order and Fitted Values.
To explore the results from the reduced quadratic model fit, contour plots of the response for each of the soap brands are given in Figure 8. Note that the response surface gives predictions on a single repeat basis, i.e. we exponentiate (6) and divide by five.
The students produced Figure 8 by randomly generating many points in the constrained region, calculating their predicted response using (6), identifying those points whose response is near some selected values and finally plotting the points.
|Water Noise Factor||Soap Noise Factor|
|Figure 8a||Figure 8b|
Figure 8: Contour Plots Showing the Predicted Number of Bubbles Per Repeat
Left panel: Ivory soap, Right panel: Joy soap
Asterisk in plot gives the maximum prediction.
Based on Figure 8, the students noted that for Ivory soap, the maximum predicted number of bubbles per repeat was 4.3 at the component proportions (S, W, G) = (0.31, 0.69, 0). While for Joy, the maximum was much higher at 16.8, and occurred at the component proportions (S, W, G) = (0.31, 0.60, 0.09). These optimal bubble solutions are shown in Figure 8 using an asterisk. Given the fitted model (6), it is also possible to optimize the response more formally using a constrained optimization routine.
In the example, the students also looked for a bubble solution that was robust across the two different soap brands. The best way to do this is not clear. Looking at the two contour plots in Figure 8 they can use their judgment to compromise. For instance, component proportions near (S, W, G) = (0.3, 0.65, 0.05) are predicted to give, on average, close to 15 bubbles for Joy and 4 for Ivory. There are many other possible ways to compromise. For instance, they could have instead tried to maximize the minimum average response. Another alternative is to use formal optimization, perhaps to minimize the larger-the-better loss, (Steiner and Hamada 1997, equation (5)), where and are the mean and variance of the response over the noise factor soap brand. However, this criterion was developed with the normal regression model in mind. For this experiment, minimizing this loss leads to a solution that drives the variance down at the expense of the mean, and results in a undesirable bubble solution with small variability but also a small mean for both types of soap. An interesting study for the students would be to develop a more meaningful criterion for the Poisson regression model.
Note that in cases where a continuous normal response is chosen, the standard modeling could be used in place of the generalized linear model. Also, without the noise factors there would be a quarter as many runs and no need to consider robustness.
As another alternative to using generalized linear models, the students could do a Bayesian analysis with the Poisson regression model (Gelman et al. 2004). This is relatively easy to do with WINBUGS (Spiegelhalter, Thomas, Best, and Lunn 2004). An advantage of a Bayesian approach is that the correlations between the parameter estimates (i.e. regression coefficients) are captured in the draws from the joint posterior distribution of the parameters. Moreover, the uncertainty on predicted means or predictons that are functions of the parameters are easily obtained.
Another option is to give conclusion based on a robust bubble solution that works reasonable well no matter what soap brand is used. Clearly something in between is also possible. Perhaps we are willing to make a recommendation for the soap brand since they can be easily bought, but want a bubble mixture that will work well over a variety of water hardness. In our example this does not change our conclusions since water type was found to have little influence.
In any case, the students should comment on possible limitations to their conclusions. In the example, all environmental variates were held fixed during the experiment. This may cause concern about generalizing the results. Perhaps, for other environmental conditions, a different bubble mixture would be optimal. This seems likely given our belief that temperature and humidity have a strong influence. How much our results are affected by measurement or mixing errors is also a worry. Note also that, in any case, with only two types of dish soap in the experiment any proposed “robust” solution is only robust across Joy and Ivory, not (necessarily) across all brands of dish soap available.
As part of the conclusion, the students could also comment on the choices made in the problem or plan stage. For instance, setting the goal to maximize the average number of bubbles may not have been the best choice if the proposed best bubble solution only produced many very small bubbles. This would probably not meet with the approval of any children testing our proposed solution. In the example, the students concluded that spring and tap water were not the best choices for their two water types. They had intended on comparing two water types with very different mineral content. However, it is likely that both the tap and spring water were quite hard, i.e. had substantial mineral content. This could have been the reason the quadratic model suggested water type was not important. A better choice may have been to use distilled water as one of the two water types.
To strengthen their conclusions, very eager students can conduct an additional simple experiment to confirm that the proposed optimal solution produces close to the predicted number of bubbles on average. If the results can be validated in this way, we would have more confidence in the fitted model and their conclusions in general. Another possibility is to suggest this as further work (for the next students!).
To make the project even larger in scope, the presented experiment could be seen as the first in a series of experiments to look for a better bubble solution. In other words, the students or class could be asked to run a series of experiments and use response surface methods. Though it is not clear how best to do this in the context of a crossed mixture by noise array. Each new experiment results in another application of PPDAC and would be expected to build on the knowledge gained in previous experiments. For example, each new experiment could further restrict (or move) the constrained experimental region.
In addition to these areas, students are also exposed to group collaboration and obtain valuable experience in report writing and/or presenting. In the project outlined in this article, a number of statistical/mathematical software packages were used. Although it is not required that students use these particular programs, the experience gained through use of the computer will greatly contribute to their learning. Throughout this article we made a number of suggestions regarding specific issues that may benefit students in planning future experiments. More ideas for projects involving mixture models can be found in Sahrmann, Piepel and Cornell (1987), Anderson (1997) and Piepel and Piepel (2000). Further examples in experimental design in general are given in Hunter (1977).
|Proportions||Number of Bubbles in Repeat|
For soap brand: –1 = Joy and +1 = Ivory
For water type: –1 = spring and +1 = tap
glm(formula = y1 ~ -1 + W + S + G + WS + WG + SG + Wn + Sn + Gn + WSn + WGn + SGn, family = poisson) %n corresponds to NS Model: poisson, link: log Df Deviance Resid. Df Resid. Dev NULL 48 8701.7 W 1 7033.0 47 1668.7 S 1 909.1 46 759.5 G 1 79.2 45 680.4 WS 1 24.9 44 655.5 WG 1 7.8 43 647.6 SG 1 7.5 42 640.1 Wn 1 433.3 41 206.8 Sn 1 52.5 40 154.3 Gn 1 0.3 39 154.0 WSn 1 1.3 38 152.7 WGn 1 1.6 37 151.0 SGn 1 31.0 36 120.0 Deviance Residuals: Min 1Q Median 3Q Max -2.7643 -1.0658 -0.1526 0.8362 4.6545 Coefficients: Estimate Std. Error z-value Pr(>|z|) W 2.3938 0.1703 — — S -3.3553 2.1339 — — G 17.7969 10.3578 — — WS 13.9705 3.5458 3.940 8.15e-05 *** WG -17.5638 12.3313 -1.424 0.1544 SG -5.2577 12.5849 -0.418 0.6761 Wn -0.1587 0.1703 -0.932 0.3515 Sn 2.2537 2.1339 1.056 0.2909 Gn 47.2846 10.3578 4.565 4.99e-06 *** WSn -5.2169 3.5458 -1.471 0.1412 WGn -53.1718 12.3313 -4.312 1.62e-05 *** SGn -69.7290 12.5849 -5.541 3.01e-08 *** Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for poisson family taken to be 1) Null deviance: 8701.69 on 48 degrees of freedom Residual deviance: 120.03 on 36 degrees of freedom AIC: 385.12 Number of Fisher Scoring iterations: 4
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Stefan H. Steiner
Department of Statistics
University of Waterloo
Los Alamos National Laboratory
Los Alamos, NM
Bethany J. Giddings White
Department of Statistics
University of Waterloo
Los Altos, CA
Department of Statistics
University of Waterloo
Department of Mathematics
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