Design of Experiments - DOE with Centre Point

[tahirawan11]

Hi,

I have performed a 2^3 Full factorial DOE with 2 replicates and two centre points (total 18 runs). The ANOVA results @ aplha value 0.05 are given as below (ANOVA pic also attached):

Estimated Effects and Coefficients for Nr of dry spots (coded units)

Term Effect Coef SE Coef T P

Constant 3.563 0.4370 8.15 0.000
Flow rate 0.875 0.437 0.4370 1.00 0.343
Vacuum during infusion 1.125 0.562 0.4370 1.29 0.230
Mould temp. -2.625 -1.312 0.4370 -3.00 0.015
Flow rate*Vacuum infusion 0.375 0.187 0.4370 0.43 0.678
Flow rate*Mould temp. 1.625 0.812 0.4370 1.86 0.096
Vacuum infusion*Mould temp. -1.125 -0.563 0.4370 -1.29 0.230
Flow rate*Vacuum infusion* -0.375 -0.187 0.4370 -0.43 0.678
Mould temp.
Ct Pt 1.438 1.3110 1.10 0.301


S = 1.74801 PRESS = 108.978
R-Sq = 67.11% R-Sq(pred) = 0.00% R-Sq(adj) = 37.87%


Analysis of Variance for Nr of dry spots (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 35.6875 35.6875 11.8958 3.89 0.049
2-Way Interactions 3 16.1875 16.1875 5.3958 1.77 0.223
3-Way Interactions 1 0.5625 0.5625 0.5625 0.18 0.678
Curvature 1 3.6736 3.6736 3.6736 1.20 0.301
Residual Error 9 27.5000 27.5000 3.0556
Pure Error 9 27.5000 27.5000 3.0556
Total 17 83.6111


As the 'P' value for 'Curvature' is more than 0.05. Can i say that there is no curvature in my response variable and a linear model is good enough to describe the relationship between the variables and i do not need to proceed with a Response Surface Modelling DOE?

ANOVA also gives me 'Pure error', can anyone tell me what is the practicle meaning of 'Pure error' result in my case. Is it too big or low?

I also made 'Main effect plots' and 'Interaction plot' and the figures are attached. I dont understand why Mintab does not connect the line between the 'High value' and 'Low value' of main effect through the 'Centre point'?

thanks in advance

/tahir

 

[Tim Folkerts]

Re: DOE with Centre point

As the 'P' value for 'Curvature' is more than 0.05. Can i say that there is no curvature in my response variable and a linear model is good enough to describe the relationship between the variables and i do not need to proceed with a Response Surface Modelling DOE?

Or at least you can say there is no conclusive evidence that there is curvature, which is not quite the same thing

ANOVA also gives me 'Pure error', can anyone tell me what is the practicle meaning of 'Pure error' result in my case. Is it too big or low?

According to minitab help ...

Lack of fit (DOE)
If your design contains replicates (multiple runs at identical, but distinct combinations of factor settings), Minitab may calculate a pure error test for lack-of-fit. The error term can be partitioned into two parts - pure error (error within replicates) and lack-of-fit error, which represents degrees of freedom that are not in the model (e.g., higher-order interaction terms). You need both types of error to perform the lack-of-fit test. Depending on the model, sometimes you only have one of the types of error, but not both.


So pure error is a measure of how well the replicates correspond. In this case, the "pure error" is a pretty noticeable fraction of the total error.

This is backed up by the relatively low adjusted R^2. Only 38% of the variation is explained by your model, which means much of the error is due to the center points and/or the lack or repeatability in the replicates.

I think the main reason that the center point is not more clearly significant is that the replications have so much variabliity themselves.

I also made 'Main effect plots' and 'Interaction plot' and the figures are attached. I dont understand why Mintab does not connect the line between the 'High value' and 'Low value' of main effect through the 'Centre point'?

I think Minitab is simply trying to show you what the experiment predicts using a linear model (the straight lines on the plots). The fact that the center points do not fall close to the line is another indication that the model does not fully explain the variability.


Tim F

 

[Miner]

Re: DOE with Centre point

As Tim said, there is no conclusive evidence for curvature, and this does not mean the same thing as saying there is no curvature. Another consideration: there may be no curvature within the design space, but curvature outside of the design space. The design space is the volume within the factor levels.

You R^2 values are low because your model is over-fitted (i.e., contains too many terms). Remove the center point term first, then the 3-way interaction, then the 2-way interactions, re-evaluating your model between each iteration. Remove terms with the highest p-value first until you are left with only the terms with a significant p-value (i.e., p <= alpha).

After the model is simplified re-evaluate the R^2 values. If they are still low, there are three possibilities. One is that you have missing control factors; two, the process has significant noise factors; three, there is significant measurement error (what was your MSA as % Process Variation).

 

[Miner]

Re: Design of Expriments - DOE with Centre point

Estimated Effects and Coefficients for Nr of dry spots (coded units)

Does "Nr of Dry spots" mean number of dry spots?

If it does, you should perform a Freeman-Tukey transform on your count data prior to performing the analysis. Use Calc > Calculator > Function > Transform Count

When you finish simplifying the model, please attach your results including the residuals plot.

 

[tahirawan11]

Re: Design of Expriments - DOE with Centre point

Hi,

Yes, the 'Nr of dry spots' means 'Number of dry spots' and the purpose of the experiment is to minimize the response variable . The response variable is measured visually and i have not performed a MSA yet as it is very easy to see the 'Dry spots' visually and i don't suspect a large measurement error. The process in question is 'Transfer Moulding' process and the dry spots are visible in the end product after the curing process has finished.

I have transformed the count data and re-analysed the data. After reducing the model to include only significant terms i get an even lower R^2 values which indicates the model is not good for future use. The residuals are also not normally distributed and their variances also doesn't seems to be constant but i don't see any significant time factor effect. The pure error (5.27206) is still very significant in the total error term (8.12932). I have also noticed a 2-factor interaction (Flow rate * Mould temp.) is also almost significant; so if i take a higher risk (alpha = 0.1) then it will also become significant.

After analysing the results i discussed with engineers but we fail to identify any more 'Control factors' or 'Noise factors'. I am thinking of running a Taguchi L8 experiment and putting 'Mould temp' as a noise factor and try to find the parametrise settings where the process is less sensitive to variation (Less pure error). Any thoughts on that???

Many thanks.

 

[Miner]

Re: Design of Expriments - DOE with Centre point

Your interpretation is correct. Mold Temperature is very significant. I would also include the 2-way interaction. Your alpha risk is only 5.2%, not 10%.

The two terms explain 20% of the variation. The step is to identify what factors are still in the error term. Since you are using count data, the resolution is 1. If you are counting a 100 spots, this is 1%. If you are counting 5 spots, this is 20%. If you are dealing with a small number, this may account for your large error term. You could deal with this by running more repeats and counting the total number of spots for all repeats.

Some noise factors to consider are batch to batch material properties, setup to setup, etc.

 

[Renzoquim]

Hi Tahirawan11,

Center points are used for two reasons: get an estimate of curvature and provide an estimate and some degrees of freedom for an error term. Now as a guideline, the number of center points runs should be at least as large as the number of factors in your experiment.
According to your results, the p-value says that is unlikely that the null hypothesis is false (Ho: there is no curvature). However, my opinion is to delete the center points and no significant interactions as well prior to running the balanced ANOVA in Minitab.

 

[Miner]

Welcome to the cove Renzoquim,

Good first post. I want to clarify a few points in your comment. Center points do not really estimate the curvature because this implies that they can estimate the magnitude of the curvature. They can detect the presence of curvature, but cannot determine which factor(s) are responsible.

Regarding the number of center points, the guidance I typically give is to use 2-3 center points per block to detect curvature, or 5-7 center points to detect curvature and estimate pure error.

If the center point is not significant, remove them from the model.