How to reconcile prediction profiler confidence intervals with parameter estimates for mixture designI have an mixture experiment with four components. It's a custom design of 16 runs with no replication that started with a model that included interactions of two factors and three components. The mixture constraint is that the four components sum to 1.0. The reduced model from the data has only the four main effects as statistically significant (p<.0001).
The prediction profiler plots show 95% confidence intervals that imply the "slope" of the effect of some significant factors could be zero; the gray band encompasses a horizontal line. It also shows the factor having the smallest effect has the largest slope.
The profiler also shows the slope of the effect of component A changing as I change the amount of component C, even though there are no interactions in the model. A has a slope of 0.05 units when C is at 0.1 and it has a slope of 0.02 units when C is 0.2. Of course the amounts of the other components change to stay within the inference space, but why the change in slope?
The attached graphic shows what I'm seeing. It also shows a Factor Settings dialog that has the min and max settings that were used in the mixture experiment.
The scatterplots and the prediction profiler plots looks similar, they make it look like factor B is insignificant. But if I leave B out of the model, the residual errors show that to be a mistake.
How do I interpret the slope and confidence intervals on the prediction profiler for this situation? Or maybe how do I relate the parameter estimates to the prediction profiler?
DOERe: How to reconcile prediction profiler confidence intervals with parameter estimates for mixture designIt looks like the problem is that you are not analyzing the data with a Scheffe mixture model. Look at the Analysis of Variance table. You will see a message there that says "Tested against reduced model: Y=0". That means you are fitting a no-intercept model. What you SHOULD see for a mixture model is "Tested against reduced model: Y=mean".
So, what went wrong? Most likely someone did some rounding when entering the values of your four components. Create a column that is the sum of your four components and you will likely see something that is different than 1. The sum must be 1 EXACTLY for JMP to analyze the data as a mixture. Once that is fixed, try re-fitting the model. Come back here if you still have questions at that point.
Re: How to reconcile prediction profiler confidence intervals with parameter estimates for mixture designThanks Dan, it turns out that one of the rows did not sum to 1.0. When I repaired it (changed one proportion from 0.47 to 0.48), then the result made me more confused.
The sums of squares (sequential and adjusted) changed hardly at all, including the error SS in the ANOVA. But, Model SS in ANOVA is nowhere near the same. The total sums of squares for the sequential SS don't sum to the value in the ANOVA in the repaired analysis like they did in the original. Somehow 1.87 units of SS have gone somewhere that I don't see.
The confidence intervals in the response profiler are the same, but not in the Y Actual by Y Predicted plot. The parameter estimates are nearly unchanged.
The attached images show the differences, one is entitled "Response Y original" the other is "Response Y repaired". You can see that the message in the ANOVA table is either "Y=0" or "Y=mean".
Re: How to reconcile prediction profiler confidence intervals with parameter estimates for mixture designYou will only need to be concerned with your Y repaired.
The Scheffe mixture model cannot be interpreted like a typical regression model. It looks like there is no intercept in the model, but there is. The intercept is included with the main effects. The interpretation is what matters. For example, the main effect parameter estimates are actually the height of the response surface at the pure component vertex of the respective components. In other words, it is the "effect" of the component plus the overall mean. That is why the ANOVA has the message "Tested against Y=mean". So the type I SS will not add up properly because of the mean being involved in all of the SS of the main effects.
The confidence bands on the observed versus predicted plot are correct for the Scheffe model. They are much wider because of the inherent multicollinearity caused by a mixture. When factors are correlated (as they are in a mixture), that leads to an inflation in the variance of the parameters. Thus, the error bands will be much wider on the model fit. The prediction profiler will look better because that is a confidence interval on the prediction, not the significance of the parameters.
Re: How to reconcile prediction profiler confidence intervals with parameter estimates for mixture designI'm starting to understand, I get that the model scales the value of the component in such a way that there is effectively an intercept. However, it also appears that the sensitivity of the model to a component is proportional to the parameter estimate. I suppose I need to find some papers that describe the statistics of the significance for Scheffe models.
There is a JMP Help somewhere that says the confidence interval on the Prediction Profiler is confidence on the mean, not on an individual prediction. If it's not related to significance of the parameters, or to the sensitivity of the model to changes in the component, what is the utility of confidence intervals on the Prediction Profiler?
What is the slope on the prediction profiler trying to convey? Why is the slope for component D greater than any of the others, but its parameter estimate is the smallest? Why do the slopes of A and B look insignificant but their parameter estimates are larger than D?
I suppose what I need is a Methods and Formulas reference for this platform and I can work it out from there. The math behind the scenes is a little opaque.
