Determination of expiration date from a non-linear fitting

[#1]

Dear Team,

Does the degradation tool for determining the expiration date rely only on a linear regression?
Is it possible to determine the expiry date using a non-linear model?

If possible how should I do?
Or does the ICH require using only a linear model?

[#2]

What exactly do you mean by "non-linear"? Do you mean curved (like a polynomial or regression using some other transformation of Time)? For example, you can apply ICH Q1E with sqrt(Time) rather than Time, and you can use JMP's degradation tool. And, strictly speaking, that is still a linear model. "Linear" in the statistical sense refers to being linear in the parameters in the model, not the "X" variable.     

[#3]

DEAR MRB, thanks for your reply.

By heaving of data set as showed in picture is correct to fit it with a linear regression? The degradation tool fit the data with a linear regression?

The data attached is made by just graphing tool. 

By using a cubic fitting looks that the fitting have e better correlation with the numbers.

Do you think that i'm thinking something wrong? Or i'm doing confusion...?

Thanks for you help.

[#4]

Good question(s). As you know, determination of expiration is about extrapolation...not interpolation.  For the purposes of interpolation, such a fit may be appropriate.  However, extrapolating with such a model is ill-advised. Perhaps, starting  around Time Point = 7, it doesn't drop any more.  So, all degradation occurs prior to 7 months.  So a piecewise model may be appropriate; one straight line going down until Time = 7 or so, then another straight line (looks fairly horizontal) starting at Time=7.  If you can then argue, scientifically, that this is the expected behavior (ie, all the degradation occurs by Time = 7), then you still have a  linear model. In the "Fit model"  platform, you will have two  Model Effects: Time, and T7=max(0, Time-7) (create another variable in your data table called T7). Now, it does get more complicated since it appears (as indicated by the three different plotting symbols) you have three batches. If so, Batch will have to be one of the effects as well. And then the interaction of Batch with Time and Batch with T7.  Looking at the plot, I suspect the interaction terms are unnecessary (but ICH Q1E requires that assessment). And, no...though the model is still linear (in the statistical sense) you won't be able to use JMP's degradation platform since that platform limits you to a subset of linear models. You will have to use Fit Model platform .For interpretation of such a model (I'll simplify and assume you have no interactions, so you have Batch, Time, and T2 in the model). the model is then B0 + Batch + B1*Time + B2*max(0, T-7).   In the parameter estimates results, the Term for Time is the initial slope (B1), and the Term for Time + The Term for T7 (B2) is the slope after Time=7. i.e., first slope=B1, second slope = B1+B2. And, FWIW,  I've used Time = 7 here as the "knot". The optimum knot may not be exactly 7...but it looks like it is close-ish to 7. If you wanted a model that also estimated the knot, that is a non-linear model that is beyond the scope of what can be easily discussed in such a forum. However, you may try a few different knots and choose the one that minimizes the Root Mean Squared Error in the Summary of Fit section of the fit model output.               

[#5]

Please, 

Can you explain how the T7 variable should be done?
The time point variable is 1, 3,6,9,10,12.....
The T7? 1,3,6?. And leave blank cells for the months after 7?
And in this case, do I have to change or delete something on the Time point variable?

[#6]

Presumably you have columns for batch, Time, and  Deg. Just create a new variable called T7.  The function is Maximum( 0, Time - 7 ).. See my attached table and run the scripts saved therein; the data is just made up but it will give you the idea of where to start.

[#7]

Dear MRB,

I got every point you described.

I used the Fit model tool and i obtained a model.

The model suggest consider only Batch, Time and T7. The interactions Batch*Time and Batch*T7 are not significant relevant, then Removed from the model.

Now the last step is how I can extrapolate the expiration date. 

How I can evaluate when the model met my spec? Considering the 0,25 confidence suggested by the guidelines?

[#8 · Accepted solution]

See attached for something a bit more thorough.  I saved the prediction and 95% lower bound formulas (see the the save columns option from the red triangle in the fit model results)  added some points to the end of the data set to extrapolate via graph builder. The knot in my example is 6.2 (yours may be closer to 7) and I set an arbitrary spec of 500.   p-value for Batch, in my example, is less than 0.25 so it must be in the model. 

[#9]

Thanks So much!