Re: Rounding errors

From: "diane r mould" <drmould@attglobal.net> Subject: Re: Rounding errors Date: Fri, 23 Mar 2001 17:14:47 -0500 Dear All Below are further explanations for using the BAND matrix. I dont think that there is much in the manual on this topic. Bill Bachman reported: "The only reference I have been able to find is in NONMEM V Supplemental Guide, p.1: "2. Band Symmetric Matrices An intial estimate of a diagonal block of either the OMEGA or SIGMA matrices may have a band symmetric form, in which case the final estimate had the same from." >Below is the NONMEM record for this, > assuming that ETA(1) describes the variance on V, ETA(2) on CL and ETA(3) on > KA. > > $OMEGA BLOCK(3) > .1 > .01 .1 > 0 .01 .1 > > HOW DOES NONMEM KNOW THAT THE OFF-DIAGONAL ELEMENT IS TO BE FIXED TO ZERO? > I THOUGHT YOU COULDN'T USE THE FIXED OPTION ON INDIVIDUAL ELEMENTS IN A > BLOCK. DOESN'T NONMEM BALK AT THIS THINKING IT IS A PARAMETER WITH ZERO FOR > A STARTING VALUE? I THOUGHT ALL ELEMENTS WITHIN A BLOCK MUST BE NON-ZERO. In NONMEM, my understanding is that coding a BLOCK statement as follows: $OMEGA BLOCK(3) .1 0 .1 0 0 .1 effectively fixes the off-diagonal elements to zero. NONMEM is 'smart' enough to know that this is a BAND matrix. This is analagous to writing $OMEGA .1 .1 .1 which does the same thing - it fixes the off-diagonal elements to zero. Therefore using a zero as an intial estimate in the BLOCK fixes the value, NONMEM will not estimate that element. The assumption that all the associated variance covariance terms are zero is not necessarily appropriate, and has been a source of discussion on several previous occasions in this forum. As to using the FIX option on individual elements in the BLOCK - you are quite correct in this. A statement such as the following FIXES all the records within the BLOCK: $OMEGA BLOCK(3) .1 .01 .1 .01 .01 .1 FIX However, using the BAND approach, you can selectively FIX elements to zero. > However, the elements in the BAND matrix > must, of course, be symmetrical. Therefore a BAND matrix of > > $OMEGA BLOCK(3) > .1 > .01 .1 > 0 0 .1 > > will produce an error from NONMEM. > > ACTUALLY THE ABOVE OMEGA BLOCK IS STILL SYMMETRICAL SO I'M NOT SURE WHY THIS > EXAMPLE SHOULD BOMB BUT NOT THE FIRST EXAMPLE WITH ONLY ONE OFF-DIAGONAL SET > TO ZERO. Sorry all - I apologize for not describing this more concisely. Perhaps it is more clear to say that the lower triangular elements have been set to zero. To extend the example that I gave previously into a 4x4 BLOCK: $OMEGA BLOCK(4) .1 .01 .1 0 .01 .1 0 0 .01 .1 would run successfully. I think that it would be good to just try this out and experiment a bit. > $OMEGA BLOCK(2) > .1 > .01 .1 > $OMEGA .1 > > YES, THIS IS STANDARD BLOCKING OF AN OMEGA MATRIX (IE., ELEMENTS BETWEEN > BLOCKS ARE UNCORRELATED BUT ELEMENTS WITHIN A BLOCK ARE CORRELATED). Yes, exactly. > I have found the use of the BAND matrix to be quite useful at times, > particularly if the model is to be used later for simulation work for > example. > > YES, IT IS IMPORTANT TO GIVE AS MUCH ATTENTION TO THE COVARIANCE STRUCTURE > AS THE FIXED-EFFECTS STRUCTURAL MODEL IF YOU KNOW THE MODEL WILL BE USED FOR > SIMULATION PURPOSES. Agreed. Given the fact that models are frequently built and then later used for simulation work, I am beginning to wonder too if we should distinguish between a simulation model and a descriptive one. ============================================================================ As a separate note to the other NMUSERS, Leonid Gibiansky wrote in with another way to handle this aspect: I used to express the off-diagonal elements in terms of the THETA parameters, keeping the OMEGA matrix diagonal and FIXED to 1, something like MYETA1 = THETA(1)*ETA(1) MYETA2 = THETA(2)*ETA(2)+THETA(3)*ETA(1) MYETA3 = THETA(4)*ETA(3)+THETA(5)*ETA(2)+THETA(6)*ETA(1) $OMEGA BLOCK(3) FIXED 10 0 10 0 0 10 Then you can control correlation, and OMEGA elements are expressed in terms of THETA(1)-THETA(6) Alternatively, you may introduce THETAs so that they are equal to OMEGA elements, and re-express coefficients in the expressions for MYETAs. Yet another alternative is to introduce THETAs as diagonal elements of OMEGA and correlation coefficients. However, for high dimension of OMEGA matrix this is not feasible, and the representation above is the only way to control correlation that I know.