Ah. Here some advice acquired in my undergraduate engineering courses may be pertinent. When fitting an empirical curve whose shape one knows approximately, it is efficient to choose design points more closely spaced in the vicinity of critical values like maxima, minima, and points of inflection; and more widely spaced in regions where the curve in question is linear (just to check that it IS at least roughly linear in those regions, and to estimate the linear domain with some precision); when the precision of one's measurement is roughly the same throughout the domain of the predictor(s), as you say is the case. (When precision varies along the domain, these desiderata are somewhat modified by wanting to take more design points where precision is poor, and fewer where the data have high precision.) (For your sigmoidal curves, I'd guess that there aren't any maxima or minima to worry about, but that there are at least two points of inflection to be modelled.)
When the form of the curve is known exactly, usually from underlying mathematical theory, the same general considerations apply, but now the question is, what design points (values of X, generally, in X-Y space) would be most helpful in (a) estimating the parameters of the formal model and (b) distinguishing between regimes for which the model has different parameters. (If you think of a linear model for [over?]-simplicity, there are two parameters, one of which locates the response curve (a straight line) vertically, and the other reflects the slope of the line. Design points near the middle of the interesting/useful domain (of values of X) are usually most efficient in estimating location; points near the extremes of the domain are most efficient in estimating slope. If the sigmoidal model for your curves is well-established in form, quite possibly much the same considerations apply; if the functional form, or some but not all of the parameters of the model, are still more or less unknown, I'd be inclined to concentrate my points more around the rapidly-changing regimes, while including some others as a sort of check on the less-rapidly-changing regions. At some level, you're really asking "Which points can I expect to show me the best separation between models that I want to distinguish between, which points will give me the best available precision in estimating the parameters I'm currently looking at, and which points will help me diagnose a badly fitting model?" We may agree that today (or perhaps yesterday) it might have seemed reasonable to try to fit sigmoidal data with a cubic, or perhaps quintic, function in X; tomorrow it may become obvious that "Oh! That ought to be a logistic function!", which might lead to other choices for apparently-optimal design points.) Good luck! -- DFB. On Wed, 26 May 2004, Xinmiao wrote in part: > ... I'm trying to design an experiment in which behavioral/neuronal > response v.s. stimulus strength curves are to be measured. We know > that the tuning curves are sigmoidal/linear, and my question was how I > should spread out the sampling points along the stimulus dimension, > say 4 or 7 or even more? We've also known that the error-variance, at > least for neuronal responses, should be approximately same as mean, > the square root of which shouldn't differ much between different > independent variable (stimulus strength). ... <snip, the rest> ------------------------------------------------------------ Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
