Edited to note that this answer is incorrect as I have mentioned in my later comment.

Interesting question. I think it should be possible to use a non-linear regression model to fit the domain limits and the same time as the sigmoidal relationship. For example, the stata nl command nl "fits an arbitrary nonlinear regression function by least squares ". https://www.stata.com/manuals13/rnl.pdf

It can easily be used to fit a beta regression using logit link with an arbitrary multiplicative scaling of the link function and an additive constant, which would allow for variable limits.

Thanks for your help, Garrett. Would the "least squares" fit not assume normal errors?

Hi Jochen,

You're right, my answer above doesn't actually make sense, especially since nl can't fit a beta regression by least squares anyway. A good example of why you should think before you type.

However I still think that this kind of approach could work if the data are homoskedatic on the logit scale.

As you mention, the data shown have "low variance close to the [estimated] limits, larger variance in between", this will can be the case for Y even if logit(Y) or probit(Y) is homoskedatic , so the first step is to checksee if you can find a transform such as logit(Y) or probit(Y) which gives an outcome with constant variance. If it does, then an ordinary logit/probit model with a multiplcative scaling parameter can still work.

I think I get your point, and that might work. I just would use a fit with a logit link instead of working with transformed response values. I have seen that -particularily in logit models - this can make a considerable difference, especially if there are interactions involved.

However, my problem still remains: in a first step I would need to estimate the upper and lower limits, to scale the data so that the scaled date would be homoscedastic on the logit scale. Apart from the fact that I dont't know how to estimate these limits*, I don't know how the uncertainty in this estmation will propagate to the estimation of the parameters of the logit-model (on the scaled data).

* a non-linear fit of a sigmoid model estimates the upper bound assuming normal errors. If I use that, scaling will produce values outside of (0,1), what shurely is not correct (I hope you understand what I try to say).

One time I modeled that kind of data.

My approach was based on sigmoid functions with three parameters

We estimated parameters of functions, and than calculated another parameters of process like yours.

Let see screenshots. Vmax - is only one of additional parameters that can estimated after fitting basic equation.

Of course the errors ( and CI) were calculated taking into account propagation error.

I can provide a detailed description if necessary.

Thank you Olga. The problem with your approach is that these are conventional non-linear regressions that assume normal distributed errors with constant varariance. Consider you have (noisy) data that really extends to the limits. The point is that is will approach the limits but not exceed them, nut the model fit assumes normal errors everywhere.

I attached a plot showing what I mean for a simulated data set with clear upper and lower limits. The functional relationship is 4-parameter logistic, the black line shows the "true" model, the red line is the nonlinear fit. The dotted red lines show the confidence band (obtained by bootstrap). This does not fit to the assumptions. The band should be much wider around x = 0 (in the middle) and be much thinner around x close to -10 and +10 (at the left and right side).

Yes, Jochen , I realized the problem. I deal with heteroskedastic process ( tumor growth), and provide normal distibution of error thougth logarithm.

looking at the plot... (a) may be piecewise regression will be more appopriate aproach ; (b) refine/revision/transform variables ; (c) use ordered factor variable instead of continous

Instead of a forehead functional model try to "wash the elephant by part" :)

Thank you, I will keep that in mind. Right now I am still more interested if such problems do have a (possible) solution.

I think that the suspicion of limits should not be simply based on the distribution unless the distribution shows a cliff-effect (horizontal or near-horizontal end(s) of the CDF).

The real question is the data generation function. The extreme(s) may be

a) truncated. The sampling method means that those who are beyond a threshold are excluded. Example : elevation of enzymes in acute coronary heart disease. Patients with really massive heart attacks will die before their enzymes reach a peak value. Or studies of mild hypertension which only recruit those whose BP exceeds one threshold but falls below another.

b) censored. The sample isn't attenuated, but the measurement cannot read above or below a threshold value. For example, in measuring bacteria concentration in drinking water, assays typically cannot measure above about 2k bugs per 100 ml. This causes a spike of values at the exact upper limit.

c) truly bounded. Most health outcome scales are bounded at zero and an upper limit that represents the highest score.

Truncated and censored regression models are well known. With truly bounded data, one interesting suggestion is to convert the score bounds (not the observed high and low values) into the interval 0/1 and then use a logistic approach. This has the plus of allowing you to model any quantile of the distribution.

Ronán Michael Conroy, I totally agree. It's in fact c) what I am talking about, and myproblem is, that the bounds are not known (even not the lower bound which may be >0). Without knowing these bounds it's not possible to rescale the values (I think this is termed "normalizing" if X is scaled to the domain (0,1)) and to use a logistic approach (or beta regression). If I have to estimate the bounds, it's not clear to me how the uncertainty about thelimits propagated to the logistic (or beta regression) model.

In case c), prior knowledge is vital. And you clearly don't have enough information, Jochen Wilhelm , to place these.

I would hesitate to infer the bounds based on a CDF as it looks at a rough guess that the low extreme of the Y variable has an unusually high density, suggesting to me that this may be a floor effect, and that a censored regression approach might be better.

Why do we end up with mystery datasets like this? Surely the investigator should know their part of the process and be able to tell us why the data look like they do!

Just kidding. There's a lovely Thai expression for "that's never going to happen" (ชาติหน้าตอนบ่าย ๆ) which translates as "one fine afternoon in your next reincarnation".

Olga Krasko "Instead of a forehead functional model try to "wash the elephant by part" :)"

What a great expression!

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a MCMC-powered bayesian approach has been suggested for such 4-parameter logistic regression problems ; see section 4.4 below

http://bictel.ulg.ac.be/ETD-db/collection/available/ULgetd-12192012-155142/unrestricted/thesis.pdf

(kind of a huge sponge to wash the elephant in just one move !)

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Thanks Fabrice. I think this is goind into the right direction. I will have to find out how the variance function can be changed. In the proposed model the variance is a power function of the mean. In the case I am looking for the variance is an inverse scaled beta function of the mean (scaled to the estimated upper and lower bounds, at which the variance will be close to 0). I am not even evry sure if this idea is correct... that will take some time. Aren't there mathematicians who could help? This sponge seems to wash me away faster than it cleans the elephant... :)

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i am sure i do not get this variance thing !

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assume the simplest problem

f(x ; x0) = 1 / (1+exp(x-x0))

and i want to fit that to data (xi, yi)

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depending on my assumptions on the data generation process, i could consider two simple noise models

.

yi = 1 / (1+exp(x-x0)) + ei

(some kind of measurement noise and i cannot assume that the observations will lie in [0, 1])

.

yi = 1 / (1+exp((x-x0) + ei))

(the uncertainty is on the regression of the log odd ratio and the squashing function reduces the observations domain to [0, 1] deterministically)

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in the second case, applying the delta method shows that the variance on y will have the behaviour you require (from what i understood ...) ; if s² is the variance noise :

s²(x) = [f(x)(1-f(x))]²s²

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i suppose the second case seems a little odd in a dose response scenario since it translates into a noise on the dose which may be just ruled out by the data generation process

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but, then, in the first case, i do not see why the squashing of the variance should be required for extreme doses (assuming i have a measurement noise)

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another potentially intersting resource :

https://pdfs.semanticscholar.org/8517/79099a662b91c8153a9a40e069f0276eecbc.pdf

A Better Lemon Squeezer? Maximum-Likelihood Regression With Beta-Distributed Dependent Variables

Michael Smithson

The Australian National University

Jay Verkuilen

University of Illinois at Urbana–Champaign

Uncorrectable skew and heteroscedasticity are among the “lemons” of psychological data, yet many important variables naturally exhibit these properties. For scales with a lower and upper bound, a suitable candidate for models is the beta distribution, which is very flexible and models skew quite well. The authors present maximum-likelihood regression models assuming that the dependent variable is conditionally beta distributed rather than Gaussian. The approach models both means (location) and variances (dispersion) with their own distinct sets of predictors (continuous and/or categorical), thereby modeling heteroscedasticity. The location submodel link function is the logit and thereby analogous to logistic regression, whereas the dispersion submodel is log linear. Real examples show that these models handle the independent observations case readily. The article discusses comparisons between beta regression and alternative techniques, model selection and interpretation, practical estimation, and software.

... but this does not solve the problem of modeling the bondary values

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Assuming your goal is prediction you might try Generalized Additive Models. some good resources are here; https://www.google.com/search?q=generalized+additive+models+in+r&oq=generalized+additive+models+in+r&aqs=chrome.0.0l8.34293j1j8&sourceid=chrome&ie=UTF-8

Best wishes, David Booth

Yes, but the question remains: what error model to chose and how do the (required) estimatesoftheupper and lower bounds impact the uncertainty (of the predcition). I could certainly ignore the existence of boundaries, but the aim was to particularily consider them.