Which statistical test for change in a nonlinear regression model?

4 min readJul 23, 2022

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Hi guys,

I have a rather fundamental question regarding the analysis of my data involving nonlinear fitting and I hope it is appropriate to post it here. For the sake of brevity I will not provide the whole code and only summarize the essential steps, but of course I can add any details you request.

I have some data which represents some response to a stimulus as a function of the distance to the stimulation site. The data shows, as to be expected, a decay in the response variable, which may be best approximated by a sigmoidal fit. So I applied the BOLTZMANN equation to the data and let MATLAB predict confidence bounds for new observations:

`% Define model function (BOLTZMANN);f = @(beta0,conds)beta0(1) + ((beta0(2)-beta0(1)) ./ (1+exp((beta0(3) - conds) ./ beta0(4))));% Find initialization parameters:resp50 = (max(resp) + min(resp))/2;x50 = 5000; %Educated guessinidat = [0,max(resp),resp50,x50];% Estimate the fitted function:[beta,res,jac,covb] = nlinfit(conds',fliplr(resp),f,inidat);% Fit the function:xfit = linspace(min(conds),max(conds),100);[yfit,delta,n,df,varpred] = nlpredci(f,xfit,beta,res,'Covar',covb,'PredOpt','observation'); %Function edited, see belowyfit = fliplr(yfit);delta = fliplr(delta');varpred = fliplr(varpred');`

Behold the plotted result (Embedding this image did not work.)

I am now adressing the question, how far I can get off the reference site until responses are to be regarded non-maximum. I.e. up from which distance are my (predicted) responses signicantly different to the maximum a 0 mm? I did not find a pre-described solution to such a question, so I developed a little bit naively my own approach, and I would like to ask you to tell me if it is appropriate or if there is some superior method.

My idea was simply to run multiple pairwise t-tests given the statistics from the NLINFIT function (which I edited as to return sample size n, degrees of freedom v, and predicted variance varpred, so I would not have to do the calculations on my own). Thus, I iterate through the predictions unless the tested pair is significantly different:

`alpha = 0.05;for i=2:length(yfit)     testdiff = yfit(1) - yfit(i);     %Common MSE is mean of both estimated variances (s. ONLINESTATBOOK p.376):     mse = (varpred(1) + varpred(i))/2;     %Common SE:     testse = sqrt(2*mse/n); %Correct?     %Compute t-value:     t = testdiff/testse;     %Common df:     testdf = (n-1) + (n-1); %Correct?     p = tpdf(t,testdf);     if p < (alpha/(i-1)) %With BONFERRONI correction (correct?)        m = i;        break;     end  end`

As you can see, I also tried to add some BONFERRONI correction of the alpha-level to account for these multiple comparisions. I am aware, that the t-test I used may be inappropriate for correlated pairs (which is evidently the case).

According to my rule of thumb, I would expect a cut-off x-value somewhere were the confidence intervals of the fit do not intersect anymore. Surprisingly, I obtain a way earlier cut-off as you can see in the picture above.

NOTE:-

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I suggest doing paired t-tests ( ttest2 ) between your reference (at 0 mm) and data taken from stimuli at various distances, for instance 0 mm and 1 mm, 0 mm and 2 mm, etc. (My guess is that you would see significance at 10 to 15 mm, depending on whether your error bars represent standard errors or 95% confidence limits.) This is relatively common in the literature I am familiar with, and generally does not require the Bonferroni correction, because you are not also comparing 2 mm and 3 mm and others. (I also suggest consulting the statistics guidance for the journal you plan to submit your data to.)

The Boltzmann equation is interesting, but you might consider using a model more appropriate to the system you are measuring (unless you’re doing the sort of physics the Boltzmann equation describes). I suspect a reviewer would want to know the reason you chose it and how it describes your experiment. You have not described the system you are investigating, but using a regression model fit may be redundant if you are only interested in the differences between the result of stimulation at various distances from the reference site.

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