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I have a set of means and standard deviations. For each mean I can calculate a 95% confidence interval. I plot these means and confidence intervals against an independent variable and I fit a best sigmoidal curve through this data.

I want to calculate the region of 95% confidence above and below this curve (i.e. be able to make estimates of the 95% confidence interval between the measured data points.)

What should I be looking to use? Should I fit a second sigmodal curve to the upper limits of each data point's interval and a third curve to the lower limits of each interval?

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  • $\begingroup$ What is non-linear here? Are your means the proportions of 'successes' from Bernoulli trials? Do you have access to the raw data? $\endgroup$
    – gung - Reinstate Monica
    Commented May 2, 2014 at 13:57
  • $\begingroup$ My understanding is that since the sigmoid function is a non-linear function, this is an example of non-linear regression? $\endgroup$
    – user16548
    Commented May 5, 2014 at 11:29
  • $\begingroup$ Are you just running a logistic regression? $\endgroup$
    – gung - Reinstate Monica
    Commented May 5, 2014 at 13:37
  • $\begingroup$ My means are proportions of successes, but I decided I wasn't trying to do "logistic" regression as such, because I do not expect the sigmoid curve to tend to 1 as the upper limit, but rather some other value that indicates less than 100% success rate. I'm keen to try and establish what that upper limit is though, by using the regression model. And then also keen to establish an x-axis value that corresponds to this limit, presumably the I point at which the model becomes statistically insignificantly different from the limit.v $\endgroup$
    – user16548
    Commented May 6, 2014 at 14:38

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For given values of the independent varible you generate set of data and calculate a mean and confidence interval. You fit a sigmoidal curve through the mean values. Now, at each given value of the independent variable with an associated set of data, the 95% confidence interval of this set of data will serve as best estimates of the upper and lower bounds for that independent variable value. Therefore, it makes sense to fit a sigmoidal curve through the upper bounds and another through the lower bounds to get a 95% confidence interval above and below your fitted mean curve.

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  • $\begingroup$ Hi Martino, Thanks for affirming my thoughts on this. My concern is that can we assume upper and lower confidence limits will follow the same model as the means: I have found a another sub set of my data in which the sigmoidal curve for the upper limits actually intersects with the best fit curve for the means. This doesn't seem right. $\endgroup$
    – user16548
    Commented May 5, 2014 at 11:32
  • $\begingroup$ I guess you need to ask yourself why you feel a sigmoidal curve is the correct model. Does this follow from a hypothesis on the fit to the means or did you decide to fit a sigmoidal after collecting the data, eg it looked sigmoidal so you fitted a sigmoidal? Why not model the best fit with a b-spline curve? Alternatively, why not look at the fitting a sigmoidal curve to percentile intervals rather than confidence intervals - in other words - fits to the 5th percentile, the median and the 95 percentile values. This may be useful if you have small sample sizes. $\endgroup$
    – martino
    Commented May 6, 2014 at 10:40
  • $\begingroup$ These are good questions, thank you. My decision to use a sigmoidal model comes from previous studies, assessment of the data shape, and pragmatic consideration of the expected shape. I have a sample size of 23, which I understand to be enough for t-tests if the distribution is approximately normal. I'm just looking for a reason why I can expect the confidence limits to follow a similar model to the means. $\endgroup$
    – user16548
    Commented May 6, 2014 at 14:43
  • $\begingroup$ The sample size is relatively small and this may be the reason you see overlap in the fit to the 95% confidence interval. Lets say that at any value of the independent variable you expect the sample data to have the same variance as any other fixed value ie it is homoscedastic. If so, you can estimate a single value for variance using all your data. This will give you a single confidence interval that can be applied at each fixed value of the independent variable.In turn, the sigmoidal fitted to this interval will just be the mean fitted signoidal shifted up or down resulting in no overlap $\endgroup$
    – martino
    Commented May 8, 2014 at 10:30

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