When fitting a curve, how do I calculate the 95% confidence interval for my fitted parameters?

Question

I am fitting curves to my data to extract one parameter. However, I am unsure what the certainty of that parameter is and how I would calculate / express its $95$% confidence interval.

Say for a dataset containing data that exponentially decays, I fit a curve to each dataset. Then the information I want to extract is the exponent $b$. I know the values of $t$ and the value of $a$ I am not interested in (thats a variable that comes from the population, not the process Im trying to model).

I use non-linear regression to fit these parameters. However I dont know how to compute the $95$% confidence interval for any method, so broader answers are welcome as well.

$$f= a\cdot e^{-bt}$$

Once I have my value for $b$, how do I calculate its $95$% confidence interval? Thanks in advance!

Answer 1

The problem with linearizing and then using linear regression is that the assumption of a Gaussian distribution of residuals is not likely to be true for the transformed data.

It is usually better to use nonlinear regression. Most nonlinear regression programs report the standard error and confidence interval of the best-fit parameters. If yours doesn't, these equations may help.

Each standard error is computed using this equation:

SE(Pi) = sqrt[ (SS/DF) * Cov(i,i) ]

• Pi : i-th adjustable(non-constant) parameter
• SS : sum of squared residuals
• DF : degrees of freedom (the number of data points minus number of parameters fit by regression)
• Cov(i,i) : i-th diagonal element of covariance matrix
• sqrt() : square root

And here is the equation to compute the confidence interval for each parameter from the best-fit value, its standard error, and the number of degrees of freedom.

From [BestFit(Pi)- t(95%,DF)*SE(Pi)]  TO  [BestFit(Pi)+
 t(95%,DF)*SE(Pi)] 

• BestFit(Pi) is the best fit value for the i-th parameter
• t is the value from the t distribution for 95% confidence for the specified number of DF.

• DF is degrees of freedom.

  Example with Excel for 95% confidence (so alpha = 0.05) and 23 degrees of freedom: = TINV(0.05,23) DF equals degrees of freedom (the number of data points minus number of parameters fit by regression)

Answer 2

If believe an appropriate model for your data is:

$f = ae^{-bt}$

Then you can take a log transform your response data such that an appropriate model is:

$f' = a' -bt$

with $f' = ln(f)$ and $a' = ln(a)$. The transformed data can be fit using simple linear regression and an estimate for the intercept and slope along with standard errors obtained. If the critical t value and standard error are applied to the parameter estimate, a confidence interval for that parameter estimate can be formed. In R:

# Rough simulated data set.
set.seed(1)
a <- 50; b <- 0.2; n <- 25
x <- 1:n
y <- a*(exp(-b * x))
y <- y + rnorm(n, sd=0.25)
y <- ifelse(y>0, y, 0.1)
plot(x,y)

# Linearise:
y2 <- log(y)
plot(x,y2)

# Fit model to transformed data
model <- lm(y2 ~ x)
summary(model)
confint(model)

# Or:
param <- summary(model)$coefficients[, 1]; se <- summary(model)$coefficients[, 2]
param + qt(0.975, 23) * se
param - qt(0.975, 23) * se

If you are using the model for predicting you should be sure to check that the assumptions of SLR have been met - iid $~N(0,\sigma^2)$.