Is the relationship a curve rather than a straight line? (nonlinear regression, four fixed formulas)

In brief

Is the relationship a curve rather than a straight line? Nonlinear regression fits one of four fixed formulas: Michaelis-Menten, exponential, logistic or power. The target is a number, predicted from exactly one number. The formula must be justified by the subject.

How it works in Statable

"New analysis", then choose the card "Can something be predicted?".
Choose the target column and exactly one predictor column with numbers.
"More options" → "Model" → "Nonlinear regression (fixed formula)", then choose the formula.
The result shows the parameters with intervals; there is no R².

The four formulas

Michaelis-Menten: y = a · x / (b + x). The curve rises and approaches an upper limit a (saturation).
Exponential: y = a · exp(b · x). Growth or decline by the same percentage per step.
Logistic: y = a / (1 + exp((c − x) / b)). An S-shaped curve with upper limit a and midpoint at c.
Power: y = a · x^b. The curve rises or falls with the power b.

There is no free formula. The formula is a decision based on subject matter and should not be chosen by which one fits best.

What the table looks like

One row per case, a target column with numbers and exactly one predictor column with numbers. Rows with a gap in one of the two columns are left out; the number is shown in the result. Depending on the formula, the calculation needs positive values for the starting values: for Michaelis-Menten and power x and y above 0, for exponential and logistic y above 0. Otherwise Statable blocks the calculation and says in how many cases this does not hold.

Reading the result

For each parameter (a, b, with the logistic formula also c) the estimate b, SE, t, p and two 95% intervals are shown: the Wald interval (with the t distribution and the degrees of freedom df given) and the profile interval. If the profile interval cannot be calculated, a sentence says so, and the Wald interval remains.
The model: n, df, the residual sum of squares (RSS), the residual standard deviation σ and AIC.
The picture shows the points and the fitted curve; a table shows the fitted values and the residuals.

When the calculation fails

If the calculation finds no result (no convergence), Statable shows no numbers but a sentence. Then the formula probably does not fit the data, or the starting values are not enough. Fitting a straight line as a Michaelis-Menten curve fails, for example.

Trap: what the numbers do not say

There is no R². In nonlinear models the apparent R² cannot be read as in linear regression; that is why Statable gives none.
The formula must be justified by subject matter. That a curve fits well does not yet show that the relationship runs that way.
The calculation finds a local optimum. With other starting values a different solution can result.
Statements for x outside the range of the data are not covered: the curve can run quite differently there.
The p values and intervals are approximations. The Wald interval and the profile interval can differ considerably. The residuals should be inspected.
If R reports a warning during the calculation, it is shown in the result; the numbers are then to be read with caution.

Requirements and limits

The cases are independent of each other, and the spread around the curve is roughly equal. The table of the residuals helps with checking.
More cases and more different x values than parameters are needed.

Reporting the result

nonlinear regression (Michaelis-Menten), a = 212.68, b = 0.06, RSS = 1195.45, df = 10

Typical examples

How fast does a reaction run depending on the concentration (Michaelis-Menten)?
How does a stock grow over time (exponential or logistic)?

For specialists

R (stats): nls(formula, start = s, control = nls.control(maxiter = 200, tol = 1e-7, minFactor = 1/4096)). The starting values s come automatically from linear auxiliary regressions. The profile interval is confint() on the nls model. Checked on Puromycin (Michaelis-Menten), ChickWeight (exponential), DNase (logistic) and cars (power) against independent calculations: parameters, standard errors and residual sum of squares agree.