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 Key concepts: Growth equations

What are growth equations used for?

Growth equations are used in many situations: growth of bacterial cultures, growth of organisms, adaptation of technology or ideas among a population, growth of economies...

We provide a few growth equations that might be useful as a starting point. We can't provide much help with these.

Variations

Different parameterization. This means the other equation has the same number of parameters, and generates the same family of curves, but the parameters have different meanings.

Different definition of Y. Our equations assume that Y quantifies the value that is growing. Sometimes you'll see equations defined where Y is the logarithm of the ratio of current population value divided by the initial value.

Differential equations. The equations built in to Prism define Y as a function of X and parameters. You'll sometimes see growth equations defined as the derivative of Y as a function of X.

t or X? We use X to define the independent variable, which for growth equations is time. You'll often see equations using t instead.

More parameters. Some versions of growth equations have more parameters, giving the curve more inflection points.

It is easy to clone our equations and modify as you need.

No accounting for autocorrelation

If you fit a growth model using Prism's nonlinear regression, it works like fitting any other model. The analysis assumes that each point has an ideal Y value predicted by the model with the addition or subtraction of random error. In other words, nonlinear regression assumes that each residual is independent of the others. If one residual is positive, the next one is equally likely to be positive or negative.

With growth studies, sometimes the assumption of independence is clearly violated. If you are following one individual and growth at one time point is a bit higher than predicted by the model, then the growth at the next time point will probably be higher too. The residuals of the two points are not independent. They are sort of cumulative.

Methods exist for fitting models when there are correlated residuals, but Prism doesn't include them.