Independence of the errors

The normal linear model assumes that the errors are uncorrelated with each other, but correlated errors sometimes arise.

Correlated errors may arise in an experiment in a greenhouse where adjacent plants will be grown in similar conditions (light, moisture, air flow). An unusually high growth rate for one plant may be associated with environmental conditions that also cause unusually high growth rates in adjacent plants.

Correlated errors are most common when the observations are made sequentially in time. This is called serial correlation.

Assessing serial correlation

Strong serial correlation may be visible in a plot of residuals against time. A more formal test uses a test statistic called the Durbin-Watson statistic, d. Writing the successive residuals as e1, e2, ..., en,

If successive residuals are similar, d will be close to zero. An approximate p-value can be obtained from a computer, special statistical tables or with a simulation such as that below.

Warning

If a linear model is used for a time series, but the relationship is actually nonlinear, successive residuals also tend to be similar and the Durbin-Watson statistic will also be small.

An unusually small Durbin-Watson statistic can be caused by either serial correlation or nonlinearity.