Autocorrelation

Many parametric statistical procedures (e.g., ANOVA, linear regression) assume that the errors of the models used in the analysis are independent of one another (i.e., the errors are not correlated). When this assumption is not met in the context of time-series research designs, the errors are said to be autocorrelated or dependent. Because time-series designs involve the collection of data from a single participant at many points in time rather than from many participants at one point in time, the assumption of independent errors inherent in many parametric statistical analyses may not be met. When this occurs, the outcome of these analyses and the conclusions drawn from them are likely to be misleading unless some corrective action is taken.

The error in a time-series linear model usually refers to an observed value Yt (i.e., a dependent variable score observed in a theoretical process at time t) minus the predicted value Ŷt (based on parameters in the model). When actual sample data are involved (instead of theoretical process data), the predicted values are based on the estimates of the parameters in the model, and the difference Yt − Ŷt is called a residual. Hence, a residual is an estimate of an error. For example, if a researcher proposes an ANOVA model for a two-phase interrupted time-series design, the residual is defined as an observed value in a realization (i.e., a sample) of the process minus the mean of the relevant phase. If the sign and size of the residuals are unrelated to the sign and size of the residuals that follow them, there will be no autocorrelation, and this implies that the errors of the model are independent. If, however, positive residuals tend to be followed in time by positive residuals and negative residuals tend to be followed by negative residuals, the autocorrelation will be positive; this is evidence that the independence assumption is violated. Similarly, if positive residuals tend to be followed by negative residuals and negative residuals tend to be followed by positive residuals, the autocorrelation will be negative, and once again, this is evidence that the independence assumption is violated. Autocorrelated errors are especially likely to occur when (a) the time between observations is very short, (b) the outcome behavior changes very slowly, (c) important predictor variables are left out of the model, or (d) the functional form (e.g., linear) of the relationship between the predictors and the outcome is incorrectly specified.

How Autocorrelation Is Measured

Although one can measure autocorrelation in many different ways, the most frequently encountered method involves the computation of a single coefficient called the lag-1 autocorrelation coefficient. This autocorrelation coefficient represents the correlation between the residuals at their associated time t and those same residuals shifted ahead by one unit of time. The sample coefficient computed on actual data is denoted as r1 whereas the population (or process) parameter is denoted as ρ1. Like most two-variable correlation coefficients, the autocorrelation coefficient must fall between −1.0 and +1.0. The conventional formula for computing the sample coefficient is

...