Quality & Quantity 35: 365–387, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Autocorrelation and Bias in Short Time Series: An
and ROSER BONO
Department of Methodology of Behavioral Sciences, Psychology Faculty, University of Barcelona,
Abstract. The conventional ﬁrst-order autocorrelation coefﬁcient r
generates an empirical bias
when it is applied to short time series. The properties of this estimator have been examined with a
Monte Carlo simulation study using the MATLAB program (version 5.2). This study also analyzes
the function of the empirical bias with the polynomic regression and derives a polynomic ﬁtting
model for different sample sizes. In this way, a new estimator that has been corrected by the absolute
value of the ﬁtting model (r
) is proposed. Having analyzed the statistical properties of the estimator
, it is shown that the empirical bias generated by r
is less in relationship to r
+. The results
of the study make it possible to verify that the mean squared error associated to the estimator r
less than that of r
. Thus, the coefﬁcient r
is recommended to estimate the lag-one autocorrelation
coefﬁcient in samples under 50 observations.
Key words: autocorrelation, Monte Carlo simulation, short time series, statistical power.
The presence of serial dependence or autocorrelation in short time series (STS)
is a common fact in behavioral data. The residual of autocorrelated data is not
randomly distributed since it is affected by systematic components. One of the most
common systematic components of autocorrelation is known as autoregression
(Hartmann et al., 1980; Jones et al., 1977). Thus, many works use the terms of
ﬁrst-order autocorrelation ((1) and lag-one autoregression indistinctly.
A ﬁrst attempt to analyze behavioral data consisted in applying t or F conven-
tional statistics (Gentile et al., 1972; Shine and Bower, 1971). The initial proposal
was strongly criticized since autocorrelation biases the estimated value of these
statistics when the effect of the treatment is analyzed (Hartmann, 1974; Keselman
and Leventhal, 1974; Kratochwill et al., 1974; Thorensen and Elashoff, 1974).
According to whether the autocorrelation coefﬁcient is positive or negative the F
value is overestimated or underestimated (Scheffé, 1959). The alternative to the
conventional statistics models was the time-series analysis since it made it possible
to know the structure of serial dependency (Glass et al., 1975; Gottman, 1981;
Corresponding author: Jaume Arnau, Facultad de Psicolog
ıa, Universidad de Barcelona, De-
partamento de Metodolog ˛aa de las Ciencias del Comportamiento, C/Passeig de la Vall d’Hebron,
171, 08035 Barcelona, Spain. Phone: 34-3-312 50 79; Fax: 34-3-402 13 59; e-mail: email@example.com