Cross correlation bias in lag analysis of aquatic time series

Cross-correlation analysis is the most valuable and widely used statistical tool for evaluating the strength and direction of time-lagged relationship s between ecological variables. Although it is well understood that tempora l autocorrelation can inflate estimates of cross correlations and cause hig h rates of incorrectly concluding that lags exist among time series (i.e. t ype I error), in this study we show that a problem we term intra-multiplici ty can cause substantial bias in crosscorrelation analysis even in the abse nce of autocorrelation. Intra-multiplicity refers to the numerous time lags examined and cross-correlation coefficients computed within a pair of time series during cross-correlation analysis. We show using Monte Carlo simula tions that intra-multiplicity can spuriously inflate estimates of cross cor relations by identifying incorrect time lags. Further, unlike autocorrelati on, which generally identifies lags close to the true lag, intra-multiplici ty can erroneously identify lags anywhere in the time series and commonly r esults in a direction change of the correlation (i.e. positive or negative) . Using Monte Carlo simulations we develop formulas that quantify the bias introduced by intra-multiplicity as a function of sample size, true cross c orrelation between the series, and the number of time lags examined. A prio ri these formulas enable researchers to determine the sample size needed to minimize the biases introduced by intra-multiplicity. A posteriori the for mulas can be used to predict the expected bias and type I error rate associ ated with the data at hand, as well as the maximum number of time lags that can be analyzed to minimize the effects of intra-multiplicity. We examine the relationship between commercial catch of chum salmon and surface temper atures of the North Pacific (1925-1992) to illustrate the problems of intra -multiplicity in fisheries studies and the application of our formulas. The se analyses provide a more robust framework to assess the temporal relation ships between ecological variables.