Violation of Independence – Part I

Abstract

This chapter explains how correlation structures can be added to the linear regression and additive model. The mixed effects models from Chapters 4 and 5 can also be extended with a temporal correlation structure. The title of this chapter contains 'Part I', suggesting that there is also a Part II. Indeed, that is the next chapter. In part I, we use regularly spaced time series, whereas in the next chapter, irregular spaced time series, spatial data, and data along an age gradient are analysed. We use a bird time series data set previously analysed in Reed et al. (2007). In the first section , we start with only one species and show how the linear regression model can be extended with a residual temporal correlation structure. In the second section, we use the same approach for a multivariate time series. In Section 6.3, the owl data are used again. 6.1 Temporal Correlation and Linear Regression Reed et al. (2007) analysed abundances of three bird species measured at three islands in Hawaii. The data were annual abundances from 1956 to 2003. Here, we use one of these time series, moorhen abundance on the island of Kauai, to illustrate how to deal with violation of independence. A time series plot is given in Fig. 6.1. We applied a square root transformation to stabilise the variance, but strictly speaking , this is unnecessary as methods discussed earlier (Chapter 4) can be used to model the heterogeneity present in the original series. However, we do not want to over-complicate matters at this stage by mixing different concepts in the same model. The following R code imports the data and makes a plot of square-root-transformed moorhen numbers. > library(AED); data(Hawaii) > Hawaii$Birds plot(Hawaii$Year, Hawaii$Birds, xlab = "Year", ylab = "Moorhen abundance on Kauai") Note that there is a general increase since the mid 1970s. Reed et al. (2007) used a dummy variable to test the effects of the implementation of new management