The R Book

Testing for a Trend in the Time Series

It is important to know whether these data provide any evidence for global warming. The trend part of the figure indicates a fluctuating increase, but is it significant? The mean temperature in the last 9 years was 0.71° C higher degrees than in the first 10 years:

ys<-factor(1+(yr>1996))
tapply(upper,ys,mean)

       1         2
14.62056  15.32978

We cannot test for a trend with linear regression because of the massive temporal pseudoreplication. Suppose we tried this:

model1<-lm(upper~index+sin(time*2*pi)+cos(time*2*pi))
summary(model1)

Coefficients:
                      Estimate   Std.Error  t value  Pr(>|t|)
(Intercept)          1.433e+01   8.136e-02  176.113    <2e-16  ***
index                1.807e-04   2.031e-05    8.896    <2e-16  ***
sin(time * 2 * pi)  -2.518e+00   5.754e-02  -43.758    <2e-16  ***
cos(time * 2 * pi)  -7.240e+00   5.749e-02 -125.939    <2e-16  ***

It would suggest (wrongly, as we shall see) that the warming was highly significant (index p value < 2 × 10⁻¹⁶ for a slope of 0.000 180 7 degrees of warming per day, leading to a predicted increase in mean temperature of 1.254 degrees over the 6940 days of the time series).

Since there is so much temporal pseudoreplication we should use a mixed model (lmer, p. 640), and because we intend to compare two models with different fixed effects we use the method of maximum likelihood ("ML" rather than "REML"). The explanatory variable for any trend is index, and we fit the model with and without this variable, allowing for different intercepts for the different years ...