Section 9.2: Motivation
161
936-point extratropical Northern Hemisphere grid without an apriori expectation of the result. The height data is at least cross-correlated 1 , the ratio
of sampie size to dimension of the height field is small (even if the bulk of
its meaningful variance is compressed into EOFs [Chapter 13] this dimension
would still be at least 15 to 20), and the calculations are part of a "fishing
expedition" (Le. aposteriori). Correlation maps were also produced with the
SOl leading the winter heights by one and two seasons and at all three leads
(0-2 seasons) for JJA as weIl, thereby increasing the number of correlations
and interdependencies between them to be evaluated. All correlations were
tested at the 5% level with a Students-t test.
9.2.1 Local VS. Field Significance
The results of the lag-O winter correlations are shown in Figure 9.1. The
most basic problem facing the analyst given these results is whether or not a
null hypothesis that the field of correlations arose by chance can be rejected
with confidence, i.e. to decide whether or not there is sufficient evidence to
believe the relationship between SOl and winter DJF extratropical Northern
Hemisphere 700 hpa heights was non-accidental.
The fact that there are a number of areas where the correlations have
local significance and that they ac count for more than 5% of the grid area is
insufficient in itself to reject the null hypothesis at the 5% level and presume
there is field significance. First, if the null hypothesis was everywhere true
there would still be multiple opportunities for local significance to occur by
chance. Moreover, the expected number of these occurrences would be 5% of
all of the tests so the prob ability would be 0.5 that more than this number
would occur by accident.
In fact even if all of the tests were independent there would still be a
substantial probability (from binomial considerations) that noticeably more
or less than 5% of the 5% local,tests would pass by accident (under the null
hypothesis). The per cent of area locally significant at the 5% level that is
equalled or exceeded 5% of the time under the null hypothesis is given by the
curve in Figure 9.2, and for 936 points is equal to more than 6%. Thus, to
reject the null hypothesis at the 5% level in the absence of spatial correlation
requires local significance over more than 6% ofthe grid. In Figure 9.111.4%
of the area is covered by local tests significant at the 5% level.
The effect of spatial correlation in the height data is to lower the "effective"
number of tests and to broaden the distribution of out comes , leading to more
stringent criteria for field significance. The winter lag-O result would be field
significant only if this unknown effective number of tests exceeded about 52
(the intercept of 11.4% with the binomially-derived curve in Figure 9.2). In
the present case there is some uncertainty that this is the case, therefore a
1 All of the annual time series used in this analysis have some serial correlation that
was treated heuristically in the significance tests. For the present this will be ignored. In
Section 9.4 more solidly-based strategies tor serial correlation will be presented.
161
936-point extratropical Northern Hemisphere grid without an apriori expectation of the result. The height data is at least cross-correlated 1 , the ratio
of sampie size to dimension of the height field is small (even if the bulk of
its meaningful variance is compressed into EOFs [Chapter 13] this dimension
would still be at least 15 to 20), and the calculations are part of a "fishing
expedition" (Le. aposteriori). Correlation maps were also produced with the
SOl leading the winter heights by one and two seasons and at all three leads
(0-2 seasons) for JJA as weIl, thereby increasing the number of correlations
and interdependencies between them to be evaluated. All correlations were
tested at the 5% level with a Students-t test.
9.2.1 Local VS. Field Significance
The results of the lag-O winter correlations are shown in Figure 9.1. The
most basic problem facing the analyst given these results is whether or not a
null hypothesis that the field of correlations arose by chance can be rejected
with confidence, i.e. to decide whether or not there is sufficient evidence to
believe the relationship between SOl and winter DJF extratropical Northern
Hemisphere 700 hpa heights was non-accidental.
The fact that there are a number of areas where the correlations have
local significance and that they ac count for more than 5% of the grid area is
insufficient in itself to reject the null hypothesis at the 5% level and presume
there is field significance. First, if the null hypothesis was everywhere true
there would still be multiple opportunities for local significance to occur by
chance. Moreover, the expected number of these occurrences would be 5% of
all of the tests so the prob ability would be 0.5 that more than this number
would occur by accident.
In fact even if all of the tests were independent there would still be a
substantial probability (from binomial considerations) that noticeably more
or less than 5% of the 5% local,tests would pass by accident (under the null
hypothesis). The per cent of area locally significant at the 5% level that is
equalled or exceeded 5% of the time under the null hypothesis is given by the
curve in Figure 9.2, and for 936 points is equal to more than 6%. Thus, to
reject the null hypothesis at the 5% level in the absence of spatial correlation
requires local significance over more than 6% ofthe grid. In Figure 9.111.4%
of the area is covered by local tests significant at the 5% level.
The effect of spatial correlation in the height data is to lower the "effective"
number of tests and to broaden the distribution of out comes , leading to more
stringent criteria for field significance. The winter lag-O result would be field
significant only if this unknown effective number of tests exceeded about 52
(the intercept of 11.4% with the binomially-derived curve in Figure 9.2). In
the present case there is some uncertainty that this is the case, therefore a
1 All of the annual time series used in this analysis have some serial correlation that
was treated heuristically in the significance tests. For the present this will be ignored. In
Section 9.4 more solidly-based strategies tor serial correlation will be presented.