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Significance of Correlations

Values

The table below lists the correlation values that are significant at 4 significance levels and specified degrees of freedom. To perform a 1-tailed test, simply use the values in the list. For example, the 95% significance level correlation value for 40 years is listed under the .950 column for 40 degrees of freedom and is .264. For a 2-tailed test, for the significance level you want (for example 95%) use the value for the
level+((1.000-level)/2.0) significance level.
That is, .950+(1.000-.950)/2 or .975 level
For 40 years, a correlation of +/- .3124. would be significant. These significance levels are local. For a resolution of 144x73 gridpoints, one would assume at least .05X(144*73)=526 grids would be significant by chance at the one-sided 95% level.

Caveats

Note that determining actual field significance is trickier than this due to the grids being correlated in space so that Monte Carlo or similar tests are usually run.

For some variables (like SST), there is year to year correlation at a region so that the number of years will be greater than the actual degrees of freedom meaning a higher correlation value is needed for significance.
(Linear) correlation makes certain assumptions about the data which can lead to spurious high correlations (or, hide real relationships). For example, the assumption is that the data is normally distributed which doesn't always hold for variables like precipitation. Also, since the method maximizes linear relationships, variables in quadrature may appear to have 0 correlation when in fact the relationship between them is exactly defined.

NOTE, CORRELATION DOES NOT IMPLY CAUSATION!!

Two variables a and b may be highly correlated but the correlation could mean a causes b, b causes a, the correlation is due to a third factor related to both a and b, or, the correlation could simply arise by chance. The user should be cautious when interpreting results.

A good discussion on the significance of correlations can be found at Graphpad's software website. Another discussion cxan be found Here.

References

Livezey, R.E. and W.Y. Chen, 1983: Statistical field significance and it's determination by Monte Carlo Techniques. Mon. Wea. Review., 111, 46-59.

Diaconis, P. and B. Efron, 1983: Computer Intensive methods in statistics, Sci. Am., 248, 116-130.

                Significance Level
Degrees of   .950   .975   .990   .995
 Freedom
    2        1.000  1.000  1.000  1.000
    3        0.920  0.954  0.977  0.986
    4        0.833  0.891  0.936  0.956
    5        0.758  0.829  0.889  0.919
    6        0.697  0.774  0.844  0.880
    7        0.646  0.727  0.802  0.843
    8        0.605  0.685  0.764  0.808
    9        0.570  0.650  0.729  0.775
   10        0.540  0.619  0.699  0.746
   11        0.514  0.592  0.671  0.719
   12        0.491  0.567  0.647  0.695
   13        0.471  0.546  0.624  0.672
   14        0.453  0.526  0.604  0.652
   15        0.437  0.509  0.585  0.633
   16        0.423  0.493  0.568  0.615
   17        0.410  0.478  0.552  0.599
   18        0.398  0.465  0.538  0.584
   19        0.387  0.453  0.524  0.570
   20        0.377  0.441  0.512  0.557
   21        0.367  0.431  0.500  0.545
   22        0.358  0.421  0.489  0.533
   23        0.350  0.411  0.479  0.522
   24        0.343  0.403  0.469  0.512
   25        0.336  0.395  0.460  0.503
   26        0.329  0.387  0.451  0.493
   27        0.322  0.380  0.443  0.485
   28        0.316  0.373  0.436  0.476
   29        0.311  0.366  0.428  0.469
   30        0.305  0.360  0.421  0.461
   31        0.300  0.354  0.415  0.454
   32        0.295  0.349  0.408  0.447
   33        0.291  0.343  0.402  0.441
   34        0.286  0.338  0.396  0.434
   35        0.282  0.333  0.391  0.428
   36        0.278  0.329  0.385  0.423
   37        0.274  0.324  0.380  0.417
   38        0.271  0.320  0.375  0.412
   39        0.267  0.316  0.370  0.407
   40        0.264  0.312  0.366  0.402
   41        0.260  0.308  0.361  0.397
   42        0.257  0.304  0.357  0.392
   43        0.254  0.300  0.353  0.388
   44        0.251  0.297  0.349  0.384
   45        0.248  0.294  0.345  0.379