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# rtest

Determines the statistical significance of a linear correlation coefficient.

## Prototype

```	function rtest (
r       : numeric,
Nr      : integer,
opt [1] : integer
)

return_val [dimsizes(r)] :  float or double
```

## Arguments

r

Scalar or array of any dimensionality containing the linear correlation coefficients (-1 <= r <= +1).

Nr

Array of the same dimensionality as r or else a scalar. Contains the number of observations used to determine the coefficients. If Nr is a scalar and r is an array of any dimension, then the scalar Nr value will be used for all tests. Nr must be at least three but should be at least eight.

opt

Currently not used. Set to zero.

## Return value

The two-tailed probability value (0 to 1, inclusive). The return value will have the same dimensionality as r. The return type will be double if r is of type double, and float otherwise.

## Description

This function determines the statistical significance of the linear correlation coefficient (Pearson's r). The null hypothesis is that the two variables used to calculate r are independent (i.e. r=0.0). It's assumed that the variables used to calculate r are normally distributed.

The probability value(s) returned by rtest are two-sided (ie, non-directional) and represent estimates of the statistical significance. Commonly, values of 0.10, 0.05, and 0.01 are used as critical levels. Note: the user should specify the critical significance level prior to the calculation.

The t-statistic used

```              t = r*sqrt((Nr-2)/(1-r^2))
```
is reasonable for Nr > 8 [Climate Change, WMO Technical Note No.79, 1971, p66]. Further, the t-statistic is appropriate "even if the binormal assumption is not well substantiated" [Numerical Recipies, Press et al, 1986, p486].

Note: Beginning with NCL version 4.3.0, if Nr < 3 then the return values will be set to the appropriate _FillValue.

## Examples

Example 1

Consider testing the following at the 0.05 level.

```
siglvl = 0.05            ; a-priori specified sig level
pr     = rtest(0.2, 100, 0)    ; ===> pr=0.046
if (pr.lt.siglvl) then
print("rtest(0.2,100,0) is significant at the "+siglvl+" significance level)
else
print("rtest(0.2,100,0) is NOT significant at the "+siglvl+" significance level)
end if

; array of correlations with different sampling sizes

n      = (/23,13,10,6,8,22,14,24,21,16,26,17,5,28,15,18,11,12,15/)
r      = fspan(-0.9,0.9,19)
tval   = r/sqrt((1.0-r^2)/(n-2))
prob   = rtest(r,n,0)
yes_no = where(prob.lt.siglvl, True, False)

print(r+"   "+n+"   "+tval+"   prob+"   "+yes_no)
```
The (edited) results are:
```          r    N      tval      prob      yes_no
-----------------------------------------
(0)     -0.9   23   -9.46183   5.05827e-09 True
(1)     -0.8   13   -4.42217   0.00102483  True
(2)     -0.7   10   -2.77241   0.0242063   True
(3)     -0.6   6    -1.5       0.208       False
(4)     -0.5   8    -1.41421   0.207031    False
(5)     -0.4   22   -1.9518    0.0651067   False
(6)     -0.3   14   -1.08941   0.297367    False
(7)     -0.2   24   -0.957427  0.348756    False
(8)     -0.1   21   -0.438086  0.666264    False
(9)      0     16    0.0       1.0         False
(10)     0.1   26    0.492366  0.626935    False
(11)     0.2   17    0.790569  0.441516    False
(12)     0.3   5     0.544705  0.623838    False
(13)     0.4   28    2.22539   0.0349395   True
(14)     0.5   15    2.08167   0.0576988   False
(15)     0.6   18    3         0.0084795   True
(16)     0.7   11    2.94059   0.016471    True
(17)     0.8   12    4.21637   0.00178184  True
(18)     0.9   15    7.44453   4.87198e-06 True

```

Example 2

Assume x is a one-dimensional array with no missing data. Use the esacr function to estimate the lag-one autocorrelation.

```
siglvl= 0.10            ; a-priori specified sig level
mxlag = 1
acr   = esacr(x,mxlag)  ; auto correlation coef acr(0)=1  ,  acr(1)=lag_1
Nx    = dimsizes(x)
prob  = rtest(acr(1), Nx, 0)
if (prob.lt.siglvl) then
[do something]
end if
```

prob will be a scalar containing the significance, and will range between zero and one. Let's say acr = 0.569 and Nx = 10. Then prob will be = 0.086. The conclusion is that the null hypothesis is rejected and that there is correlation between the two variables.

If x has missing data (x@_FillValue), then use NCL functions num and ismissing:

```  Nx    = num(.not.ismissing(x))
prob  = rtest(acr(1), Nx, 0)
```

Example 3

Assume x and y are one-dimensional arrays with no missing data. Use the esccr function:

```  siglvl= 0.025             ; a-priori specified sig level
mxlag = 0
ccr   = esccr(x,y,mxlag)  ; cross correlation coef
Nr    = dimsizes(x)
prob  = rtest(ccr, Nr, 0)
if (prob.lt.siglvl) then  ; if significant 'do something'
[do something]
end if
```
prob will be a scalar containing the significance. It will range between zero and one.

Example 4

Assume x is dimensioned ntim x nlat x mlon and has named dimensions "time", "lat", "lon". Further assume x contains monthly mean data for ten years (ntim=120). Determine the lag correlations up to lag 3. Use dimension reordering to do the auto-correlations over time. Then:

```  mxlag = 3
acr   = esacr( x(lat|:,lon|:,time|:), mxlag )  ; acr(nlat,mlon,mxlag)

do lag=1,mxlag
prob  = rtest(acr(:,:,lag), ntim, 0)        ; prob(nlat,mlon)
:                                   ; whatever,
:                                   ; say, plot prob
end do
```

will yield prob dimensioned nlat x mlon at each lag.