h = rjtest (x) performs the Ryan-Joiner test of the null hypothesis that the sample in the vector x comes from a normal distribution with unknown mean and variance, against the alternative that it does not come from a normal distribution. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise. x must be a vector of real values with at least four non-missing elements; NaN values are treated as missing and removed.

The test statistic is the correlation between the sorted sample and the approximate expected normal order statistics of Blom, $$ R = \frac{\sum_{i=1}^n x_{(i)} b_i} {\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2 \sum_{i=1}^n b_i^2}}, \qquad b_i = \Phi^{-1}\left( \frac{i - 3/8}{n + 1/4} \right). $$ R lies in (0,1], and small values of R are evidence against normality. R^2 is the statistic W' of the Shapiro-Francia test in swtest. To test for a lognormal distribution, test log (x).

The following Name-Value pairs are supported:

Without 'MonteCarloTolerance' the critical value at the significance levels 0.01, 0.05 and 0.1 is the approximation of Ryan and Joiner, $$ c_\alpha (n) = \min \left( b_0 + \frac{b_1}{\sqrt{n}} + \frac{b_2}{n} + \frac{b_3}{n^2}, \; 1 \right), $$ with the coefficients

At other significance levels the critical value is interpolated between these three with a monotone piecewise cubic (pchip), extended below 0.01 along the line through the values at 0.01 and 0.05, and above 0.1 along the line to 1 at significance level 1. The p-value is interpolated linearly in R through the points (0, 0), (c01, 0.01), (c05, 0.05), (c10, 0.1) and (1, 1), where c01, c05 and c10 are the three critical values, so away from those levels it is a rough guide only. The approximation is calibrated for samples of up to 100 values, and a larger sample gives a warning; use 'MonteCarloTolerance' for such samples, or for an accurate p-value.

[h, p] = rjtest (…) also returns the p-value p of the test, the probability of observing a statistic as small as rjstat under the null hypothesis.

[h, p, rjstat, critval] = rjtest (…) also returns the test statistic rjstat and the critical value critval at significance level alpha. The null hypothesis is rejected when rjstat < critval.

A sample whose values are all equal has no defined statistic and is refused, as is a sample holding an infinite value, where MATLAB returns NaN for p, rjstat and critval and 1 for h. The option names are not case sensitive but must be given in full, bar 'MonteCarloTol', where MATLAB accepts any shortened name.

References:

1. T. A. Ryan and B. L. Joiner. Normal probability plots and tests for normality. Technical report, Statistics Department, The Pennsylvania State University, 1976.
2. G. Blom. Statistical Estimates and Transformed Beta-Variables. Wiley, New York, 1958.

See also: swtest, adtest, jbtest, kstest, lillietest

Source Code: rjtest