Update wide vs narrow code
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function [H, pValue, W] = swtest(x, alpha)
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%SWTEST Shapiro-Wilk parametric hypothesis test of composite normality.
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% [H, pValue, SWstatistic] = SWTEST(X, ALPHA) performs the
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% Shapiro-Wilk test to determine if the null hypothesis of
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% composite normality is a reasonable assumption regarding the
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% population distribution of a random sample X. The desired significance
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% level, ALPHA, is an optional scalar input (default = 0.05).
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%
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% The Shapiro-Wilk and Shapiro-Francia null hypothesis is:
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% "X is normal with unspecified mean and variance."
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%
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% This is an omnibus test, and is generally considered relatively
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% powerful against a variety of alternatives.
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% Shapiro-Wilk test is better than the Shapiro-Francia test for
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% Platykurtic sample. Conversely, Shapiro-Francia test is better than the
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% Shapiro-Wilk test for Leptokurtic samples.
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%
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% When the series 'X' is Leptokurtic, SWTEST performs the Shapiro-Francia
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% test, else (series 'X' is Platykurtic) SWTEST performs the
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% Shapiro-Wilk test.
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%
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% [H, pValue, SWstatistic] = SWTEST(X, ALPHA)
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%
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% Inputs:
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% X - a vector of deviates from an unknown distribution. The observation
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% number must exceed 3 and less than 5000.
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%
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% Optional inputs:
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% ALPHA - The significance level for the test (default = 0.05).
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%
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% Outputs:
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% SWstatistic - The test statistic (non normalized).
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%
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% pValue - is the p-value, or the probability of observing the given
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% result by chance given that the null hypothesis is true. Small values
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% of pValue cast doubt on the validity of the null hypothesis.
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%
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% H = 0 => Do not reject the null hypothesis at significance level ALPHA.
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% H = 1 => Reject the null hypothesis at significance level ALPHA.
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Copyright (c) 17 March 2009 by Ahmed Ben Sada %
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% Department of Finance, IHEC Sousse - Tunisia %
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% Email: ahmedbensaida@yahoo.com %
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% $ Revision 3.0 $ Date: 18 Juin 2014 $ %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% References:
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%
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% - Royston P. "Remark AS R94", Applied Statistics (1995), Vol. 44,
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% No. 4, pp. 547-551.
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% AS R94 -- calculates Shapiro-Wilk normality test and P-value
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% for sample sizes 3 <= n <= 5000. Handles censored or uncensored data.
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% Corrects AS 181, which was found to be inaccurate for n > 50.
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% Subroutine can be found at: http://lib.stat.cmu.edu/apstat/R94
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%
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% - Royston P. "A pocket-calculator algorithm for the Shapiro-Francia test
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% for non-normality: An application to medicine", Statistics in Medecine
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% (1993a), Vol. 12, pp. 181-184.
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%
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% - Royston P. "A Toolkit for Testing Non-Normality in Complete and
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% Censored Samples", Journal of the Royal Statistical Society Series D
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% (1993b), Vol. 42, No. 1, pp. 37-43.
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%
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% - Royston P. "Approximating the Shapiro-Wilk W-test for non-normality",
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% Statistics and Computing (1992), Vol. 2, pp. 117-119.
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%
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% - Royston P. "An Extension of Shapiro and Wilk's W Test for Normality
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% to Large Samples", Journal of the Royal Statistical Society Series C
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% (1982a), Vol. 31, No. 2, pp. 115-124.
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%
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%
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% Ensure the sample data is a VECTOR.
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%
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if numel(x) == length(x)
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x = x(:); % Ensure a column vector.
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else
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error(' Input sample ''X'' must be a vector.');
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end
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%
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% Remove missing observations indicated by NaN's and check sample size.
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%
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x = x(~isnan(x));
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if length(x) < 3
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error(' Sample vector ''X'' must have at least 3 valid observations.');
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end
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if length(x) > 5000
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warning('Shapiro-Wilk test might be inaccurate due to large sample size ( > 5000).');
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end
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if max(x) == min(x)
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H = 0;
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pValue = 0;
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W = 0;
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return;
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end
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%
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% Ensure the significance level, ALPHA, is a
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% scalar, and set default if necessary.
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%
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if (nargin >= 2) && ~isempty(alpha)
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if ~isscalar(alpha)
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error(' Significance level ''Alpha'' must be a scalar.');
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end
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if (alpha <= 0 || alpha >= 1)
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error(' Significance level ''Alpha'' must be between 0 and 1.');
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end
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else
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alpha = 0.05;
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end
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% First, calculate the a's for weights as a function of the m's
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% See Royston (1992, p. 117) and Royston (1993b, p. 38) for details
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% in the approximation.
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x = sort(x); % Sort the vector X in ascending order.
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n = length(x);
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mtilde = norminv(((1:n)' - 3/8) / (n + 1/4));
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weights = zeros(n,1); % Preallocate the weights.
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if kurtosis(x) > 3
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% The Shapiro-Francia test is better for leptokurtic samples.
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weights = 1/sqrt(mtilde'*mtilde) * mtilde;
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%
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% The Shapiro-Francia statistic W' is calculated to avoid excessive
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% rounding errors for W' close to 1 (a potential problem in very
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% large samples).
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%
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W = (weights' * x)^2 / ((x - mean(x))' * (x - mean(x)));
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% Royston (1993a, p. 183):
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nu = log(n);
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u1 = log(nu) - nu;
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u2 = log(nu) + 2/nu;
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mu = -1.2725 + (1.0521 * u1);
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sigma = 1.0308 - (0.26758 * u2);
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newSFstatistic = log(1 - W);
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%
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% Compute the normalized Shapiro-Francia statistic and its p-value.
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%
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NormalSFstatistic = (newSFstatistic - mu) / sigma;
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% Computes the p-value, Royston (1993a, p. 183).
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pValue = 1 - normcdf(real(NormalSFstatistic), 0, 1);
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else
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% The Shapiro-Wilk test is better for platykurtic samples.
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c = 1/sqrt(mtilde'*mtilde) * mtilde;
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u = 1/sqrt(n);
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% Royston (1992, p. 117) and Royston (1993b, p. 38):
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PolyCoef_1 = [-2.706056 , 4.434685 , -2.071190 , -0.147981 , 0.221157 , c(n)];
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PolyCoef_2 = [-3.582633 , 5.682633 , -1.752461 , -0.293762 , 0.042981 , c(n-1)];
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% Royston (1992, p. 118) and Royston (1993b, p. 40, Table 1)
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PolyCoef_3 = [-0.0006714 , 0.0250540 , -0.39978 , 0.54400];
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PolyCoef_4 = [-0.0020322 , 0.0627670 , -0.77857 , 1.38220];
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PolyCoef_5 = [0.00389150 , -0.083751 , -0.31082 , -1.5861];
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PolyCoef_6 = [0.00303020 , -0.082676 , -0.48030];
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PolyCoef_7 = [0.459 , -2.273];
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weights(n) = polyval(PolyCoef_1 , u);
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weights(1) = -weights(n);
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if n > 5
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weights(n-1) = polyval(PolyCoef_2 , u);
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weights(2) = -weights(n-1);
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count = 3;
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phi = (mtilde'*mtilde - 2 * mtilde(n)^2 - 2 * mtilde(n-1)^2) / ...
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(1 - 2 * weights(n)^2 - 2 * weights(n-1)^2);
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else
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count = 2;
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phi = (mtilde'*mtilde - 2 * mtilde(n)^2) / ...
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(1 - 2 * weights(n)^2);
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end
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% Special attention when n = 3 (this is a special case).
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if n == 3
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% Royston (1992, p. 117)
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weights(1) = 1/sqrt(2);
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weights(n) = -weights(1);
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phi = 1;
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end
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%
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% The vector 'WEIGHTS' obtained next corresponds to the same coefficients
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% listed by Shapiro-Wilk in their original test for small samples.
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%
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weights(count : n-count+1) = mtilde(count : n-count+1) / sqrt(phi);
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%
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% The Shapiro-Wilk statistic W is calculated to avoid excessive rounding
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% errors for W close to 1 (a potential problem in very large samples).
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%
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W = (weights' * x) ^2 / ((x - mean(x))' * (x - mean(x)));
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%
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% Calculate the normalized W and its significance level (exact for
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% n = 3). Royston (1992, p. 118) and Royston (1993b, p. 40, Table 1).
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%
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newn = log(n);
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if (n >= 4) && (n <= 11)
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mu = polyval(PolyCoef_3 , n);
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sigma = exp(polyval(PolyCoef_4 , n));
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gam = polyval(PolyCoef_7 , n);
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newSWstatistic = -log(gam-log(1-W));
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elseif n > 11
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mu = polyval(PolyCoef_5 , newn);
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sigma = exp(polyval(PolyCoef_6 , newn));
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newSWstatistic = log(1 - W);
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elseif n == 3
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mu = 0;
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sigma = 1;
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newSWstatistic = 0;
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end
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%
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% Compute the normalized Shapiro-Wilk statistic and its p-value.
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%
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NormalSWstatistic = (newSWstatistic - mu) / sigma;
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% NormalSWstatistic is referred to the upper tail of N(0,1),
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% Royston (1992, p. 119).
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pValue = 1 - normcdf(NormalSWstatistic, 0, 1);
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% Special attention when n = 3 (this is a special case).
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if n == 3
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pValue = 6/pi * (asin(sqrt(W)) - asin(sqrt(3/4)));
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% Royston (1982a, p. 121)
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end
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end
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%
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% To maintain consistency with existing Statistics Toolbox hypothesis
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% tests, returning 'H = 0' implies that we 'Do not reject the null
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% hypothesis at the significance level of alpha' and 'H = 1' implies
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% that we 'Reject the null hypothesis at significance level of alpha.'
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%
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H = (alpha >= pValue);
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