-- | -- Module : Statistics.Test.WilcoxonT -- Copyright : (c) 2010 Neil Brown -- License : BSD3 -- -- Maintainer : bos@serpentine.com -- Stability : experimental -- Portability : portable -- -- The Wilcoxon matched-pairs signed-rank test is non-parametric test -- which could be used to whether two related samples have different -- means. -- -- WARNING: current implementation contain serious bug and couldn't be -- used with samples larger than 1023. -- <https://github.com/bos/statistics/issues/18> module Statistics.Test.WilcoxonT ( -- * Wilcoxon signed-rank matched-pair test wilcoxonMatchedPairTest , wilcoxonMatchedPairSignedRank , wilcoxonMatchedPairSignificant , wilcoxonMatchedPairSignificance , wilcoxonMatchedPairCriticalValue -- * Data types , TestType(..) , TestResult(..) ) where -- -- -- -- Note that: wilcoxonMatchedPairSignedRank == (\(x, y) -> (y, x)) . flip wilcoxonMatchedPairSignedRank -- The samples are zipped together: if one is longer than the other, both are truncated -- The value returned is the pair (T+, T-). T+ is the sum of positive ranks (the -- These values mean little by themselves, and should be combined with the 'wilcoxonSignificant' -- function in this module to get a meaningful result. -- ranks of the differences where the first parameter is higher) whereas T- is -- the sum of negative ranks (the ranks of the differences where the second parameter is higher). -- to the the length of the shorter sample. import Control.Applicative ((<$>)) import Data.Function (on) import Data.List (findIndex) import Data.Ord (comparing) import Prelude hiding (sum) import Statistics.Function (sortBy) import Statistics.Sample.Internal (sum) import Statistics.Test.Internal (rank, splitByTags) import Statistics.Test.Types (TestResult(..), TestType(..), significant) import Statistics.Types (Sample) import qualified Data.Vector.Unboxed as U wilcoxonMatchedPairSignedRank :: Sample -> Sample -> (Double, Double) wilcoxonMatchedPairSignedRank a b = (sum ranks1, negate (sum ranks2)) where (ranks1, ranks2) = splitByTags $ U.zip tags (rank ((==) `on` abs) diffs) (tags,diffs) = U.unzip $ U.map (\x -> (x>0 , x)) -- Attack tags to distribution elements $ U.filter (/= 0.0) -- Remove equal elements $ sortBy (comparing abs) -- Sort the differences by absolute difference $ U.zipWith (-) a b -- Work out differences -- | The coefficients for x^0, x^1, x^2, etc, in the expression -- \prod_{r=1}^s (1 + x^r). See the Mitic paper for details. -- -- We can define: -- f(1) = 1 + x -- f(r) = (1 + x^r)*f(r-1) -- = f(r-1) + x^r * f(r-1) -- The effect of multiplying the equation by x^r is to shift -- all the coefficients by r down the list. -- -- This list will be processed lazily from the head. coefficients :: Int -> [Integer] coefficients 1 = [1, 1] -- 1 + x coefficients r = let coeffs = coefficients (r-1) (firstR, rest) = splitAt r coeffs in firstR ++ add rest coeffs where add (x:xs) (y:ys) = x + y : add xs ys add xs [] = xs add [] ys = ys -- This list will be processed lazily from the head. summedCoefficients :: Int -> [Double] summedCoefficients n | n < 1 = error "Statistics.Test.WilcoxonT.summedCoefficients: nonpositive sample size" | n > 1023 = error "Statistics.Test.WilcoxonT.summedCoefficients: sample is too large (see bug #18)" | otherwise = map fromIntegral $ scanl1 (+) $ coefficients n -- | Tests whether a given result from a Wilcoxon signed-rank matched-pairs test -- is significant at the given level. -- -- This function can perform a one-tailed or two-tailed test. If the first -- parameter to this function is 'TwoTailed', the test is performed two-tailed to -- check if the two samples differ significantly. If the first parameter is -- 'OneTailed', the check is performed one-tailed to decide whether the first sample -- (i.e. the first sample you passed to 'wilcoxonMatchedPairSignedRank') is -- greater than the second sample (i.e. the second sample you passed to -- 'wilcoxonMatchedPairSignedRank'). If you wish to perform a one-tailed test -- in the opposite direction, you can either pass the parameters in a different -- order to 'wilcoxonMatchedPairSignedRank', or simply swap the values in the resulting -- pair before passing them to this function. wilcoxonMatchedPairSignificant :: TestType -- ^ Perform one- or two-tailed test (see description below). -> Int -- ^ The sample size from which the (T+,T-) values were derived. -> Double -- ^ The p-value at which to test (e.g. 0.05) -> (Double, Double) -- ^ The (T+, T-) values from 'wilcoxonMatchedPairSignedRank'. -> Maybe TestResult -- ^ Return 'Nothing' if the sample was too -- small to make a decision. wilcoxonMatchedPairSignificant test sampleSize p (tPlus, tMinus) = case test of -- According to my nearest book (Understanding Research Methods and Statistics -- by Gary W. Heiman, p590), to check that the first sample is bigger you must -- use the absolute value of T- for a one-tailed check: OneTailed -> (significant . (abs tMinus <=) . fromIntegral) <$> wilcoxonMatchedPairCriticalValue sampleSize p -- Otherwise you must use the value of T+ and T- with the smallest absolute value: TwoTailed -> (significant . (t <=) . fromIntegral) <$> wilcoxonMatchedPairCriticalValue sampleSize (p/2) where t = min (abs tPlus) (abs tMinus) -- | Obtains the critical value of T to compare against, given a sample size -- and a p-value (significance level). Your T value must be less than or -- equal to the return of this function in order for the test to work out -- significant. If there is a Nothing return, the sample size is too small to -- make a decision. -- -- 'wilcoxonSignificant' tests the return value of 'wilcoxonMatchedPairSignedRank' -- for you, so you should use 'wilcoxonSignificant' for determining test results. -- However, this function is useful, for example, for generating lookup tables -- for Wilcoxon signed rank critical values. -- -- The return values of this function are generated using the method detailed in -- the paper \"Critical Values for the Wilcoxon Signed Rank Statistic\", Peter -- Mitic, The Mathematica Journal, volume 6, issue 3, 1996, which can be found -- here: <http://www.mathematica-journal.com/issue/v6i3/article/mitic/contents/63mitic.pdf>. -- According to that paper, the results may differ from other published lookup tables, but -- (Mitic claims) the values obtained by this function will be the correct ones. wilcoxonMatchedPairCriticalValue :: Int -- ^ The sample size -> Double -- ^ The p-value (e.g. 0.05) for which you want the critical value. -> Maybe Int -- ^ The critical value (of T), or Nothing if -- the sample is too small to make a decision. wilcoxonMatchedPairCriticalValue sampleSize p = case critical of Just n | n < 0 -> Nothing | otherwise -> Just n Nothing -> Just maxBound -- shouldn't happen: beyond end of list where m = (2 ** fromIntegral sampleSize) * p critical = subtract 1 <$> findIndex (> m) (summedCoefficients sampleSize) -- | Works out the significance level (p-value) of a T value, given a sample -- size and a T value from the Wilcoxon signed-rank matched-pairs test. -- -- See the notes on 'wilcoxonCriticalValue' for how this is calculated. wilcoxonMatchedPairSignificance :: Int -- ^ The sample size -> Double -- ^ The value of T for which you want the significance. -> Double -- ^ The significance (p-value). wilcoxonMatchedPairSignificance sampleSize rnk = (summedCoefficients sampleSize !! floor rnk) / 2 ** fromIntegral sampleSize -- | The Wilcoxon matched-pairs signed-rank test. The samples are -- zipped together: if one is longer than the other, both are -- truncated to the the length of the shorter sample. -- -- For one-tailed test it tests whether first sample is significantly -- greater than the second. For two-tailed it checks whether they -- significantly differ -- -- Check 'wilcoxonMatchedPairSignedRank' and -- 'wilcoxonMatchedPairSignificant' for additional information. wilcoxonMatchedPairTest :: TestType -- ^ Perform one-tailed test. -> Double -- ^ The p-value at which to test (e.g. 0.05) -> Sample -- ^ First sample -> Sample -- ^ Second sample -> Maybe TestResult -- ^ Return 'Nothing' if the sample was too -- small to make a decision. wilcoxonMatchedPairTest test p smp1 smp2 = wilcoxonMatchedPairSignificant test (min n1 n2) p $ wilcoxonMatchedPairSignedRank smp1 smp2 where n1 = U.length smp1 n2 = U.length smp2