-- | -- 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. module Statistics.Test.WilcoxonT ( -- * Wilcoxon signed-rank matched-pair test wilcoxonMatchedPairTest , wilcoxonMatchedPairSignedRank , wilcoxonMatchedPairSignificant , wilcoxonMatchedPairSignificance , wilcoxonMatchedPairCriticalValue -- * Data types , TestType(..) , TestResult(..) ) where import Control.Applicative ((<$>)) import Data.Function (on) import Data.List (findIndex) import Data.Ord (comparing) import qualified Data.Vector.Unboxed as U import Statistics.Types (Sample) import Statistics.Function (sortBy) import Statistics.Test.Types import Statistics.Test.Internal -- | The Wilcoxon matched-pairs signed-rank test. -- -- The value returned is the pair (T+, T-). T+ is the sum of positive ranks (the -- 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). -- These values mean little by themselves, and should be combined with the 'wilcoxonSignificant' -- function in this module to get a meaningful result. -- -- The samples are zipped together: if one is longer than the other, both are truncated -- to the the length of the shorter sample. -- -- Note that: wilcoxonMatchedPairSignedRank == (\(x, y) -> (y, x)) . flip wilcoxonMatchedPairSignedRank wilcoxonMatchedPairSignedRank :: Sample -> Sample -> (Double, Double) wilcoxonMatchedPairSignedRank a b = ( U.sum ranks1 , negate $ U.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 -> [Int] 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 = map fromIntegral . scanl1 (+) . coefficients -- | 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-tailed test (see description above). -> 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