{-# LANGUAGE FlexibleContexts, Rank2Types, ScopedTypeVariables #-} -- | Student's T-test is for assessing whether two samples have -- different mean. This module contain several variations of -- T-test. It's a parametric tests and assumes that samples are -- normally distributed. module Statistics.Test.StudentT ( studentTTest , welchTTest , pairedTTest , module Statistics.Test.Types ) where import Statistics.Distribution hiding (mean) import Statistics.Distribution.StudentT import Statistics.Sample (mean, varianceUnbiased) import Statistics.Test.Types import Statistics.Types (mkPValue,PValue) import Statistics.Function (square) import qualified Data.Vector.Generic as G import qualified Data.Vector.Unboxed as U import qualified Data.Vector.Storable as S import qualified Data.Vector as V -- | Two-sample Student's t-test. It assumes that both samples are -- normally distributed and have same variance. Returns @Nothing@ if -- sample sizes are not sufficient. studentTTest :: (G.Vector v Double) => PositionTest -- ^ one- or two-tailed test -> v Double -- ^ Sample A -> v Double -- ^ Sample B -> Maybe (Test StudentT) studentTTest test sample1 sample2 | G.length sample1 < 2 || G.length sample2 < 2 = Nothing | otherwise = Just Test { testSignificance = significance test t ndf , testStatistics = t , testDistribution = studentT ndf } where (t, ndf) = tStatistics True sample1 sample2 {-# INLINABLE studentTTest #-} {-# SPECIALIZE studentTTest :: PositionTest -> U.Vector Double -> U.Vector Double -> Maybe (Test StudentT) #-} {-# SPECIALIZE studentTTest :: PositionTest -> S.Vector Double -> S.Vector Double -> Maybe (Test StudentT) #-} {-# SPECIALIZE studentTTest :: PositionTest -> V.Vector Double -> V.Vector Double -> Maybe (Test StudentT) #-} -- | Two-sample Welch's t-test. It assumes that both samples are -- normally distributed but doesn't assume that they have same -- variance. Returns @Nothing@ if sample sizes are not sufficient. welchTTest :: (G.Vector v Double) => PositionTest -- ^ one- or two-tailed test -> v Double -- ^ Sample A -> v Double -- ^ Sample B -> Maybe (Test StudentT) welchTTest test sample1 sample2 | G.length sample1 < 2 || G.length sample2 < 2 = Nothing | otherwise = Just Test { testSignificance = significance test t ndf , testStatistics = t , testDistribution = studentT ndf } where (t, ndf) = tStatistics False sample1 sample2 {-# INLINABLE welchTTest #-} {-# SPECIALIZE welchTTest :: PositionTest -> U.Vector Double -> U.Vector Double -> Maybe (Test StudentT) #-} {-# SPECIALIZE welchTTest :: PositionTest -> S.Vector Double -> S.Vector Double -> Maybe (Test StudentT) #-} {-# SPECIALIZE welchTTest :: PositionTest -> V.Vector Double -> V.Vector Double -> Maybe (Test StudentT) #-} -- | Paired two-sample t-test. Two samples are paired in a -- within-subject design. Returns @Nothing@ if sample size is not -- sufficient. pairedTTest :: forall v. (G.Vector v (Double, Double), G.Vector v Double) => PositionTest -- ^ one- or two-tailed test -> v (Double, Double) -- ^ paired samples -> Maybe (Test StudentT) pairedTTest test sample | G.length sample < 2 = Nothing | otherwise = Just Test { testSignificance = significance test t ndf , testStatistics = t , testDistribution = studentT ndf } where (t, ndf) = tStatisticsPaired sample {-# INLINABLE pairedTTest #-} {-# SPECIALIZE pairedTTest :: PositionTest -> U.Vector (Double,Double) -> Maybe (Test StudentT) #-} {-# SPECIALIZE pairedTTest :: PositionTest -> V.Vector (Double,Double) -> Maybe (Test StudentT) #-} ------------------------------------------------------------------------------- significance :: PositionTest -- ^ one- or two-tailed -> Double -- ^ t statistics -> Double -- ^ degree of freedom -> PValue Double -- ^ p-value significance test t df = case test of -- Here we exploit symmetry of T-distribution and calculate small tail SamplesDiffer -> mkPValue \$ 2 * tailArea (negate (abs t)) AGreater -> mkPValue \$ tailArea (negate t) BGreater -> mkPValue \$ tailArea t where tailArea = cumulative (studentT df) -- Calculate T statistics for two samples tStatistics :: (G.Vector v Double) => Bool -- variance equality -> v Double -> v Double -> (Double, Double) {-# INLINE tStatistics #-} tStatistics varequal sample1 sample2 = (t, ndf) where -- t-statistics t = (m1 - m2) / sqrt ( if varequal then ((n1 - 1) * s1 + (n2 - 1) * s2) / (n1 + n2 - 2) * (1 / n1 + 1 / n2) else s1 / n1 + s2 / n2) -- degree of freedom ndf | varequal = n1 + n2 - 2 | otherwise = square (s1 / n1 + s2 / n2) / (square s1 / (square n1 * (n1 - 1)) + square s2 / (square n2 * (n2 - 1))) -- statistics of two samples n1 = fromIntegral \$ G.length sample1 n2 = fromIntegral \$ G.length sample2 m1 = mean sample1 m2 = mean sample2 s1 = varianceUnbiased sample1 s2 = varianceUnbiased sample2 -- Calculate T-statistics for paired sample tStatisticsPaired :: (G.Vector v (Double, Double), G.Vector v Double) => v (Double, Double) -> (Double, Double) {-# INLINE tStatisticsPaired #-} tStatisticsPaired sample = (t, ndf) where -- t-statistics t = let d = G.map (uncurry (-)) sample sumd = G.sum d in sumd / sqrt ((n * G.sum (G.map square d) - square sumd) / ndf) -- degree of freedom ndf = n - 1 n = fromIntegral \$ G.length sample