{-# LANGUAGE ViewPatterns #-}-- | Calculation of confidence intervalsmoduleStatistics.ConfidenceInt(poissonCI ,poissonNormalCI ,binomialCI ,naiveBinomialCI -- * References-- $references)whereimportStatistics.Distribution importStatistics.Distribution.ChiSquared importStatistics.Distribution.Beta importStatistics.Types -- | Calculate confidence intervals for Poisson-distributed value-- using normal approximationpoissonNormalCI ::Int->Estimate NormalErr DoublepoissonNormalCI :: Int -> Estimate NormalErr Double
poissonNormalCI Int
n |Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0=[Char] -> Estimate NormalErr Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.ConfidenceInt.poissonNormalCI negative number of trials"|Bool
otherwise=Double -> Double -> Estimate NormalErr Double
forall a. a -> a -> Estimate NormalErr a
estimateNormErr Double
n' (Double -> Double
forall a. Floating a => a -> a
sqrtDouble
n' )wheren' :: Double
n' =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
n -- | Calculate confidence intervals for Poisson-distributed value for-- single measurement. These are exact confidence intervalspoissonCI ::CL Double->Int->Estimate ConfInt DoublepoissonCI :: CL Double -> Int -> Estimate ConfInt Double
poissonCI cl :: CL Double
cl @(CL Double -> Double
forall a. CL a -> a
significanceLevel ->Double
p )Int
n |Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0=[Char] -> Estimate ConfInt Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.ConfidenceInt.poissonCI: negative number of trials"|Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==Int
0=Double -> (Double, Double) -> CL Double -> Estimate ConfInt Double
forall a. Num a => a -> (a, a) -> CL Double -> Estimate ConfInt a
estimateFromInterval Double
m (Double
0,Double
m2 )CL Double
cl |Bool
otherwise=Double -> (Double, Double) -> CL Double -> Estimate ConfInt Double
forall a. Num a => a -> (a, a) -> CL Double -> Estimate ConfInt a
estimateFromInterval Double
m (Double
m1 ,Double
m2 )CL Double
cl wherem :: Double
m =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
n m1 :: Double
m1 =Double
0.5Double -> Double -> Double
forall a. Num a => a -> a -> a
*ChiSquared -> Double -> Double
forall d. ContDistr d => d -> Double -> Double
quantile (Int -> ChiSquared
chiSquared (Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
n ))(Double
p Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Double
2)m2 :: Double
m2 =Double
0.5Double -> Double -> Double
forall a. Num a => a -> a -> a
*ChiSquared -> Double -> Double
forall d. ContDistr d => d -> Double -> Double
complQuantile (Int -> ChiSquared
chiSquared (Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
2))(Double
p Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Double
2)-- | Calculate confidence interval using normal approximation. Note-- that this approximation breaks down when /p/ is either close to 0-- or to 1. In particular if @np < 5@ or @1 - np < 5@ this-- approximation shouldn't be used.naiveBinomialCI ::Int-- ^ Number of trials->Int-- ^ Number of successes->Estimate NormalErr DoublenaiveBinomialCI :: Int -> Int -> Estimate NormalErr Double
naiveBinomialCI Int
n Int
k |Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<=Int
0Bool -> Bool -> Bool
||Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0=[Char] -> Estimate NormalErr Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.ConfidenceInt.naiveBinomialCI: negative number of events"|Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>Int
n =[Char] -> Estimate NormalErr Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.ConfidenceInt.naiveBinomialCI: more successes than trials"|Bool
otherwise=Double -> Double -> Estimate NormalErr Double
forall a. a -> a -> Estimate NormalErr a
estimateNormErr Double
p Double
σ wherep :: Double
p =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
k Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
n σ :: Double
σ =Double -> Double
forall a. Floating a => a -> a
sqrt(Double -> Double) -> Double -> Double
forall a b. (a -> b) -> a -> b
$Double
p Double -> Double -> Double
forall a. Num a => a -> a -> a
*(Double
1Double -> Double -> Double
forall a. Num a => a -> a -> a
-Double
p )Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
n -- | Clopper-Pearson confidence interval also known as exact-- confidence intervals.binomialCI ::CL Double->Int-- ^ Number of trials->Int-- ^ Number of successes->Estimate ConfInt DoublebinomialCI :: CL Double -> Int -> Int -> Estimate ConfInt Double
binomialCI cl :: CL Double
cl @(CL Double -> Double
forall a. CL a -> a
significanceLevel ->Double
p )Int
ni Int
ki |Int
ni Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<=Int
0Bool -> Bool -> Bool
||Int
ki Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0=[Char] -> Estimate ConfInt Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.ConfidenceInt.binomialCI: negative number of events"|Int
ki Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>Int
ni =[Char] -> Estimate ConfInt Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.ConfidenceInt.binomialCI: more successes than trials"|Int
ki Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==Int
0=Double -> (Double, Double) -> CL Double -> Estimate ConfInt Double
forall a. Num a => a -> (a, a) -> CL Double -> Estimate ConfInt a
estimateFromInterval Double
eff (Double
0,Double
ub )CL Double
cl |Int
ni Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==Int
ki =Double -> (Double, Double) -> CL Double -> Estimate ConfInt Double
forall a. Num a => a -> (a, a) -> CL Double -> Estimate ConfInt a
estimateFromInterval Double
eff (Double
lb ,Double
0)CL Double
cl |Bool
otherwise=Double -> (Double, Double) -> CL Double -> Estimate ConfInt Double
forall a. Num a => a -> (a, a) -> CL Double -> Estimate ConfInt a
estimateFromInterval Double
eff (Double
lb ,Double
ub )CL Double
cl wherek :: Double
k =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
ki n :: Double
n =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
ni eff :: Double
eff =Double
k Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Double
n lb :: Double
lb =BetaDistribution -> Double -> Double
forall d. ContDistr d => d -> Double -> Double
quantile (Double -> Double -> BetaDistribution
betaDistr Double
k (Double
n Double -> Double -> Double
forall a. Num a => a -> a -> a
-Double
k Double -> Double -> Double
forall a. Num a => a -> a -> a
+Double
1))(Double
p Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Double
2)ub :: Double
ub =BetaDistribution -> Double -> Double
forall d. ContDistr d => d -> Double -> Double
complQuantile (Double -> Double -> BetaDistribution
betaDistr (Double
k Double -> Double -> Double
forall a. Num a => a -> a -> a
+Double
1)(Double
n Double -> Double -> Double
forall a. Num a => a -> a -> a
-Double
k ))(Double
p Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Double
2)-- $references---- * Clopper, C.; Pearson, E. S. (1934). "The use of confidence or-- fiducial limits illustrated in the case of the-- binomial". Biometrika 26: 404–413. doi:10.1093/biomet/26.4.404---- * Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban-- (2001). "Interval Estimation for a Binomial Proportion". Statistical-- Science 16 (2): 101–133. doi:10.1214/ss/1009213286. MR 1861069.-- Zbl 02068924.