Testovanie normality
Lenka Filová
data(iris)
head(iris)## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 5.1 3.5 1.4 0.2 setosa
## 2 4.9 3.0 1.4 0.2 setosa
## 3 4.7 3.2 1.3 0.2 setosa
## 4 4.6 3.1 1.5 0.2 setosa
## 5 5.0 3.6 1.4 0.2 setosa
## 6 5.4 3.9 1.7 0.4 setosa
plotids <- with(iris, plot3d(Sepal.Length, Sepal.Width, Petal.Length, type="s", col=as.numeric(Species), size=as.numeric(Petal.Width)))
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Funkcia mvn
Univerzálna funkcia na testovanie jedno- aj viacrozmernej normality, detekciu odľahlých pozorovaní a grafické znázornenie testov.
mvn(data = iris[-4], subset = "Species", mvnTest = "hz",univariateTest = "AD", univariatePlot = "histogram", multivariatePlot = "qq", multivariateOutlierMethod = "adj", showOutliers = TRUE, showNewData = TRUE, bc=TRUE)library(MVN)## sROC 0.1-2 loadedres <- mvn(iris[,-5], multivariateOutlierMethod = "adj", showOutliers = TRUE, showNewData = TRUE)
Test viacrozmernej normality
res$multivariateNormality## Test Statistic p value Result
## 1 Mardia Skewness 67.4305087780622 4.75799820400835e-07 NO
## 2 Mardia Kurtosis -0.230112114480973 0.818004651478034 YES
## 3 MVN <NA> <NA> NOTest jednorozmernej normality
res$univariateNormality## Test Variable Statistic p value Normality
## 1 Shapiro-Wilk Sepal.Length 0.9761 0.0102 NO
## 2 Shapiro-Wilk Sepal.Width 0.9849 0.1012 YES
## 3 Shapiro-Wilk Petal.Length 0.8763 <0.001 NO
## 4 Shapiro-Wilk Petal.Width 0.9018 <0.001 NODeskriptívne štatistiky
res$Descriptives## n Mean Std.Dev Median Min Max 25th 75th Skew
## Sepal.Length 150 5.843333 0.8280661 5.80 4.3 7.9 5.1 6.4 0.3086407
## Sepal.Width 150 3.057333 0.4358663 3.00 2.0 4.4 2.8 3.3 0.3126147
## Petal.Length 150 3.758000 1.7652982 4.35 1.0 6.9 1.6 5.1 -0.2694109
## Petal.Width 150 1.199333 0.7622377 1.30 0.1 2.5 0.3 1.8 -0.1009166
## Kurtosis
## Sepal.Length -0.6058125
## Sepal.Width 0.1387047
## Petal.Length -1.4168574
## Petal.Width -1.3581792Odľahlé pozorovania
res$multivariateOutliers## Observation Mahalanobis Distance Outlier
## 16 16 57.630 TRUE
## 15 15 51.827 TRUE
## 34 34 51.665 TRUE
## 33 33 46.571 TRUE
## 17 17 43.523 TRUE
## 6 6 36.532 TRUE
## 19 19 36.061 TRUE
## 23 23 36.035 TRUE
## 20 20 35.559 TRUE
## 47 47 34.862 TRUE
## 11 11 34.711 TRUE
## 37 37 34.549 TRUE
## 49 49 34.116 TRUE
## 22 22 32.859 TRUE
## 38 38 31.825 TRUE
## 5 5 31.628 TRUE
## 41 41 30.801 TRUE
## 45 45 30.505 TRUE
## 18 18 29.758 TRUE
## 1 1 29.748 TRUE
## 28 28 28.867 TRUE
## 32 32 28.787 TRUE
## 29 29 28.286 TRUE
## 44 44 27.088 TRUE
## 36 36 26.659 TRUE
## 40 40 26.261 TRUE
## 7 7 25.886 TRUE
## 8 8 25.829 TRUE
## 21 21 25.298 TRUE
## 50 50 25.206 TRUE
## 27 27 24.391 TRUE
## 12 12 24.295 TRUE
## 3 3 23.499 TRUE
## 43 43 22.958 TRUE
## 25 25 22.188 TRUE
## 48 48 22.048 TRUE
## 24 24 21.955 TRUE
## 14 14 21.731 TRUE
## 10 10 20.267 TRUE
## 30 30 20.173 TRUE
## 35 35 20.076 TRUE
## 2 2 20.031 TRUE
## 46 46 19.793 TRUE
## 39 39 19.569 TRUE
## 13 13 19.523 TRUE
## 4 4 19.270 TRUE
## 31 31 18.615 TRUE
## 26 26 17.921 TRUE
## 9 9 17.104 TRUENové dáta bez odľahlých pozorovaní
res$newData## Sepal.Length Sepal.Width Petal.Length Petal.Width
## 100 5.7 2.8 4.1 1.3
## 101 6.3 3.3 6.0 2.5
## 102 5.8 2.7 5.1 1.9
## 103 7.1 3.0 5.9 2.1
## 104 6.3 2.9 5.6 1.8
## 105 6.5 3.0 5.8 2.2
## 106 7.6 3.0 6.6 2.1
## 107 4.9 2.5 4.5 1.7
## 108 7.3 2.9 6.3 1.8
## 109 6.7 2.5 5.8 1.8
## 110 7.2 3.6 6.1 2.5
## 111 6.5 3.2 5.1 2.0
## 112 6.4 2.7 5.3 1.9
## 113 6.8 3.0 5.5 2.1
## 114 5.7 2.5 5.0 2.0
## 115 5.8 2.8 5.1 2.4
## 116 6.4 3.2 5.3 2.3
## 117 6.5 3.0 5.5 1.8
## 118 7.7 3.8 6.7 2.2
## 119 7.7 2.6 6.9 2.3
## 120 6.0 2.2 5.0 1.5
## 121 6.9 3.2 5.7 2.3
## 122 5.6 2.8 4.9 2.0
## 123 7.7 2.8 6.7 2.0
## 124 6.3 2.7 4.9 1.8
## 125 6.7 3.3 5.7 2.1
## 126 7.2 3.2 6.0 1.8
## 127 6.2 2.8 4.8 1.8
## 128 6.1 3.0 4.9 1.8
## 129 6.4 2.8 5.6 2.1
## 130 7.2 3.0 5.8 1.6
## 131 7.4 2.8 6.1 1.9
## 132 7.9 3.8 6.4 2.0
## 133 6.4 2.8 5.6 2.2
## 134 6.3 2.8 5.1 1.5
## 135 6.1 2.6 5.6 1.4
## 136 7.7 3.0 6.1 2.3
## 137 6.3 3.4 5.6 2.4
## 138 6.4 3.1 5.5 1.8
## 139 6.0 3.0 4.8 1.8
## 140 6.9 3.1 5.4 2.1
## 141 6.7 3.1 5.6 2.4
## 142 6.9 3.1 5.1 2.3
## 143 5.8 2.7 5.1 1.9
## 144 6.8 3.2 5.9 2.3
## 145 6.7 3.3 5.7 2.5
## 146 6.7 3.0 5.2 2.3
## 147 6.3 2.5 5.0 1.9
## 148 6.5 3.0 5.2 2.0
## 149 6.2 3.4 5.4 2.3
## 150 5.9 3.0 5.1 1.8
## 42 4.5 2.3 1.3 0.3
## 51 7.0 3.2 4.7 1.4
## 52 6.4 3.2 4.5 1.5
## 53 6.9 3.1 4.9 1.5
## 54 5.5 2.3 4.0 1.3
## 55 6.5 2.8 4.6 1.5
## 56 5.7 2.8 4.5 1.3
## 57 6.3 3.3 4.7 1.6
## 58 4.9 2.4 3.3 1.0
## 59 6.6 2.9 4.6 1.3
## 60 5.2 2.7 3.9 1.4
## 61 5.0 2.0 3.5 1.0
## 62 5.9 3.0 4.2 1.5
## 63 6.0 2.2 4.0 1.0
## 64 6.1 2.9 4.7 1.4
## 65 5.6 2.9 3.6 1.3
## 66 6.7 3.1 4.4 1.4
## 67 5.6 3.0 4.5 1.5
## 68 5.8 2.7 4.1 1.0
## 69 6.2 2.2 4.5 1.5
## 70 5.6 2.5 3.9 1.1
## 71 5.9 3.2 4.8 1.8
## 72 6.1 2.8 4.0 1.3
## 73 6.3 2.5 4.9 1.5
## 74 6.1 2.8 4.7 1.2
## 75 6.4 2.9 4.3 1.3
## 76 6.6 3.0 4.4 1.4
## 77 6.8 2.8 4.8 1.4
## 78 6.7 3.0 5.0 1.7
## 79 6.0 2.9 4.5 1.5
## 80 5.7 2.6 3.5 1.0
## 81 5.5 2.4 3.8 1.1
## 82 5.5 2.4 3.7 1.0
## 83 5.8 2.7 3.9 1.2
## 84 6.0 2.7 5.1 1.6
## 85 5.4 3.0 4.5 1.5
## 86 6.0 3.4 4.5 1.6
## 87 6.7 3.1 4.7 1.5
## 88 6.3 2.3 4.4 1.3
## 89 5.6 3.0 4.1 1.3
## 90 5.5 2.5 4.0 1.3
## 91 5.5 2.6 4.4 1.2
## 92 6.1 3.0 4.6 1.4
## 93 5.8 2.6 4.0 1.2
## 94 5.0 2.3 3.3 1.0
## 95 5.6 2.7 4.2 1.3
## 96 5.7 3.0 4.2 1.2
## 97 5.7 2.9 4.2 1.3
## 98 6.2 2.9 4.3 1.3
## 99 5.1 2.5 3.0 1.1Grafické znázornenie
mvn(iris[ ,-5], multivariatePlot = "qq")
## $multivariateNormality
## Test Statistic p value Result
## 1 Mardia Skewness 67.4305087780622 4.75799820400835e-07 NO
## 2 Mardia Kurtosis -0.230112114480973 0.818004651478034 YES
## 3 MVN <NA> <NA> NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Shapiro-Wilk Sepal.Length 0.9761 0.0102 NO
## 2 Shapiro-Wilk Sepal.Width 0.9849 0.1012 YES
## 3 Shapiro-Wilk Petal.Length 0.8763 <0.001 NO
## 4 Shapiro-Wilk Petal.Width 0.9018 <0.001 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th Skew
## Sepal.Length 150 5.843333 0.8280661 5.80 4.3 7.9 5.1 6.4 0.3086407
## Sepal.Width 150 3.057333 0.4358663 3.00 2.0 4.4 2.8 3.3 0.3126147
## Petal.Length 150 3.758000 1.7652982 4.35 1.0 6.9 1.6 5.1 -0.2694109
## Petal.Width 150 1.199333 0.7622377 1.30 0.1 2.5 0.3 1.8 -0.1009166
## Kurtosis
## Sepal.Length -0.6058125
## Sepal.Width 0.1387047
## Petal.Length -1.4168574
## Petal.Width -1.3581792mvn(iris[ ,1:2], multivariatePlot = "persp")
## $multivariateNormality
## Test Statistic p value Result
## 1 Mardia Skewness 9.46144098216622 0.0505456076692466 YES
## 2 Mardia Kurtosis -0.853178029438543 0.393560585232763 YES
## 3 MVN <NA> <NA> YES
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Shapiro-Wilk Sepal.Length 0.9761 0.0102 NO
## 2 Shapiro-Wilk Sepal.Width 0.9849 0.1012 YES
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th Skew
## Sepal.Length 150 5.843333 0.8280661 5.8 4.3 7.9 5.1 6.4 0.3086407
## Sepal.Width 150 3.057333 0.4358663 3.0 2.0 4.4 2.8 3.3 0.3126147
## Kurtosis
## Sepal.Length -0.6058125
## Sepal.Width 0.1387047mvn(iris[ ,1:2], multivariatePlot = "contour")
## $multivariateNormality
## Test Statistic p value Result
## 1 Mardia Skewness 9.46144098216622 0.0505456076692466 YES
## 2 Mardia Kurtosis -0.853178029438543 0.393560585232763 YES
## 3 MVN <NA> <NA> YES
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Shapiro-Wilk Sepal.Length 0.9761 0.0102 NO
## 2 Shapiro-Wilk Sepal.Width 0.9849 0.1012 YES
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th Skew
## Sepal.Length 150 5.843333 0.8280661 5.8 4.3 7.9 5.1 6.4 0.3086407
## Sepal.Width 150 3.057333 0.4358663 3.0 2.0 4.4 2.8 3.3 0.3126147
## Kurtosis
## Sepal.Length -0.6058125
## Sepal.Width 0.1387047mvn(iris[ ,-5], univariatePlot = "histogram")
## $multivariateNormality
## Test Statistic p value Result
## 1 Mardia Skewness 67.4305087780622 4.75799820400835e-07 NO
## 2 Mardia Kurtosis -0.230112114480973 0.818004651478034 YES
## 3 MVN <NA> <NA> NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Shapiro-Wilk Sepal.Length 0.9761 0.0102 NO
## 2 Shapiro-Wilk Sepal.Width 0.9849 0.1012 YES
## 3 Shapiro-Wilk Petal.Length 0.8763 <0.001 NO
## 4 Shapiro-Wilk Petal.Width 0.9018 <0.001 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th Skew
## Sepal.Length 150 5.843333 0.8280661 5.80 4.3 7.9 5.1 6.4 0.3086407
## Sepal.Width 150 3.057333 0.4358663 3.00 2.0 4.4 2.8 3.3 0.3126147
## Petal.Length 150 3.758000 1.7652982 4.35 1.0 6.9 1.6 5.1 -0.2694109
## Petal.Width 150 1.199333 0.7622377 1.30 0.1 2.5 0.3 1.8 -0.1009166
## Kurtosis
## Sepal.Length -0.6058125
## Sepal.Width 0.1387047
## Petal.Length -1.4168574
## Petal.Width -1.3581792mvn(iris[ ,-5], univariatePlot = "scatter")
## $multivariateNormality
## Test Statistic p value Result
## 1 Mardia Skewness 67.4305087780622 4.75799820400835e-07 NO
## 2 Mardia Kurtosis -0.230112114480973 0.818004651478034 YES
## 3 MVN <NA> <NA> NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Shapiro-Wilk Sepal.Length 0.9761 0.0102 NO
## 2 Shapiro-Wilk Sepal.Width 0.9849 0.1012 YES
## 3 Shapiro-Wilk Petal.Length 0.8763 <0.001 NO
## 4 Shapiro-Wilk Petal.Width 0.9018 <0.001 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th Skew
## Sepal.Length 150 5.843333 0.8280661 5.80 4.3 7.9 5.1 6.4 0.3086407
## Sepal.Width 150 3.057333 0.4358663 3.00 2.0 4.4 2.8 3.3 0.3126147
## Petal.Length 150 3.758000 1.7652982 4.35 1.0 6.9 1.6 5.1 -0.2694109
## Petal.Width 150 1.199333 0.7622377 1.30 0.1 2.5 0.3 1.8 -0.1009166
## Kurtosis
## Sepal.Length -0.6058125
## Sepal.Width 0.1387047
## Petal.Length -1.4168574
## Petal.Width -1.3581792Porovnanie rôznych testov pre viacrozmernú normalitu
mvn(iris[,-5], mvnTest = "mardia")$multivariateNormality## Test Statistic p value Result
## 1 Mardia Skewness 67.4305087780622 4.75799820400835e-07 NO
## 2 Mardia Kurtosis -0.230112114480973 0.818004651478034 YES
## 3 MVN <NA> <NA> NOmvn(iris[,-5], mvnTest = "hz")$multivariateNormality## Test HZ p value MVN
## 1 Henze-Zirkler 2.336394 0 NOmvn(iris[,-5], mvnTest = "royston")$multivariateNormality## Test H p value MVN
## 1 Royston 50.39667 3.098229e-11 NOVezmeme iba jeden druh kosatcov:
setosa <- iris[1:50, 1:4]
mvn(setosa, mvnTest = "mardia")$multivariateNormality## Test Statistic p value Result
## 1 Mardia Skewness 25.6643445196298 0.177185884467652 YES
## 2 Mardia Kurtosis 1.29499223711606 0.195322907441934 YES
## 3 MVN <NA> <NA> YESmvn(setosa, mvnTest = "hz")$multivariateNormality## Test HZ p value MVN
## 1 Henze-Zirkler 0.9488453 0.04995356 NOmvn(setosa, mvnTest = "royston")$multivariateNormality## Test H p value MVN
## 1 Royston 31.51803 2.187653e-06 NOTransformácie na normálne rozdelenie
Box-Cox transformácia
Priamo v príkaze mvn:
mvn(iris[,-5], mvnTest = "mardia", bc=TRUE)$multivariateNormality## Test Statistic p value Result
## 1 Mardia Skewness 35.7910554915828 0.0162739088388607 NO
## 2 Mardia Kurtosis -0.806554908104519 0.419922961559335 YES
## 3 MVN <NA> <NA> NOMocninová transformácia
library(car)## Loading required package: carDatap1 <- powerTransform(setosa)
summary(p1)## bcPower Transformations to Multinormality
## Est Power Rounded Pwr Wald Lwr Bnd Wald Upr Bnd
## Sepal.Length 0.4166 1 -2.6421 3.4752
## Sepal.Width 1.2729 1 -0.1900 2.7358
## Petal.Length 0.7286 1 -0.8958 2.3531
## Petal.Width 0.0244 0 -0.5009 0.5498
##
## Likelihood ratio test that transformation parameters are equal to 0
## (all log transformations)
## LRT df pval
## LR test, lambda = (0 0 0 0) 3.922044 4 0.41666
##
## Likelihood ratio test that no transformations are needed
## LRT df pval
## LR test, lambda = (1 1 1 1) 12.9728 4 0.011409(tY1 <- bcPower(with(setosa, cbind(Sepal.Length,Sepal.Width,Petal.Length,Petal.Width)),coef(p1, round=TRUE)))## Sepal.Length^1 Sepal.Width^1 Petal.Length^1 Petal.Width^0
## [1,] 4.1 2.5 0.4 -1.6094379
## [2,] 3.9 2.0 0.4 -1.6094379
## [3,] 3.7 2.2 0.3 -1.6094379
## [4,] 3.6 2.1 0.5 -1.6094379
## [5,] 4.0 2.6 0.4 -1.6094379
## [6,] 4.4 2.9 0.7 -0.9162907
## [7,] 3.6 2.4 0.4 -1.2039728
## [8,] 4.0 2.4 0.5 -1.6094379
## [9,] 3.4 1.9 0.4 -1.6094379
## [10,] 3.9 2.1 0.5 -2.3025851
## [11,] 4.4 2.7 0.5 -1.6094379
## [12,] 3.8 2.4 0.6 -1.6094379
## [13,] 3.8 2.0 0.4 -2.3025851
## [14,] 3.3 2.0 0.1 -2.3025851
## [15,] 4.8 3.0 0.2 -1.6094379
## [16,] 4.7 3.4 0.5 -0.9162907
## [17,] 4.4 2.9 0.3 -0.9162907
## [18,] 4.1 2.5 0.4 -1.2039728
## [19,] 4.7 2.8 0.7 -1.2039728
## [20,] 4.1 2.8 0.5 -1.2039728
## [21,] 4.4 2.4 0.7 -1.6094379
## [22,] 4.1 2.7 0.5 -0.9162907
## [23,] 3.6 2.6 0.0 -1.6094379
## [24,] 4.1 2.3 0.7 -0.6931472
## [25,] 3.8 2.4 0.9 -1.6094379
## [26,] 4.0 2.0 0.6 -1.6094379
## [27,] 4.0 2.4 0.6 -0.9162907
## [28,] 4.2 2.5 0.5 -1.6094379
## [29,] 4.2 2.4 0.4 -1.6094379
## [30,] 3.7 2.2 0.6 -1.6094379
## [31,] 3.8 2.1 0.6 -1.6094379
## [32,] 4.4 2.4 0.5 -0.9162907
## [33,] 4.2 3.1 0.5 -2.3025851
## [34,] 4.5 3.2 0.4 -1.6094379
## [35,] 3.9 2.1 0.5 -1.6094379
## [36,] 4.0 2.2 0.2 -1.6094379
## [37,] 4.5 2.5 0.3 -1.6094379
## [38,] 3.9 2.6 0.4 -2.3025851
## [39,] 3.4 2.0 0.3 -1.6094379
## [40,] 4.1 2.4 0.5 -1.6094379
## [41,] 4.0 2.5 0.3 -1.2039728
## [42,] 3.5 1.3 0.3 -1.2039728
## [43,] 3.4 2.2 0.3 -1.6094379
## [44,] 4.0 2.5 0.6 -0.5108256
## [45,] 4.1 2.8 0.9 -0.9162907
## [46,] 3.8 2.0 0.4 -1.2039728
## [47,] 4.1 2.8 0.6 -1.6094379
## [48,] 3.6 2.2 0.4 -1.6094379
## [49,] 4.3 2.7 0.5 -1.6094379
## [50,] 4.0 2.3 0.4 -1.6094379summary(p2 <- powerTransform(cbind(Sepal.Length,Sepal.Width,Petal.Length,Petal.Width) ~ Species, iris))## bcPower Transformations to Multinormality
## Est Power Rounded Pwr Wald Lwr Bnd Wald Upr Bnd
## Sepal.Length -0.3957 0.0 -1.1615 0.3702
## Sepal.Width 0.6381 1.0 -0.0817 1.3578
## Petal.Length 0.3669 0.5 0.1880 0.5459
## Petal.Width 0.5776 0.5 0.4661 0.6891
##
## Likelihood ratio test that transformation parameters are equal to 0
## (all log transformations)
## LRT df pval
## LR test, lambda = (0 0 0 0) 124.7563 4 < 2.22e-16
##
## Likelihood ratio test that no transformations are needed
## LRT df pval
## LR test, lambda = (1 1 1 1) 84.18866 4 < 2.22e-16testTransform(p2, c(0, 1, 0.5, 0.5))## LRT df pval
## LR test, lambda = (0 1 0.5 0.5) 5.563751 4 0.23418tY2 <- bcPower(with(iris, cbind(Sepal.Length,Sepal.Width,Petal.Length,Petal.Width)),coef(p2, round=TRUE))
scatterplotMatrix( ~ tY2|Species, iris)