In this chapter we present goodness-of-fit tests for the Gaussian distribution. In Section 11.1tests based on the empirical distribution function (EDF) are treated. A good resource for thiskind of test is Stephens (1986). We start with the Kolmogorov–Smirnov test. It evaluatesthe greatest vertical distance between the EDF and the theoretical cumulative distributionfunction (CDF). If both, or one parameter are estimated from the sample the distribution ofthe test statistic changes and the test is called the Lilliefors test on normality.
Section 11.2 deals with tests not based on the EDF such as the Jarque–Bera test whichcompares observed and expected moments of the normal distribution.
11.1 Tests based on the EDF
11.1.1 Kolmogorov–Smirnov test (Lilliefors test for normality)
Description: Tests if a sample is sampled from a normal distribution with parameter𝜇 and 𝜎2.
Assumptions: • Data are measured at least on an ordinal scale.• The sample random variables X1, … ,Xn are identically, indepen-dently distributed with observations x1, … , xn and a continuousdistribution function F(x).
Fn(x) is the EDF of the sample andΦ is the CDF of the standard normal distribution
Test decision: Reject H0 if for the observed value d of D(A) d ≥ d1−𝛼(B) d+ ≥ d+
1−𝛼(C) d− ≥ d−
1−𝛼The critical values d1−𝛼 , d
+1−𝛼 , d
−1−𝛼 can be found, for example, inMiller
(1956).
p-values: (A) p = P(D ≥ d)(B) p = P(D+ ≥ d+)(C) p = P(D− ≥ d−)
Annotations: • This test evaluates the greatest vertical distance between the EDFand the CDF of the standard normal distribution.
• The test statistic D is the maximum of D+ and D−: D =max(D+,D−).
• If the sample mean and variance are estimated from the sample thedistribution of the test statistic changes and different critical valuesare needed. Lilliefors published tables with corrected values (Lil-liefors 1967) and the test is also known as the Lilliefors test fornormality.
• SAS and R use different methods to calculate p-values. Hence,results may differ.
Example: To test the hypothesis that the systolic blood pressure of a certain populationis distributed according to a normal distribution. A dataset of 55 subjects is sampled(dataset in Table A.1).
Test --Statistic--- -----p Value------Kolmogorov–Smirnov D 0.117254 Pr > D 0.0587
*** Variant 2 ****Fitted Normal Distribution for mmhgParameters for Normal Distribution
Parameter Symbol EstimateMean Mu 130Std Dev Sigma 19.16691
Goodness-of-Fit Tests for Normal Distribution
Test ----Statistic----- ------p Value-----Kolmogorov–Smirnov D 0.11725352 Pr > D >0.250
Remarks:
• SAS only calculates D = max(D+,D−) as test statistic.
• Variant 1 calculates the Lilliefors test for normality by using the sample mean andsample variance for standardizing the sample. The keyword normal enables thistest.
• With the variant 2 the original Kolmogorov–Smirnov test with the option nor-mal of the histogram statement can be calculated; values for the mean andvariance have to be provided. Here 𝜇 = 130 and 𝜎 = 19.16691 are chosen.
• The syntax is normal(normal-options). If normal-options is not given or nor-mal(mu=EST sigma=EST) is given the same test is calculated as with variant1. The following normal-options are valid: mu=value where value is the mean 𝜇of the normal distribution and sigma=value where value is the standard devia-tion 𝜎 of the normal distribution. Note, these values are the true parameters of thenormal distribution to test against not the sample parameters. This can be seenin the above example. In both variants the same D-statistic is calculated but thep-values are different.
R code
# Calculate mean and standard deviationm<-mean(blood_pressure$mmhg)s<-sd(blood_pressure$mmhg)
• R only computes the Kolmogorov–Smirnov test, so if the parameters are esti-mated from the sample as in the above example the p-values are incorrect.
• In the case of ties a warning is prompted that the reported p-values may beincorrect.
• pnorm indicates that it is tested for the normal distribution.
• mean=value is optional. The value specifies the mean of the normal distributionto test for. The default is 0, if mean=value is not specified.
• sd=value is optional. The value specifies the standard deviation of the normaldistribution to test for. The default is 1, if sd=value is not specified.
• alternative=“value” is optional and defines the type of alternative hypoth-esis: “two.sided”=the CDFs of F(x) and Φ( x−𝜇
𝜎) differ (A); “greater”=the CDF
of F(x) lies above that of Φ( x−𝜇𝜎) (B); “less”=the CDF of F(x) lies below that of
Φ( x−𝜇𝜎) (C). Default is “two.sided”.
• exact=value is optional. If value is TRUE, no ties are present and the samplesize is less than 100 an exact p-value is calculated. The default is NULL, that is,no exact p-values.
11.1.2 Anderson–Darling test
Description: Tests if a sample is sampled from a normal distribution with parameter𝜇 and 𝜎2.
Assumptions: • Data are measured at least on an ordinal scale.• The random variables X1, … ,Xn are identically, independently dis-tributed with observations x1, … , xn and a continuous distributionfunction F(x).
Hypotheses: H0 ∶ F(x) = Φ( x−𝜇𝜎) ∀x vs
H1 ∶ F(x) ≠ Φ( x−𝜇𝜎) for at least one x
Test statistic: A2 = −n − 1
n
n∑i=1
(2i − 1)[ln (pi) + ln (1 − pn−i+1)]
where pi = Φ(
X(i)−XS
), i = 1, … , n,
and X(1), … ,X(n) the sample in ascending order.
TESTS ON NORMALITY 143
Test decision: Reject H0 if for the observed value a2 of A2
a2 ≥ a𝛼Critical values a𝛼 can be found, for example, in table 4.2 of Stephens(1986).
p-values: p = P(A2 ≥ a2)
Annotations: • The test statistic A2 was proposed by Anderson and Darling (1952).• Stephens (1986) also treats the case that either 𝜇 or 𝜎 or both are
unknown. They are estimated by X and s2 =∑n
i (Xi − X)2∕(n − 1).For the most common case that both are unknown the test statisticis modified as A2∗ = (1.0 + 0.75∕n + 2.25∕n2)A2. For the modifiedtest statistic A2∗ critical values are given in table 4.7 of Stephens(1986).
• Formulas of approximate p-values can also be found in Stephens(1986).
Example: To test the hypothesis that the systolic blood pressure of a certain populationis distributed according to a normal distribution. A dataset of 55 subjects is sampled(dataset in Table A.1).
Test --Statistic--- -----p Value------Anderson–Darling A-Sq 0.888742 Pr > A-Sq 0.0224
*** Variant 2 ****Fitted Normal Distribution for mmhgParameters for Normal Distribution
Parameter Symbol EstimateMean Mu 130Std Dev Sigma 19.16691
Goodness-of-Fit Tests for Normal Distribution
144 STATISTICAL HYPOTHESIS TESTING WITH SAS AND R
Test ----Statistic----- ------p Value-----Anderson-Darling A-Sq 0.88874206 Pr > A-Sq >0.250
Remarks:
• SAS computes A2 and not A2∗.
• Variant 1 calculates the Anderson–Darling test using the samplemean and samplevariance to standardize the sample. The keyword normal enables this test.
• With the variant 2 using the option normal of the histogram statement thetest with known theoretical 𝜇 and 𝜎 is computed.
• The syntax is normal(normal-options). If normal-options is not given thesame test is calculated as with variant 1. The following normal-options are valid:mu=value where value is the mean 𝜇 and sigma=value where value is thestandard deviation 𝜎. Thereby, versions of the test are available for 𝜇 or 𝜎 orboth known. Note, these values are the true parameters of the normal distributionto test for not the sample parameters. In all variants the same A2 statistic iscalculated but the p-values are different. This can be seen in the above example.
R code
# Get number of observationsn<-length(blood_pressure$mmhg)
# Standardize the blood pressurem<-mean(blood_pressure$mmhg)s<-sd(blood_pressure$mmhg)z<-(blood_pressure$mmhg-m)/s
# z1 is the array of the ascending sorted valuesz1<-sort(z)
# z2 is the array of the descending sorted valuesz2<-sort(z,decreasing=TRUE)
# Calculate the test statisticAD<-(1/n)*sum((1-2*seq(1:n))*(log(pnorm(z1))+
log(1-pnorm(z2))))-n
# Calculate modified test statisticAD_mod<-(1.0+0.75/n+2.25/n∧2)*AD
# Calculate approximative p-values according table 4.9# from Stephens (1986)if (AD_mod<=0.200)
• This example uses sample moments for 𝜇 and 𝜎 and the modified test statisticA2∗. The approximate p-value is calculated according to Stephens (1986). Theapproximation can be used for samples of size n ≥ 8.
11.1.3 Cramér–von Mises test
Description: Tests if a sample is sampled from a normal distribution with parameter𝜇 and 𝜎2.
Assumptions: • Data are measured at least on an ordinal scale.• The random variables X1, … ,Xn are identically, independently dis-tributed with observations x1, … , xn and a continuous distributionfunction F(x).
Hypotheses: H0 ∶ F(x) = Φ( x−𝜇𝜎) ∀x vs
H1 ∶ F(x) ≠ Φ( x−𝜇𝜎) for at least one x
Test statistic: W2 = 1
12n+
n∑i=1
(pi −
2i−12n
)2
where pi = Φ(
X(i)−XS
), i = 1, … , n,
and X(1), … ,X(n) the sample in ascending order.
Test decision: Reject H0 if for the observed value 𝑤2 of W2
𝑤2 ≥ 𝑤1−𝛼Critical values𝑤1−𝛼 can be found, for example, in Pearson and Hartley(1972).
p-values: p = P(W2 ≥ 𝑤2)
146 STATISTICAL HYPOTHESIS TESTING WITH SAS AND R
Annotations: • The test was independently introduced by Cramér (1928) and vonMises (1931).
• Stephens (1986) also treats the case that either 𝜇 or 𝜎 or both are
unknown. They are estimated by X and s2 =∑n
i (Xi − X)2∕(n − 1).For the most common case that both are unknown the test statistic ismodified asW2∗ = (1 + 0.5∕n) ∗ W2. For the modified test statisticW2∗ critical values are given in table 4.7 of Stephens (1986).
• Formulas of approximate p-values can also be found in Stephens(1986).
Example: To test the hypothesis that the systolic blood pressure of a certain populationis distributed according to a normal distribution. A dataset of 55 subjects is sampled(dataset in Table A.1).
• Variant 1 calculates the Cramér–von Mises test by using the sample mean andsample variance to standardize the sample. The keyword normal enables thistest.
• With the variant 2 using the option normal of the histogram statement thetest with known theoretical 𝜇 and 𝜎 is computed.
• The syntax is normal(normal-options). If normal-options is not given thesame test is calculated as with variant 1. The following normal-options are valid:mu=value where value is the mean 𝜇 and sigma=value where value is thestandard deviation 𝜎. Thereby, versions of the test are available for 𝜇 or 𝜎 orboth known. Note, these values are the true parameters of the normal distributionto test for not the sample parameters. In all variants the same W2 statistic iscalculated but the p-values are different. This can be seen in the above example.
R code
# Get number of observationsn<-length(blood_pressure$mmhg)
# Standardize the blood pressurem<-mean(blood_pressure$mmhg)s<-sd(blood_pressure$mmhg)z<-(blood_pressure$mmhg-m)/s
# Sort the samplez<-sort(z)
# Calculate the test statisticW_sq<-1/(12*n)+sum((pnorm(z)-(2*seq(1:n)-1)/(2*n))∧2)
# Calculate approximative p-values according to table 4.9# from Stephens (1986)W<-(1 + 0.5/n) * W_sqif (W<0.0275)
• This example uses sample moments for 𝜇 and 𝜎 and the modified test statisticW2∗. The approximate p-value is calculated according to Stephens (1986). Theapproximation can be used for sample sizes ≥ 7.
11.2 Tests not based on the EDF
11.2.1 Shapiro–Wilk test
Description: Tests if a sample is sampled from a normal distribution.
Assumptions: • Data are measured on a metric scale.• The random variables X1, … ,Xn are identically, independently dis-tributed with observations x1, … , xn and a continuous distributionfunction F(x).
• The mean 𝜇 and variance 𝜎 are unknown.
Hypotheses: H0 ∶ F(x) = Φ( x−𝜇𝜎) ∀x vs
H1 ∶ F(x) ≠ Φ( x−𝜇𝜎) for at least one x
Test statistic: W =
( n∑i=1aiX(i)
)2
n∑i=1
(X(i) − X)2
with coefficients (a1, … , an) =m′V−1√
m′V−1V−1m,
where m′ = (m1, … ,mn) is the mean vector and V is the covariancematrix of standard normal order statisticsand X(1), … ,X(n) is the sample in ascending order.
Test decision: Reject H0 if for the observed value 𝑤 of W𝑤 ≤ 𝑤𝛼
Critical values 𝑤𝛼 for n ≤ 50 can be found, for example, in Shapiroand Wilk (1965).
p-values: p = P(W ≤ 𝑤)
TESTS ON NORMALITY 149
Annotations: • The test statistic W was proposed by Shapiro and Wilk (1965).• For the test statistic it holds that 0 < W ≤ 1.• The distribution of the test statistic W depends on the sample size.Shapiro and Wilk (1965) derived approximate values of the coef-ficients as well as percentage points of the null distribution of thetest statistic for sample sizes up to n = 50. Royston (1982, 1992)developed approximations of these values for sample sizes up ton = 5000.
• The Shapiro–Wilk test is a powerful test, especially in samples withsmall sample sizes Shapiro et al. (1968).
Example: To test the hypothesis that the systolic blood pressure of a certain populationis distributed according to a normal distribution. A dataset of 55 subjects is sampled(dataset in Table A.1).
• R calculates the Shapiro–Wilk test only for sample sizes ≤5000.
• For sample sizes ≥4 the p-values are calculated based on Royston (1995).
11.2.2 Jarque–Bera test
Description: Tests if a sample is sampled from a normal distribution.
Assumptions: • Data are measured on a metric scale.• The random variables X1, … ,Xn are identically, independently dis-tributed with observations x1, … , xn and a continuous distributionfunction F(x).
• The mean 𝜇 and variance 𝜎 are unknown.
Hypotheses: H0 ∶ F(x) = Φ( x−𝜇𝜎) ∀x vs
H1 ∶ F(x) ≠ Φ( x−𝜇𝜎) for at least one x
Test statistic: JB = n(
𝛾21
6+ (𝛾2−3)2
24
)with 𝛾1 =
1
n
n∑i=1
(Xi − X)3(1
n
n∑i=1
(Xi − X)2)3∕2
and 𝛾2 =
1
n
n∑i=1
(Xi − X)4(1
n
n∑i=1
(Xi − X)2)2
Test decision: Reject H0 if for the observed value jb of JBjb ≥ 𝜒2
1−𝛼;2
p-values: p = 1 − P(JB ≤ jb)
Annotations: • This test was introduced by Jarque and Bera (1987) as a Lagrangemultiplier test with the alternative hypothesis covering any other dis-tribution from the Pearson family of distributions.
• For the calculation of the test statistic the sample skewness 𝛾1 andsample kurtosis 𝛾2 are used. If the data are normally distributed theskewness is zero and the kurtosis is three, so large values of the teststatistic JB are arguing against the null hypothesis.
TESTS ON NORMALITY 151
• The test statistic JB is asymptotically 𝜒2-distributed with twodegrees of freedom.
• Critical values, which are obtained by Monte Carlo simulations andshould be used for small sample sizes, can be found in Jarque andBera (1987) or Dep and Sefton (1996).
Example: To test the hypothesis that the systolic blood pressure of a certain populationis distributed according to a normal distribution. A dataset of 55 subjects is sampled(dataset in Table A.1).
# Output resultscat("Jarque–Bera Test \n\n","JB ","p-value","\n","---------------------","\n",jb," ",p_value,"\n")
R output
Jarque–Bera Test
JB p-value---------------------2.627909 0.2687552
Remarks:
• There is no R function to calculate the Jarque–Bera test directly.
• This implementation of the test uses the 𝜒2-distribution with two degrees of free-dom to calculate the p-values. Because this is the asymptotic distribution of thetest statistic, the p-value is for low sample sizes only a rough approximation.
References
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TESTS ON NORMALITY 153
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