Testing Homoscedasticity, Multivariate Normality, and Missing Completely at Random

Description

The main purpose of this package is to test whether the missing data mechanism, in a set of incompletely observed data, is one of missing completely at random (MCAR). As a by-product, however, this package can impute incomplete data, is able to perform a test to determine whether data have a multivariate normal distribution or whether the covariances for several populations are equal. The test of MCAR follows the methodology proposed by Jamshidian and Jalal (2010). It is based on testing equality of covariances between groups consisting of identical missing data patterns. The data are imputed, using two options of normality and distribution free, and the test of equality of covariances between groups with identical missing data patterns is performed also with options of assuming normality (Hawkins test) or non-parametrically. The user, can optionally use her own method of data imputation as well. Multiple imputation is an option as a diagnostic tool to help identify cases or variables that contribute to rejection of MCAR, when the MCAR test is rejecetd (See Jamshidian and Jalal, 2010 for details). As explained in Jamshidian, Jalal, and Jansen (2014), this package can also be used for imputing missing data, test of multivariate normality, and test of equality of covariances between several groups when data are complete.

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Jamshidian, M. and Jalal, S. (2010). “Tests of homoscedasticity, normality, and missing at random for incomplet multivariate data,” Psychometrika, 75, 649-674, doi:10.1007/s11336-010-9175-3.

Jamshidian, M. Jalal, S., and Jansen, C. (2014). “ MissMech: An R Package for Testing Homoscedasticity, Multivariate Normality, and Missing Completely at Random (MCAR),” Journal of Statistical Software, 56(6), 1-31. https://www.jstatsoft.org/v56/i06/, doi:10.18637/jss.v056.i06."

See Also

Useful links:

• https://github.com/indenkun/MissMech

• Report bugs at https://github.com/indenkun/MissMech/issues

K-Sample Anderson Darling Test

Description

This is a non-parametric K-sample test that tests equality of distribution of a variable between k populations based on samples from each of the populations.

Usage

AndersonDarling(data, number.cases)

Arguments

Details

The data is a vector including all the k samples to be used for the test. The j-th element of number.cases is the number of cases in sample j (included in data), for j= 1,...,k.

Value

Note

The test does not adjust for tie observations.

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Scholz, F.W. and Stephens, M.A. (1987). ”K-Sample Anderson-Darling Tests,”Journal of the American Statistical Association, 82, 918-924, doi:10.2307/2288805.

Examples

#---- Example 1
set.seed(50)
n1 <- 30
n2 <- 45
n3 <- 60
v1 <- rnorm(n1)
v2 <- runif(n2)
v3 <- rnorm(n3, 2, 3)
AndersonDarling(data = c(v1, v2, v3), number.cases=c(n1, n2, n3))

#---- Example 2
set.seed(50)
n1 <- 30
n2 <- 45
n3 <- 60
v1 <- rt(n1,4)
v2 <- rt(n2,4)
v3 <- rt(n3,4)
AndersonDarling(data=c(v1, v2, v3), number.cases=c(n1, n2, n3))

Hessian of the observed datat Multivariate Normal Log-Likelihood with Incomplete Data

Description

The Hessian of the normal-theory observed data log-likelihood function, evaluated at a given value of the mean vector and the covariance matrix, when data are incomplete. The output is a symmetric matrix with rows/columns corresponding to elements in the mean vector and lower diagonal of the covariance matrix.

Usage

Ddf(data, mu, sig)

Arguments

Details

While mu is a vector, it has to be input as a matrix object. See example nelow.

Value

Note

There must be no rows in data that contain no observations.

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Jamshidian, M. and Bentler, P. M. (1999). “ML estimation of mean and covariance structures with missing data using complete data routines.” Journal of Educational and Behavioral Statistics, 24, 21-41, doi:10.2307/1165260.

Examples

set.seed <- 50
n <- 200
p <- 4
pctmiss <- 0.2
y <- matrix(rnorm(n * p),nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA
mu <- c(0,0,0,0)
sig <- matrix(c(1,0,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1),4,4)
Ddf(data=y, as.matrix(mu), sig)

Removes groups with identical missing data patterns having at most a given number of cases

Description

Removes groups of missing data patterns with number of cases less than or equal to a specified value (ncases).

Usage

DelLessData(data, ncases = 0)

Arguments

Value

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

Examples

set.seed <- 50
n <- 200
p <- 4
pctmiss <- 0.2
y <- matrix(rnorm(n * p),nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA
out <- DelLessData(data=y, ncases = 4)
dim(y)
dim(out$data)

Test Statistic for the Hawkins Homoscedasticity Test

Description

Produces the F_ij's and A_ij's that are used in the Hawkins' test of homogeneoity of covariances. See Hawkins (1981) and Jamshidian and Jalal (2010) for more details.

Usage

Hawkins(data, spatcnt)

Arguments

Value

Note

There must be no rows in data that contain no observations.

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Hawkins, D. M. (1981). “A new test for multivariate normality and homoscedasticity,” Technometrics, 23, 105-110, doi:10.2307/1267983.

Jamshidian, M. and Jalal, S. (2010). “Tests of homoscedasticity, normality, and missing at random for incomplete multivariate data,” Psychometrika, 75, 649-674, doi:10.1007/s11336-010-9175-3.

Examples

set.seed <- 50
n <- 200
p <- 4
pctmiss <- 0.2
y <- matrix(rnorm(n * p),nrow = n)
spatcnt <- c(20, 50, 70, 200)
h <- Hawkins(data=y, spatcnt)

Parametric and Non-Parameric Imputation

Description

This function imputes the data using two methods.

method 'Normal' - Imputes the data assuming that the data come from a multivariate normal distribution with mean mu and covariance sig. If mu or sig are not inputted, then their maximum likelihood estimate is used. The imputed values are based on the conditional distribution of the missing given the observed and mu and sigma; see Jamshidian and Jalal (2010) for more details.

method 'Dist.Free' - This method imputes the data nonparametrically using the method of Sirvastava and Dolatabadi (2009). Also see Jamshidian and Jalal (2010).

Usage

Impute(data, mu = NA, sig = NA, imputation.method = "Normal", resid = NA)

Arguments

Details

This routine uses OrderMissing to order data accordinng to missing data patterns. The output consists of imputed data both in its original order as well as post ordering by OrderMissing.

Value

Note

In the above descriptions "original data" refers to the input data after deletion of the rows consisting of all NA's (if any) .

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Srivastava, M. S. and Dolatabadi, M. (2009). “Multiple imputation and other resampling scheme for imputing missing observations,” Journal of Multivariate Analysis, 100, 1919-1937, doi:10.1016/j.jmva.2009.06.003.

Jamshidian, M. and Jalal, S. (2010). “Tests of homoscedasticity, normality, and missing at random for incomplete multivariate data,” Psychometrika, 75, 649-674, doi:10.1007/s11336-010-9175-3.

Examples

set.seed <- 50
n <- 200
p <- 4
pctmiss <- 0.2
y <- matrix(rnorm(n * p),nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA

yimp1 <- Impute(data=y, mu = NA, sig = NA, imputation.method = "Normal", resid = NA)
yimp2 <- Impute(data=y, mu = NA, sig = NA, imputation.method = "Dist.Free", resid = NA)

Evaluating Legendre's Polynomials of Degree 1, 2, 3, or 4

Description

This function evaluates the values of Legendre polynomials of degrees 1 to 4 on [0,1] at a value(s) x.

Usage

LegNorm(x)

Arguments

Value

The returned list has the following elements:

Note

Legendre's polynomias on [0,1] are calculated.

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

David, F. N. (1939). “On Neyman's "smooth" test for goodness of fit: I. Distribution of the criterion Psi2 when the hypothesis tested is true,” Biometrika, 31, 191-199, doi:10.1093/biomet/31.1-2.191.

Examples

p <- LegNorm(c(5.2,11,15))
p$p3

ML Estimates of Mean and Covariance Based on Incomplete Data

Description

Normal theory maximum likelihood estimates of mean and covariance matrix is obtained when data are incomplete, using EM algorithm (see Jamshidian and Bentler 1999). If the option Hessian is set to TRUE, then the observed information containing the standard errors of the parameter estimates is also computed.

Usage

Mls(data, mu = NA, sig = NA, tol = 1e-06, Hessian = FALSE)

Arguments

Value

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Jamshidian, M. and Bentler, P. M. (1999). “ML estimation of mean and covariance structures with missing data using complete data routines.” Journal of Educational and Behavioral Statistics, 24, 21-41, doi:10.2307/1165260.

Examples

set.seed <- 50
n <- 200
p <- 4
pctmiss <- 0.2 # Generate 20\% missing data
y <- matrix(rnorm(n * p),nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA
ml <- Mls(data=y, mu = NA, sig = NA, tol = 1e-6, Hessian=FALSE)
ml

Order Missing Data Pattern

Description

This function rearranges the data based on their missing data patterns. Morever, missing data patterns consisting of fewer than the user specified number in del.lesscases is deleted from the dataset.

Usage

OrderMissing(data, del.lesscases = 0)

Arguments

Value

An object of class orderpattern - a list including elements:

Note

If you run the following command line: out <- OrderMissing(data=y, del.lesscases=0), then y[out$caseorder,] is equal to the output data (i.e. out$data). Also out$data[order(out$caseorder),] is equal to the original data y. Note that if del.lesscases is greater than zero, the command out$data[order(out$caseorder),] will result in a data set that has no case order correspondnce to the original data.

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

Examples

set.seed <- 50
n <- 200
p <- 4
pctmiss <- 0.2
y <- matrix(rnorm(n * p),nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA
out <- OrderMissing(y, del.lesscases = 0)
a <- out$caseorder
z = out$data
y[a,] # Reverting the original data to the new output order
z[order(a),] # Reverting the ordered datat to the original order

Testing Homoscedasticity, Multivariate Normality, and Missing Completely at Random

Description

The main purpose of this package is to test whether the missing data mechanism, for an incompletely observed data set, is one of missing completely at random (MCAR). As a by product, however, this package has the capabilities of imputing incomplete data, performing a test to determine whether data have a multivariate normal distribution, performing a test of equality of covariances for groups, and obtaining normal-theory maximum likelihood estimates for mean and covariance when data are incomplete. The test of MCAR follows the methodology proposed by Jamshidian and Jalal (2010). It is based on testing equality of covariances between groups having identical missing data patterns. The data are imputed, using two options of normality and distribution free, and the test of equality of covariances between groups with identical missing data patterns is performed also with options of assuming normality (Hawkins test) or non-parametrically. Users can optionally use their own method of data imputation as well. Multiple imputation is an additional feature of the program that can be used as a diagnostic tool to help identify cases or variables that contribute to rejection of MCAR, when the MCAR test is rejecetd (See Jamshidian and Jalal, 2010 for details). As explained in Jamshidian, Jalal, and Jansen (2014), this package can also be used for imputing missing data, test of multivariate normality, and test of equality of covariances between several groups when data are completly observed.

Usage

TestMCARNormality(
  data,
  del.lesscases = 6,
  imputation.number = 1,
  method = "Auto",
  imputation.method = "Dist.Free",
  nrep = 10000,
  n.min = 30,
  seed = 110,
  alpha = 0.05,
  imputed.data = NA
)

Arguments

Details

Theoretical, technical and prcatical details about this program and its uses can be found in Jamshidian and Jalal (2010) and Jamshidian, Jalal, and Jansen (2014).

Value

Note

Note 1: In the above descriptions "original data" refers to the input data after deletion of the rows consisting of all NA's (if any)

Note 2: The normal theory maximum likelihood estimate of mean and covariance is obtained using the EM algorithm, as described in Jamshidian and Bentler (1999). The standard errors for these estimates, based on the observed information matrix, can be obtained via the function Ddf, included in this package.

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Jamshidian, M. and Bentler, P. M. (1999). “ML estimation of mean and covariance structures with missing data using complete data routines.” Journal of Educational and Behavioral Statistics, 24, 21-41, doi:10.2307/1165260.

Jamshidian, M. and Jalal, S. (2010). “Tests of homoscedasticity, normality, and missing at random for incomplete multivariate data,” Psychometrika, 75, 649-674, doi:10.1007/s11336-010-9175-3.

Jamshidian, M. Jalal, S., and Jansen, C. (2014). “MissMech: An R Package for Testing Homoscedasticity, Multivariate Normality, and Missing Completely at Random (MCAR),” Journal of Statistical Software, 56(6), 1-31, doi:10.18637/jss.v056.i06.

Examples

#-- Example 1: Data are MCAR and normally distributed

n <- 300
p <- 5
pctmiss <- 0.2
set.seed(1010)
y <- matrix(rnorm(n * p),nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA
out <- TestMCARNormality(data=y)
print(out)

# --- Prints the p-value for both the Hawkins and the nonparametric test
summary(out)

# --- Uses more cases
out1 <- TestMCARNormality(data=y, del.lesscases = 1)
print(out1)

#---- performs multiple imputation
Out <- TestMCARNormality (data = y, imputation.number = 10)
summary(Out)
boxplot(Out)

#-- Example 2: Data are MCAR and non-normally distributed (t distributed with d.f. = 5)

n <- 300
p <- 5
pctmiss <- 0.2
set.seed(1010)
y <- matrix(rt(n * p, 5), nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA
out <- TestMCARNormality(data=y)
print(out)

# Perform multiple imputation
Out_m <- TestMCARNormality (data = y, imputation.number = 20)
boxplot(Out_m)

#-- Example 3: Data are MAR (not MCAR), but are normally distributed

n <- 300
p <- 5
r <- 0.3
mu <- rep(0, p)
sigma <- r * (matrix(1, p, p) - diag(1, p))+ diag(1, p)
set.seed(110)
eig <- eigen(sigma)
sig.sqrt <- eig$vectors %*%  diag(sqrt(eig$values)) %*%  solve(eig$vectors)
sig.sqrt <- (sig.sqrt + sig.sqrt) / 2
y <- matrix(rnorm(n * p), nrow = n) %*%  sig.sqrt
tmp <- y
for (j in 2:p){
  y[tmp[, j - 1] > 0.8, j] <- NA 
}
out <- TestMCARNormality(data = y, alpha =0.1)
print(out)

#-- Example 4: Multiple imputation; data are MAR (not MCAR), but are normally distributed

n <- 300
p <- 5
pctmiss <- 0.2
set.seed(1010)
y <- matrix (rnorm(n * p), nrow = n)
missing <- matrix(runif(n * p), nrow = n) < pctmiss
y[missing] <- NA
Out <- OrderMissing(y)
y <- Out$data
spatcnt <- Out$spatcnt
g2 <- seq(spatcnt[1] + 1, spatcnt[2])
g4 <- seq(spatcnt[3] + 1, spatcnt[4])
y[c(g2, g4), ] <- 2 * y[c(g2, g4), ]
out <- TestMCARNormality(data = y, imputation.number = 20)
print(out)
boxplot(out)

# Removing Groups 2 and 4
y1= y[-seq(spatcnt[1]+1,spatcnt[2]),]
out <- TestMCARNormality(data=y1,imputation.number = 20)
print(out)
boxplot(out)

#-- Example 5: Test of homoscedasticity for complete data

n <- 50
p <- 5
r <- 0.4
sigma <- r * (matrix(1, p, p) - diag(1, p)) + diag(1, p)
set.seed(1010)
eig <- eigen(sigma)
sig.sqrt <- eig$vectors %*%  diag(sqrt(eig$values)) %*%  solve(eig$vectors)
sig.sqrt <- (sig.sqrt + sig.sqrt) / 2
y1 <- matrix(rnorm(n * p), nrow = n) %*%  sig.sqrt
n <- 75
p <- 5
y2 <- matrix(rnorm(n * p), nrow = n)
n <- 25
p <- 5
r <- 0
sigma <- r * (matrix(1, p, p) - diag(1, p)) + diag(2, p)
y3 <- matrix(rnorm(n * p), nrow = n) %*%  sqrt(sigma)
ycomplete <- rbind(y1 ,y2 ,y3)
y1 [ ,1] <- NA
y2[,c(1 ,3)] <- NA
y3 [ ,2] <- NA
ygroup <- rbind(y1, y2, y3)
out <- TestMCARNormality(data = ygroup, method = "Hawkins", imputed.data = ycomplete)
print(out)

# ---- Example 6, real data

data(agingdata)
TestMCARNormality(agingdata, del.lesscases = 1)

Test of Goodness of Fit (Uniformity)

Description

This routine tests whether the values in a vector x is distributed as uniform (0,1). The Neyman's smooth test of fit, as described by Ladwina (1994) is used. The p-values are obtained based on a resampling method from uniform (0,1).

Usage

TestUNey(x, nrep = 10000, sim = NA, n.min = 30)

Arguments

Value

Author(s)

Mortaza Jamshidian, Siavash Jalal, and Camden Jansen

References

Ledwina, T. (1994). “Data-driven version of neyman's smooth test of fit,” Journal of the American Statistical Association, 89, 1000-1005, doi:10.2307/2290926.

Examples

# Example 1
x <- runif(100)
TestUNey(x, nrep = 10000, sim = NA)

# Example 2
x <- runif(30,2,5)
x <- (x-min(x))/(max(x)-min(x))
TestUNey(x, nrep = 10000, sim = NA)

# Example 3
x <- c(0.6,0.6,0.5,0.7,0.3,0.4,0.5,0.4,0.2,0.4,0.2,0.5,0.7,0.1,0.7,0.1,0.5,0.5,0.4,0.6,0.3)
TestUNey(x, nrep = 10000, sim = NA)

Montpetit and Bergeman Longitudinal Study on Aging Data

Description

The data consist of 521 cases and 7 variables with 280 of the cases being complete. The variables are Education, Income, Perceived satisfaction of social support, Social coping, Total life events scale, Depression scale, and Self-rated help.

Usage

agingdata

Format

A data frame with 521 observations on the following 7 variables.

References

Montpetit A, Bergeman CS (2007). “Dimensions of control: Mediational analyses of the stress Chealth relationship.” Personality and Individual Differences, 43, 2237 - 2248, doi:10.1016/j.paid.2007.07.003.