Comparing Two Groups with Various Descriptive Statistics

Description

The ' groupcompare' package performs various statistical tests to compare two groups. It calculates descriptive statistics and quantile statistics and then conducts some normality tests and variance homogeneity tests. Based on these assumptions checks, it provides results from t-test or Wilcoxon rank sum test, permutation tests, and bootstrap confidence intervals.

Details

The main function ‘groupcompare' of the package is designed to compare two independent or paired groups using various statistical tests. It calculates descriptive statistics and quantile statistics. Then it performs Shapiro-Wilk normality tests, variance homogeneity test (Levene’s test), t-test, Wilcoxon signed-rank sum test (or Mann-Whitney U test), permutation tests, and bootstrap confidence intervals.

groupcompare

The main function which compares descriptive statistics of two groups using a variety of statistical tests.

bivarplot

Generates various plots to visualize and compare the distribution and characteristics of two variables.

bootstrap

Calculates bootstrap confidence intervals for the descriptive statistics or any statistic implemented in a custom function.

calchubermeandif

Computes the difference between Huber???s M-estimator of location of two groups in long data format.

calcquantdif

Calculates the differences between specified quantiles for grouped data.

calcquantile

Calculates the quantiles (percentiles) for a given vector of data at specified fractions.

calcstatdif

Calculates the differences in multiple statistics (mean, median, IQR, variance) for grouped data.

ci2df

Converts a list of confidence intervals into a data frame.

descstats

Calculates the common and robust descriptive statistics.

ghdist

Generates a random sample from the g-and-h (gh) distribution with specified parameters.

groupdata

A data set contains seven data frames with two variables from various distributions.

hdqe

Computes the Harrell-Davis quantile estimator for given quantile levels.

intnorm

Performs an inverse normal transformation on non-normally distributed data.

levene.test

Performs the Levene test to check the homogeneity of variances across groups.

long2wide

Converts long-format data to wide-format data by splitting based on groups.

permtest

Performs a permutation test on long-format data to evaluate differences between two groups using a specified test statistic.

quail

A data frame containing daily weight gains (in grams) of two quail breeds during a fattening period.

wide2long

Converts wide-format data to long-format data.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

bootstrap, permtest, ghdist, bivarplot

Examples

# Sample dataset in long format
set.seed(123)
group1 <- rnorm(30, mean=50, sd=2)
group2 <- rnorm(30, mean=51, sd=3)
df <- data.frame(value=c(group1, group2), group=rep(c("A", "B"), each=30))

# Compare the groups using various descriptive statistics
result <-  groupcompare(df, cl=0.95, alternative="two.sided",
  q=c(0.25, 0.5, 0.75), qt=0, R=500, out=FALSE, verbose=TRUE)
result

# Compare the groups using Huber's M-estimator of location with bootstrap
bshubermean <- bootstrap(df, statistic=calchubermeandif, 
  alternative="two.sided", alpha=0.05, R=500)
bshubermean

permhubermean <- permtest(df, statistic=calchubermeandif, 
  alternative="two.sided", R=500)
permhubermean$pval

Plots for Two Variables

Description

Generates various plots to visualize and compare the distribution and characteristics of two variables.

Usage

bivarplot(ds)

Arguments

Details

This function generates a series of plots to compare two variables, including density plots, ECDF plots, boxplots, violin plots, QQ plots, symmetry plots, and empirical shift plots. If data is in long format, the function uses the second column of ds for the group label.

Value

Generates a series of plots to the current graphical device.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

Examples

# Create data for two independent groups, each with 50 observations
set.seed(1)
df1 <- data.frame(value = rnorm(100), group = rep(1:2, each = 50))
head(df1)

# Plots for visualization
bivarplot(df1)

# Convert to long data to wide data
df2 <- long2wide(df1)
head(df2)

# Plots for visualization
bivarplot(df2)

Bootstrap for Various Statistics

Description

Conducts a bootstrap procedure to calculate confidence intervals for various statistics.

Usage

bootstrap(x, statistic, alternative="two.sided", alpha = 0.05, 
  Q = c(0.25, 0.5, 0.75), qt = 7, R = 3000)

Arguments

Details

This function performs a bootstrap procedure to calculate confidence intervals for various statistics. It is mainly used to evaluate the differences between various statistics for two groups based on a specified function. The function calculates confidence intervals using different methods, including normal, basic, percentile, and bias-corrected and accelerated (BCa) intervals. It allows users to pass custom statistics (via statistic) that include parameters like quantiles (via Q) and types of quantiles (qt), making it versatile for non-standard use cases. In this way, the function extends the capabilities of boot::boot.ci (R's suggested package) by supporting more advanced statistical needs and customizable interval calculations. This function also incorporates the argument alternative can be set to "greater" or "less" for one-tailed confidence intervals, whereas boot::boot.ci primarily focuses on two-tailed intervals.Therefore, it is particularly useful for specialized applications. The data can be provided in long format, and the test uses a specified number of bootstrap replicates to calculate the empirical distribution of the test statistic under the null hypothesis.

Value

A list containing the data frames for the following components for each statistic:

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

permtest

Examples

# Load the quail dataset 
data(quail)

# Bootstrap for the difference of various basic statistics 
# Increase R for real-world applications

bootres <- bootstrap(quail, statistic = calcstatdif, R=200)
bootres

# Arrange the results as a data frame
ci2df(bootres)

# Bootstrap for the differences of quantiles 
bootresq <- bootstrap(quail, statistic = calcquantdif, R=200)
bootresq

# Arrange the results as a data frame
ci2df(bootresq)

The difference between Huber's M-estimator of location

Description

Computes the difference between Huber's M-estimator of location of two groups in long data format.

Usage

calchubermeandif(x, indices, ...)

Arguments

Details

This function demonstrates the structure of a user-defined statistic function to use with bootstrap and permutation test. The function calculates the difference between Huber's M-estimator of location of two groups using the iterative weighted mean method. Huber's M-estimator of location is robust to outliers and is computed using an iterative re-weighting procedure. The internal function follows:

• Initialize with the median of the data.

• Compute weights based on deviations from the current mean.

• Update the mean iteratively until convergence is reached.

Value

A numeric value representing the difference between Huber's M-estimator of location of the two groups.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

median

Examples

df <- data.frame(value = rnorm(100), group = rep(1:2, each = 50))
bivarplot(df)

# Bootstrap confidence intervals for the difference of 
# Huber's M-estimator of location of two groups
# Increase R for real-world applications
bshubermean <- bootstrap(df, statistic=calchubermeandif, alpha=0.05,
   alternative="less", R=200)
bshubermean

Quantile Differences

Description

Calculates the differences between specified quantiles for grouped data.

Usage

calcquantdif(x, indices, Q=seq(0.1, 0.9, 0.1), qt=7)

Arguments

Details

This function calculates the differences between specified quantiles for groups defined by the second column of the input data. It uses the specified quantile type to compute the quantiles. Types of quantiles are:

• 0: Harrell-Davis estimator (not available in stats::quantile function).

• 1: Inverse of the empirical distribution function.

• 2: Similar to Type 1 but with averaging at discontinuities.

• 3: Empirical distribution with sampling.

• 4: Linear interpolation of the empirical distribution function.

• 5: Linear interpolation of the expectations for the order statistics.

• 6: Linear interpolation of the modes for the order statistics.

• 7: The default in the stats::quantile function.

• 8: Median-unbiased estimator.

• 9: Normal-unbiased estimator.

For the details on types, see the quantile and hdqe function.

Value

A numeric vector containing the differences between the specified quantiles for each group.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

References

Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician, 50, 361-365. <doi:10.2307/2684934>.

See Also

quantile, hdqe, calcstatdif

Examples

df <- data.frame(value = rnorm(100), group = rep(1:2, each = 50))

# Difference between the medians
mediandif <- calcquantdif(df, Q=0.5)
mediandif

# Differences between the quartiles
quantdifs <- calcquantdif(df, Q=c(0.25, 0.5, 0.75))
quantdifs

# Differences between the percentiles from P10 to P90 using the method 5
quants <- seq(0.1, 0.9, 0.1)
quantdifs <- calcquantdif(df, Q=quants, qt=5)
quantdifs

# Differences between the percentiles from P10 to P90 using the method Harrell-Davis
quants <- seq(0.1, 0.9, 0.1)
quantdifs <- calcquantdif(df, Q=quants, qt=0)
quantdifs

Sample Quantiles

Description

Calculates the quantiles (percentiles) for a given vector of data at specified fractions.

Usage

calcquantile(x, indices, Q = seq(0.1, 0.9, 0.1), qt = 7)

Arguments

Details

This function calculates the quantiles at specified fractions of the given data set. If qt is 0, the hdqe function is used.

• 0: Harrell-Davis estimator (not available in stats::quantile function).

• 1: Inverse of the empirical distribution function.

• 2: Similar to Type 1 but with averaging at discontinuities.

• 3: Empirical distribution with sampling.

• 4: Linear interpolation of the empirical distribution function.

• 5: Linear interpolation of the expectations for the order statistics.

• 6: Linear interpolation of the modes for the order statistics.

• 7: The default in the stats::quantile function.

• 8: Median-unbiased estimator.

• 9: Normal-unbiased estimator.

For the details on types, see the quantile and hdqe functions.

Value

Returns a numeric vector containing the calculated quantiles.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

References

Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician 50, 361–365. <doi:10.2307/2684934>.

See Also

quantile, hdqe

Examples

x <- rnorm(100)
calcquantile(x)
calcquantile(x, qt=9)
calcquantile(x, qt = 0)

Differences of Basic Descriptive Statistics

Description

Calculates the differences in multiple statistics (mean, median, IQR, variance) for grouped data.

Usage

calcstatdif(x, indices, ...)

Arguments

Details

This function calculates the differences in multiple statistics (mean, median, interquartile range (IQR), and variance) for groups defined by the second column of the input data. The function is used in permutation test and bootstrapping as a statistical function.

Value

A named numeric vector containing the differences in the specified statistics for each group. The names of the vector elements are "MEAN", "MED", "IQR", "VAR", "SKEW", and "KURT" for mean, median, interquartile range, variance, skewness, and kurtosis, respectively.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

calcquantdif

Examples

# Generate example data 
set.seed(1199)  
grp1 <- rnorm(20, 45, 5)
grp2 <- c(rnorm(10, 45, 10), rnorm(10, 52, 20))
df <- data.frame(cbind(grp1=grp1, grp2=grp2))
head(df)
bivarplot(df)

# Reshape the data into long format
df <- wide2long(df)
head(df)

# Differences between the basic stats
calcstatdif(df)

Convert List to Data Frame

Description

Converts a list of confidence intervals into a data frame.

Usage

ci2df(x)

Arguments

Details

This function takes a list of confidence intervals and converts it into a data frame. Each row represents a method, and each column represents a statistic. Confidence intervals are formatted as strings in the form [lower, upper].

Value

A data frame with rows representing methods and columns representing statistics, containing the confidence intervals.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

bootstrap

Examples

ciresults <- list(
  method1 = data.frame(lower = c(-0.1, 0.2), upper = c(0.3, 0.4), row.names = c("stat1", "stat2")),
  method2 = data.frame(lower = c(0.2, 0.3), upper = c(0.4, 0.5), row.names = c("stat1", "stat2"))
)
ciresults

cidf <- ci2df(ciresults)
cidf

Descriptive Statistics

Description

This function calculates various descriptive statistics for a given numeric vector. These statistics include measures of central tendency, dispersion, skewness, kurtosis, and some robust estimators.

Usage

descstats(x, trim = 0.1, k = 1.5)

Arguments

Details

In order to determine an appropriate k value for th Huber M-estimator some experiments might be needed. In the literature, commonly used k values typically range from 1.5 to 2. Users can start by choosing any value within this range. However, to determine an appropriate k within a given range, it is also selected by performing Huber estimations for each k value within this range, as shown in the example below. In the output, the estimated Huber M-estimator values can be checked on a plot. Select k values where a smooth trend or plateau is reached. If the Huber M-estimator values stabilize after a certain k value, that k value may be appropriate. Finally, if there are outliers and you want to reduce their impact, you can use smaller k values.

Value

A list containing the computed descriptive statistics, including:

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

Examples

set.seed(123)
x <- rnorm(100, mean=50, sd=5)
descriptives <- descstats(x)
as.data.frame(descriptives)

descriptives$mean
descriptives$se

# Determining the appropriate k in a given set of different k values. 
# This parameter is used to calculate the Huber M-estimator of the location
# Array of k values for testing
k <- seq(0, 5, by = 0.1)
k <- k[k> 0]
result <- sapply(k, function(y) descstats(x, k = y)$hubermean)
names(result) <- paste0("k=", k)
result

plot(k, result, type = "b", col = "blue", pch = 19, ylab = "Huber's mean")

descstats(x, k=2, trim=0.05)$hubermean

Random Sample from G-H Distribution

Description

Generates a random sample from the G-and-H (GH) distribution with specified parameters.

Usage

ghdist(n = 30, A = 0, B = 1, g = 0, h = 0)

Arguments

Details

The gh distribution is a flexible distribution defined by four parameters: A (location), B (scale), g (skewness), and h (kurtosis). The parameter B must be positive, and h must be zero or positive. This function generates random samples from the gh distribution using these parameters.

The GH distribution was introduced by John W. Tukey in 1977 as a way to model data with varying degrees of skewness and kurtosis. The distribution is defined by transforming standard normal random variables using the g and h parameters to control skewness and kurtosis, respectively.

Value

A numeric vector of length n containing the generated random samples.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

References

Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley.

See Also

rnorm

Examples

 set.seed(50) 
 A <- ghdist(100, 50, 2, g=0, h=0)
 B <- ghdist(100, 50, 3, g=0.5, h=0.5)
 ds <- data.frame(A=A, B=B)
 head(ds)
 bivarplot(ds)

Comparing Two Groups with Various Descriptive Statistics

Description

Performs a variety of statistical tests and generates plots to compare two groups.

Usage

 groupcompare (ds, paired=FALSE, cl=0.95, alternative="two.sided",
   qtest=TRUE, q=seq(0.1, 0.9, by=0.1), qt=7, lognorm=TRUE, 
   R=3000, plots=TRUE, out=FALSE, verbose=TRUE)

Arguments

Details

This function calculates descriptive statistics and performs a t-test or Wilcoxon test on two groups of data, depending on the variance homogeneity test and normality test. It also generates various plots, including density plots, ECDF plots, boxplots, violin plots, QQ plots, symmetry plots, and empirical shift plots, to visualize the data.

Value

A list that contains the results of the statistical tests and the bivariate plots, if plots is TRUE.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

bootstrap, permtest, bivarplot

Examples

   # Test with default values
   df <- data.frame(grp1 = rnorm(50), grp2 = rnorm(50))
   df <- wide2long(df)
   result <-  groupcompare(df)
   print(result)

   # Test with daily weight gain of two quail breeds
   data(quail)
   result <-  groupcompare(quail, alternative="two.sided", cl=0.95,
     R=200, plots=TRUE, out=FALSE, verbose=TRUE)
   print(result)

   # Test with milk trace mineral data
   filepath <- system.file("extdata", "milkcomp.csv", package = "groupcompare")
   milktrace <- read.table(filepath, header=TRUE, sep=",")
   head(milktrace) 
   milkzinc <- as.data.frame(cbind(milktrace$Zn, milktrace$grp))
   colnames(milkzinc) <- c("zn","grp")
   head(milkzinc)
   tail(milkzinc)

   result <-  groupcompare(milkzinc, cl=0.99, alternative="greater",
      R=200, plots=TRUE, out=FALSE, verbose=TRUE)
   print(result)
 

Synthetic Data for Two Independent Groups

Description

This data set contains seven data frames with two variables from various distributions.

Usage

 data(groupdata)

Format

A list containing 7 data frames:

df1

Normally distributed, equal means, and equal variances. Columns: A, B.

df2

Normally distributed, unequal means, and unequal variances. Columns: A, B.

df3

Normally distributed, equal means, and unequal variances. Columns: A, B.

df4

A normal, B right-skewed, equal means, and unequal variances. Columns: A, B.

df5

A normal, B right-skewed, equal means, and unequal variances. Columns: A, B.

df6

A normal, B right-skewed, equal means, and unequal variances. Columns: A, B.

df7

A normal, B right-skewed, equal means, and unequal variances. Columns: A, B.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

Examples

  data(groupdata)
  df1 <- groupdata$df1
  head(df1)
  bivarplot(df1)

  df4 <- groupdata$df4
  head(df4)
  bivarplot(df4)

Harrell-Davis Quantile Estimator

Description

Computes the Harrell-Davis quantile estimator for given quantile levels.

Usage

hdqe(x, Q = c(0.25, 0.5, 0.75))

Arguments

Details

The function computes the Harrell-Davis quantile estimator, which estimates data quantiles by calculating a weighted average of order statistics. The weights are based on the beta distribution.

Value

A numeric vector containing estimated quantiles.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

Examples

set.seed(1)
x <- sample(1:10, 50, replace=TRUE)
quantile(x, probs=c(0.25, 0.5, 0.75), type=1) # quantiles with Type 1 in stats::quantile
quantile(x, probs=c(0.25, 0.5, 0.75), type=7) # quantiles with Type 7 (default) in stats::quantile
hdqe(x, Q=c(0.25, 0.5, 0.75)) # quantiles with Harrell-Davis Estimator

Inverse Normal Transformation Function

Description

This function performs an inverse normal transformation on non-normally distributed data. Using the provided rank information, it can also revert the transformed data back to its original scale.

Usage

intnorm(x)

Arguments

Value

A numeric vector of the transformed values.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

Examples

set.seed(1)
xnonnormal <- c(rexp(25, rate = 1), rexp(25, rate = 0.2))
xnormal <- intnorm(xnonnormal)

# Plot density and perform the Shapiro-Wilk test for non-normal and normalized data
opar <- par(mfrow = c(1, 2))

# Non-normal data plot
density_xnonnormal <- density(xnonnormal)
plot(density_xnonnormal, main = "Density Plot of Non-Normal Data", 
     xlab = "Value", ylab = "Density", col = "blue", lwd=2)
polygon(density_xnonnormal, col = rgb(1, 0, 0, 0.3))
shapirotest1 <- shapiro.test(xnonnormal)
mtext(paste("p-value:", round(shapirotest1$p.value, 4)), 
      side = 3, line = 0.5, at = mean(xnonnormal), col = "black")
shapirotest1

# Normalized data plot
density_xnormal <- density(xnormal)
plot(density_xnormal, main = "Density Plot of Inverse Normalized Data", 
     xlab = "Value", ylab = "Density", col = "blue", lwd=2)
polygon(density_xnormal, col =  rgb(1, 0, 0, 0.3)) 
shapirotest2 <- shapiro.test(xnormal)
mtext(paste("p-value:", round(shapirotest2$p.value, 4)), 
      side = 3, line = 0.5, at = mean(xnormal), col = "black")
shapirotest2

par(opar)

Levene Test for Homogeneity of Variances

Description

This function performs the Levene test to check the homogeneity of variances across groups.

Usage

levene.test(ds)

Arguments

Details

The function calculates the absolute deviations from the group means, performs an ANOVA on these deviations, and returns the p-value of the Levene test.

Value

A single numeric value named Levene.p, representing the p-value of the Levene test.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

Examples

df <- data.frame(values = c(1.1, 2.3, 2.5, 3.7, 1.2, 2.1, 3.4, 3.9),
  groups = c("A", "A", "B", "B", "A", "A", "B", "B"))
levene.test(df)

Convert Long-Format Data to Wide-Format Data

Description

Converts long-format data to wide-format data by splitting based on groups.

Usage

long2wide(x)

Arguments

Details

This function converts long-format data to wide format by splitting it based on unique groups in the second column. The resulting data frame has columns for each group, where each column contains the values of the first column for the corresponding group.

Value

A data frame with columns corresponding to unique groups and rows containing the values for each group.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

wide2long

Examples

# Normal distributed groups of data with different means and different variances
set.seed(21)
obs1 <- rnorm(20, 50, 5)
obs2 <- rnorm(20, 55, 3)
obs <- c(obs1, obs2)
grp <- c(rep(as.factor("A"), 20), rep(as.factor("B"), 20))
ds1 <- data.frame(obs = obs, group = grp)
head(ds1)

# Reshape data to long format
ds2 <- long2wide(ds1)
head(ds2)
bivarplot(ds2)

Permutation Test on Long-Format Data

Description

Performs a permutation test on long-format data to evaluate differences between two groups using a specified test statistic.

Usage

permtest(x, y = NULL, statistic, alternative = "two.sided",
   Q = seq(0.1, 0.9, by = 0.1), qt = 7, R = 10000)

Arguments

Details

The function allows researchers to perform robust statistical testing by utilizing permutations. This approach does not rely on distributional assumptions and is particularly useful when the sample size is small or the data distribution is unknown. The test generates an empirical distribution of the test statistic by repeatedly permuting group labels and recalculating the statistic, providing p-values based on these permutations.

Value

A list containing the following components:

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

bootstrap

Examples

# Group1 normal, Group 2 right skewed with equal means and different variances
set.seed(1199)  
grp1 <- rnorm(20, 45, 5)
grp2 <- c(rnorm(10, 45, 10), rnorm(10, 52, 20))
df <- data.frame(cbind(grp1=grp1, grp2=grp2))
head(df)
bivarplot(df)

# Reshape the data into long format
df <- wide2long(df)
head(df)

# Differences between the basic stats
calcstatdif(df)

# Permutation test for the differences between the basic stats of two groups
result <- permtest(df, statistic = calcstatdif, alternative = "two.sided", R = 500)
result$pval

# Permutation with custom statistics
# A custom function to compute the differences between the group means
meancomp <- function(x, ...){
   meandif <- diff(rev(tapply(x[,1], x[,2], mean)))
   return(meandif)
}

# Permutation test with meancomp function
result <- permtest(x = df, statistic = meancomp, alternative="less", R=500)
result$pval

Daily Weight Gains of Quail Breeds

Description

A data frame containing daily weight gains (in grams) of two quail breeds during a fattening period.

Usage

data(quail)

Format

A data frame with 16 observations on the following 2 variables:

dwg

A numeric vector containing daily weight gains (g).

genotype

A factor with levels A and B indicating breeds or genotypes.

Details

The dataset is used to compare the daily weight gain performances of different breeds of quails.

Examples

data(quail)
head(quail)

Convert Wide-Format Data to Long-Format Data

Description

Converts wide-format data to long-format data.

Usage

wide2long(x, y = NULL, grpnames=NULL)

Arguments

Details

This function converts wide-format data to long format by combining two columns of data into a single column and creating a grouping variable to distinguish the original data sources.

Value

A data frame with two columns: Obs containing the combined values of x and y, and group indicating the original data source.

Author(s)

Zeynel Cebeci, A. Firat Ozdemir, Engin Yildiztepe

See Also

long2wide

Examples

 # Normal distributed groups of data with equal means and different variances
 set.seed(2)
 grp1 <- rnorm(20, 50, 5)
 grp2 <- rnorm(20, 50, 9)
 ds1 <- data.frame(A=grp1, B=grp2)
 head(ds1)
 bivarplot(ds1)

 # Convert to long data
 ds2 <- wide2long(ds1)
 head(ds2)