test.z: z-Test

Description

This function performs one-sample, two-sample, and paired-sample z-tests and provides descriptive statistics, effect size measure, and a plot showing error bars for (difference-adjusted) confidence intervals with jittered data points.

Usage

test.z(x, ...)
# S3 method for default
test.z(x, y = NULL, sigma = NULL, sigma2 = NULL, mu = 0,
       paired = FALSE, alternative = c("two.sided", "less", "greater"),
       conf.level = 0.95, hypo = TRUE, descript = TRUE, effsize = FALSE,
       plot = FALSE, point.size = 4, adjust = TRUE, error.width = 0.1,
       xlab = NULL, ylab = NULL, ylim = NULL, breaks = ggplot2::waiver(),
       line = TRUE, line.type = 3, line.size = 0.8, jitter = TRUE,
       jitter.size = 1.25, jitter.width = 0.05, jitter.height = 0,
       jitter.alpha = 0.1, title = "", subtitle = "Confidence Interval",
       digits = 2, p.digits = 4, as.na = NULL, check = TRUE,
       output = TRUE, ...)
# S3 method for formula
test.z(formula, data, sigma = NULL, sigma2 = NULL,
       alternative = c("two.sided", "less", "greater"), conf.level = 0.95,
       hypo = TRUE, descript = TRUE, effsize = FALSE,
       plot = FALSE, point.size = 4, adjust = TRUE, error.width = 0.1,
       xlab = NULL, ylab = NULL, ylim = NULL, breaks = ggplot2::waiver(),
       jitter = TRUE, jitter.size = 1.25, jitter.width = 0.05, jitter.height = 0,
       jitter.alpha = 0.1, title = "",  subtitle = "Confidence Interval",
       digits = 2, p.digits = 4, as.na = NULL, check = TRUE,
       output = TRUE, ...)

Value

Returns an object of class misty.object, which is a list with following entries:

call: function call
type: type of analysis
sample: type of sample, i.e., one-, two-, or paired sample
formula: formula of the current analysis
data: data frame specified in data
plot: ggplot2 object for plotting the results
args: specification of function arguments
result: result table

Arguments

x: a numeric vector of data values.
...: further arguments to be passed to or from methods.
y: a numeric vector of data values.
sigma: a numeric vector indicating the population standard deviation(s). In case of two-sample z-test, equal standard deviations are assumed when specifying one value for the argument sigma; when specifying two values for the argument sigma, unequal standard deviations are assumed. Note that either argument sigma or argument sigma2 is specified.
sigma2: a numeric vector indicating the population variance(s). In case of two-sample z-test, equal variances are assumed when specifying one value for the argument sigma2; when specifying two values for the argument sigma, unequal variance are assumed. Note that either argument sigma or argument sigma2 is specified.
mu: a numeric value indicating the population mean under the null hypothesis. Note that the argument mu is only used when computing a one-sample z-test.
paired: logical: if TRUE, paired-sample z-test is computed.
alternative: a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".
hypo: logical: if TRUE, null and alternative hypothesis are shown on the console.
descript: logical: if TRUE, descriptive statistics are shown on the console.
effsize: logical: if TRUE, effect size measure Cohen's d is shown on the console.
conf.level: a numeric value between 0 and 1 indicating the confidence level of the interval.
plot: logical: if TRUE, a plot showing error bars for confidence intervals is drawn.
point.size: a numeric value indicating the size aesthetic for the point representing the mean value.
adjust: logical: if TRUE (default), difference-adjustment for the confidence intervals in a two-sample design is applied.
error.width: a numeric value indicating the horizontal bar width of the error bar.
xlab: a character string specifying the labels for the x-axis.
ylab: a character string specifying the labels for the y-axis.
ylim: a numeric vector of length two specifying limits of the limits of the y-axis.
breaks: a numeric vector specifying the points at which tick-marks are drawn at the y-axis.
line: logical: if TRUE (default), a horizontal line is drawn at mu for the one-sample t-test or at 0 for the paired-sample t-test.
line.type: an integer value or character string specifying the line type for the line representing the population mean under the null hypothesis, i.e., 0 = blank, 1 = solid, 2 = dashed, 3 = dotted, 4 = dotdash, 5 = longdash, 6 = twodash.
line.size: a numeric value indicating the size aesthetic for the line representing the population mean under the null hypothesis.
jitter: logical: if TRUE (default), jittered data points are drawn.
jitter.size: a numeric value indicating the size aesthetic for the jittered data points.
jitter.width: a numeric value indicating the amount of horizontal jitter.
jitter.height: a numeric value indicating the amount of vertical jitter.
jitter.alpha: a numeric value indicating the opacity of the jittered data points.
title: a character string specifying the text for the title for the plot.
subtitle: a character string specifying the text for the subtitle for the plot.
digits: an integer value indicating the number of decimal places to be used for displaying descriptive statistics and confidence interval.
p.digits: an integer value indicating the number of decimal places to be used for displaying the p-value.
as.na: a numeric vector indicating user-defined missing values, i.e. these values are converted to NA before conducting the analysis.
check: logical: if TRUE, argument specification is checked.
output: logical: if TRUE, output is shown on the console.
formula: in case of two sample z-test (i.e., paired = FALSE), a formula of the form y ~ group where group is a numeric variable, character variable or factor with two values or factor levels giving the corresponding groups.
data: a matrix or data frame containing the variables in the formula formula.

Author

Takuya Yanagida takuya.yanagida@univie.ac.at

Details

Cohen's d reported when argument effsize = TRUE is based on the population standard deviation specified in sigma or the square root of the population variance specified in sigma2. In a one-sample and paired-sample design, Cohen's d is the mean of the difference scores divided by the population standard deviation of the difference scores (i.e., equivalent to Cohen's \(d_z\) according to Lakens, 2013). In a two-sample design, Cohen's d is the difference between means of the two groups of observations divided by either the population standard deviation when assuming and specifying equal standard deviations or the unweighted pooled population standard deviation when assuming and specifying unequal standard deviations.

References

Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 1-12. https://doi.org/10.3389/fpsyg.2013.00863

Rasch, D., Kubinger, K. D., & Yanagida, T. (2011). Statistics in psychology - Using R and SPSS. John Wiley & Sons.

Examples

Run this code

dat1 <- data.frame(group = c(1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2),
                   x = c(3, 1, 4, 2, 5, 3, 2, 3, 6, 4, 3, NA))

#--------------------------------------
# One-Sample Design

# Two-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
test.z(dat1$x, sigma = 1.2, mu = 3)

# Two-sided one-sample z-test
# population mean = 3, population variance = 1.44
test.z(dat1$x, sigma2 = 1.44, mu = 3)

# One-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
test.z(dat1$x, sigma = 1.2, mu = 3, alternative = "greater")

# Two-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
# convert value 3 to NA
test.z(dat1$x, sigma = 1.2, mu = 3, as.na = 3)

# Two-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
# print Cohen's d
test.z(dat1$x, sigma = 1.2, mu = 3, effsize = TRUE)

# Two-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
# do not print hypotheses and descriptive statistics
test.z(dat1$x, sigma = 1.2, mu = 3, hypo = FALSE, descript = FALSE)

# Two-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
# print descriptive statistics with 3 digits and p-value with 5 digits
test.z(dat1$x, sigma = 1.2, mu = 3, digits = 3, p.digits = 5)

if (FALSE) {
# Two-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
# plot results
test.z(dat1$x, sigma = 1.2, mu = 3, plot = TRUE)

# Load ggplot2 package
library(ggplot2)

# Save plot, ggsave() from the ggplot2 package
ggsave("One-sample_z-test.png", dpi = 600, width = 3, height = 6)

# Two-sided one-sample z-test
# population mean = 3, population standard deviation = 1.2
# extract plot
p <- test.z(dat1$x, sigma = 1.2, mu = 3, output = FALSE)$plot
p

# Extract data
plotdat <- data.frame(test.z(dat1$x, sigma = 1.2, mu = 3, output = FALSE)$data[[1]])

# Extract results
result <- test.z(dat1$x, sigma = 1.2, mu = 3, output = FALSE)$result

# Draw plot in line with the default setting of test.z()
ggplot(plotdat, aes(0, x)) +
  geom_point(data = result, aes(x = 0L, m), size = 4) +
  geom_errorbar(data = result, aes(x = 0L, y = m, ymin = m.low, ymax = m.upp),
                width = 0.2) +
  scale_x_continuous(name = NULL, limits = c(-2, 2)) +
  scale_y_continuous(name = NULL) +
  geom_hline(yintercept = 3, linetype = 3, size = 0.8) +
  labs(subtitle = "Two-Sided 95% Confidence Interval") +
  theme_bw() + theme(plot.subtitle = element_text(hjust = 0.5),
                     axis.text.x = element_blank(),
                     axis.ticks.x = element_blank())
}

#--------------------------------------
# Two-Sample Design

# Two-sided two-sample z-test
# population standard deviation (SD) = 1.2, equal SD assumption
test.z(x ~ group, sigma = 1.2, data = dat1)

# Two-sided two-sample z-test
# population standard deviation (SD) = 1.2 and 1.5, unequal SD assumption
test.z(x ~ group, sigma = c(1.2, 1.5), data = dat1)

# Two-sided two-sample z-test
# population variance (Var) = 1.44 and 2.25, unequal Var assumption
test.z(x ~ group, sigma2 = c(1.44, 2.25), data = dat1)

# One-sided two-sample z-test
# population standard deviation (SD) = 1.2, equal SD assumption
test.z(x ~ group, sigma = 1.2, data = dat1, alternative = "greater")

# Two-sided two-sample z-test
# population standard deviation (SD) = 1.2, equal SD assumption
# print Cohen's d
test.z(x ~ group, sigma = 1.2, data = dat1, effsize = TRUE)

# Two-sided two-sample z-test
# population standard deviation (SD) = 1.2, equal SD assumption
# do not print hypotheses and descriptive statistics,
# print Cohen's d
test.z(x ~ group, sigma = 1.2, data = dat1, descript = FALSE, hypo = FALSE)

# Two-sided two-sample z-test
# population mean = 3, population standard deviation = 1.2
# print descriptive statistics with 3 digits and p-value with 5 digits
test.z(x ~ group, sigma = 1.2, data = dat1, digits = 3, p.digits = 5)

if (FALSE) {
# Two-sided two-sample z-test
# population standard deviation (SD) = 1.2, equal SD assumption
# plot results
test.z(x ~ group, sigma = 1.2, data = dat1, plot = TRUE)

# Load ggplot2 package
library(ggplot2)

# Save plot, ggsave() from the ggplot2 package
ggsave("Two-sample_z-test.png", dpi = 600, width = 4, height = 6)

# Two-sided two-sample z-test
# population standard deviation (SD) = 1.2, equal SD assumption
# extract plot
p <- test.z(x ~ group, sigma = 1.2, data = dat1, output = FALSE)$plot
p
}

#-----------------

group1 <- c(3, 1, 4, 2, 5, 3, 6, 7)
group2 <- c(5, 2, 4, 3, 1)

# Two-sided two-sample z-test
# population standard deviation (SD) = 1.2, equal SD assumption
test.z(group1, group2, sigma = 1.2)

#--------------------------------------
# Paired-Sample Design

dat2 <- data.frame(pre = c(1, 3, 2, 5, 7),
                   post = c(2, 2, 1, 6, 8), stringsAsFactors = FALSE)

# Two-sided paired-sample z-test
# population standard deviation of difference score = 1.2
test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE)

# Two-sided paired-sample z-test
# population variance of difference score = 1.44
test.z(dat2$pre, dat2$post, sigma2 = 1.44, paired = TRUE)

# One-sided paired-sample z-test
# population standard deviation of difference score = 1.2
test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE,
       alternative = "greater")

# Two-sided paired-sample z-test
# population standard deviation of difference score = 1.2
# convert value 1 to NA
test.z(dat2$pre, dat2$post, sigma = 1.2, as.na = 1, paired = TRUE)

# Two-sided paired-sample z-test
# population standard deviation of difference score = 1.2
# print Cohen's d
test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE, effsize = TRUE)

# Two-sided paired-sample z-test
# population standard deviation of difference score = 1.2
# do not print hypotheses and descriptive statistics
test.z(dat2$pre, dat2$post, sigma = 1.2, mu = 3, paired = TRUE,
       hypo = FALSE, descript = FALSE)

# Two-sided paired-sample z-test
# population standard deviation of difference score = 1.2
# print descriptive statistics with 3 digits and p-value with 5 digits
test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE,
       digits = 3, p.digits = 5)

if (FALSE) {
# Two-sided paired-sample z-test
# population standard deviation of difference score = 1.2
# plot results
test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE, plot = TRUE)

# Load ggplot2 package
library(ggplot2)

# Save plot, ggsave() from the ggplot2 package
ggsave("Paired-sample_z-test.png", dpi = 600, width = 3, height = 6)

# Two-sided paired-sample z-test
# population standard deviation of difference score = 1.2
# extract plot
p <- test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE, output = FALSE)$plot
p

# Extract data
plotdat <- data.frame(test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE,
                      output = FALSE)$data)

# Difference score
plotdat$diff <- plotdat$y - plotdat$x

# Extract results
result <- test.z(dat2$pre, dat2$post, sigma = 1.2, paired = TRUE,
                  output = FALSE)$result

# Draw plot in line with the default setting of test.t()
ggplot(plotdat, aes(0, diff)) +
  geom_point(data = result, aes(x = 0, m.diff), size = 4) +
  geom_errorbar(data = result,
                aes(x = 0L, y = m.diff, ymin = m.low, ymax = m.upp), width = 0.2) +
   scale_x_continuous(name = NULL, limits = c(-2, 2)) +
   scale_y_continuous(name = "y") +
   geom_hline(yintercept = 0, linetype = 3, size = 0.8) +
   labs(subtitle = "Two-Sided 95% Confidence Interval") +
   theme_bw() + theme(plot.subtitle = element_text(hjust = 0.5),
                      axis.text.x = element_blank(),
                      axis.ticks.x = element_blank())
}

Run the code above in your browser using DataLab