1 Test Statistic for Sign Test for One Sample in R
2 Simple Sign Test for One Sample in R
3 Sign Test for One Sample: Critical Value in R
4 Two-tailed Sign Test for One Sample in R
5 One-tailed Sign Test for One Sample in R
6 Sign Test for One Sample: Test Statistics & P-values in R

Here, we discuss the sign test for one sample in R with interpretations, including, test statistics, p-values, and confidence intervals.

The sign test for one sample in R can be performed with the SIGN.test() function from the "BSDA" package.

The sign test for one sample can be used to test whether the median of the population an independent random sample comes from is equal to a certain value (which is stated in the null hypothesis) or not. It is a non-parametric alternative to the one sample t-test.

In the sign test for one sample, the test statistic is based on the signs of the observed values minus the null hypothesis median. It is the number of observations greater than the null hypothesis median, and inference is based on binomial distribution.

Sign Test for One Sample Tests & Hypotheses
Question	Is the median equal to $m_0$?	Is the median greater than $m_0$?	Is the median less than $m_0$?
Form of Test	Two-tailed	Right-tailed test	Left-tailed test
Null Hypothesis, $H_0$	$m = m_0$	$m = m_0$	$m = m_0$
Alternate Hypothesis, $H_1$	$m \neq m_0$	$m > m_0$	$m < m_0$

Install and Load "BSDA" Package:

To use the SIGN.test() function form the "BSDA" package, first install the package, then load it as follows:

install.packages("BSDA")

library(BSDA)

Sample Steps to Run a Sign Test for One Sample:

# Create the data for the sign test for one sample

data = c(1.3, 0.5, -0.4, 1.4, -1.5, 0.5)

# Run the sign test for one sample with specifications

SIGN.test(data, alternative = "two.sided",
          md = 0, conf.level = 0.95)


    One-sample Sign-Test

data:  data
s = 4, p-value = 0.6875
alternative hypothesis: true median is not equal to 0
95 percent confidence interval:
 -1.39  1.39
sample estimates:
median of x 
        0.5 

Achieved and Interpolated Confidence Intervals: 

                  Conf.Level L.E.pt U.E.pt
Lower Achieved CI     0.7812  -0.40   1.30
Interpolated CI       0.9500  -1.39   1.39
Upper Achieved CI     0.9688  -1.50   1.40

Table of Some Sign Test for One Sample Arguments in R
Argument	Usage
x	Sample data values
md	Population median value in null hypothesis
alternative	Set alternate hypothesis as "greater", "less", or the default "two.sided"
conf.level	Level of confidence for the test and confidence interval (default = 0.95)

Creating a Sign Test for One Sample Object:

# Create data
data = rnorm(100)

# Create object
sgnt_object = SIGN.test(data, alternative = "two.sided",
                        md = 0, conf.level = 0.95)

# Extract a component
sgnt_object$statistic

 s 
52

Table of Some Sign Test for One Sample Object Outputs in R
Test Component	Usage
sgnt_object$statistic	Test-statistic; number of values greater than the median
sgnt_object$p.value	P-value
sgnt_object$estimate	Point estimate or sample median
sgnt_object$conf.int	Confidence interval

1 Test Statistic for Sign Test for One Sample in R

The sign test for one sample has test statistics, $S$, of the form:

\[S = \sum_{i=1}^n I_{(x_i-m_0)>0},\]

which is the number of observations greater than the null hypothesis median.

For $n_d$ which is the number of non-zero $(x_i - m_0)$, inference on $S$ is based on the binomial distribution with $\text{size} = n_d$ and $\text{prob} = 0.5$.

$x_i's$ are the sample values,

$m_0$ is the population median value to be tested and set in the null hypothesis,

$I_{(x_i-m_0)>0}$ is $1$ when $(x_i-m_0)>0$ and $0$ otherwise, and

$n$ is the sample size.

2 Simple Sign Test for One Sample in R

Enter the data by hand.

data = c(-1.4, 0.5, -0.5, -0.7, -1.9,
         -0.1, -1.6, -3.3, -0.8, 1.8,
         -1.2, 1.2, -3.2, -0.1, 1.0)

For the following null hypothesis $H_0$, and alternative hypothesis $H_1$, with the level of significance $\alpha=0.05$.

$H_0:$ the population median is equal to 0 ($m = 0$).

$H_1:$ the population median is not equal to 0 ($m \neq 0$, hence the default two-sided).

Because the level of significance is $\alpha=0.05$, the level of confidence is $1 - \alpha = 0.95$.

The SIGN.test() function has the default alternative as "two.sided", the default median as 0, and the default level of confidence as 0.95, hence, you do not need to specify the "alternative", "md", and "conf.level" arguments in this case.

SIGN.test(data, alternative = "two.sided",
          md = 0, conf.level = 0.95)

Or:

SIGN.test(data)


    One-sample Sign-Test

data:  data
s = 4, p-value = 0.1185
alternative hypothesis: true median is not equal to 0
95 percent confidence interval:
 -1.5643663  0.3930989
sample estimates:
median of x 
       -0.7 

Achieved and Interpolated Confidence Intervals: 

                  Conf.Level  L.E.pt  U.E.pt
Lower Achieved CI     0.8815 -1.4000 -0.1000
Interpolated CI       0.9500 -1.5644  0.3931
Upper Achieved CI     0.9648 -1.6000  0.5000

The sample median, $\tilde x$, is -0.7,

test statistic, $S$, is 4,

the p-value, $p$, is 0.1185,

the interpolated 95% confidence interval is [-1.5643663, 0.3930989].

Interpretation:

Note that for SIGN.test() in R, the two methods may disagree for some edge cases, as p-value is based on binomial distribution, and confidence interval is based on interpolation applying binomial distribution.

P-value: With the p-value ($p = 0.1185$) being greater than the level of significance 0.05, we fail to reject the null hypothesis that the population median is equal to 0. This is the same as the "two.sided" test for $\text{Binomial($n_d$, 0.5)}$, that is, binom.test(4, 15, 0.5, "two.sided")$p.value $= 0.1184692$.
Confidence Interval: With the null hypothesis median value ($m = 0$) being inside the confidence interval, $[-1.5643663, 0.3930989]$, we fail to reject the null hypothesis that the population median is equal to 0.

3 Sign Test for One Sample: Critical Value in R

To get the critical value for a sign test for one sample in R, you can use the qbinom() function for binomial distribution to derive the quantile associated with the given level of significance value $\alpha$.

With $n_d$ as the number of non-zero $(x_i - m_0)$ as above.

For two-tailed test with level of significance $\alpha$. The critical values are: qbinom($\alpha/2$, $n_d$, 0.5) - 1, and qbinom($1-\alpha/2$, $n_d$, 0.5) + 1.

For one-tailed test with level of significance $\alpha$. The critical value is: for left-tailed, qbinom($\alpha$, $n_d$, 0.5) - 1; and for right-tailed, qbinom($1-\alpha$, $n_d$, 0.5) + 1.

Example:

For $\alpha = 0.05$, $n_d = 65$.

Two-tailed:

qbinom(0.025, 65, 0.5) - 1; qbinom(0.975, 65, 0.5) + 1

[1] 24

[1] 41

One-tailed:

#Left
qbinom(0.05, 65, 0.5) - 1

[1] 25

#Right
qbinom(0.95, 65, 0.5) + 1

[1] 40

4 Two-tailed Sign Test for One Sample in R

Using the Indometh$conc data from the "datasets" package with 10 sample observations from 66 observations below:

Indometh$conc

 [1] 1.50 0.07 0.36 2.72 0.22 0.59 2.05 0.39 0.84 0.09

For the following null hypothesis $H_0$, and alternative hypothesis $H_1$, with the level of significance $\alpha=0.1$.

$H_0:$ the population median is equal to 0.15 ($m = 0.15$).

$H_1:$ the population median is not equal to 0.15 ($m \neq 0.15$, hence the default two-sided).

Because the level of significance is $\alpha=0.1$, the level of confidence is $1 - \alpha = 0.9$.

SIGN.test(Indometh$conc, alternative = "two.sided",
          md = 0.15, conf.level = 0.9)


    One-sample Sign-Test

data:  Indometh$conc
s = 43, p-value = 0.01866
alternative hypothesis: true median is not equal to 0.15
90 percent confidence interval:
 0.1980247 0.5017279
sample estimates:
median of x 
       0.34 

Achieved and Interpolated Confidence Intervals: 

                  Conf.Level L.E.pt U.E.pt
Lower Achieved CI     0.8911  0.200 0.4800
Interpolated CI       0.9000  0.198 0.5017
Upper Achieved CI     0.9360  0.190 0.5900

Interpretation:

P-value: With the p-value ($p = 0.01866$) being less than the level of significance 0.1, we reject the null hypothesis that the population median is equal to 0.15.
$S$ T-statistic: With $n_d = 66$, and test statistics value ($S = 43$) being greater than or equal to $\text{qbinom(0.95, 66, 0.5)+1}=41$, or being within $41 \text{ to } 66$, we reject the null hypothesis that the population median is equal to 0.15.
Confidence Interval: With the null hypothesis median value ($m = 0.15$) being outside the confidence interval, $[0.1980247, 0.5017279]$, we reject the null hypothesis that the population median is equal to 0.15.

5 One-tailed Sign Test for One Sample in R

Right Tailed Test

Using the Indometh$conc data from the "datasets" package with 10 sample observations from 66 observations below:

Indometh$conc

 [1] 1.50 0.48 0.70 0.64 0.08 0.07 0.13 0.11 0.06 0.09

For the following null hypothesis $H_0$, and alternative hypothesis $H_1$, with the level of significance $\alpha=0.1$.

$H_0:$ the population median is equal to 0.2 ($m = 0.2$).

$H_1:$ the population median is greater than 0.2 ($m > 0.2$, hence one-sided).

Because the level of significance is $\alpha=0.1$, the level of confidence is $1 - \alpha = 0.9$.

SIGN.test(Indometh$conc, alternative = "greater",
          md = 0.2, conf.level = 0.9)


    One-sample Sign-Test

data:  Indometh$conc
s = 39, p-value = 0.06802
alternative hypothesis: true median is greater than 0.2
90 percent confidence interval:
 0.2226711       Inf
sample estimates:
median of x 
       0.34 

Achieved and Interpolated Confidence Intervals: 

                  Conf.Level L.E.pt U.E.pt
Lower Achieved CI     0.8661 0.2300    Inf
Interpolated CI       0.9000 0.2227    Inf
Upper Achieved CI     0.9124 0.2200    Inf

Interpretation:

P-value: With the p-value ($p = 0.06802$) being less than the level of significance 0.1, we reject the null hypothesis that the population median is equal to 0.2.
$S$ T-statistic: $n_d = 66 - 1 = 65$ because one observation is $0.2$. With $n_d = 65$, and test statistics value ($S = 39$) being greater than or equal to $\text{qbinom(0.9, 65, 0.5)+1}=39$, or being within $39 \text{ to } 65$, we reject the null hypothesis that the population median is equal to 0.2.
Confidence Interval: With the null hypothesis median value ($m = 0.2$) being outside the confidence interval, $[0.2226711, \infty)$, we reject the null hypothesis that the population median is equal to 0.2.

Left Tailed Test

For the following null hypothesis $H_0$, and alternative hypothesis $H_1$, with the level of significance $\alpha=0.05$.

$H_0:$ the population median is equal to 0.4 ($m = 0.4$).

$H_1:$ the population median is less than 0.4 ($m < 0.4$, hence one-sided).

Because the level of significance is $\alpha=0.05$, the level of confidence is $1 - \alpha = 0.95$.

SIGN.test(Indometh$conc, alternative = "less",
          md = 0.4, conf.level = 0.95)


    One-sample Sign-Test

data:  Indometh$conc
s = 28, p-value = 0.1605
alternative hypothesis: true median is less than 0.4
95 percent confidence interval:
      -Inf 0.5017279
sample estimates:
median of x 
       0.34 

Achieved and Interpolated Confidence Intervals: 

                  Conf.Level L.E.pt U.E.pt
Lower Achieved CI     0.9456   -Inf 0.4800
Interpolated CI       0.9500   -Inf 0.5017
Upper Achieved CI     0.9680   -Inf 0.5900

Interpretation:

P-value: With the p-value ($p = 0.1605$) being greater than the level of significance 0.05, we fail to reject the null hypothesis that the population median is equal to 0.4.
$S$ T-statistic: $n_d = 66 - 1 = 65$ because one observation is $0.4$. With $n_d = 65$, and test statistics value ($S = 28$) being greater than $\text{qbinom(0.05, 65, 0.5)-1}=25$, or not being within $0 \text{ to } 25$, we fail to reject the null hypothesis that the population median is equal to 0.4.
Confidence Interval: With the null hypothesis median value ($m = 0.4$) being inside the confidence interval, $(-\infty, 0.5017279]$, we fail to reject the null hypothesis that the population median is equal to 0.4.

6 Sign Test for One Sample: Test Statistics & P-values in R

Here for a sign test for one sample, we show how to get the test statistics (or S-value), and p-values from the SIGN.test() function in R, or by written code.

data_os = Indometh$conc
sgnt_object = SIGN.test(data_os, alternative = "two.sided",
                        md = 0.3, conf.level = 0.9)
sgnt_object


    One-sample Sign-Test

data:  data_os
s = 34, p-value = 0.8043
alternative hypothesis: true median is not equal to 0.3
90 percent confidence interval:
 0.1980247 0.5017279
sample estimates:
median of x 
       0.34 

Achieved and Interpolated Confidence Intervals: 

                  Conf.Level L.E.pt U.E.pt
Lower Achieved CI     0.8911  0.200 0.4800
Interpolated CI       0.9000  0.198 0.5017
Upper Achieved CI     0.9360  0.190 0.5900

To get the test statistic or S-value:

\[S = \sum_{i=1}^n I_{(x_i-m_0)>0},\]

which is the number of observations greater than the null hypothesis median.

sgnt_object$statistic

 s 
34

# to remove name S
unname(sgnt_object$statistic)

[1] 34

Same as:

md = 0.3; diffs = data_os - md
sum(diffs>0)

[1] 34

To get the p-value:

with $n_d$ as the number of non-zero $(x_i - m_0)$.

For $X \sim Binomial(n_d, 0.5)$

Two-tailed:

For $S = n_d/2$, $Pvalue = 1$.

For $S > n_d/2$, $Pvalue = 2 * P(X \geq S) = 2 * \sum_{x=S}^{n_d} P(X=x)$.

For $S < n_d/2$, $Pvalue = 2 * P(X \leq S) = 2 * \sum_{x=0}^{S} P(X=x)$.

One-tailed:

For right-tail, $Pvalue = P(X \geq S) = \sum_{x=S}^{n_d} P(X=x)$ or for left-tail, $Pvalue = P(X \leq S) = \sum_{x=0}^{S} P(X=x)$.

sgnt_object$p.value

[1] 0.804317

Same as:

md = 0.3; diffs = data_os - md
nd = sum(diffs!=0); nd

[1] 65

Note that the p-value depends on the $\text{test statistics}$ ($S = 34 > n_d/2 = 32.5$). We also use the distribution function pbinom() for the binomial distribution in R.

2*sum(dbinom(34:65, 65, 0.5))

[1] 0.804317

2*(1-pbinom(33, 65, 0.5))

[1] 0.804317

One-tailed example:

# Right tailed
1-pbinom(33, 65, 0.5)
# Left tailed
pbinom(34, 65, 0.5)

Question	Is the median equal to \(m_0\)?	Is the median greater than \(m_0\)?	Is the median less than \(m_0\)?
Form of Test	Two-tailed	Right-tailed test	Left-tailed test
Null Hypothesis, \(H_0\)	\(m = m_0\)	\(m = m_0\)	\(m = m_0\)
Alternate Hypothesis, \(H_1\)	\(m \neq m_0\)	\(m > m_0\)	\(m < m_0\)

Sign Tests for One Sample in R

Install and Load "BSDA" Package:

Sample Steps to Run a Sign Test for One Sample:

Creating a Sign Test for One Sample Object:

1 Test Statistic for Sign Test for One Sample in R

2 Simple Sign Test for One Sample in R

Interpretation:

3 Sign Test for One Sample: Critical Value in R

4 Two-tailed Sign Test for One Sample in R

Interpretation:

5 One-tailed Sign Test for One Sample in R

Right Tailed Test

Interpretation:

Left Tailed Test

Interpretation:

6 Sign Test for One Sample: Test Statistics & P-values in R

To get the test statistic or S-value:

To get the p-value: