The one-sample sign test compares the number of observations greater than or less than the default value without accounting for the magnitude of the difference between each observation and the default value. The test is similar in purpose to the one-sample Wilcoxon signed-rank test, but looks specifically at the median value, and is not affected by the distribution of the data.

The test is conducted with functions in the *DescTools*
package, the *nonpar *package, or the *BSDA* package. These
functions produce a *p*-value for the hypothesis, as well as the median
and confidence interval of the median for the data.

##### Appropriate data

• One-sample data

• Data are ordinal, interval, or ratio

##### Hypotheses

• Null hypothesis: The median of the population from which the sample was drawn is equal to the default value.

• Alternative hypothesis (two-sided): The median of the population from which the sample was drawn is not equal to the default value.

##### Interpretation

Reporting significant results as e.g. “Likert scores were significantly different from a default value of 3” is acceptable. As is e.g. “Median Likert scores were significantly different from a default value of 3”

### Packages used in this chapter

The packages used in this chapter include:

• BSDA

• DescTools

• rcompanion

• nonpar

The following commands will install these packages if they are not already installed:

if(!require(BSDA)){install.packages("BSDA")}

if(!require(DescTools)){install.packages("DescTools")}

if(!require(rcompanion)){install.packages("rcompanion")}

if(!require(nonpar)){install.packages("nonpar")}

### One-sample sign test example

For appropriate plots and summary statistics, see the *One-sample
Wilcoxon Signed-rank Test* chapter.

Data = read.table(header=TRUE, stringsAsFactors=TRUE, text="

Speaker Rater Likert

'Maggie Simpson' 1 3

'Maggie Simpson' 2 4

'Maggie Simpson' 3 5

'Maggie Simpson' 4 4

'Maggie Simpson' 5 4

'Maggie Simpson' 6 4

'Maggie Simpson' 7 4

'Maggie Simpson' 8 3

'Maggie Simpson' 9 2

'Maggie Simpson' 10 5

")

### Check the data frame

library(psych)

headTail(Data)

str(Data)

summary(Data)

#### Sign test with the DescTools package

Note that *Data$Likert* is the one-sample data, and *mu=3*
indicates the default value to compare to.

library(DescTools)

SignTest(Data$Likert,

mu = 3)

One-sample Sign-Test

S = 7, number of differences = 8, p-value = 0.07031

### Note the p-value in the output above

alternative hypothesis: true median is not equal to 3

97.9 percent confidence interval:

3 5

sample estimates:

median of the differences

4

### Median value and confidence interval

#### Sign test with the nonpar package

Note that *Data$Likert* is the one-sample data, and *m=3*
indicates the default value to compare to. At the time of writing, it appears
that the *exact=FALSE* option actually produces the exact test.

library(nonpar)

signtest(Data$Likert, m=3, conf.level=0.95, exact=FALSE)

Exact Sign Test

The p-value is 0.07032

The 95 % confidence interval is [ 2 , 4 ].

#### Sign test with the BSDA package

Note that *Data$Likert* is the one-sample data, and *md=3*
indicates the default value to compare to.

library(BSDA)

SIGN.test(Data$Likert,

md = 3)

One-sample Sign-Test

s = 7, p-value = 0.07031

alternative hypothesis: true median is not equal to 3

### Note the p-value in the output above

95 percent confidence interval:

3.000000 4.675556

sample estimates:

median of x

4

### Median value and confidence interval

#### Effect size statistics

One way to assess the effect size after a one-sample sign test is to use a dominance statistic. This statistic simply looks at the proportion of observations greater than the default median value minus the proportion of observations less than the default median value. A value of 1 would indicate that all observations are greater than the default median, and a value of –1 would indicate that all observations are less than the default median. A value of 0 indicates that the number of observations greater than the default median are equal to the number that are less than the default median.

A VDA-like statistic can be calculated as *Dominance / 2 +
0.5*. This statistic varies from 0 to 1, with 0.5 being equivalent to a
dominance value of 0.

Note that neither of these statistics take into account values tied to the default median value.

library(rcompanion)

oneSampleDominance(Data$Likert, mu=3)

n Median mu Less Equal Greater Dominance VDA

1 10 4 3 0.1 0.2 0.7 0.6 0.8

oneSampleDominance(Data$Likert, mu=3, ci=TRUE)

n Median mu Less Equal
Greater Dominance lower.ci upper.ci VDA lower.vda.ci upper.vda.ci

1 10 4 3 0.1 0.2 0.7 0.6 0.2 0.9 0.8
0.6 0.95

#### Manual calculations

Likert = c(3, 4, 5, 4, 4, 4, 4, 3, 2, 5)

Greater = sum(Likert > 3)

NotMedian = sum(Likert != 3)

binom.test(Greater, NotMedian)

Exact binomial test

number of successes = 7, number of trials = 8, p-value = 0.07031

MU = 3

N = length(Likert)

GreaterProp = sum(Likert > MU) / N

GreaterProp

0.7

LesserProp = sum(Likert < MU) / N

LesserProp

0.1

EqualProp = sum(Likert == MU) / N

EqualProp

0.2