# Binomial test for a single proportion - overview

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Binomial test for a single proportion
Mann-Whitney-Wilcoxon test
Independent variableIndependent/grouping variable
NoneOne categorical with 2 independent groups
Dependent variableDependent variable
One categorical with 2 independent groupsOne of ordinal level
Null hypothesisNull hypothesis
H0: $\pi = \pi_0$

Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
• H0: the population median for group 1 is equal to the population median for group 2
Else:
Formulation 1:
• H0: the population scores in group 1 are not systematically higher or lower than the population scores in group 2
Formulation 2:
• H0: P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.
Alternative hypothesisAlternative hypothesis
H1 two sided: $\pi \neq \pi_0$
H1 right sided: $\pi > \pi_0$
H1 left sided: $\pi < \pi_0$
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
• H1 two sided: the population median for group 1 is not equal to the population median for group 2
• H1 right sided: the population median for group 1 is larger than the population median for group 2
• H1 left sided: the population median for group 1 is smaller than the population median for group 2
Else:
Formulation 1:
• H1 two sided: the population scores in group 1 are systematically higher or lower than the population scores in group 2
• H1 right sided: the population scores in group 1 are systematically higher than the population scores in group 2
• H1 left sided: the population scores in group 1 are systematically lower than the population scores in group 2
Formulation 2:
• H1 two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1)
• H1 right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1)
• H1 left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 1)
AssumptionsAssumptions
• Sample is a simple random sample from the population. That is, observations are independent of one another
• Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
Test statisticTest statistic
$X$ = number of successes in the sampleTwo different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
The second type of test statistic is the Mann-Whitney $U$ statistic:
• $U = W - \dfrac{n_1(n_1 + 1)}{2}$
where $n_1$ is the sample size of group 1.

Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.
Sampling distribution of $X$ if H0 were trueSampling distribution of $W$ and of $U$ if H0 were true
Binomial($n$, $P$) distribution.

Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).

Sampling distribution of $W$:
For large samples, $W$ is approximately normally distributed with mean $\mu_W$ and standard deviation $\sigma_W$ if the null hypothesis were true. Here \begin{aligned} \mu_W &= \dfrac{n_1(n_1 + n_2 + 1)}{2}\\ \sigma_W &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} Hence, for large samples, the standardized test statistic $$z_W = \dfrac{W - \mu_W}{\sigma_W}\\$$ follows approximately the standard normal distribution if the null hypothesis were true. Note that if your $W$ value is based on group 2, $\mu_W$ becomes $\frac{n_2(n_1 + n_2 + 1)}{2}$.

Sampling distribution of $U$:
For large samples, $U$ is approximately normally distributed with mean $\mu_U$ and standard deviation $\sigma_U$ if the null hypothesis were true. Here \begin{aligned} \mu_U &= \dfrac{n_1 n_2}{2}\\ \sigma_U &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} Hence, for large samples, the standardized test statistic $$z_U = \dfrac{U - \mu_U}{\sigma_U}\\$$ follows approximately the standard normal distribution if the null hypothesis were true.

For small samples, the exact distribution of $W$ or $U$ should be used.

Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated.
Significant?Significant?
Two sided:
• Check if $X$ observed in sample is in the rejection region or
• Find two sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
Right sided:
• Check if $X$ observed in sample is in the rejection region or
• Find right sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
Left sided:
• Check if $X$ observed in sample is in the rejection region or
• Find left sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
For large samples, the table for standard normal probabilities can be used:
Two sided:
Right sided:
Left sided:
n.a.Equivalent to
-If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2).
Example contextExample context
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$?Do men tend to score higher on social economic status than women?
SPSSSPSS
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
• Put your dichotomous variable in the box below Test Variable List
• Fill in the value for $\pi_0$ in the box next to Test Proportion
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
• Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
• Continue and click OK
JamoviJamovi
Frequencies > 2 Outcomes - Binomial test
• Put your dichotomous variable in the white box at the right
• Fill in the value for $\pi_0$ in the box next to Test value
• Under Hypothesis, select your alternative hypothesis
T-Tests > Independent Samples T-Test
• Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Mann-Whitney U
• Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questions