Binomial test for a single proportion overview

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Binomial test for a single proportion	Friedman test	McNemar's test
Independent variable	Independent/grouping variable	Independent variable
None	One within subject factor ($\geq 2$ related groups)	2 paired groups
Dependent variable	Dependent variable	Dependent variable
One categorical with 2 independent groups	One of ordinal level	One categorical with 2 independent groups
Null hypothesis	Null hypothesis	Null 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.	H₀: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups Usually the related groups are the different measurement points. 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.	Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options: First score of pair is 0, second score of pair is 0 First score of pair is 0, second score of pair is 1 (switched) First score of pair is 1, second score of pair is 0 (switched) First score of pair is 1, second score of pair is 1 The null hypothesis H₀ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) = P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the null hypothesis are: H₀: $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones for the first paired group and $\pi_2$ is the population proportion of ones for the second paired group H₀: for each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
H₁ two sided: $\pi \neq \pi_0$ H₁ right sided: $\pi > \pi_0$ H₁ left sided: $\pi < \pi_0$	H₁: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups	The alternative hypothesis H₁ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are: H₁: $\pi_1 \neq \pi_2$ H₁: for each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)
Assumptions	Assumptions	Assumptions
Sample is a simple random sample from the population. That is, observations are independent of one another	Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic	Test statistic	Test statistic
$X$ = number of successes in the sample	$Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$ Here $N$ is the number of 'blocks' (usually the subjects - so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated.	$X^2 = \dfrac{(b - c)^2}{b + c}$ Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0.
Sampling distribution of $X$ if H0 were true	Sampling distribution of $Q$ if H₀ were true	Sampling distribution of $X^2$ 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).	If the number of blocks $N$ is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom. For small samples, the exact distribution of $Q$ should be used.	If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$.
Significant?	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 $\alpha$	If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$	For test statistic $X^2$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$ If $b + c$ is small, the table for the binomial distribution should be used, with as test statistic $b$: Check if $b$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $b$ and check if it is equal to or smaller than $\alpha$
n.a.	n.a.	Equivalent to
-	-	Stuart-Maxwell test, with a categorical dependent variable consisting of two independent groups Cochran's Q test, with two related groups Two sided sign test: $b = W$, and $X^2 = z^2$
Example context	Example context	Example context
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$?	Is there a difference in depression level between measurement point 1 (pre-intervention), measurement point 2 (1 week post-intervention), and measurement point 3 (6 weeks post-intervention)?	Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?
SPSS	SPSS	SPSS
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 > K Related Samples... Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables Under Test Type, select the Friedman test	Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the McNemar test
Jamovi	Jamovi	Jamovi
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	ANOVA > Repeated Measures ANOVA - Friedman Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures	Frequencies > Paired Samples - McNemar test Put one of the two paired variables in the box below Rows and the other paired variable in the box below Columns
Practice questions	Practice questions	Practice questions

Binomial test for a single proportion - overview