Sign test - overview
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Sign test | Pearson correlation | $z$ test for the difference between two proportions | Goodness of fit test | Cochran's Q test | Chi-squared test for the relationship between two categorical variables |
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Independent variable | Variable 1 | Independent/grouping variable | Independent variable | Independent/grouping variable | Independent /column variable | |
2 paired groups | One quantitative of interval or ratio level | One categorical with 2 independent groups | None | One within subject factor ($\geq 2$ related groups) | One categorical with $I$ independent groups ($I \geqslant 2$) | |
Dependent variable | Variable 2 | Dependent variable | Dependent variable | Dependent variable | Dependent /row variable | |
One of ordinal level | One quantitative of interval or ratio level | One categorical with 2 independent groups | One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with 2 independent groups | One categorical with $J$ independent groups ($J \geqslant 2$) | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
| H0: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level. | H0: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. |
| H0: $\pi_1 = \pi_2 = \ldots = \pi_I$
Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$ | H0: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
| H1 two sided: $\rho \neq \rho_0$ H1 right sided: $\rho > \rho_0$ H1 left sided: $\rho < \rho_0$ | H1 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ |
| H1: not all population proportions are equal | H1: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
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Assumptions | Assumptions of test for correlation | Assumptions | Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | |
$W = $ number of difference scores that is larger than 0 | Test statistic for testing H0: $\rho = 0$:
| $z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$ | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells. | If a failure is scored as 0 and a success is scored as 1:
$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells. | |
Sampling distribution of $W$ if H0 were true | Sampling distribution of $t$ and of $z$ if H0 were true | Sampling distribution of $z$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $Q$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | |
The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true. | Sampling distribution of $t$:
| Approximately the standard normal distribution | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | |
Significant? | Significant? | Significant? | Significant? | Significant? | Significant? | |
If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
| $t$ Test two sided:
| Two sided:
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| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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n.a. | Approximate $C$% confidence interval for $\rho$ | Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$ | n.a. | n.a. | n.a. | |
- | First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
| Regular (large sample):
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n.a. | Properties of the Pearson correlation coefficient | n.a. | n.a. | n.a. | n.a. | |
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Equivalent to | Equivalent to | Equivalent to | n.a. | Equivalent to | n.a. | |
Two sided sign test is equivalent to
| OLS regression with one independent variable:
| When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels. | - | Friedman test, with a categorical dependent variable consisting of two independent groups. | - | |
Example context | Example context | Example context | Example context | Example context | Example context | |
Do people tend to score higher on mental health after a mindfulness course? | Is there a linear relationship between physical health and mental health? | Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$? | Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks? | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | |
SPSS | SPSS | SPSS | SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Correlate > Bivariate...
| SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Descriptive Statistics > Crosstabs...
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Jamovi | Jamovi | Jamovi | Jamovi | Jamovi | Jamovi | |
Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
| Regression > Correlation Matrix
| Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples - $\chi^2$ test of association
| Frequencies > N Outcomes - $\chi^2$ Goodness of fit
| Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
| Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |