# Marginal Homogeneity test / Stuart-Maxwell test - overview

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Marginal Homogeneity test / Stuart-Maxwell test | Marginal Homogeneity test / Stuart-Maxwell test | One sample $z$ test for the mean |
You cannot compare more than 3 methods |
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Independent variable | Independent variable | Independent variable | |

2 paired groups | 2 paired groups | None | |

Dependent variable | Dependent variable | Dependent variable | |

One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with $J$ independent groups ($J \geqslant 2$) | One quantitative of interval or ratio level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

H_{0}: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$ | H_{0}: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$ | H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | |

Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. | H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. | H_{1} two sided: $\mu \neq \mu_0$H _{1} right sided: $\mu > \mu_0$H _{1} left sided: $\mu < \mu_0$
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Assumptions | Assumptions | Assumptions | |

- Sample of pairs is a simple random sample from the population of pairs. That is, pairs 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
| - Scores are normally distributed in the population
- Population standard deviation $\sigma$ is known
- Sample is a simple random sample from the population. That is, observations are independent of one another
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Test statistic | Test statistic | Test statistic | |

Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand. | Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand. | $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$. | |

Sampling distribution of the test statistic if H_{0} were true | Sampling distribution of the test statistic if H_{0} were true | Sampling distribution of $z$ if H_{0} were true | |

Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Standard normal distribution | |

Significant? | Significant? | Significant? | |

If we denote the test statistic as $X^2$:
- 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 we denote the test statistic as $X^2$:
- 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 $\alpha$
| Two sided:
- Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
- Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
- Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
- Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
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n.a. | n.a. | $C\%$ confidence interval for $\mu$ | |

- | - | $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). The confidence interval for $\mu$ can also be used as significance test. | |

n.a. | n.a. | Effect size | |

- | - | Cohen's $d$:Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$ | |

n.a. | n.a. | Visual representation | |

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Example context | Example context | Example context | |

Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best? | Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best? | Is the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$ | |

SPSS | SPSS | n.a. | |

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 Marginal Homogeneity 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 Marginal Homogeneity test
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Practice questions | Practice questions | Practice questions | |