# Marginal Homogeneity test / Stuart-Maxwell test - overview

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Marginal Homogeneity test / Stuart-Maxwell test | Marginal Homogeneity test / Stuart-Maxwell test | Paired sample $t$ test |
You cannot compare more than 3 methods |
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Independent variable | Independent variable | Independent variable | |

2 paired groups | 2 paired groups | 2 paired groups | |

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 of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair. | |

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
| - Difference scores are normally distributed in the population
- Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores 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. | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ 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 $t$ 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 | $t$ distribution with $N - 1$ degrees of freedom | |

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 $t$ observed in sample is at least as extreme as critical value $t^*$ or
- Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
- Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
- Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
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n.a. | n.a. | $C\%$ confidence interval for $\mu$ | |

- | - | $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N-1}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). 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 of the difference scores and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$ | |

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

- | - | ||

n.a. | n.a. | Equivalent to | |

- | - | - One sample $t$ test on the difference scores.
- Repeated measures ANOVA with one dichotomous within subjects factor.
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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 difference between the mental health scores before and after an intervention different from $\mu_0 = 0$? | |

SPSS | SPSS | SPSS | |

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
| Analyze > Compare Means > Paired-Samples T Test...
- Put the two paired variables in the boxes below Variable 1 and Variable 2
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n.a. | n.a. | Jamovi | |

- | - | T-Tests > Paired Samples T-Test
- Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line
- Under Hypothesis, select your alternative hypothesis
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Practice questions | Practice questions | Practice questions | |