Goodness of fit test - overview
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Goodness of fit test | Spearman's rho | Kruskal-Wallis test |
You cannot compare more than 3 methods |
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Independent variable | Variable 1 | Independent/grouping variable | |
None | One of ordinal level | One categorical with $I$ independent groups ($I \geqslant 2$) | |
Dependent variable | Variable 2 | Dependent variable | |
One categorical with $J$ independent groups ($J \geqslant 2$) | One of ordinal level | One of ordinal level | |
Null hypothesis | Null hypothesis | Null hypothesis | |
| H0: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H0: there is no monotonic relationship between the two variables in the population. | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
| H1 two sided: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
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Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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. | $t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | |
Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $H$ if H0 were true | |
Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Approximately the $t$ distribution with $N - 2$ degrees of freedom | For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used. | |
Significant? | Significant? | Significant? | |
| Two sided:
| For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
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Example context | Example context | Example context | |
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 $\pi_{high} = 0.2$? | Is there a monotonic relationship between physical health and mental health? | Do people from different religions tend to score differently on social economic status? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
| Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
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Jamovi | Jamovi | Jamovi | |
Frequencies > N Outcomes - $\chi^2$ Goodness of fit
| Regression > Correlation Matrix
| ANOVA > One Way ANOVA - Kruskal-Wallis
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Practice questions | Practice questions | Practice questions | |