ANCOVA - overview
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ANCOVA | Spearman's rho | Mann-Whitney-Wilcoxon test |
You cannot compare more than 3 methods |
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Independent variables | Variable 1 | Independent/grouping variable | |
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates) | One of ordinal level | One categorical with 2 independent groups | |
Dependent variable | Variable 2 | Dependent variable | |
One quantitative of interval or ratio level | One of ordinal level | One of ordinal level | |
THIS TABLE IS YET TO BE COMPLETED | 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 both populations:
Formulation 1:
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n.a. | 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 both populations:
Formulation 1:
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n.a. | Assumptions | Assumptions | |
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n.a. | Test statistic | Test statistic | |
- | $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. | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | |
n.a. | Sampling distribution of $t$ if H0 were true | Sampling distribution of $W$ and of $U$ if H0 were true | |
- | Approximately the $t$ distribution with $N - 2$ degrees of freedom | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | |
n.a. | Significant? | Significant? | |
- | Two sided:
| For large samples, the table for standard normal probabilities can be used: Two sided:
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n.a. | n.a. | Equivalent to | |
- | - | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | |
n.a. | Example context | Example context | |
- | Is there a monotonic relationship between physical health and mental health? | Do men tend to score higher on social economic status than women? | |
n.a. | SPSS | SPSS | |
- | Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
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n.a. | Jamovi | Jamovi | |
- | Regression > Correlation Matrix
| T-Tests > Independent Samples T-Test
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Practice questions | Practice questions | Practice questions | |