ANCOVA - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
ANCOVA | Kruskal-Wallis test | One sample Wilcoxon signed-rank test |
You cannot compare more than 3 methods |
---|---|---|---|
Independent variables | Independent/grouping variable | Independent variable | |
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates) | One categorical with $I$ independent groups ($I \geqslant 2$) | None | |
Dependent variable | Dependent variable | 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 | |
- | 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:
| H0: $m = m_0$
Here $m$ is the population median, and $m_0$ is the population median according to the null hypothesis. | |
n.a. | Alternative hypothesis | Alternative hypothesis | |
- | 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:
| H1 two sided: $m \neq m_0$ H1 right sided: $m > m_0$ H1 left sided: $m < m_0$ | |
n.a. | Assumptions | Assumptions | |
- |
|
| |
n.a. | Test statistic | Test statistic | |
- | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | Two different types of test statistics can be used, but both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic.
In order to compute each of the test statistics, follow the steps below:
| |
n.a. | Sampling distribution of $H$ if H0 were true | Sampling distribution of $W_1$ and of $W_2$ if H0 were true | |
- | 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. | Sampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1 - \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately the standard normal distribution if the null hypothesis were true. Sampling distribution of $W_2$: If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately the standard normal distribution if the null hypothesis were true. If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated. | |
n.a. | Significant? | Significant? | |
- | For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
| For large samples, the table for standard normal probabilities can be used: Two sided:
| |
n.a. | Example context | Example context | |
- | Do people from different religions tend to score differently on social economic status? | Is the median mental health score of office workers different from $m_0 = 50$? | |
n.a. | SPSS | SPSS | |
- | Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| Specify the measurement level of your variable on the Variable View tab, in the column named Measure. Then go to:
Analyze > Nonparametric Tests > One Sample...
| |
n.a. | Jamovi | Jamovi | |
- | ANOVA > One Way ANOVA - Kruskal-Wallis
| T-Tests > One Sample T-Test
| |
Practice questions | Practice questions | Practice questions | |