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 | Two sample $t$ test - equal variances assumed | Ordinal logistic regression |
You cannot compare more than 3 methods |
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Independent variables | Independent/grouping variable | Independent variables | |
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates) | One categorical with 2 independent groups | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | |
Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One quantitative of interval or ratio level | One of ordinal level | |
THIS TABLE IS YET TO BE COMPLETED | Null hypothesis | THIS TABLE IS YET TO BE COMPLETED | |
- | H0: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | - | |
n.a. | Alternative hypothesis | n.a. | |
- | H1 two sided: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | - | |
n.a. | Assumptions | n.a. | |
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n.a. | Test statistic | n.a. | |
- | $t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$. | - | |
n.a. | Pooled standard deviation | n.a. | |
- | $s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$ | - | |
n.a. | Sampling distribution of $t$ if H0 were true | n.a. | |
- | $t$ distribution with $n_1 + n_2 - 2$ degrees of freedom | - | |
n.a. | Significant? | n.a. | |
- | Two sided:
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n.a. | $C\%$ confidence interval for $\mu_1 - \mu_2$ | n.a. | |
- | $(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2 - 2}$ 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 $\mu_1 - \mu_2$ can also be used as significance test. | - | |
n.a. | Effect size | n.a. | |
- | Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other. | - | |
n.a. | Visual representation | n.a. | |
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n.a. | Equivalent to | n.a. | |
- | One way ANOVA with an independent variable with 2 levels ($I$ = 2):
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n.a. | Example context | n.a. | |
- | Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women. | - | |
n.a. | SPSS | n.a. | |
- | Analyze > Compare Means > Independent-Samples T Test...
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n.a. | Jamovi | n.a. | |
- | T-Tests > Independent Samples T-Test
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Practice questions | Practice questions | Practice questions | |