ANCOVA overview

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ANCOVA	Kruskal-Wallis test	Cochran's Q test
Independent variables	Independent/grouping variable	Independent/grouping 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 ($I \geqslant 2$)	One within subject factor ($\geq 2$ related groups)
Dependent variable	Dependent variable	Dependent variable
One quantitative of interval or ratio level	One of ordinal level	One categorical with 2 independent groups
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: H₀: the population medians for the $I$ groups are equal Else: Formulation 1: H₀: the population scores in any of the $I$ groups are not systematically higher or lower than the population scores in any of the other groups Formulation 2: H₀: P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.	H₀: $\pi_1 = \pi_2 = \ldots = \pi_I$ Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$
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: H₁: not all of the population medians for the $I$ groups are equal Else: Formulation 1: H₁: the poplation scores in some groups are systematically higher or lower than the population scores in other groups Formulation 2: H₁: for at least one pair of groups: P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)	H₁: not all population proportions are equal
n.a.	Assumptions	Assumptions
-	Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another	Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
n.a.	Test statistic	Test statistic
-	$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ Here $N$ is the total sample size, $R_i$ is the sum of ranks in group $i$, and $n_i$ is the sample size of group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N (N + 1)} \times \sum \frac{R^2_i}{n_i}$ and then subtract $3(N + 1)$. Note: if ties are present in the data, the formula for $H$ is more complicated.	If a failure is scored as 0 and a success is scored as 1: $Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups.
n.a.	Sampling distribution of $H$ if H₀ were true	Sampling distribution of $Q$ 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.	If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom
n.a.	Significant?	Significant?
-	For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$: 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 the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$: 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 $\alpha$
n.a.	n.a.	Equivalent to
-	-	Friedman test, with a categorical dependent variable consisting of two independent groups.
n.a.	Example context	Example context
-	Do people from different religions tend to score differently on social economic status?	Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks?
n.a.	SPSS	SPSS
-	Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples... Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum Continue and click OK	Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples... Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables Under Test Type, select Cochran's Q test
n.a.	Jamovi	Jamovi
-	ANOVA > One Way ANOVA - Kruskal-Wallis Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable	Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to: ANOVA > Repeated Measures ANOVA - Friedman Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures
Practice questions	Practice questions	Practice questions

ANCOVA - overview