Kruskal-Wallis test - overview

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Kruskal-Wallis test
Chi-squared test for the relationship between two categorical variables
One sample $z$ test for the mean
You cannot compare more than 3 methods
Independent/grouping variableIndependent /column variableIndependent variable
One categorical with $I$ independent groups ($I \geqslant 2$)One categorical with $I$ independent groups ($I \geqslant 2$)None
Dependent variableDependent /row variableDependent variable
One of ordinal levelOne categorical with $J$ independent groups ($J \geqslant 2$)One quantitative of interval or ratio level
Null hypothesisNull hypothesisNull 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: there is no association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
  • H0: the distribution of the dependent variable is the same in each of the $I$ populations
If there is one random sample of size $N$ from the total population:
  • H0: the row and column variables are independent
H0: $\mu = \mu_0$

Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.
Alternative hypothesisAlternative hypothesisAlternative 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: there is an association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
  • H1: the distribution of the dependent variable is not the same in all of the $I$ populations
If there is one random sample of size $N$ from the total population:
  • H1: the row and column variables are dependent
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
AssumptionsAssumptionsAssumptions
  • 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 size is large enough for $X^2$ to be approximately chi-squared distributed under the null hypothesis. Rule of thumb:
    • 2 $\times$ 2 table: all four expected cell counts are 5 or more
    • Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more
  • There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population
  • Scores are normally distributed in the population
  • Population standard deviation $\sigma$ is known
  • Sample is a simple random sample from the population. That is, observations are independent of one another
Test statisticTest statisticTest 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.
$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size.

The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$.
Sampling distribution of $H$ if H0 were trueSampling distribution of $X^2$ if H0 were trueSampling distribution of $z$ 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.

Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedomStandard normal distribution
Significant?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 $\alpha$
  • 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 $\alpha$
Two sided: Right sided: Left sided:
n.a.n.a.$C\%$ confidence interval for $\mu$
--$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval).

The confidence interval for $\mu$ can also be used as significance test.
n.a.n.a.Effect size
--Cohen's $d$:
Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$
n.a.n.a.Visual representation
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One sample z test
Example contextExample contextExample context
Do people from different religions tend to score differently on social economic status? Is there an association between economic class and gender? Is the distribution of economic class different between men and women?Is the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$
SPSSSPSSn.a.
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 > Descriptive Statistics > Crosstabs...
  • Put one of your two categorical variables in the box below Row(s), and the other categorical variable in the box below Column(s)
  • Click the Statistics... button, and click on the square in front of Chi-square
  • Continue and click OK
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JamoviJamovin.a.
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
Frequencies > Independent Samples - $\chi^2$ test of association
  • Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns
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Practice questionsPractice questionsPractice questions