Chisquared test for the relationship between two categorical variables  overview
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Chisquared test for the relationship between two categorical variables  Friedman test  Two sample $t$ test  equal variances assumed 
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Independent /column variable  Independent/grouping variable  Independent/grouping variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  One within subject factor ($\geq 2$ related groups)  One categorical with 2 independent groups  
Dependent /row variable  Dependent variable  Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  One of ordinal level  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  Null hypothesis  
H_{0}: 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:
 H_{0}: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups
Usually the related groups are the different measurement points. 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.  H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
H_{1}: 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:
 H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups  H_{1} two sided: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  
Assumptions  Assumptions  Assumptions  


 
Test statistic  Test statistic  Test statistic  
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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.  $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
Here $N$ is the number of 'blocks' (usually the subjects  so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated.  $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
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.  n.a.  Pooled standard deviation  
    $s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$  
Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  
Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used.  $t$ distribution with $n_1 + n_2  2$ degrees of freedom  
Significant?  Significant?  Significant?  
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 Two sided:
 
n.a.  n.a.  $C\%$ confidence interval for $\mu_1  \mu_2$  
    $(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
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.  n.a.  Effect size  
    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.  n.a.  Visual representation  
    
n.a.  n.a.  Equivalent to  
    One way ANOVA with an independent variable with 2 levels ($I$ = 2):
 
Example context  Example context  Example context  
Is there an association between economic class and gender? Is the distribution of economic class different between men and women?  Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?  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.  
SPSS  SPSS  SPSS  
Analyze > Descriptive Statistics > Crosstabs...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Compare Means > IndependentSamples T Test...
 
Jamovi  Jamovi  Jamovi  
Frequencies > Independent Samples  $\chi^2$ test of association
 ANOVA > Repeated Measures ANOVA  Friedman
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  Practice questions  