Chisquared test for the relationship between two categorical variables  overview
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Chisquared test for the relationship between two categorical variables  $z$ test for a single proportion 


Independent /column variable  Independent variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  None  
Dependent /row variable  Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  One categorical with 2 independent groups  
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}: $\pi = \pi_0$
$\pi$ is the population proportion of 'successes'; $\pi_0$ is the population proportion of successes according to the null 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} two sided: $\pi \neq \pi_0$ H_{1} right sided: $\pi > \pi_0$ H_{1} left sided: $\pi < \pi_0$  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where 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  $z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
$p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis.  
Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  
Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  Approximately the standard normal distribution  
Significant?  Significant?  
 Two sided:
 
n.a.  Approximate $C\%$ confidence interval for $\pi$  
  Regular (large sample):
 
n.a.  Equivalent to  
 
 
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 the proportion of smokers amongst office workers different from $\pi_0 = .2$? Use the normal approximation for the sampling distribution of the test statistic.  
SPSS  SPSS  
Analyze > Descriptive Statistics > Crosstabs...
 Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 
Jamovi  Jamovi  
Frequencies > Independent Samples  $\chi^2$ test of association
 Frequencies > 2 Outcomes  Binomial test
 
Practice questions  Practice questions  