Goodness of fit test  overview
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Goodness of fit test  Chisquared test for the relationship between two categorical variables 


Independent variable  Independent /column variable  
None  One categorical with $I$ independent groups ($I \geqslant 2$)  
Dependent variable  Dependent /row variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  One categorical with $J$ independent groups ($J \geqslant 2$)  
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:
 
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:
 
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells.  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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.  
Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  
Significant?  Significant?  

 
Example context  Example context  
Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$?  Is there an association between economic class and gender? Is the distribution of economic class different between men and women?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 Analyze > Descriptive Statistics > Crosstabs...
 
Jamovi  Jamovi  
Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 Frequencies > Independent Samples  $\chi^2$ test of association
 
Practice questions  Practice questions  