Goodness of fit test  overview
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Goodness of fit test  McNemar's test  Two sample $z$ test 


Independent variable  Independent variable  Independent/grouping variable  
None  2 paired groups  One categorical with 2 independent groups  
Dependent variable  Dependent variable  Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  One categorical with 2 independent groups  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  Null hypothesis  
 Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
 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  
 The alternative hypothesis H_{1} is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
 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 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.  $X^2 = \dfrac{(b  c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0.  $z = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ 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 $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1  \bar{y}_2$ 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 $\mu_2 > \mu_1$.  
Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  If $b + c$ is large enough (say, > 20), approximately the chisquared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$.  Standard normal distribution  
Significant?  Significant?  Significant?  
 For test statistic $X^2$:
 Two sided:
 
n.a.  n.a.  $C\%$ confidence interval for $\mu_1  \mu_2$  
    $(\bar{y}_1  \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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 $\mu_1  \mu_2$ can also be used as significance test.  
n.a.  n.a.  Visual representation  
    
n.a.  Equivalent to  n.a.  
 
   
Example context  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 $\pi_{high} = 0.2$?  Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?  Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women.  
SPSS  SPSS  n.a.  
Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
   
Jamovi  Jamovi  n.a.  
Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 Frequencies > Paired Samples  McNemar test
   
Practice questions  Practice questions  Practice questions  