Goodness of fit test  overview
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Goodness of fit test  Paired sample $t$ test  MannWhitneyWilcoxon 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 quantitative of interval or ratio level  One of ordinal level  
Null hypothesis  Null hypothesis  Null hypothesis  
 H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
 H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
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.  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.  Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.  
Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $W$ and of $U$ if H_{0} were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  $t$ distribution with $N  1$ degrees of freedom  Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated.  
Significant?  Significant?  Significant?  
 Two sided:
 For large samples, the table for standard normal probabilities can be used: Two sided:
 
n.a.  $C\%$ confidence interval for $\mu$  n.a.  
  $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ 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 $\mu$ can also be used as significance test.    
n.a.  Effect size  n.a.  
  Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$    
n.a.  Visual representation  n.a.  
    
n.a.  Equivalent to  Equivalent to  
 
 If there are no ties in the data, the two sided MannWhitneyWilcoxon test is equivalent to the KruskalWallis test with an independent variable with 2 levels ($I$ = 2).  
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 $\pi_{high} = 0.2$?  Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?  Do men tend to score higher on social economic status than women?  
SPSS  SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 Analyze > Compare Means > PairedSamples T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
 
Jamovi  Jamovi  Jamovi  
Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 TTests > Paired Samples TTest
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  Practice questions  