Goodness of fit test  overview
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Goodness of fit test  Two sample $t$ test  equal variances assumed  KruskalWallis test 


Independent variable  Independent/grouping variable  Independent/grouping variable  
None  One categorical with 2 independent groups  One categorical with $I$ independent groups ($I \geqslant 2$)  
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}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
 H_{1} two sided: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ 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.  $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
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 $\mu_2 > \mu_1$.  $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  
n.a.  Pooled standard deviation  n.a.  
  $s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$    
Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $H$ if H_{0} were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  $t$ distribution with $n_1 + n_2  2$ degrees of freedom  For large samples, approximately the chisquared distribution with $I  1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.  
Significant?  Significant?  Significant?  
 Two sided:
 For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 
n.a.  $C\%$ confidence interval for $\mu_1  \mu_2$  n.a.  
  $(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
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.  Effect size  n.a.  
  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.  Visual representation  n.a.  
    
n.a.  Equivalent to  n.a.  
  One way ANOVA 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 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.  Do people from different religions tend to score differently on social economic status?  
SPSS  SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 Analyze > Compare Means > IndependentSamples T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 
Jamovi  Jamovi  Jamovi  
Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 TTests > Independent Samples TTest
 ANOVA > One Way ANOVA  KruskalWallis
 
Practice questions  Practice questions  Practice questions  