ANCOVA: overview
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ANCOVA  One sample $t$ test for the mean 


Independent variables  Independent variable  
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)  None  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  
THIS TABLE IS YET TO BE COMPLETED  Null hypothesis  
  $\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to H0  
n.a.  Alternative hypothesis  
  Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  
n.a.  Assumptions  
 
 
n.a.  Test statistic  
  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $s$ is the sample standard deviation, $N$ is the sample size. 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 $\mu_0$  
n.a.  Sampling distribution of $t$ if H0 were true  
  $t$ Distribution with $N  1$ degrees of freedom  
n.a.  Significant?  
  Two sided:
 
n.a.  $C\%$ confidence interval for $\mu$  
  $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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  
  Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0$  
n.a.  Visual representation  
  
n.a.  Example context  
  Is the average mental health score of office workers different from $\mu_0$ = 50?  
Pratice questions  Pratice questions  