ANCOVA: overview
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ANCOVA  Two sample $t$ test  equal variances assumed 


Independent variables  Independent variable  
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)  One categorical with 2 independent groups  
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  
  $\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2  
n.a.  Alternative hypothesis  
  Two sided: $\mu_1 \neq \mu_2$ Right sided: $\mu_1 > \mu_2$ Left sided: $\mu_1 < \mu_2$  
n.a.  Assumptions  
 
 
n.a.  Test statistic  
  $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
$\bar{y}_1$ 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, $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to H0. 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 $\mu_2 > \mu_1$  
n.a.  Pooled standard deviation  
  $s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$  
n.a.  Sampling distribution of $t$ if H0 were true  
  $t$ distribution with $n_1 + n_2  2$ degrees of freedom  
n.a.  Significant?  
  Two sided:
 
n.a.  $C\%$ confidence interval for $\mu_1  \mu_2$  
  $(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
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  
  Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1  \bar{y}_2}{s_p}$$ Indicates how many standard deviations $s_p$ the two sample means are removed from each other  
n.a.  Visual representation  
  
n.a.  Equivalent to  
  One way ANOVA with an independent variable with 2 levels ($I$ = 2):
OLS regression with one categorical independent variable with 2 levels:
 
n.a.  Example context  
  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.  
n.a.  SPSS  
  Analyze > Compare Means > IndependentSamples T Test...
 
n.a.  Jamovi  
  TTests > Independent Samples TTest
 
Practice questions  Practice questions  