ANCOVA: overview

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ANCOVA
One sample $t$ test for the mean
Independent variablesIndependent variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)None
Dependent variableDependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio level
THIS TABLE IS YET TO BE COMPLETEDNull hypothesis
-$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
n.a.Alternative hypothesis
-Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
n.a.Assumptions
-
  • Scores are normally distributed in the population
  • Sample is a simple random sample from the population. That is, observations are independent of one another
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: Right sided: Left 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 $t_{N-1}$ 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
-
One sample t test
n.a.Example context
-Is the average mental health score of office workers different from $\mu_0$ = 50?
n.a.SPSS
-Analyze > Compare Means > One-Sample T Test...
  • Put your variable in the box below Test Variable(s)
  • Fill in the value for $\mu_0$ in the box next to Test Value
n.a.Jamovi
-T-Tests > One Sample T-Test
  • Put your variable in the box below Dependent Variables
  • Under Hypothesis, fill in the value for $\mu_0$ in the box next to Test Value, and select your alternative hypothesis
Practice questionsPractice questions