ANCOVA - overview

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ANCOVA
Multilevel analysis
Paired sample $t$ test
You cannot compare more than 3 methods
Independent variablesIndependent variablesIndependent variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables, plus at least one random factor2 paired groups
Dependent variableDependent variableDependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio levelOne quantitative of interval or ratio level
THIS TABLE IS YET TO BE COMPLETEDTHIS TABLE IS YET TO BE COMPLETEDNull hypothesis
--H0: $\mu = \mu_0$

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.
n.a.n.a.Alternative hypothesis
--H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
n.a.n.a.Assumptions
--
  • Difference scores are normally distributed in the population
  • Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another
n.a.n.a.Test statistic
--$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 $\mu_0$.
n.a.n.a.Sampling distribution of $t$ if H0 were true
--$t$ distribution with $N - 1$ degrees of freedom
n.a.n.a.Significant?
--Two sided: Right sided: Left sided:
n.a.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.n.a.Effect size
--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.n.a.Visual representation
--
Paired sample t test
n.a.n.a.Equivalent to
--
  • One sample $t$ test on the difference scores.
  • Repeated measures ANOVA with one dichotomous within subjects factor.
n.a.n.a.Example context
--Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?
n.a.n.a.SPSS
--Analyze > Compare Means > Paired-Samples T Test...
  • Put the two paired variables in the boxes below Variable 1 and Variable 2
n.a.n.a.Jamovi
--T-Tests > Paired Samples T-Test
  • Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line
  • Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questionsPractice questions