ANCOVA: overview

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ANCOVA
Two sample $t$ test - equal variances assumed
Independent variablesIndependent 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 variableDependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio level
THIS TABLE IS YET TO BE COMPLETEDNull 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
-
  • Within each population, the scores on the dependent variable are normally distributed
  • The standard deviation of the scores on the dependent variable is the same in both populations: $\sigma_1 = \sigma_2$
  • Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
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: Right sided: Left 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
-
Two sample t test - equal variances assumed
n.a.Equivalent to
-One way ANOVA with an independent variable with 2 levels ($I$ = 2):
  • two sided pooled sample $t$ test equivalent to ANOVA $F$ test when $I$ = 2
  • pooled sample $t$ test equivalent to $t$ test for contrast when $I$ = 2
  • pooled sample $t$ test equivalent to $t$ test multiple comparisons when $I$ = 2

OLS regression with one categorical independent variable with 2 levels:
  • two sided pooled sample $t$ test equivalent to $F$ test regression model
  • pooled sample $t$ test equivalent to $t$ test for regression coefficient $\beta_1$
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 > Independent-Samples T Test...
  • Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
  • Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
  • Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
  • Continue and click OK
n.a.Jamovi
-T-Tests > Independent Samples T-Test
  • Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
  • Under Tests, select Student's (selected by default)
  • Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questions