Paired sample t test: overview

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Paired sample $t$ test
Independent variable
2 paired groups
Dependent variable
One quantitative of interval or ratio level
Null hypothesis
$\mu = \mu_0$
$\mu$ is the unknown population mean of the difference scores; $\mu_0$ is the population mean of the difference scores according to H0, which is usually 0
Alternative hypothesis
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
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
Population of difference scores can be conceived of as the difference scores we would find if we would apply our study (e.g., applying an intervention and measuring pre-post scores) to all individuals in the population.
Test statistic
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to H0, $s$ is the sample standard deviation of the difference scores, $N$ 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 $\mu_0$
Sampling distribution of $t$ if H0 were true
$t$ distribution with $N - 1$ degrees of freedom
Significant?
Two sided: Right sided: Left sided:
$C\%$ 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.
Effect size
Cohen's $d$:
Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0$
Visual representation
Paired sample t test
Equivalent to
One sample $t$ test on the difference scores
Repeated measures ANOVA with one dichotomous within subjects factor
Example context
Is the average difference between the mental health scores before and after an intervention different from $\mu_0$ = 0?
SPSS
Analyze > Compare Means > Paired-Samples T Test...
  • Put the two paired variables in the boxes below Variable 1 and Variable 2
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 questions