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The t-test for dependent groups

The $t$ -test for dependent groups is used when the measurement of an analysed variable you do twice, each time in different conditions (but you should assume, that variances of the variable in both measurements are pretty close to each other). We want to check how big is the difference between the pairs of measurements ( $d_i=x_{1i}-x_{2i}$ ). This difference is used to verify the hypothesis informing us that the mean of the difference in the analysed population is 0.

Basic assumptions:

measurement on an interval scale,
normality of distribution of measurements $d_i$ (or the normal distribution for an analysed variable in each measurement),
a dependent model.

Hypotheses:

$\begin{array}{cc} \mathcal{H}_0: & \mu_0=0,\\ \mathcal{H}_1: & \mu_0\ne0, \end{array}$

where:

$\mu_0$ , – mean of the differences $d_i$ in a population.

The test statistic is defined by:

$\begin{displaymath} t=\frac{\overline{d}}{sd_d}\sqrt{n}, \end{displaymath}$

where:

$\overline{d}$ – mean of differences $d_i$ in a sample,

$sd_d$ – standard deviation of differences $d_i$ in a sample,

$n$ – number of differences $d_i$ in a sample.

Test statistic has the t-Student distribution with $n-1$ degrees of freedom.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

Note

standard deviation of the difference is defined by:

$\begin{displaymath} sd_d=\displaystyle{\sqrt{\frac{\sum_{i=1}^{n}(d_i-\overline{d})^2}{n-1}}}, \end{displaymath}$

standard error of the mean of differences is defined by:

$\begin{displaymath} SEM_{d}=\displaystyle{\frac{SD_d}{\sqrt{n}}}. \end{displaymath}$

Standardized effect size.

The Cohen's d determines how much of the variation occurring is the difference between the averages, while taking into account the correlation of the variables.

$\begin{displaymath} d=\frac{dz}{\sqrt{1-r_p}}, \end{displaymath}$ .

where:

$dz=\left|\frac{\overline{d}}{sd_d}\right|$ ,

$r_p$ – the Pearson product-moment correlation coefficient.

When interpreting an effect, researchers often use general guidelines proposed by Cohen ¹⁾ defining small (0.2), medium (0.5) and large (0.8) effect sizes.

The settings window with the t-test for dependent groups can be opened in Statistics menu→Parametric tests→t-test for dependent groups or in ''Wizard''.

Note

Calculations can be based on raw data or data that are averaged like: arithmetic mean of difference, standard deviation of difference and sample size.

EXAMPLE(BMI.pqs file)

A clinic treating eating disorders studied the effect of a recommended „diet A” on weight change. A sample of 120 obese patients were put on the diet. Their BMI levels were measured twice: before the diet and after 180 days of the diet. To test the effectiveness of the diet, the obtained BMI measurements were compared.

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & $Mean BMI values do not change with diet,$\\ \mathcal{H}_1: & $Mean BMI values change as a result of diet.$ \end{array}$

Comparing $p<0.0001$ with a significance level $\alpha=0.05$ we find that the mean BMI level changed significantly. Before the diet, it was higher by less than 2 units on average.

The study was able to use the Student's t-test for dependent groups because the distribution of the difference between pairs of measurements was a normal distribution (Lilliefors test, $p=0.0837$ ).

¹⁾

Cohen J. (1988), Statistical Power Analysis for the Behavioral Sciences, Lawrence Erlbaum Associates, Hillsdale, New Jersey

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The t-test for dependent groups

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