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Dependent Samples t test

2.1. Dependent Samples t test

Dependent Samples t Test

First, it goes by many names...
- Dependent Samples t Test
- Matched-Pairs t Test
- Repeated Measures t Test
- t Test for Dependent Means
- t Test for Related Samples
Essentially, it compares two sample means which are known to be related in some way.

Types of Related Samples

There are three types of related samples which are appropriate for the Dependent Samples t Test.

Natural Pairs: Comparing the scores of two groups of subjects which are related naturally.
- E.g., Twins (twin 1 vs. twin 2 on scores of trait anxiety).
Matched Pairs: Pairs two groups of subjects on some characteristic that is likely to be important to the variable which you are studying.
- E.g., Married couples (husbands vs. wives on ratings of marital satisfaction).
Repeated Measures: Same people at two different times of measure (of the same variable).
- E.g., Pretest/post test (patients' life satisfaction ratings before and after treatment for depression).

Gist: There is some sort of known meaningful relationship between the two groups of scores.

Similiar to previous tests, but...

Now we are testing a Difference Score.
- Instead of a mean; we are using the mean of differences.
When dealing with pairs of scores, we simply subtract one set of the pair from the other to get our difference scores for each pair.
- $X - Y = D$ for the difference scores.
- It is the difference scores (D) with which we conduct the t test.
The population of difference scores has and we again estimate the standard deviation ().
- If there is no difference between the pairs, then the mean of the difference scores will be equal to zero.
- Use the following notation:
$\mu_D = 0$ or $\mu_1 - \mu_2 = 0$

2.2. Dependent Samples t Test Example

NHST Example

We are interested in relationship satisfaction of young adults before and after they go off to college/university which separates them from their sweat-heart.
- Relationship satisfaction rating: 0 - 50 range, higher score indicates greater satisfaction.
Collect a sample of young adults which are about to be separated from their boy- or girl- friend by going to college.
Have them rate their satisfaction with the relationship.
Then, after they have spent the first semester at colleges/universities (away from their sweat-heart), they again rate their relationship satisfaction.

Step 1: State the Null and Alternative Hypotheses

Define the populations: Relationship Satisfaction = RS

Population 1: RS prior to college separation.
Population 2: RS after the first semester.

State the Hypotheses:

The null hypothesis is: there will be no difference between the means of the ratings (before college vs. after the first semester).
$H_0: \mu_1 - \mu_2 \leq 0$ or $\mu_D \leq 0$
The alternative hypothesis is: there will be a significant decrease in the ratings.
$H_1: \mu_1 - \mu_2 > 0$ or $\mu_D > 0$

Note the directional alternative hypothesis; pay careful attention to how we specified the hypotheses in symbols.

If the alternative hypothesis expected an increase, then $H_1: \mu_1 - \mu_2 < 0$ or $\mu_D < 0$

Step 2: Comparison Distribution (estimate $\sigma$ with $S_M$ ).

Pair	Before	After	$D$	$\overline{D}$	$(D - \overline{D})$	$\left(D - \overline{D}\right)^2$
1	40	32	8	8	0	0
2	38	31	7	8	-1	1
3	36	30	6	8	-2	4
4	42	31	11	8	3	9
4			32			$SOS = 14$

$\overline{D} = \sum(D)/n_D = 32/4 = 8$

$S^2 = SOS/df = 14/n_D - 1 = 14/3 = 4.67$

$S_M^2 = \frac{S^2}{n_D} = \frac{4.67}{4} = 1.1675$

$S_M = \sqrt{S_M^2} = \sqrt{1.1675} = 1.08$

Step 3: Determine the critical value

Deja-vu...?
Determine the critical t value.
Significance level = .05, one-tailed test, $df = n_D - 1 = 3$ .
Look in the appropriate column of the t table.
- One-tailed, significance level = .05
http://www.math.unb.ca/~knight/utility/t-table.htm
Critical t value is 2.353
Which means, we need $t_{calc}$ to be greater than |2.353| (absolute value of 2.353) to find a significant difference.

Same critical value we used with the One Sample t Test.

Step 4: Calculate t

The general t formula:

$t = \frac{\overline{X} - \mu}{S_M}$

Remember we are dealing with the difference scores (D) but the t formula is essentially the same as above.
From previous calculations, we have: $\overline{D} = 8, \mu_D = 0, S_M = 1.08$

$t = \frac{\overline{D} - \mu_D}{S_M} = \frac{8 - 0}{1.08} = 7.41$

So, our $t_{calc} = 7.41$ which is fairly large.

Step 5: Compare and make a decision.

Since $t_{calc} = 7.41 > 2.353 = t_{crit}$ we reject the null hypothesis.
The initial interpretation is... There was a significant decrease in ratings of relationship satisfaction from before college separation ( ) to after the first semester of college separation ( ), (one-tailed).
- One could also state the results as; $t(3) = 7.41, p < .05$ because .05 was our significance level.
- The 3 above is the degrees of freedom (df).
At this point in the course, you should be able to calculate the means and standard deviations of each group of scores; however the exact $p$ value was obtained by verifying the results above in SPSS (see below).

Image M8_001

2.3. Effect Size

Effect Size

Recall, Cohen's d:

$d = \frac{\overline{X} - \mu}{\sigma}$

Here, we do not know so we use to estimate (not because it is influenced heavily by sample size).
- Effect sizes are not influenced by sample size.
Also, because we are dealing with difference scores, we use slightly different symbols in the formula:

$d = \frac{\overline{D} - \mu_D}{S}$

Effect Size continued

Cohen's d for difference scores:

$d = \frac{\overline{D} - \mu_D}{S}$

We have $S^2 = 4.67$ from above, which leads to an estimate of $\sigma$ :

$S = \sqrt{4.67} = 2.16$

Which in turn, leads to d:

$d = \frac{\overline{D} - \mu_D}{S} = \frac{8 - 0}{2.16} = 3.70$

A large effect size.

2.4. Using Delta for Power

Using Delta ( $\delta$ ) for Statistical Power

As was done in the one sample t test situation, here again we calculate $\delta$ as a combination of sample size and effect size and use it to look up the power in the $\delta$ table.
However, because we now have two samples, the formula is slightly different.

$\delta = d * \sqrt{\frac{n}{2}}$

So, for our current example, we get:

$\delta = d * \sqrt{\frac{n}{2}} = 3.7 * \sqrt{\frac{4}{2}} = 3.7 * \sqrt{2} = 3.7 * 1.414 = 5.233$

Since our $\delta$ table shows that anything with a $\delta > 5.00$ equates to power greater than 0.99, we can safely assume we have at least a power of 0.99.

Using Delta to calculate appropriate sample sizeThe more useful way to use is for calculating adequate sample size during the planning of the study. First, we need to calculate for a desired power with a one-tailed test at .05 significance level. , now we can calculate the sample size for a given effect size . 2.5. Calculating a Confidence Interval with Using essentially the same procedures we used with the one sample t test, we can calculate the lower limit (LL) and upper limit (UL). Recall, the general formulas for a confidence interval are: LL = (-crit)*(SE) + mean and UL = (+crit)*(SE) + mean When in the Dependent Samples situation, we simply use the difference score mean. Interpretation of In this example, we calculated a 95% confidence interval () because our critical value was based on a significance level of .05. If we drew an infinite number of samples of young adults' relationship satisfaction ratings, 95% of those samples' difference score means would be between 5.459 and 10.541. Remember, the population difference score mean is fixed (but unknown); while each sample has its own difference score mean (samples fluctuate). 2.6. Summary of Section 2 Dependent Samples t Test Usage? Fortunately... The Dependent samples t Test is quite powerful (i.e., statistical power). Repeated measures designs (or matched pairs, dependent samples, etc.) have more power due to less variance between the groups of scores. The same individuals being tested at different times. But, there is no control group for comparison, so the results are somewhat limited. The t test in general is robust to errors. Robust (in this situation) means, even with moderate departures from normality we can be confident in our results. Summary of Section 2 Section 2 covered the following topics: The Dependent Samples t Test. An NHST example of the Dependent Samples t Test. Cohen's d Effect Size Use of delta () for Statistical Power Use of delta () for calculating a-priori sample size Calculation of Confidence Intervals. Next: Independent Up: Module 8: Introduction to Previous: One Sample Contents jds0282 2010-10-15