**A Priori Sample Size for Dependent Samples t-test
**

Sample Size Calculation for Dependent Samples *t-*tests are not as simple as sample size calculation for the independent samples *t*-test. While the sample size requirement is smaller because the two samples are related or correlated, the calculation is somewhat complicated. In order to calculate the minimum sample size for the dependent samples *t*-test being used in your Master's thesis, Ph.D. thesis, Master's dissertation, or Ph.D. dissertation, you are going to need some information or have a good idea of values for key pieces of information.

**What is my thesis power analysis or dissertation power analysis for a dependent samples t-test or paired samples t-test?
**

For the purposes of this example, I will refer to a Jacob Cohen book, *Statistical Power Analysis for the Behavioral Sciences.* We are going to define power analysis for the dependent samples *t-*test as the sample size necessary to…

- Achieve a Power of 0.80
- Detect a reasonable difference between the groups with a medium effect size of 0.50
- Detect a significant difference between the groups at a 0.05 level of significance

Important to understand is that each of the four items mentioned (Sample Size, Power, Effect Size, and Level of Significance) are a function on one another, meaning that changing any one of these things is going to change the value of the other three. So we have determined values for Power (0.80), Effect Size (0.50), and Level of Significance (0.05), and we are trying to figure out how many people we need to actually achieve all of these values. Get help with your thesis power analysis or dissertation power analysis

**How is sample size calculation for a dependent samples t-test or paired samples t-test different than that of an independent samples t-test?
**

Before, we found the sample size for an independent samples *t-*test by looking at a simple table. This time, the groups are related and we need to take that into account in our sample size calculation. Before we continue, however, I would like to tell you the good news… The sample size requirements for the independent samples *t-*test are much larger than that of the dependent samples *t-*test. If you have calculated the independent samples *t-*test sample size and power analysis for your dissertation or thesis, can obtain that many pairs of participants, and are not under a lot of pressure from your college or university to justify the sample size, then...

**Stop Here**.

*t-*test, your sample size is not going to be measure in participants, but in pairs of participants. Get help with your thesis power analysis or dissertation power analysis

**How do I calculate the sample size for dependent samples t-tests or paired samples t-test?
**

Since we have established that we are conducting a dependent samples *t-*test, and are assuming the scores for the pairs are related or correlated, we need to know the degree to which they are correlated. Since this is an a priori power analysis for a dissertation or thesis, we could not possibly know the exact correlation between pairs in the sample we are going to obtain and must estimate the degree to which these pairs are going to be correlated.

If this is a dissertation or thesis following other research designs and there is empirical research information available, then look for a correlation coefficient from some of these other studies. For instance, if four of the studies you have read found a strong correlation (0.90) between childhood obesity prior to dieting and after dieting, then we can assume that we are going to find the same thing in our study. We would assume that the correlation would be 0.90. Get help with your thesis power analysis or dissertation power analysis

**What is the equation for calculating sample size for dependent samples t-test?
**

Here it is in all of its mathematical glory…

n= n_(.10)/(100d^(2 ) )+ 1

where, **n _{.10}** = 1571 and is the necessary sample size for the given

**a**or α and Power,

**d**= 0.645 and is the ES index for

**t**tests of means in standard unit calculated by the equation:

d= d_(4^' )/√(1-r)

where **
**= the medium effect size of 0.50 and

**= 0.40**

*r***is the estimation of the correlation within the pairs.**

**This yields**

**d =**0.645 = 0.05/√ (1 – 0.40).

**OR**

Just tell them you need 39 pairs of participants or scores. Better yet, get professional help with your thesis or dissertation power analysis. Let us give you a customized power analysis for your Master's thesis, Ph.D. thesis, Master's dissertation, or Ph.D. dissertation.