# Effect Size for Dependent Sample t test

An * effect size *is a measure of the strength of a phenomenon, conveying the estimated magnitude of a relationship without making any statement about the true relationship.

*measure(s) play an important role in*

**Effect size***and statistical*

**meta-analysis***. So reporting effect size in thesis, reports or research reports can be considered as a good practice, especially when presenting some empirical results/ findings, because it measures the practical importance of a significant finding. In a simple way, we can say that*

**power analyses***is a way of quantifying the size of the difference between the two groups.*

**effect size*** Effect size *is usually computed after rejecting the

*in*

**null hypothesis***So if the null hypothesis is not rejected (i.e. accepted) then effect size has little meaning.*

**statistical hypothesis****testing procedure.**There are different formulas for different statistical tests to measure the * effect size*. In general, the

*can be computed in two ways.*

**effect size**- As the
difference between two means**standardized** - As the
**effect size****correlation (correlation**between the independent variables classification and the individual scores on the dependent variable).

**The effect size for dependent sample t-test**

The effect size of paired sample t-test (dependent sample t-test) known as** *** Cohen’s d (effect size)* ranging from $-\infty$ to $\infty$ evaluated the degree measured in

*standard deviation*units that the mean of the difference scores is equal to zero. If the value of

*d*equals 0, then it means that the difference scores is equal to zero. However larger the

*d*value from 0, more is the effect size.

* The effect size *for

*dependent sample t-test*can be computed by using

\[d=\frac{\overline{D}-\mu_D}{SD_D}\]

Note that both the * Pooled Mean (D) *and

*are reported in SPSS output under paired differences.*

**standard deviation**Let the * effect size*,

*d*= 2.56 which means that the sample means difference and the population mean difference is 2.56 standard deviations apart. The sign has no effect on the size of an effect i.e. -2.56 and 2.56 are equivalent

*.*

**effect sizes**The *d* statistics can also be computed from obtained *t* value and the number of paired observation by Ray and Shadish’s (1996) such as

\[d=\frac{t}{\sqrt{N}}\]

The value of *d* is usually categorized as small, medium and large. With * cohen’s d*:

- d=0.2 to 0.5 small effect
- d=0.5 to 0.8, medium effect
- d= 0.8 and higher, large effect.

**Computing Effect Size from $R^2$**

**Computing Effect Size from $R^2$**

Another method of computing the effect size is with r-squared ($r^2$), i.e.

\[r^2=\frac{t^2}{t^2+df}\]

It can be categorized in small, medium and large effect as

- $r^2=0.01$, small effect
- $r^2=0.09$, medium effect
- $r^2=0.25$, large effect.

The *non‐significant* results of t-test indicate that we fail to reject the hypothesis that the two conditions have equal means in the population. The larger the value of $r^2$ indicates the larger effect (effect size), while a large effect size with a* non‐significant* result suggests that the study should be replicated with a larger *sample size*.

So larger the value of *effect size* computed from either method indicates a very large effect, meaning that means are likely very different.

**References:**

- Ray, J. W., & Shadish, W. R. (1996). How interchangeable are different estimators of effect size?
*Journal of Consulting and Clinical Psychology*,*64*, 1316-1325. (see also “Correction to Ray and Shadish (1996)”,*Journal of Consulting and Clinical Psychology*,*66*, 532, 1998) - Kelley, Ken; Preacher, Kristopher J. (2012). “On Effect Size”.
*Psychological Methods***17**(2): 137–152. doi:10.1037/a0028086.