Wednesday, May 28, 2014

2-Sample Unpooled t-Test Effect Size in Excel 2010 and Excel 2013

This is one of the following six articles on 2-Independent-Sample Unpooled t-Tests in Excel

2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013

Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test

Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test

2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools

Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013

Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

Two-Independent-Sample Unpooled t-Test Effect Size in Excel

Effect size in a t-Test is a convention of expressing how large the difference between two groups is without taking into account the sample size and whether that difference is significant.

Effect size of Hypotheses Tests of Mean is usually expressed in measures of Cohen’s d. Cohen’s d is a standardized way of quantifying the size of the difference between the two groups. This standardization of the size of the difference (the effect size) enables classification of that difference in relative terms of “large,” “medium,” and “small.”

A large effect would be a difference between two groups that is easily noticeable with the measuring equipment available. A small effect would be a difference between two groups that is not easily noticed.

Effect size for a two-independent-sample, unpooled t-Test is a method of expressing the distance between the difference between sample mean, x_bar1-x_bar2, and the Constant in a standardized form that does not depend on the sample size.

Remember that the Test Statistic (the t Value) for a two-independent-sample t-Test (both pooled and unpooled t-Tests) calculated by the following formula:

(Click On Image To See a Larger Version)

Pooled t-Test formulas are used when the variances of both independent sample groups are similar. The rule of thumb is that the pooled t-Test formulas are used if the sample standard deviation of one of the groups is no more than twice as large as the sample standard deviation of the other group. Unpooled t-Test formulas are used the difference between the sample standard deviations is larger.

The t Value in a pooled t-Test is calculated as follows:

(Click On Image To See a Larger Version)

The Standard Error for a pooled t-Test is calculated as follows:

(Click On Image To See a Larger Version)

Effect Size for a pooled t-Test is calculated as follows:

(Click On Image To See a Larger Version)

In this case spooled can be derived from SEpooled with the following calculation:

(Click On Image To See a Larger Version)

Effect Size for an unpooled t-Test is calculated as follows:

(Click On Image To See a Larger Version)

sunpooled for purposes of calculating Effect Size can be derived from SE in the same way that spooled can as follows:

(Click On Image To See a Larger Version)

The Standard Error for a unpooled t-Test is calculated as follows:

(Click On Image To See a Larger Version)

sunpooled can therefore be calculated as follows:

(Click On Image To See a Larger Version)

With algebraic manipulation, the formula for sunpooled can be shortened to the following formula:

(Click On Image To See a Larger Version)

sunpooled = 18.905

The t Value specifies the number of Standard Errors that the differences between sample means, x_bar1-x_bar2, is from the Constant. The t Value is dependent upon the sample size, n. The t Value determines whether the test has achieved statistical significance and is dependent upon sample size. Achieving statistical significance means that the Null Hypothesis (H0: x_bar1-x_bar2 = Constant = 0) has been rejected.

The Effect Size, d, for a two-independent-sample, unpooled t-Test is a very similar measure that does not directly depend on sample size and has the following formula:

(Click On Image To See a Larger Version)

sunpooled pools the sample standard deviations based upon the proportion of combined samples that each of the sample sizes n1 and n2 represent and not the absolute values of n1 and n2. sunpooled is therefore not directly dependent on sample sizes n1 and n2.

A test’s Effect Size can be quite large even though the test does not achieve statistical significance due to small sample size.

The d measured here is Cohen’s d for a two-independent-sample, unpooled t-Test. The Effect Size is a standardized measure of size of the difference that the t-Test is attempting to detect. The Effect Size for a two-independent-sample, unpooled t-Test is a measure of that difference in terms of the number of sample standard deviations. Note that sample size has no effect on Effect Size. Effect size values for the two-independent-sample, unpooled t-Test are generalized into the following size categories:

d = 0.2 up to 0.5 = small Effect Size

d = 0.05 up to 0.8 = medium Effect Size

d = 0.8 and above = large Effect Size

In this example, the Effect Size is calculated as follows:

d = |x_bar1 - x_bar2 – Constant| / sunpooled = |46.55 – 42.24 – 0| / 18.905 = 0.228

An Effect Size of d = 0.228 is considered to be a small effect.

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