# Ordinal vs Ordinal paired

## Part 3b: Effect size

In the impression and visualisation sections, we noticed that before seeing the commercial the scores were fairly evenly distributed among the categories, but after the commercial the first category seems to have a relatively high amount of cases. This was all based on the sample data. In the test section we've shown that this change is significant.

As a finishing touch we should also report an effect size measure. The Rosenthal correlation is a generic one that simply divides the standardized test statistics, by the square root of the sample size (Rosenthal, 1991, p. 19), which I will refer to as the Rosenthal correlation coefficient (as to differentiate it with other correlation coefficients). The z-value in the example is -4.25 and the number of pairs used 54, so we get r = -4.25 / SQRT(54) = -0.58

In the example it shows to be -0.58. Unfortunately there is no formal way to determine if 0.58 (you can ignore the negative sign) is high or low (although almost everyone would agree this is high), and the rules of thumb floating around on the internet vary quite a lot, often depending on the field (e.g. biology, medicine, business, etc.). Click here to see a table with various other rule of thumbs for the interpretation.

One such rule of thumb is from Bartz (1999, p. 184) shown in Table 1.

Rosenthal Correlation | Interpretation |
---|---|

0.00 < 0.20 |
very low |

0.20 < 0.40 |
low |

0.40 < 0.60 |
moderate |

0.60 < 0.80 |
strong |

0.80 < 1.00 |
very strong |

In this example we can therefor speak of a relatively strong effect size.

A Wilcoxon Signed-ranks test indicated people tend to like the brand more before seeing the commercial (*Mdn* = 3) than after seeing it (*Mdn* = 2), *Z* = 4.25, *p* < .001, *r* = .58.

**Click here to see how you can perform the test with an Online calculator, or Manually.**

**Online calculator**

Enter the standardized test statistic, and the sample size:

**Manually (Formula and Example)**

**Formula**

The formula to determine the Rosenthal Correlation Coefficient is:

In this formula *Z* is the Z-statistic, which in the case of a Wilcoxon one-sample test is the adjusted W statistic, and *n* the sample size.

**Example**

Note: Continuation of the example used in the test section example.

In the test section we already calculated the Z value for this example. That was:

Now we can determine the Rosenthal correlation:

We're now done with all our analyses, so let's combine everything for the report on the next page.

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