Cramer’s V

In the previous articles, we saw different forms of correlation, this time , we delve into the realm of statistical analysis, focusing on Cramer’s V.

Cramer’s V is a statistical measure used to assess the strength of association between two categorical variables in a contingency table. It’s particularly useful for tables larger than 2x2, meaning it can be applied to variables with more than two categories.

What the difference between Cramer’s V and Phi Cofficient :

• Phi Coefficient is specifically used for 2x2 contingency tables, meaning it’s applicable only when both variables are binary (each having two categories) but Cramer’s V is a more general measure used for contingency tables of any size. It’s applicable for tables larger than 2x2, making it suitable for variables with more than two categories.
• The formula for Cramer’s V is also based on the Chi-square statistic but adjusted for the size of the contingency table.

Note : Both Cramer's V and the Phi coefficient range from 0 to 1, where 0 indicates no association and 1 indicates a perfect association.

You can find more information about the Chi-square test in the article below

How do we compute it?

• V = Cramer’s V coefficient
• χ2 = Chi-square statistic from the chi-square test
• n = Total number of observations
• r = Number of rows in the contingency table
• c = Number of columns in the contingency table

Solving the Cramer’s V Puzzle with Detective Conan

Detective Conan a curious detective decided to explore whether a suspect’s job influenced the outcome of a case. To do this, he gathered data from various episodes, focusing on two main aspects: the occupation of the suspects and whether they were found guilty or innocent.

He created a table, where he listed the jobs down one side ; artists, businesspeople, doctors, and teachers. Along the top, he marked two outcomes: ‘Guilty’ and ‘Innocent.’ Under each job, he carefully tallied how many suspects were found guilty and how many were innocent.

With his table complete, our detective had a new challenge: figuring out if the job really mattered in the outcome of the cases.

This is where Cramer’s V, a magical formula to measure the strength of connection between two things, came into play. He knew this formula would tell him if the suspect’s job and the case outcome were just coincidentally aligned or if there was a real connection.

First, he needed a special number called the Chi-square statistic, a clue to how much the outcomes differed from what one would expect by chance.

He remembered the formula from the previous article ??

O is the observed frequency and E is the expected frequency under the null hypothesis (no association between the variables).

The expected frequency for each cell in a contingency table is calculated as:

• RT is Row Total
• CT is Column Total
• N is the Grand total

1/ Expected Frequencies

For the ‘Artist’ and ‘Guilty’: E = (15 x 40)/70 and it’s equal to 8.57

Note : We calculate E for each cell in the table.

2/ Chi-square Calculation : Now that we have the Expected values we need to plug them in the formula and sum all the results to get the Chi-square statistic which is approximately 7.05

Now that we has the Chi-square statistic and the total number of cases, which added up to 70.

Using the Cramer’s V formula the detective worked his way through the calculation.

• The top part of the fraction was the Chi-square number (7.05) divided by the total cases (70).
• The bottom part was the smaller number between the job types minus one (4–1=3) and the outcomes minus one (2–1=1). In this case, it was 1.
• So it’s √(7.05/70) and Cramer’s V is approximately 0.317

Indicating then, a moderate association between a suspect’s occupation and the case outcome. But rest assured, in the real world, your occupation won’t lead you into a “Detective Conan” episode!

It’s always fun to explore fictional scenarios with a bit of humor and creativity, especially when they intersect with the world of statistics and data analysis. I really think it’s easier to understand

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