Measures of Association – Tetrachoric and Kendall Tall Correlation.

**Highlights:**

- Measures of association and types
- Tetrachoric Correlation
- Kendall Tall Correlation

Measures of association deals with finding relationship between attributes of variables that are not capable of direct quantitative measurement but can only be inferred by the presence or absence of other variables.

Three typres of measures of association includes;

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**Positive Association:** in this, two variables are either present or absent.

**Independent association:** the present of one variable does not in any way interfere or affect the presence of other variable.

**Negative Association:** the presence of one variable causes the absence of the other variable. The technique that can be used to measure association is Tetrachoric correlation.

**Tetrachoric Correlation:** here we find the relationship between two d nominal unobservable variable or trait measured or represented by an ordinal variable. The two nominal unobservable variable have an assumption of underline continuity e.g attitude , anger, interest, emotion etc.

This correlation is given as

Ytet = COS(180/(1+Sqr(BC/AD).

**Example:** suppose a researcher is interested in measuring the relationship between students; responses to two items on a symbolic reasoning test.

Item 1 | Item 2 |

1 | 0 |

0 | 0 |

1 | 1 |

0 | 1 |

1 | 1 |

1 | 1 |

0 | 0 |

0 | 1 |

0 | 1 |

1 | 0 |

1 | 1 |

1 | 1 |

0 | 0 |

Solution

This will be represented in a two by two matrix.

Correct response = 1

Incorrect response = 0, just like we did with the Phi coefficient.

Item2 | ||||

Item1 |
Correct(1) | Incorrect(0) | Total | |

Correct(1) | 5 (A) | 2(B) | 7 | |

Incorrect(0) | 3(C) | 3(D) | 6 | |

Total | 8 | 5 |

## Tetrachoric and Kendall Tall Correlation – Measures of Association

To get the value of the first cell which is 5, you go to the table 1 above and count anywhere you see (1,1) or counting the numbers of (1,1). That is it shoes those that responded to both item1 and item2 correctly.

Now those that responded to item1 correctly and item2 incorrectly is (1,0) and the values can be gotten by counting the numbers of (1,0) in table one above which is 2. Likewise those that responded to item1 incorrectly and item2 correctly is 3, gotten through same procedures.

Also those that responded to item1 incorrectly and item2 incorrectly is 3, this can be gotten by counting the number of times (0,0) in table1 above.

Applying the formula

Ytet = COS(180/(1+Sqr(BC/AD).

BC = 2 x 3 = 6

AD = 5 x 3 = 15

Ytet = COS(180/(1+Sqr(6/15)

= COS(180/1.63)

= -0.35

This shows a negative moderate relationship between student responses to item1 and item2. It implies that a reasonable number of students who responded to item1 correctly tended to an incorrect response to item2. Also, most of the student or individuals who responded to item1 incorrectly tended to have a correct response to item2.

**Note:** If BC/AD is greater than 1, the value of Ytet will be positive and if less than 1 then the value of Ytet will be negative.

**Kendall Tall**

This technique is used when finding the relationship between the ranks or rating of judges by taking note of their relative ordering.

Example: Assuming we are interested in finding the relationship between rank of 10 students as assigned by two judges.

### Point To Note With Kendall Tall

you can’t have a tie here unlike spearman rank correlation.

Here we have concordant and disconcordant.

**Concordant: **it talks about agreement between the judges.

**Disconcordant: **talks about disagreement between the rank of judges.

Judge1 | Judge2 |

2 | 3 |

1 | 4 |

6 | 1 |

8 | 2 |

3 | 5 |

7 | 9 |

5 | 7 |

9 | 8 |

4 | 10 |

10 | 6 |

Solution

The first thing to do is to arrange judge1 in ascending order along its corresponding values for judge2.

Judge1 | Judge2 | Concordant | Disconcordant |

1 | 4 | 6 | 3 |

2 | 3 | 6 | 2 |

3 | 5 | 5 | 2 |

4 | 10 | 0 | 6 |

5 | 7 | 2 | 3 |

6 | 1 | 4 | 0 |

7 | 9 | 0 | 3 |

8 | 2 | 2 | 0 |

9 | 8 | 0 | 1 |

10 | 6 | 0 | 0 |

Total | 25 | 20 |

**How to find the Concordant and Disconcordant Columns**

To find the concordant, go to the second column and check for the first value which is 4, then count the positions that are suppose to come after 4 naturally and record it. So from the table, how many positions are supposed to come after 4 naturally? Observe that 5,6,7,8,9 and 10 are suppose to come after 4, so just count it which is 6 and record it under concordant.

The disconcordant for that first value(4) is gotten by counting the number of positions that are normally supposed to come before 4, count it and record it under disconcordant. Now how many positions are supposed to come before 4? Observe that 1, 2 and 3 are normally supposed to be before 4, so the number of your disconcordant is 3, since three positions are supposed to be before 4. Now this takes care of both the concordant and disconcordant for the first value 4.

##### Procedure for determining concordant and disconcordant

Concordant and disconcordant for the second value which is 3 is gotten through same procedure.

Now, to find the concordant, go to the second column and check for the second value which is 3, then count the positions that are suppose to come after 3 naturally and record it. So from the table, how many positions are supposed to come after 3 naturally? Observe that ,5,10,7,9,8 and 6 are suppose to come after 3, so just count it which is 6 positions and record it under concordant.

The disconcordant for that second value(3) is gotten by counting the number of positions that are normally supposed to come before 3, count it and record it under disconcordant. Now how many positions are supposed to come before 3? Observe that 1 and 2 are normally supposed to be before 3, so the number of your disconcordant is 2, since two positions are supposed to be before 3. Now this takes care of both the concordant and disconcordant for the second value 3.

So continue the process to find the concordant and disconcordant of the remaining values in the second column.

After you are done and have gotten the total for both the concordant and disconcordant, proceed to use the below formula for the calculation.

J = (C–D)/(C+D)

Where, c = Concordant and D = disconcordant

J = (25 – 20)/(25+20)

= 5/45 = 0.11

This shows a low relationship in the ranking of judges, it implies that the judges rarely agree in their ranking.

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