Biserial Correlation Lecture 58

 

Biserial Correlation

Lecture 58

The biserial correlation measures the strength of association between an artificial dichotomous variable and a continuous variable. The artificial dichotomous variable is defined by the researcher, expert or investigator, like pass/fail, weak/strong, etc.

e.g., an examiner decides that below 40% is considered a fail. Thus, above 40% will be considered a pass.

Coefficient of Biserial Correlation

A quantitative measure that measures the strength of association between an artificial dichotomous variable and a continuous variable is called biserial correlation, denoted by ρb (population) and rb⁣ (sample). The following is used to compute the biserial correlation between an artificial dichotomous and continuous variable.

Where:

X−p is the mean of the interval variable’s values associated with the dichotomous variable’s first category.

X−q is the mean of the interval variable’s values associated with the dichotomous variable’s second category.

s is the standard deviation of the variable on the interval scale.

Pp is the proportion of the interval variable values associated with the dichotomous variable’s first category.

Pq is the proportion of the interval variable values associated with the dichotomous variable’s second category.

The mean and proportion of the dichotomous variable's first “p” category:

The mean and proportion of the dichotomous variable second “q” category:

The standard deviation “s” can be obtained as:

Z is the table value against Pp and Pq and is called the height of the ordinate of the normal curve separating the proportions p and q.

Hypothesis Testing about the Association of an Artificial dichotomous and Continuous Variable

Let rb be the estimate of ρb, computed from the values of a sample selected from a bivariate population that consists of an artificial dichotomous and continuous variable. Suppose it is desired to test that there is no association between an artificial dichotomous and continuous variable.

That’s

H0: ρb = 0

versus suitable alternatives.

If the sample size is small, the following test statistics are used to test the above null hypothesis.

If the sample size is large, the following test statistics are used to test the above null hypothesis.

Example 13.20: A sociology department of a college wants to know if the grade point averages (GPAs) can predict the performance of students in the comprehensive exam that is necessary for graduation. The comprehensive exam is scored as either pass or fail. The college graduate sociology department wanted to know if the grade point averages (GPAs) of its students could be used to forecast how well they do on the comprehensive exam that is necessary to graduate. There are two possible grades for the comprehensive exam: pass and fail. Last year, sixteen students took the comprehensive exam. Five students did not pass the exam. The GPAs and exam results of the students are given below:

Participants	Exam Result	GPA
1	F	3.5
2	F	3.4
3	F	3.3
4	F	3.2
5	F	3.6
6	P	4.0
7	P	3.6
8	P	4.0
9	P	4.0
10	P	3.8
11	P	3.9
12	P	3.9
13	P	4.0
14	P	3.8
15	P	3.5
16	P	3.6

Test the hypothesis that there is no association between exam results and GPAs at a 5% significance level.

Solution: First calculate the biserial correlation and next test the hypothesis.

Participants	Exam Result	X	X^2
1	F	3.5	12.25
2	F	3.4	11.56
3	F	3.3	10.89
4	F	3.2	10.24
5	F	3.6	12.96
6	P	4.0	16.00
7	P	3.6	12.96
8	P	4.0	16.00
9	P	4.0	16.00
10	P	3.8	14.44
11	P	3.9	15.21
12	P	3.9	15.21
13	P	4.0	16.00
14	P	3.8	14.44
15	P	3.5	12.25
16	P	3.6	12.96
		59.1	219.37

Let p represent the fail category and q the pass category.

np = 5 nq = 11 n = 16

$$

Now, determine the height of the unit normal curve ordinate, y, at the point dividing Pp = 0.3125 and Pq = 0.6875

However, we will compute the value. Using the above table also provides the z-score at the point dividing Pp = 0.31 and Pq = 0.68, z = 0.49:

Now, compute the biserial correlation coefficient using

The biserial correlation shows a strong association between exam results and GPAs.

Now to test the hypothesis

i. State the null and alternative hypotheses.

$$

H0: ρb = 0 vs. H1: ρb ≠ 0 

ii. The significance level: α = 0.05

iii. The test statistic: The sample size is small.

iv. Reject H0 when |t| > 2.360

$v.\; Computation:$

vi. Remarks: The computed t value falls in the rejection area; the sample data does not provide sufficient evidence to accept the null hypothesis about the association of exam results and GPAs. Thus, it is concluded that there exists an association between exam results and GPAs.

$$

Example 13.21: An investigator sought to ascertain whether poverty and self-esteem are associated. 18 participants were categorised as either above or below the poverty line based on their income level. A 20-person survey measuring self-esteem was filled out by participants. The results of the survey are presented in the following table:

No.	Poverty Line	Survey Score
1	above	26
2	above	38
3	above	26
4	above	35
5	above	32
6	above	22
7	above	28
8	above	33
9	above	19
10	above	23
11	above	25
12	above	28
13	above	41
14	above	35
15	above	47
16	above	20
17	above	29
18	above	21
19	above	24
20	below	19
21	below	39
22	below	21
23	below	18
24	below	22
25	below	31
26	below	36
27	below	15
28	below	21
29	below	12
30	below	7
31	below	34
32	below	24
33	below	27
34	below	21
35	below	15
36	below	19
37	below	17
38	below	31
39	below	22
40	below	19

Test the hypothesis: there is a strong association between survey score and poverty line.

Solution: First calculate the biserial correlation and next test the hypothesis.

No.	Poverty	X	X^2
1	above	26	676
2	above	38	1444
3	above	26	676
4	above	35	1225
5	above	32	1024
6	above	22	484
7	above	28	784
8	above	33	1089
9	above	19	361
10	above	23	529
11	above	25	625
12	above	28	784
13	above	41	1681
14	above	35	1225
15	above	47	2209
16	above	20	400
17	above	29	841
18	above	21	441
19	above	24	576
20	below	19	361
21	below	39	1521
22	below	21	441
23	below	18	324
24	below	22	484
25	below	31	961
26	below	36	1296
27	below	15	225
28	below	21	441
29	below	12	144
30	below	7	49
31	below	34	1156
32	below	24	576
33	below	27	729
34	below	21	441
35	below	15	225
36	below	19	361
37	below	17	289
38	below	31	961
39	below	22	484
40	below	19	361
		1022	28904

The mean and proportion of the dichotomous variable first “p” above the category:

The mean and proportion of the dichotomous variable second “q” below the category:

The standard deviation “s” can be obtained as:

Z is the table value against Pp and Pq and is called height of the ordinate of the normal curve separating the proportion p = 0.475 and q = 0.525, y = 0.3982

The coefficient of biserial correlation is given below:

Now to test the hypothesis

i. State the null and alternative hypotheses.

$$

H0: ρb = 0 vs. H1: ρb ≠ 0 

ii. The significance level: α = 0.05

iii. The test statistic: The sample size large

$iv.\; Reject\; H0\; when\; |z|\; >\; 1.96$

$v.\; Computation:$

$$

vi. Remarks: The z-calculated value falls in the rejection area; the sample data does not provide sufficient evidence to accept the null hypothesis. Thus, it is concluded that there exists an association between poverty level and survey score.

• Read More: Point Biserial Correlation