Question

26. "Correlation does not equal causation" is a phrase we hear often in the media and in various college classes a) Why is this true? Why can't I just run a correlation, find a significant result, and assume a causal relationship between the variables I measured? b) If I find a significant correlation between X and Y, what can I do to establish a causal inference? Let's say I find that student GPA is significantly correlated with the amount of sleep students get each night. How could you design a test to determine whether or not sleep causes changes in students' GPA?

Answer 1

26.

a)

It is true because if some variable X is increasing along with some other variable Y during the same time period, it does not mean that the increase in one variable is causing the increase in another variable though they have strong correlation between them. For example, when your age is increasing, the price of a particular stock has also increased. It does not mean that the increase in your age is the cause of increase in stock's price or the increase in stock's price is the cause of increase in your age. But if you find the correlation coefficient between them, they will show the strong correlation.

Thus, "Correlation does not equal causation".

Also "the lack of correlation does not mean that there is no causal relationship between the variables".

b)

There are various studies or experimental methods to find whether there is a significant relationship between two variables, i.e., change in one variable is causing the change in another variable. The change may be either increase or decrease."Controlled experiments​​​​​​" are recommended to conduct to determine if there is a causal relationship between the variables. However, no method provides strong evidence about the causality because causality is not a property contained in the data.

The variable which is causing the change in another variable is said to be "independent variable", usually represented by X and the variable which is being influenced or changed by the change in independent variable is said to be "dependent variable", usually represented by Y. Sometimes both of them may influence each other. Sometimes, more than one independent variable influences the dependent variable.

One of the methods to determine the significance of the relationship between the two variables is "Chi-square test of independence". Under this method, the null hypothesis states that there is no significant relationship between the two variables (i.e., X and Y are independent) and the alternative hypothesis states that there is a significant relationship between the two variables (i.e., X and Y are dependent variables).

Given:

X =Amount of sleep (say, in hours)

Y =students' GPA

Create 2x2 contingency table:

Column variable factors:

Higher GPA (GPA $\geq$ 5.0 out of 10.0)

Lower GPA (GPA < 5.0 out of 10.0)

Row variable factors:

More amount of sleep ($\geq$ 6 hours)

Less amount of sleep (< 6 hours)

Take a sample of students and place the number of students in each box of the contingency table. For example, if 10 students has GPA of 5.0 or more and they had 6 hours or more amount of sleep, place number "10" in the first box which is common to GPA of 5.0 or more and 6 hours or more amount of sleep.

Example:

	More GPA (5 or more)	Less GPA (<5)	Total
More sleep (6 hrs or more)	10	4	14
Less sleep (<6 hrs)	6	20	26
Total	16	24	N =40

Conduct "Chi-square test of independence" to determine if the two variables, amount of sleep and GPA are independent or not independent.

You can also create 3x3 table or 2x3 table or 3x2 table or 2x4 table, etc,. depending on the problem and the requirement and carry out the above test.

Another method is to use "Chi-square test for observed and expected frequencies".

Divide the amount of sleep into some categories (say, 4 or 5 categories) and find out the total GPA of sampled students. Divide the sum by number of categories and the quotient is the expected GPA for each category. Compare these expected GPAs with those of observed GPAs in those corresponding categories. (expected total must be equal to observed total). Find out the test statistic, $\chi^2$ and compare it with the critical value of $\chi^2$ at some significance level (say, 0.05) (or use p-value method which is to be compared with the significance level) and make the conclusion.

Once again note that the causality is not a property contained in the data and no method provides any guarantee.