What is the direction of causality when two variables, A and B, have a strong linear correlation

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Distinguishing between what does or does not provide causal evidence is a key piece of data literacy. Determining causality is never perfect in the real world. However, there are a variety of experimental, statistical and research design techniques for finding evidence toward causal relationships: e.g., randomization, controlled experiments and predictive models with multiple variables. Beyond the intrinsic limitations of correlation tests (e.g., correlations cannot not measure trivariate, potentially causal relationships), it's important to understand that evidence for causation typically comes not from individual statistical tests but from careful experimental design.

Example: Heart disease, diet and exercise

For example, imagine again that we are health researchers, this time looking at a large dataset of disease rates, diet and other health behaviors. Suppose that we find two correlations: increased heart disease is correlated with higher fat diets (a positive correlation), and increased exercise is correlated with less heart disease (a negative correlation). Both of these correlations are large, and we find them reliably. Surely this provides a clue to causation, right?

In the case of this health data, correlation might suggest an underlying causal relationship, but without further work it does not establish it. Imagine that after finding these correlations, as a next step, we design a biological study which examines the ways that the body absorbs fat, and how this impacts the heart. Perhaps we find a mechanism through which higher fat consumption is stored in a way that leads to a specific strain on the heart. We might also take a closer look at exercise, and design a randomized, controlled experiment which finds that exercise interrupts the storage of fat, thereby leading to less strain on the heart.

All of these pieces of evidence fit together into an explanation: higher fat diets can indeed cause heart disease. And the original correlations still stood as we dove deeper into the problem: high fat diets and heart disease are linked!

But in this example, notice that our causal evidence was not provided by the correlation test itself, which simply examines the relationship between observational data (such as rates of heart disease and reported diet and exercise). Instead, we used an empirical research investigation to find evidence for this association.

Correlation and Causation

What are correlation and causation and how are they different?

For example, for the two variables "hours worked" and "income earned" there is a relationship between the two if the increase in hours worked is associated with an increase in income earned. If we consider the two variables "price" and "purchasing power", as the price of goods increases a person's ability to buy these goods decreases (assuming a constant income).

Correlation is a statistical measure (expressed as a number) that describes the size and direction of a relationship between two or more variables. A correlation between variables, however, does not automatically mean that the change in one variable is the cause of the change in the values of the other variable.

Causation indicates that one event is the result of the occurrence of the other event; i.e. there is a causal relationship between the two events. This is also referred to as cause and effect.

Theoretically, the difference between the two types of relationships are easy to identify — an action or occurrence can cause another (e.g. smoking causes an increase in the risk of developing lung cancer), or it can correlate with another (e.g. smoking is correlated with alcoholism, but it does not cause alcoholism). In practice, however, it remains difficult to clearly establish cause and effect, compared with establishing correlation.

The objective of much research or scientific analysis is to identify the extent to which one variable relates to another variable. For example:

• Is there a relationship between a person's education level and their health?
• Is pet ownership associated with living longer?
• Did a company's marketing campaign increase their product sales?

The coefficient's numerical value ranges from +1.0 to –1.0, which provides an indication of the strength and direction of the relationship.

If the correlation coefficient has a negative value (below 0) it indicates a negative relationship between the variables. This means that the variables move in opposite directions (ie when one increases the other decreases, or when one decreases the other increases).

If the correlation coefficient has a positive value (above 0) it indicates a positive relationship between the variables meaning that both variables move in tandem, i.e. as one variable decreases the other also decreases, or when one variable increases the other also increases.

Where the correlation coefficient is 0 this indicates there is no relationship between the variables (one variable can remain constant while the other increases or decreases).

While the correlation coefficient is a useful measure, it has its limitations:

Correlation coefficients are usually associated with measuring a linear relationship.

If, however, the tradesperson charges based on an initial call out fee and an hourly fee which progressively decreases the longer the job goes for, the relationship between hours worked and income would be non-linear, where the correlation coefficient may be closer to 0.

Causality is the area of statistics that is commonly misunderstood and misused by people in the mistaken belief that because the data shows a correlation that there is necessarily an underlying causal relationship

The use of a controlled study is the most effective way of establishing causality between variables. In a controlled study, the sample or population is split in two, with both groups being comparable in almost every way. The two groups then receive different treatments, and the outcomes of each group are assessed.

For example, in medical research, one group may receive a placebo while the other group is given a new type of medication. If the two groups have noticeably different outcomes, the different experiences may have caused the different outcomes.

Due to ethical reasons, there are limits to the use of controlled studies; it would not be appropriate to use two comparable groups and have one of them undergo a harmful activity while the other does not. To overcome this situation, observational studies are often used to investigate correlation and causation for the population of interest. The studies can look at the groups' behaviours and outcomes and observe any changes over time.

The objective of these studies is to provide statistical information to add to the other sources of information that would be required for the process of establishing whether or not causality exists between two variables.

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