Tuesday, December 5, 2023

# What Does Causation Mean In Math

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Correlation Doesnt Equal Causation: Crash Course Statistics #8

In a correlational research design, you collect data on your variables without manipulating them.

You find that physical activity level is positively correlated with self esteem: lower levels of physical activity are associated with lower self esteem, while higher levels of physical activity are associated with higher self esteem.

Correlational research is usually high in external validity, so you can generalize your findings to real life settings. But these studies are low in internal validity, which makes it difficult to causally connect changes in one variable to changes in the other.

These research designs are commonly used when its unethical, too costly, or too difficult to perform controlled experiments. They are also used to study relationships that arent expected to be causal.

You find a positive correlation between the variables: children who spend more time playing violent video games have higher rates of aggressive behavior.

## Causal Claims From Observational Studies

• Strong relationship: For example illness is four times as likely among people exposed to a possible cause as it is for those who are not exposed.
• Strong research design
• Temporal relationship: The cause must precede the effect.
• Dose-response relationship: Higher exposure leads to a higher proportion of people affected.
• Reversible association: Removal of the cause reduces the incidence of the effect.
• Consistency: Multiple studies in different locations producing similar effects
• Biological plausibility: there is a supportable biological mechanism
• Coherence with known facts.
• ## Main Properties Of Correlations

Their correlation can be classified as either:

• Weak
• Strong
• Perfect
• In the advanced blog post coming out next week, we will get into the statistical tests that you can do to determine the correlation strength, but here, well first focus on getting a better understanding of what correlation actually means and looks like.

The following graphs show the types of correlations mentioned above:

Across each column, we show first no correlation, then a weak correlation, a strong correlation, and a perfect correlation.

The first and second row shows a positive and negative linear correlation respectively.

• A positive correlation means that when one variable goes up, the other goes up.
• A negative correlation means that when one variable goes up, the other goes down.

As we can see, no correlation just shows no relationship at all: moving to the left or the right on the x-axis does not allow us to predict any change in the y-axis.

For example, there is no correlation between the weight of my cat and the price of a new computer they have no relationship to each other whatsoever.

A weak correlation means that we can see the positive or negative correlation trend when looking at the data from afar however, this trend is very weak and may disappear when you focus in a specific area.

For example, lets take the weak positive and weak negative linear correlation from above and zoom into the x region between 0 4.

This is what we may end up with:

#### Correlation Strength and Slope?

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## Correlation Vs Causation: All You Need To Know About

In this blog, we will going to share with you the difference between correlation vs Causation. Lets get started:-

Information or data in the correct hands can be immensely powerful. It is an important factor for any decision. The famous American statistician W.Edward Deming said the famous quote, In God we trust. Everyone else brings data.

Most of the time, data or information can be misconstruction or misunderstood. One of the major misunderstandings is that correlation and causation are similar.

Our world becomes more scientific day by day. Every subject or topic can measure by analysis of the data. For example, the measurement of the population of a particular country is by collecting the data by people who do surveys.

The statistics subject help in collecting the data and also help in arranging or managing the data. It helps in finding out the reasons, causes, or effects behind the changing conditions in the population. Statistics also help you in explaining correlation vs. causation. Through this blog, you will understand the difference between both.

First of all, we understand both concepts then, we will discuss the difference between correlation vs causation:

## Hills Criteria Of Causation

Determining whether a causal relationship exists requires far more in-depth subject area knowledge and contextual information than you can include in a hypothesis test. In 1965, Austin Hill, a medical statistician, tackled this question in a paper* thats become the standard. While he introduced it in the context of epidemiological research, you can apply the ideas to other fields.

Hill describes nine criteria to help establish causal connections. The goal is to satisfy as many criteria possible. No single criterion is sufficient. However, its often impossible to meet all the criteria. These criteria are an exercise in critical thought. They show you how to think about determining causation and highlight essential qualities to consider.

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## Act On The Right Correlations For Sustained Product Growth

We are always looking for patterns around us, so our default aim is to be able to explain what we see. However, unless causation can be clearly identified, it should be assumed that were only seeing correlation.

Events that seem to connect based on common sense cant be seen as causal unless youre able to prove a clear and direct connection. And, while causation and correlation can exist at the same time, correlation doesnt mean causation.

The more adept you become at identifying true correlations within your product, the better youll get at prioritizing your efforts for user engagement and retention.

## Why Doesnt Correlation Mean Causation

There are two main reasons why correlation isnt causation. These problems are important to identify for drawing sound scientific conclusions from research.

The third variable problem means that a confounding variable affects both variables to make them seem causally related when they are not. For example, ice cream sales and violent crime rates are closely correlated, but they are not causally linked with each other. Instead, hot temperatures, a third variable, affects both variables separately.

The directionality problem is when two variables correlate and might actually have a causal relationship, but its impossible to conclude which variable causes changes in the other. For example, vitamin D levels are correlated with depression, but its not clear whether low vitamin D causes depression, or whether depression causes reduced vitamin D intake.

Youll need to use an appropriate research design to distinguish between correlational and causal relationships.

Correlational research designs can only demonstrate correlational links between variables, while experimental designs can test causation.

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## How To Test For Causation In Your Product

Causal relationships dont happen by accident.

It might be tempting to associate two variables as cause and effect. But doing so without confirming causality in a robust analysis can lead to a false positive, where a causal relationship seems to exist, but actually isnt there. This can occur if you dont extensively test the relationship between a dependent and an independent variable.

False positives are problematic in generating product insights because they can mislead you to think you understand the link between important outcomes and user behaviors. For example, you might think you know which specific key activation event results in long-term user retention, but without rigorous testing you run the risk of basing important product decisions on the wrong user behavior.

## Why Correlation Does Not Imply Causation

Correlation CAN Imply Causation! | Statistics Misconceptions

Correlation and causation are terms which are mostly misunderstood and often used interchangeably. Understanding both the statistical terms is very important not only to make conclusions but more importantly, making correct conclusion at the end. In this blogpost we will understand why correlation does not imply causation.

A lot of times we have heard correlation does not cause causation or correlation does not imply causation or correlation is not causation. But what they mean actually by saying this?

You will get a clear idea once we go through this blogpost. So lets start!

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## Getting The Basics Right

Correlation is a statistical technique which tells us how strongly the pair of variables are linearly related and change together. It does not tell us why and how behind the relationship but it just says the relationship exists.

Example: Correlation between Ice cream sales and sunglasses sold.

As the sales of ice creams is increasing so do the sales of sunglasses.

Causation takes a step further than correlation. It says any change in the value of one variable will cause a change in the value of another variable, which means one variable makes other to happen. It is also referred as cause and effect.

Example: When a person is exercising then the amount of calories burning goes up every minute. Former is causing latter to happen.

So now we know what correlation and causation is, its time to understand Correlation does not imply causation! with a famous example.

Ice cream sales is correlated with homicides in New York

As the sales of ice cream rise and fall, so do the number of homicides. Does the consumption of ice cream causing the death of the people?

No. Two things are correlated doesnt mean one causes other.

Correlation does not mean causality or in our example, ice cream is not causing the death of people.

When 2 unrelated things tied together, so these can be either bound by causality or correlation.

## A Super Short Summary

Before we begin the blog post officially

I know some of you just want the quick, no fuss, one-sentence answer. So if youre here for the short answer of what the difference between causation vs correlation is, here it is:

Correlation is a relationship between two variables when one variable changes, the other variable also changes.

Causation is when there is a real-world explanation for why this is logically happening it implies a cause and effect.

So: causation is correlation with a reason.

If youre interested in reading the full explanation to properly understand the terms, the difference between them and learn from real-world examples, keep scrolling!

The days have passed where data was mainly used by researchers or accessible only to those with tremendous technical prowess. The times when getting data was a difficult ordeal that required months of manual tracking, survey design, or tracking code written from scratch are over.

Thank goodness.

In todays age, with everything under the sun being tracked and cataloged, everyone has abundant access to data. However, this abundant access can act as a large barrier between companies that become great and companies that dont.

People that know how to speak the language of data thus have a major advantage because they can wield this powerful tool.

Great product managers suggest product tests and changes based on extensive user research and product usage data.

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## Correlation And Causation Examples In Mobile Marketing

Correlations are everywhere. As conspiracy theory debunkers like to say: If you look long enough, youll see patterns.

In the same way, if you look long enough, you may begin to see cause-and-effect relationships in your mobile marketing data where there is only correlation. We try to find a reason why A and B occur at the same time.

See if you can spot which is which in these correlation and causation examples below:

• New web design implemented > > Web page traffic increasedWas the traffic increase because of the new design ? Or was traffic simply up organically at the time when the new design was released ?
• Uploaded new app store images > > Downloads increased by 2XDid downloads increase because of the new images in your app stores? Or did they just happen to occur at the same time?
• Push notification sent every Friday > > Uninstalls increase every FridayAre people uninstalling your app because of your weekly push notifications? Or is some other factor at play?
• Increase in links to your website > > Higher ranking in search engine results Does the increase in links directly cause the better search ranking? Or are they merely correlated?

To better understand correlation vs causation, lets begin by defining terms.

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## Causation And Hypothesis Tests

Before moving on to determining whether a relationship is causal, lets take a moment to reflect on why statistically significant hypothesis test results do not signify causation.

Hypothesis tests are inferential procedures. They allow you to use relatively small samples to draw conclusions about entire populations. For the topic of causation, we need to understand what statistical significance means.

When you see a relationship in sample data, whether it is a correlation coefficient, a difference between group means, or a regression coefficient, hypothesis tests help you determine whether your sample provides sufficient evidence to conclude that the relationship exists in the population. You can see it in your sample, but you need to know whether it exists in the population. Its possible that random sampling error produced the relationship in your sample.

Statistical significance indicates that you have sufficient evidence to conclude that the relationship you observe in the sample also exists in the population.

Thats it. It doesnt address causality at all.

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## What Is The Difference Between Correlation And Causation

Answer itcorrelation betweenin theCausationcausal relationship between

Then, what does a correlation coefficient tell us?

correlation coefficientcandoes

What does correlation tell us psychology?

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What does it mean to have a correlation?

correlationcancorrelation meansmeans

## An Example Of A Formal Way Of Thinking About Causality

To give you maybe a clearer and more mathematical way to look at causality, take the following example. Suppose you want to analyze the effect of hospitalization on health. Define $Y_i$ as some health measure of individual $i$ and $D_i \in \$ to indicate whether or not that individual was hospitalized. In our first attempt, suppose we look at the average difference in health of the two kinds of individuals:$$E – E.$$On first look at the data, you might notice, counter intuitively, that individuals that have been hospitalized actually have worse health than those that have not. However, going to the hospital certainly does not make people sicker. Rather, there is a selection bias. People who go to the hospital are those people that are in worse health. So this first measure does not work. Why? Because we are not interested in just the observed differences, but rather in the potential differences .

I went through your proof, and I think it is correct $). If$E = E$, then$E = E\cdot E$. Also, it works the other way. However, I don’t see where is your problem? • If$X$and$Y$are independent, then they have zero covariance. But • If$X$and$Y$have zero covariance, then they are not necessarily independent. • Example: consider following table:  Y X | -1 0 1 --+----------------------1 | 0.25 0 0.25 1 | 0 0.5 0 The values are probabilities, i.e.$P = 0.5\$ etc. Marginal probabilities for Y are 0.25, 0.5, 0.25, and 0.5 and 0.5 for X.

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## Correlation Vs Causation Example

My mother-in-law recently complained to me: Whenever I try to text message, my phone freezes. A quick look at her smartphone confirmed my suspicion: she had five game apps open at the same time plus Facebook and YouTube. The act of trying to send a text message wasnt causing the freeze, the lack of RAM was. But she immediately connected it with the last action she was doing before the freeze.

She was implying a causation where there was only a correlation.

## An Example Of Correlation Vs Causation In Product Analytics

Correlation and causality | Statistical studies | Probability and Statistics | Khan Academy

You might expect to find causality in your product, where specific user actions or behaviors result in a particular outcome.

Picture this: you just launched a new version of your mobile app. You make the key bet that user retention for your product is linked to in-app social behaviors. You ask your team to develop a new feature that allows users to join communities.

A month after you release and announce your new communities feature, adoption sits at about 20% of all users. Curious about whether communities impact retention, you create two equally-sized cohorts with randomly selected users. One cohort only has users who joined communities, and the other only has users who did not join communities.

Your analysis reveals a shocking finding: Users who joined at least one community are being retained at a rate far greater than the average user.

Nearly 90% of those who joined communities are still around on Day 1 compared to 50% of those who didnt. By Day 7, you see 60% retention in community-joiners and about 18% retention for those who were not. This seems like a massive coup.

But hold on. The rational you knows that you dont have enough information to conclude whether joining communities causes better retention. All you know is that the two are correlated.

## Necessary And Sufficient Causes

Causes may sometimes be distinguished into two types: necessary and sufficient. A third type of causation, which requires neither necessity nor sufficiency in and of itself, but which contributes to the effect, is called a “contributory cause”.

Necessary causes
If x is a necessary cause of y, then the presence of y necessarily implies the prior occurrence of x. The presence of x, however, does not imply that y will occur.
Sufficient causes
If x is a sufficient cause of y, then the presence of x necessarily implies the subsequent occurrence of y. However, another cause z may alternatively cause y. Thus the presence of y does not imply the prior occurrence of x.
Contributory causes
For some specific effect, in a singular case, a factor that is a contributory cause is one among several co-occurrent causes. It is implicit that all of them are contributory. For the specific effect, in general, there is no implication that a contributory cause is necessary, though it may be so. In general, a factor that is a contributory cause is not sufficient, because it is by definition accompanied by other causes, which would not count as causes if it were sufficient. For the specific effect, a factor that is on some occasions a contributory cause might on some other occasions be sufficient, but on those other occasions it would not be merely contributory.