Correlation

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Correlation is the statistical concept which describes the amount and type of relationship between two variables. Using correlations we can talk about whether two variables are related to each and how that relationship functions--whether it is a positive or direct relationship or a negative or inverse relationship. The detection and measurement of correlations in both nature and social science has added to our knowledge of ourselves and the physical world.

In correlational research we measure each participant by two different measurement operations (DVs).

For example, we may have a group of people and measure their height and their weight.

Or we might measure each person's golf score and the number of years that the person has played golf.

n correlational research each participant in the research is measured on two different dependent (or criterion) variables. Are these measurements unrelated to each other or are they somehow related?

For example, are the numbers which represent height somehow related to (vary with) the numbers which represent height?

The lecture graphic shows two columns of numbers. The first column represents big toe length and a second column shows IQ. Each person (little black figure) has two measurements, Big Toe Length in cm and IQ.

INTUITIVELY: I chose this rather silly example because most people can see that there ought to be no relationship between big toe length and IQ.

If the two variables DID vary together, would they vary positively (directly) or negatively (inversely).

The lecture graphic shows a column of numbers representing Height and a second column showing Weight. Each person (little black figures) has two measurements, Height and Weight.

First Question: IS THERE ANY RELATIONSHIP between the numbers in the first and second columns?

Second Question: If there is a relationship is it direct (positive) or inverse (negative)?

INTUITIVELY: I choose this example because most people can see that there ought to be a direct, positive relationship between Height and Weight.

The two variables move in the same direction. When one goes up the other goes up. When one goes down the other goes down.

When this is so we say that the relationship between the two variables is direct or positive. The correlation will be positive.

This time we will measure each person's golf score and the number of years that that person has been playing golf.

The lecture graphic shows a column of numbers representing GOLF SCORE and a second column showing YEARS OF PLAY. Each person (little black figure) has two measurements, GOLF SCORE and YEARS OF PLAY.

First Question: IS THERE ANY RELATIONSHIP between the numbers in the first and second columns?

Second Question: If there is a relationship is it direct (positive) or inverse (negative)?

We notice that as years of play goes up, the golf score goes down. These two variables move in opposite directions. As one gets larger, the other gets smaller.

INTUITIVELY: I choose this example because most people can see that there ought to be an inverse relationship between number of years of play and golf score. (Low scores in golf are better than high scores; so more practice, hopefully, should lower your score.)

As years goes up score goes down. People with more experience playing golf get lower (better) scores.

The two variables move in opposite directions. When one goes up the other goes down. When one goes down the other goes up.

When we encounter this type of relationship, we say that the two variables are inversely or negatively related. The correlation will be negative.

The correlation coefficient is a statistic (like the mean or the variance). It has a complicated formula. You enter the data in that formula and come out with a single number which is called the correlation coefficient. There are many kinds of correlation coefficients; we will only study "little r" in this class which is the most important of all of them.

The lecture graphic shows that the correlation coefficient (little r) can range from a -1 through 0 to a +1. If you get a value outside that range you have made a mistake calculating little r.

Notice that values of r less than 0 indicate a negative or inverse relationship between variables.

Notice that values of r greater than 0 indicate a positive or direct relationship between variables.

Finally, notice that a value of r = 0 indicates no relationship between variables. The farther the r value is from zero, the greater the relationship.

Let's look at some examples. Here is an example of a direct or positive relationship.
I think you'll find it to be something that's intuitively clear. Let's say you are in Liberty Park and you have a rolling outdoor food cart. You sell snacks and drinks.

The two variables we are going to measure are the temperature outside and cold drink sales. You count the number of cold drinks you sell each day. You also keep track of the high temperature each day.

Look at the graph. Each day is represented by a red dot. The day's temperature is given by distance along the horizontal axis. The day's number of cold drink sales is given by height along the vertical axis.

VISUALIZE A CORRELATION: What you discover is that when the temperature is low, cold drink sales tend to be low; as the temperature increases to moderate, cold drink sales tend to increase up to moderate levels; and finally, when the temperature is high, then the cold drink sales are also high. This relationship is shown by the pattern of red dots on the graph.

So as one variable goes up the other variable goes up. They are related to each other somehow, and they vary in the same direction. We say that they vary in a direct way or a positive way.

In this case the correlation coefficient, r, should come out to be a relatively high positive number. (Remember the highest positive correlation is +1.)

These kinds of graphs which allow us to visualize a correlation are called SCATTERPLOTS. We will talk about how to construct scatterplots from data a little later on.

In contrast then, if we had hot drink sales at the same food cart, we may get an inverse or negative relationship between outside temperature and hot drink sales. So then a graph might look like the one on the graphic.

You'll notice that the when the temperature is very low (toward the left on the horizontal axis), hot drinks sales tend to be high, and when temperature is high, the hot drink sales tend to be low.

And so there's a relationship between the two variables (or a correlation between these two variables), but in this case it's negative (or inverse) instead of positive (or direct).

In this case the correlation coefficient, r, should come out to be a relatively low negative number. (Remember the lowest negative correlation is -1.).

Let's look at two variables which might not have a relationship. Let's say cookie sales at your cart aren't related to temperature. Cookie sales might be high when the temperature is low or they might be low when the temperature is low. They might be low when the temperature is high, or they might be high when the temperature is high.

The current graph or scatterplot demonstrates a lack of relationship or a lack of correlation. And in this case then, the correlation coefficient, r, should come out somewhere in the neighborhood of zero.

Here is a graphical example of a perfect positive correlation. (I've just called the two variables x and y because in the behavioral sciences it's hard to think of a perfect relationship.) As x gets higher in value, y gets higher in value.

In a PERFECT positive relationship you could perfectly predict x from y and y from x. All the dots would fall in a straight line. The two variables appear to be sharing information completely.

In contrast then, a perfect negative relationship, (where r = -1) would also be a straight line, but in the other direction. As x gets higher in value, y gets lower.

How did we get those graphs (for example for Temperature and Cold Drink Sales) in the previous section? We'll talk about how to do it now. Those kinds of graphs are called scatterplots.

Let's use an example. A researcher is going to make two measurements on each person in a group of N = 6 people. Another way to say that is that there will be two DV's. For the first DV, the researcher will get a self-report rating of how much exercise the person does (from 1, little, to 10, lots). The rating is meant to cover the range from "complete couch potato" to "avid exercise fanatic." That's variable X.

The second DV measurement is a 1 to 10 rating of each person's health. That's Y.

The researcher has two dependent (or criterion) variables. S/he has a sample of N = 6 subjects.

Notice on the lecture graphic that Person #1 has an exercise rating (X) of 9 and a health rating (Y) of 10. Look at the graphic until you can clearly identify person # 1's data. The data for person #1 are two numbers, 9 and 10.

Each subject has two data points. Person #2 was rated as a 1 on exercise and a 3 on health. Person #3 was rated as 9 on X (exercise) and 6 on Y (health). And so forth. To keep the example simple, only six subjects are in the study. Each person is measured on two variables.

(NOTE: In the kind of experimental research we've been doing up to now we generally just measure one aspect of the person, their mental health or whatever our examples happen to be about. But this time we measure two aspects of a person, because we want to know if there's a relationship between these two aspects of a person.)

The graphic next to this text is identical to the one above. But now we are going to construct a scatterplot. A scatterplot is a way of taking the data from a correlational research project and visualizing it on a graph.

We put one of the dependent variables (say, X) on the horizontal axis and the other DV (say, Y) on the vertical axis.

That way, each person's data will be represented by a single point on the scatterplot.

Let's see what the scatterplot would look like for our data with the exercise and health rating variables. We will put the data for each of the 6 people on the graph. That will give us our scatterplot.

For example, person #1 has X = 9 and Y = 10. You can see that person on the graph as the red dot which above 9 on the horizontal (X) axis and which is straight out from 10 on the vertical (Y) axis. Repeat this graphing process with each of the other 5 people in the sample.

You can immediately see from the overall scatterplot that there's a positive relationship between X and Y. When exercise(X) is low health (Y) tends to be low. When exercise is high, health tends to be high.

That's the whole idea of scatter plot. It's a simple way of visualizing data. You'll be asked to make some scatterplots in the homework and on exams. You should also be able to read one.

Here is the formula for r. Please copy it down in your notes. There are many possible forms of this formula, but this is the one we will be using at first.

Up to this point we have developed visual (graphical) representations of the relationship between two variables called scatterplots.

Now we're going to run the data through the formula you see here and get a single number called a correlation coefficient. This formula is called the Pearson Product Moment Correlation Coefficient or the Pearson r, or simply "little r."

We've already given the range of little r. r goes from -1 through 0 to +1. We've shown examples of scatterplots with values of little r so that you have some sense how to interpret values of r.

The Pearson r is just another descriptive statistic like the mean and the standard deviation. The mean describes the center of a single set of numbers, the standard deviation describes the spread-out-ness within single set of numbers, and r describes the relationship between two sets of numbers.

If you look at the formula it appears complex. It is. The good news is that there's only one new term in the formula, sum of X times Y (the sum of XY). All the rest of the terms will be familiar from your practice in calculating the mean and the variance. These familiar terms are the sum of X or the sum of X squared and so forth.

Preliminary calculations

Here's the data from the exercise and health example. Six volunteers are measured twice each. One measure is their exercise level; the other measure is health rating. So the six people generate 12 data points. The formula for r uses N, which is the number of people. So N = 6. (Occasionally students think that N is twelve because there are twelve scores; but that is wrong). N = 6.

What I suggest doing as you learn and on exams is to make several columns. The data are given in an X column (exercise) and a Y column (health). But let's make three more columns.

Let's call the first new column X times Y (often called the cross product). The second new column will be X squared. The third new column will be Y squared. These three columns allow you to get the most important information you will need to calculate r.

CROSS PRODUCTS: (Besides the two data columns, X and Y) the first new column is X times Y (the cross product). So let's concentrate on that column. You'll notice what X times Y is just what it says, we take the first person and multiply his or her X-score times his or her Y score. For the first person we get 9 times 10 and that's 90. This is called the cross product for that person's data.

The second person in the cross-products column has an X score of 1 which multiplied by the Y score of 3. 1 times 3 is 3.

And so forth all the way down, so jot that information down in your notes. It's better learning if you do all the work in your notes first, and then check your answers against the graphic.

X SQUARED: The next column of course, is just X squared, so you just take the X value for the first subject, 9, and square it. That gives you 81. Go all the way down the column squaring the X score for each participant.

Y SQUARED: The last column is the Y scores squared. Again, it is best if you do the work first in your notes and then check your work against the graphic.

You end up with these three new columns of numbers next to your two data columns.

THE SUMS:Finally, you must sum up all five columns (the two data columns, and three new columns). You need sum of X, sum of Y, sum of XY, sum of X squared and sum of Y squared. Those 5 sums will be central to calculating the Pearson r.

Substituting into the formula

The next step is to substitute the various sums we calculated into the formula on the left.

In the previous graphic, we calculated that the sum of the cross-products (the sum of XY) is 225, the sum of X squared is 246, and the sum of Y squared is 219. We also found that the sum of X is 34 and the sum of Y is 33.

I recommend that you substitute those sums into the formula for r right now and then check your work on the next lecture graphic.

Calculating and Interpreting r

Check your substitution against the graphic to be sure that you have done it correctly.

CARRY 5 DECIMALS: Because of the severe problems rounding error can cause in this formula please carry your working calculations to 5 decimals. That way we can check your answer for correctness to 2 or 3 decimals.

DO the arithmetic. I suggest that you do the arithmetic on your own and then check your result against the detailed computations shown on the current lecture graphic.

THE CORRELATION COEFFICIENT: The final result is that r = +0.8497. From our work with visualizing correlations on scatterplots you should have a good idea about how to interpret this coefficient.

INTERPRETATION: r can only vary from -1 to +1. A plus sign indicates a positive (direct) relationship. Because +0.8497 is well toward +1 you have a sense that exercise and health ratings are directly related to a substantial degree.

Look at the scatterplot for exercise and health, think of the range of r, and form an opinion about what an r in the neighborhood of +.85 means.

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Correlational Research Issues

Let's deal with some scientific issues regarding correlational research. The first one is to be aware of the factors which affect the size of the correlation coefficient. There are a number of factors that can effect the size of the Pearson r.

These issues are methodological, that is, they are an important part of a scientific thinking and inference. But they are so intimately tied to the value you get when you calculate r, that they must be discussed in a statistics course. Remember, by the process of abduction (sideways thinking) we routinely go back and forth between our statistical models and our scientific theories.

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Truncated or restricted range are two names for the same methodological issue. Truncate means to cut something off. It's the same as restricting how far something will go. Suppose that we've decided to find the relationship between X and Y, but we've restricted the range of one of them, X.

(Spurious in this context means originating from incorrect or erroneous procedures.)

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Restricted or Truncated Range - Spuriously Low

Imagine an example in which Y is college GPA and X is college entrance exam (ACT or SAT) score. Almost everyone has to take an ACT or SAT test to get into college. We want to know if there's a relationship between college GPA and entrance exam score. If college entrance exams (such as ACT or SAT) work, a reasonable scientific hypothesis would be that there should be a positive relationship between entrance exam score and college GPA.

If you look at the full pattern of data points on the entire scatterplot in the graphic on the left, there seems to be a decent positive relationship between X and Y. As X (entrance exam score) goes up Y (GPA) tends to go up. The relationship is not perfect but it's clearly positive. For the whole graph, the actual r = +0.90.

However, suppose we select the only those dots beyond the green line on the scatterplot. Those are the dots highlighted in the circle. Another way of saying this is that we are going to truncate or restrict the range of X (entrance exam scores). We will only look at the correlation between X and Y for those X scores that are higher than the green line.

Consider the dots within the circle as a small scatterplot. Notice that within the circle it looks like there is no relationship between X and Y. In other words, in the restricted range above the green line there is no relationship between X and Y. The calculated r = +0.05 for the dots in the circle. The correlation coefficient for the dots in the restricted range (above the green line) is spuriously low compared the r computed on the whole scatterplot.

Now you may be asking yourself, why would anyone select just a small part of their data to perform a correlation on? Unfortunately, in this example, you can't help restricting the range.

Let's say that the college where the data is gathered has admission standards. One standard is that the college admits no one below a certain entrance exam score. Suppose the green line represents the college's ACT or SAT admission standards. This means that only those people who score above the green line on X (entrance exam) will be admitted to college. Therefore they will be the only ones ever to get a college GPA at that college.

The potential students to the left of the green line are just that--potential. They were not admitted to that college because their entrance exam score was too low. Therefore, they really don't have GPA's at that college. And so the data points to the left of the green line don't actually exist. How can you measure someone's GPA at a college s/he never attended?

This can be a subtle problem. It might seem sensible to go to a college and measure the relationship between GPA and entrance exam score. But it may not be obvious that admission standards have almost certainly truncated the range of X. In effect you are only measuring the relationship between X and Y in a very restricted range of X. In that restricted range, the r is spuriously low.

This is not really a statistical problem. It is a problem with how the data were collected in the realm of science.

Restricted or Truncated Range - Spuriously High

The second scatterplot shows how you could get a spuriously high value of r by restricting the range of X.

Looking at the graphic on the left, you can see that if you consider the entire relationship between X and Y on the scatterplot there appears to be no (or little) relationship between X and Y. The overall correlation between X and Y is +0.05

In contrast, if you only measured in the restricted range above the green line (shown here by the shaded area), you would believe you were seeing a strong positive relationship. In this truncated range r = +0.90, which is spuriously high.

An outlier is a score that so different from all the other scores that it does not seem to be part of the same data set.

On the lower scatterplot on the lecture graphic, the majority of the data is clustered in one spot and then there is one score far from the rest of the data. You can see from that scatterplot that that score can greatly increase the visual impression of a positive correlation. It also affects the correlation coefficient. On the lower scatterplot the correlation coefficient would be +0.70 while on the upper scatterplot, where there is no outlier, r would be +0.06.

One strange piece of data can change r and along with it your scientific conclusion.

Outliers might be theoretically important or they might just be measurement errors. For example you might trace the outlier back to a data sheet where someone misplaced a decimal point, turning a 2.17 into a 21.7. If you're very sure you have found the error then the data can be corrected.

But there is a strong and important scientific ethic that says that data should not be altered or thrown out. It is possible that an outlier, while not understandable right now, may be a hint for future theoretical breakthroughs. If you have an outlier you might report that you have one and then report the statistics, in this case r, both with and without the outlier. That way other people who read your work have the choice of deciding whether the outlier is trivial or important.

The statistical theory behind the Pearson r assumes that the relationship between X and Y is linear. (We will define linear relationship in some detail in the next lecture on regression.) Pearson r does not accurately describe relationship unless the relationship is linear.

Look at the scatterplot on the graphic on the left. The shape formed by the dots is an inverted U. It starts out (on the right of the graph) as a positive relationship but then turns downward and becomes a negative relationship (on the left part of the graph). Therefore it indicates there is a curvilinear relationship between X and Y.

One curvilinear relationship which is well known in psychology is between performance and arousal. Arousal and performance are related by an inverted U shaped curve.

Suppose we are researching performance on shooting free throws. Performance for low levels of arousal tends to be poor. If a person is not motivated at all, doesn't care, and is too relaxed and lethargic, then performance is probably going to be poor.

As motivation and arousal go up, say the person is practicing with teammates and watched by coaches, performance will go up.

And as arousal increases beyond some optimum level, performance can decline. If the motivation is extremely high, like shooting free throws in the last seconds of a game in the NBA finals, an excellent free throw shooter can inexplicably miss shots. A certain amount of tension is necessary for good performance, but being too tense can be a detriment to performance. That's why many people work on psychological control of internal state so that they can maintain a peak arousal level even though the external circumstances are changing. A world class athlete like a woman on the balance beam in the Olympics wants to be able maintain her internal state at the perfect level of arousal.

EFFECTS OF CURVILINEARITY ON r: As we said, the Pearson correlation coefficient assumes a linear relationship between two variables. If the relationship is not linear the value of r will be attenuated (moved closer to 0).

In the inverted U shaped relationship we have been discussing, the value of r would be near 0. Look at the curve. At first there is a strong positive relationship. Then there is a strong negative relationship. Little r will average these two trends and give you a value near 0. An r value near 0 should indicate a little or no relationship between variables. But there is a strong and clear relationship between arousal and performance. In effect, r = 0 in this case would be inaccurate and misleading.

When the relationship between variables is not linear, r is not a good measure of relationship.

How do you know if the relationship is linear or not? Draw the scatterplot and look at the shape formed by the dots.

In advanced statistics there are other correlational measures that measure relationship for nonlinear cases.

We are now going to summarize some issues of research design that impact (1) on proper choice of which statistical procedures to use and (2) upon the proper interpretation of statistical results. These issues will be covered in more depth in research design; but they are worth mentioning in a statistics class. Besides the worthwhileness of learning about these ideas, you will have to understand the distinction between a True Experiment and a Correlational Study to do research in Virtual Lab properly.

IV's and DV's. In the Interface to Science lecture we talked about Independent Variables (IV's) and Dependent Variables (DV's). Review IV's and DV's if you need to. Remember that terms, Dependent Variable and Criterion Variable, are often used interchangeably.

Scientists do research in many ways, using many paradigms. As the graphic indicates, three important distinctions in research studies are among True Experiments, Quasi-Experiments, and Correlational Studies.

In this course we will only need to use the distinction between a True Experiment and a simple Correlational Study. So we have crossed out some of the distinctions that you will study later in research methods. Just focus on the parts of the graphic that are not crossed out.

We won't do any more than mention that there are Quasi-experiments nor will talk about more complicated forms of correlational studies

What does stress have to do with sleep disruption? The question includes two variables, stress and sleep disruption.

To examine how this question can be pursued in both a true experiment and a correlational study, we will develop two parallel examples. In both cases sleep disruption will be a DV. But stress will be an IV in the true experiment and a DV in the correlational study.

So the same variable, stress, might be an IV or a DV, depending on how the study is run.

As you can see from the graphic, common forms of research present us with more complex cases. We have crossed out those because we will not need them in this course.

Strong Implications for causality. Why make these research distinctions? Because the type of research design has strong implications for whether or not we can conclude whether one variable causes changes in another variable.

Virtual Lab. We have not started using Virtual Lab yet. But when we do and when you read a scientific puzzle in Virtual Lab, you will have to decide if the puzzle is asking you to do experimental or correlational research. Then you will have to go and design a study (either experimental or correlational). So Virtual Lab will give you practice using this distinction.

Conclusions of causality require a true experiment. Scientists are often fascinated with causality. What causes what? If you want to discover if an IV actually causes changes in a DV, then you need to do a well-run, true experiment. What makes an experiment well run is a source of constant debate among scientists. The main issues in how to design a well-run experiment are extensive and will be covered in research methods.

Suffice it to say that before you worry about what is well-run, you need to be sure that as a minimum you are running a true experiment if you want to impute causality between variables.

Causality cannot be concluded from correlation alone. No matter how high the correlation between two variables is you cannot conclude that one of them causes the other based on a correlational study alone.

Why Not? We will now go on to the next topic where we will discuss why you cannot impute causality from correlation alone.

Causal Hypotheses. Suppose the scientific hypothesis is causal. The scientific hypothesis is that changes in the IV cause changes in the DV.

Example. Using the stress and sleep example, the causal scientific hypothesis would be that high levels of stress cause sleep disruption. The scientists want to make the point that stress causes sleep disruption.

Correlational Study. Suppose the scientist do a correlational study like the one we just discussed above. They ask a group of 20 human volunteers to rate the stress level of their lives during the last month. They also ask the volunteers to count the number of nights in the last month that their sleep is seriously disrupted.

Hypothetical Results. Suppose they get hypothetical results we saw above in the scatterplot. Suppose also that the correlation coefficient = +0.72.

A poor conclusion. Suppose they conclude from their results that stress causes sleep disruption. Based on the results of the correlational study alone, this conclusion is open to severe criticism.

Six competing hypotheses. The graphic outlines six possible hypotheses that all compete with each other to explain the correlation between stress and sleep disruption.

The graphic uses the symbol IV for the supposed causal variable (stress) even though the correlational study did not have a true IV; it had only two DV's.

Plausible Competing Hypotheses. The graphic lists 6 hypotheses. As we shall see, they are all plausible and they all compete with each other to explain the correlation found in the study.

1. Scientific Hypothesis. It can be hypothesized that stress causes changes in sleep disruption. In light of the high correlation that was found, r = +0.72, the scientific hypothesis is a plausible explanation of the results. After all, if stress does cause sleep disruption, you would expect a high correlation between stress and sleep..

2. Reverse Causality. It can be hypothesized that the supposed dependent variable (disrupted sleep) is actually the causal variable (not visa versa). That is, disrupted sleep causes people to experience stress. Reverse causality is also plausible in our example.

3. Systemic Causality. It can be hypothesized that the two variables are part of a system. As such they mutually cause each other. In the example, the more disrupted people's sleep is the higher will be their stress and the higher their stress is the the more disrupted will be their sleep. In other words, stress disrupts sleep and disrupted sleep causes stress. There is a vicious circle in which stress and disrupted sleep maintain each other. Systemic causality is plausible in our example.

4. Third variables. Perhaps sleep have nothing to do with each other. Rather, it can be hypothesized that changes in both are caused by a third variable. Maybe adrenal gland output is the third variable. (The adrenal gland puts out adrenaline, the hormone that gives a person a zap of "fight and flight" energy.) Maybe people whose adrenal glands are producing higher amounts of adrenal experience both stress and sleep disruption. The self reports of stress and the self reports of sleep disruption are correlated because they are both caused by a third variable, adrenal output. In our example, a third variable hypothesis is plausible. You can easily make up more. Maybe the third variable is environmental noise levels, which cause both stress and disrupted sleep.

5. Multiple Causality. It can be hypothesized that many variables contribute to sleep disruption (diet, exercise, pain, health, security, etc.). Stress might, in fact, cause sleep disruption but there are so many contributing factors to sleep disruption that stress alone might not have an effect unless several other causal variables are present. In our example it is plausible that while stress contributes to sleep disruption, stress alone does not cause sleep disruption. Many other predisposing factors must also be present. It is plausible to hypothesize that the effects of stress are highly conditioned by other causal agents.

6. Chance alone. It can be hypothesized that there is no link between stress and sleep disruption. The correlation we found was a quirky, chance occurrence in the data. If we ran the study again, we would not find the same results.

Chance sounds at first hearing like a strange competing hypotheses. But in fact it is a very serious challenge to any empirical support for any scientific hypothesis. A great deal of the last half of this course will focus on how scientists use statistics to argue that their results did NOT occur by chance alone. As odd as it seems when you first come across this idea, chance is considered to be a plausible explanation of results in any research.

From this list we learn that an correlation between two variables can be explained by a number of plausible competing hypotheses. The scientific hypothesis is only one of many plausible competing hypotheses. That is, if the scientific hypothesis is true and if stress does cause sleep disruption, that causal relationship would explain the correlation, r = +0.72, between stress and sleep disruption. But, as we've discussed, it is not the only explanation. There are several other classes of hypothesis that are (1) plausible and (2) also explain the correlation. To infer causality between two variables, you must argue against all other plausible competing hypotheses that other scientists come up with. This is a very difficult task and is one of the fun sources of discourse in science. It is also one of the major foci of a research methods class.

Certain criteria are generally held to be minimal requirements for the inference that one variable (let's call it the IV) causes changes in the other (let's call it the DV).

We will just quickly mention a few positive guides toward making causal inferences.

Cause precedes the effect. In logic, the cause must precede the effect. If someone wanted to infer that stress causes sleep disruption, they could not do so if the sleep disruption occurred first and the stress occurred the next day. As a minimum the stressor would have to be first, and the sleep disruption second. Cause first, effect second.

A fallacy. But that which comes before does not necessarily cause that which comes after. In logic, people talk of the "after which, therefore because of which" fallacy. Just because a person has a stressful experience during the day does not prove that the sleep disruption that happens that night is caused by the stress. It might be; or it might not. Maybe the stress is irrelevant and one of many other variables caused the sleep disruption.

Temporal Contiguity. Causes and effects are generally considered to be close in time. The cause first and then, quickly, the effect. That is sensible. Still, when the Surgeon General says that smoking causes cancer it's clear that we are talking about a causal chain that extends over many years. Presumably, smoke causes a series of intermediate causal mechanisms which eventual cause cancer.

While a high correlation does not mean causation, there must be a high correlation between changes in the causal variable and effect variable. If stress does cause sleep disruption, you would expect to find a high correlation between changes in stress level and changes in sleep.

A difficult criterion. To infer causality, a researcher must eliminate other plausible competing hypotheses. As we pointed out in the previous section other scientists will generate many plausible hypotheses which compete to explain to explain the results of research. The power of the experimental method is that it gives researchers tools for eliminating plausible competing hypotheses. For example, in the true experiment with rats, the scientists could eliminate a third variable hypothesis that noise levels are causing sleep disruption because they are housing both groups of rats in the same quiet area. They could eliminate reverse causality because they actively manipulate stress versus no stress and know that it is the stress which comes first, not the sleep disruption. And so on.

The great intellectual adventure of science includes this interplay between plausible competing hypotheses and experimental design.

PCH of Chance. Now we have gone a bit afield, beyond statistics. Still this discussion is relevant to the proper use of statistics and how they fit into the scientific adventure. We will spend almost half the course in talking about how statistics are used as a tool for eliminating the plausible competing hypothesis of chance. For now, just let this all settle at a common sense level.