{"id":504,"date":"2021-08-13T00:35:37","date_gmt":"2021-08-13T04:35:37","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/statspsych\/?post_type=chapter&p=504"},"modified":"2022-06-04T03:30:10","modified_gmt":"2022-06-04T07:30:10","slug":"chapter-10","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/statspsych\/chapter\/chapter-10\/","title":{"raw":"10. Correlation and Regression","rendered":"10. Correlation and Regression"},"content":{"raw":"

10a. Correlation<\/h1>\r\nThis chapter marks a big shift from the inferential techniques we have learned to date. Here we will be looking at relationships between two numeric variables, rather than analyzing the differences between the means of two or more experimental groups.\r\n\r\n[pb_glossary id=\"505\"]Correlation[\/pb_glossary]<\/strong> is used to test the direction and strength of the relationships between two numerical variables. We will see how scatterplots can be used to plot variable X against variable Y to detect linear relationships. The slop of the linear relationship can be positive or negative, which reveals systematic patterns in how the two variables co-relate. We will also look at the theory of correlational<\/strong> analysis, including some cautions around interpreting the results of correlational<\/strong> analyses. Thanks to the third variable problem, correlation<\/strong> does NOT equal causation, a mantra that should be familiar from your introductory psychology courses. And finally, we will try calculating correlation by partitioning [pb_glossary id=\"507\"]covariance[\/pb_glossary]<\/strong>, and put it all into practice in a hypothesis test. Later in the chapter, we will build in [pb_glossary id=\"509\"]regression[\/pb_glossary]<\/strong>, which allows us to predict the future from the past.\r\n\r\nJust like a bar graph is helpful to examine visually the differences among means, a scatterplot allows us to visualize the pattern that represents the relationship between two numeric variables, X vs. Y.\r\n\r\nIf the trend line that best indicates the linear pattern in the scatter plot has an upward slope, we consider that a positive directionality.\"\"\r\n\r\nTo find out if there appears to be a positive correlation, you can ask yourself \u201care those that score high on one variable likely to score high on the other?\u201d Here we see an example: what is the relationship between feline friendliness and number of scritches received? As you can see, when cat friendliness is high, the cuddles received is also high. There is a clear positive trend line. This make sense \u2013 people may be more likely to offer cuddles to a cat that solicits them.\r\n\r\n\"\"\r\n\r\nA downward slope indicates a negative directionality. To find out if there appears to be a negative correlation, you can ask yourself \u201care those that score high on one variable likely to score low on the other?\u201d Here you can see an example: what is the relationship between feline aloofness and the number of scritches received? There is a clear negative trend line. This makes logical sense, because people may be less likely to offer cuddles to a cat that keeps to itself.\r\n\r\nWhen we look at a scatter plot, we want to ask ourselves two questions: one about the apparent strength of the relationship between the variables, and the other about the direction of the relationship. Let us take a look at a few examples.\r\n\r\n\"\"\r\n\r\nIn graph A), if we ask \u201care variables X and Y strongly or weakly related?\u201d We would say strongly related. This is because the points on the scatter plot are in a perfect line. There is no distance between the points and the trend line. It is a perfect correlation<\/strong>. If we ask \u201cis the trend line positive or negative in slope?\u201d We would say that it is negative in slope. As scores on variable X increase, scores on variable Y do the opposite \u2013 they decrease. We might expect such a relationship if we plotted speed against time. The faster something is, the less time it takes. In the next example, graph B), if we ask \u201care variables X and Y strongly or weakly related?\u201d We would say weakly related. There is no clear linear trend that can be visually discerned \u2013 it just looks like a random scatter of dots. With no trend line, the question about directionality is irrelevant. This correlation<\/strong> is close to zero, so it is neither positive nor negative as a directional relationship. If we look at example C), the strength is not quite as perfect as in the first example, but the dots would not be very distant from a trend line through them, so this would be a fairly strong correlation<\/strong>. As scores on variable X go up, so do those on variable Y, making this a positive correlation<\/strong>. Now it is your turn. In example D), would this be a strong relationship or weak? Or somewhere in between? And do you see a positive or a negative slope to a trendline that runs through the cloud?\r\n\r\n[h5p id=\"79\"]\r\n\r\nCorrelational<\/strong> analysis seeks to answer the question \u201chow closely related are two variables.\u201d This is a very useful analytical approach when we have two numeric variables and we wish to analyze the patterns in how they co-vary. However, correlational<\/strong> analyses have limitations that it is vital to be aware of.\r\n\r\nFirst, the correlational<\/strong> method we will cover in this course is only capable of detecting linear relationships. Patterns that have a curve to them will not be captured by the correlation<\/strong> formula we will use.\r\n\r\nSecondly, correlation<\/strong> does not<\/em> equal causation. Correlational<\/strong> research designs do not allow for causal interpretations, because the third variable problem renders correlational<\/strong> analyses vulnerable to spurious results. When we measure two variables at the same time and plot them against each other, what we can do is describe<\/em> their relationship. We can even test whether the strength of their relationship is significantly different from zero. However, we cannot determine whether X causes<\/em> Y.\r\n\r\n\"\"For example, if we measured the consumption of ice cream as well as drowning deaths on a sample of days throughout the year, we might determine that there is a strong positive relationship between the two variables. Consumption of ice cream and drowning deaths are apparently closely related phenomena. But does consumption of ice cream cause drowning deaths? That seems a little far fetched. Could there be another explanation for the pattern? Is there a third variable that could in fact explain the trends in each of the two variables measured here? What might cause people to consume more ice cream as well as put themselves at greater risk for drowning? Warm weather perhaps? If we were to plot temperate against ice cream and drowning deaths, would we see positive correlations with each? Very likely. With this third variable connecting the two, it would be a logical fallacy to interpret the apparent correlation<\/strong> shown here as meaningful. But then again, is it possible that consuming ice cream could be a risk factor for drowning? Did an elder ever tell you that you should not eat right before swimming, because you might cramp up and drown? Maybe there is some truth to that.\r\n\r\nSo how could we find out whether there is a true causal relationship between two variables? In order to make cause-effect conclusions, we must use an experimental design. Two major features of experimental research designs eliminate the logical fallacies associated with correlations<\/strong>. First, an experiment makes use of random assignment of participants to conditions, because that controls for extraneous variables like the third variable of temperature in this example. And secondly, an experiment manipulates the independent variable, to establish a cause, and then measure effects.\r\n
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Requirements for cause-effect conclusions<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nA true experiment requires the following elements in order to control for extraneous variables and establish cause-effect directionality:\r\n