12: Correlations


A common theme throughout statistics is the notion that individuals will differ on different characteristics and traits, which we call variance. In inferential statistics and hypothesis testing, our goal is to find systematic reasons for differences and rule out random chance as the cause. By doing this, we are using information on a different variable – which so far has been group membership like in ANOVA – to explain this variance. In correlations, we will instead use a continuous variable to account for the variance.

• 12.1: Variability and Covariance
Because we have two continuous variables, we will have two characteristics or score on which people will vary. What we want to know is do people vary on the scores together. That is, as one score changes, does the other score also change in a predictable or consistent way? This notion of variables differing together is called covariance (the prefix “co” meaning “together”).
• 12.2: Visualizing Relations
Visualizing data remains an important first step in understanding and describing out data before we move into inferential statistics. Nowhere is this more important than in correlation. Correlations are visualized by a scatterplot, where our X variable values are plotted on the X -axis, the Y variable values are plotted on the Y -axis, and each point or marker in the plot represents a single person’s score on X and Y.
• 12.3: Three Characteristics
When we talk about correlations, there are three traits that we need to know in order to truly understand the relation (or lack of relation) between X and Y : form, direction, and magnitude. We will discuss each of them in turn.
• 12.4: Pearson’s r
There are several different types of correlation coefficients, but we will only focus on the most common: Pearson’s r. r is a very popular correlation coefficient for assessing linear relations, and it serves as both a descriptive statistic and as a test statistic. It is descriptive because it describes what is happening in the scatterplot; r will have both a sign (+/–) for the direction and a number (0 – 1 in absolute value) for the magnitude.
• 12.5: Anxiety and Depression
Anxiety and depression are often reported to be highly linked (or “comorbid”). Our hypothesis testing procedure follows the same four-step process as before, starting with our null and alternative hypotheses. We will look for a positive relation between our variables among a group of 10 people because that is what we would expect based on them being comorbid.
• 12.6: Effect Size
• 12.7: Correlation versus Causation
• 12.8: Final Considerations
• 12.E: Correlations (Exercises)

Thumbnail: Correlation shown when the two variables' ranges are unrestricted, and when the range of is restricted to the interval (0,1). (CC BY 3.0 Unported; Skbkekas via Wikipedia)

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