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Chapter 11

Reading Notes - Chapter 11

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PSYC 2030
Krista Phillips

Chapter 11: Correlating Variables What Are Different Forms of Correlations?  Researchers view variables not in isolation, but as systematically and meaningfully associated with, or related to, other variables  Correlation coefficient -> a single number that can be used to indicate the strength of association between two variables (X and Y) – this chapter elaborates on this  In particular, we describe linearity -> correlations that reflect the degree to which mutual relations between X and Y resemble a straight line  The Pearson r -> short for Karl Pearson’s product-moment correlation coefficient, is the correlation coefficient of choice in such situations  Values of r of 1.0 (positive or negative) indicate a perfect linear relation (a fixed change in one variable is always associated with a fixed change in the other variable), whereas 0 indicates that neither X nor Y can be predicated from the other by use of a linear equation  A positive r tells us that an increase in X is associated with an increase in Y, whereas a negative r indicates that an increase in X is associated with a decrease in Y  We begin by examining what different values of r might look like.  Then we go through the steps in computing the correlation coefficient when raw data have different characteristics  The common names “Pearson r”, “point-biserial r,” and “phi” listed in the table (table 11.1), communicate whether the values of X and Y are continuous or dichotomous, although the name Pearson r also is often used in a general way to refer to any correlation computer as product- moment r  Continuous variable -> means that it is possible to imagine another value falling between any two adjacent scores  Dichotomous variable -> the variable is divided into two distinct or separate parts o Ex. Someone who studies the discrimination of pitch (highness or lowness of a tone) might be interested in correlating the changes in the frequency of sound waves (X) with the differing ability of individuals to discriminate those changes (Y). Both variables are continuous, in that we can imagine a score of 1.5 between 1 and 2 or 1.55 between 1.5 and 1.6. o Suppose a researcher was interested in correlating participants’ gender with the ability to discriminate pitch o Pitch discrimination is a continuous variable, whereas gender is dichotomously coded as male and female Box 11.1 Galton, Pearson and r  In chapter 1 where we first discussed the idea of how empirical reasoning is used in behavioral research, we mentioned Francis Galton’s fascinating relational study using longevity data to test the efficacy of certain prayers  Galton was also very intuitive about statistics and he instinctively came up with a way of measuring the “co-relation” between two variables  At the time, another of his many interesting projects concerned the relationship between the traits of father and their adult sons  One day, while he was strolling around the grounds of a castle, it started to rain and Galton sought refuge in the recess of a rock by the side of the pathway  It was there, he later recalled, that while thinking about his research, the notion of statistical correlation initially flashed across his mind  Though the word correlation was already in widespread use in physics, it is believed that Galton’s initial spelling of “co-relation” might have been a way of distancing his creation from the commonly used concept  Though the statistical concept for which he is best known is correlation, Galton did not develop the idea beyond its use in some of his relational studies  The reason that r is called the Pearson r is that it was Karl Pearson who perfected Galton’s “index of co-relation” in a more mathematically sophisticated way  Correlation (r-type) indices have other useful applications besides those that are mentioned in this chapter  In the case of dichotomous variables, we might create dichotomies in what is called a median split, by dividing variables at the median point. o Researcher might report “r-type effect sizes” on more than two conditions, which we discuss in chapter 14  Other important applications are beyond the scope of this book but are illustrated in our advanced text  For example, in a partial correlation, a researcher can measure the correlation between two variables when the influence of other variables on their relation has been eliminated statistically  As correlations usually shrink in magnitude when the variability of either of the two samples being correlated shrinks, there is a statistical solution (proposed by Karl Pearson) to correct for this “restriction of variability”  Main purpose of this chapter is to give you a working knowledge of the basics of computing and interpreting correlations in the situations that you are most likely to encounter Table 11.1 Four Forms of Correlations and Their Common Names 1) Pearson r -> Two continuous variables, such as the correlation of scores on the Scholastic Assessment Test (SAT) with grade point average (GPA) after four years of college 2) Point-biserial r (rp) -> one continuous and one dichotomous variable, such as the correlation of subject’s gender with their performance on the SAT-Verbal 3) Phi coefficient (o with line down middle) -> two dichotomous variables, such as the correlation of subject’s gender with their “yes” or “no” response to a specific question 4) Spearman rho (r s) -> two ranked variables, such as the correlation of the ranking of the top 25 college basketball teams by sports writers (Associated Press ranking) with the ranking of the same teams by college coaches (USA Today ranking) How Are Correlations Visualized in Scatter Plots?  In addition to the graphics described in the preceding chapter, another informative visual display is called a scatter plot (or scatter diagram)  It takes its name from looking like a cloud of scattered dots  Each dot represents the intersection of a line extended from a point on the X axis (the horizontal axis, or abscissa) and a line extended from a point on the Y axis (the vertical axis, or ordinate)  For now, we will concentrate on the raw scores (the X 1and the X2scores) of these 10 students on the two exams  Figure 11.1 displays the scores shown in table 11.2 in a scatter plot  Imagine a straight line through the dots  The higher the correlation is, the more tightly clustered along the line are the dots in a scatter plot (and therefore, the better is the linear predictability)  The cloud of dots slopes up for positive correlations and slopes down for negative correlations, and the linearity becomes clearer as the correlation becomes higher  From this information, what would you guess is the value of the Pearson r represented by the data? How Is a Product-Moment Correlation Calculated?  There are many useful formulas for calculating different forms of the product-moment correlation coefficient (r)  The following formula (which defines Pearson r conceptually) can be used in most situations: o R x= sum of Z Zy/ N  This formula indicates that the linear correlation between two variables (X and Y) is equal to the sum of the products of the z scores (the standard scores) of X and Y divided by the number (N) of pairs of X and Y scores  The name product-moment correlation came from the idea that the z scores (in the numerator) are distances from the mean (also called moments) that are multiplied by each other (Z Zy) to form “products”  To use this formula we begin by transforming raw scores (X and Y scores, in some case X1 and X2) to z scores by following the procedure described in the previous chapter o This is to say we calculate the mean (M) and the standard deviation of each column of X and Y scores and then substitute the calculated values in the (X-M)/ standard deviation formula where X is any student’s score  Rxy is rounded to two decimal places = .90  It’s easier to just use a computer program to calculate r or calculator  Can also calculate the Pearson r from raw scores rather than z scores using another formula, which is easier than the conceptual formula How Is Dummy Coding Used in Correlation?  Point-biserial correlation (pb) -> another case of the product-moment r. the point means that the scores for one variable are points on a continuum, and the biserial means that scores for the other variable are dichotomous  in many cases, the dichotomous scores may be arbitrarily applied numerical values, such as 0 and 1, or -1 and +1  the quantification of two levels of a dichotomous variable is called dummy coding when numerical values such as 0 and 1 are used to indicate the two distinct parts  dummy coding is a tremendously useful method because it allows us to quantify any variable that can be represented as dichotomous (also called binary, meaning there are two parts or two categories)  for example, suppose you have performed an experiment in which there were two groups (an experimental and a control group) and you want to correlate group membership with scores on the dependent variable  to indicate each participant’s group membership, you code 1 for experimental group and 0 for control group  another dichotomous independent variable that is typically recast into 1s and 0s is gender  not only dichotomous independent variables can be dummy-coded this way, but also dichotomous dependent variables can be recast into 1s and os such as success rate (1 = succeed vs. 0 =fail) Box 11.2 Linearity and Nonlinearity  Pearson r is a measure of linearity  though r is clos
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