Group Differences Results

Warner, R. M. (2021). Applied sta s cs II: Mul variable and Mul variate Techniques (3rd ed.). Los Angeles,CA: Sage Publica ons. ISBN: 9781544398723. CHAPTER 3 STATISTICAL CONTROL: WHAT CAN HAPPEN WHEN YOU ADD A THIRD VARIABLE? 3.1 WHAT IS STATISTICAL CONTROL? Bivariate correla on (Pearson’s r) is an index of the strength of the linear rela onship between one independent variable (X1) and one dependent variable (Y). This chapter moves beyond the two-variable research situa on to ask, “Does our understanding of the nature and strength of the predic ve rela onship between a predictor variable, X1, and a dependent variable, Y, change when we take a third variable, X2, into account in our analysis? If so, how does it change?” This introduces one of the most important concepts in sta s cs: the idea of sta s cal control. When we examine the associa on between each pair of variables (such as X1 and Y) in the context of a larger analysis that includes one or more addi onal variables, our understanding of the way X1 and Y are related o en changes. For example, Judge and Cable (2004) reported that salary (Y) is predictable from height (X1). An obvious ques on can be raised: Does this correla on occur because on average, women earn less than men, and are shorter than men? In this example, sex would be an X2 control variable. This chapter introduces concepts involved in sta s cal control using two simple analyses. The first is evalua on of an X1, Y associa on for separate groups on the basis of X2 scores. For example, Judge and Cable (2004) controlled for sex by repor ng correla ons between height (X1) and salary (Y) separately for male and female groups. Second, the par al correla on between X1 and Y, controlling for X2, is another way of assessing whether controlling for X2 changes the apparent rela onship between other variables. Analyses in later chapters, such as mul ple regression, implement sta s cal control by including control variables in the regression equa on. It is easier to grasp sta s cal control concepts if we start with the simplest possible three-variable situa on. You will then be able to see how this works in analyses that include more than three variables. For a sta s cal analysis to make sense, it must be based on a properly specified model. “Properly specified” means that all the variables that should be included in the analysis are included and that no variables that should not be included in the analysis are included. Model refers to a specific analysis, such as a mul ple regression equa on to predict Y from X1 and X2. We can never know for certain whether we have a properly specified model. A well-developed theory can be a helpful guide when you choose variables. When researchers choose control variables, they usually choose them because other researchers have used these control variables in past research, or because the X2 control variable is thought to be correlated with X1 and/or Y. Later analyses (mul ple regression) make it possible to include more than one control variable. When control (X2) variables make sense, and are based on a widely accepted theory, then the descrip on of the X1, Y associa on when X2 is controlled is usually preferred to a descrip on of the associa on of X1 and Y (r1Y) by itself.

Sta s cal control can help rule out some poten al rival explana ons. Recall the requirements for making an inference that X causes Y. There must be a reasonable theory to explain how X might cause Y. Scores on X and Y must be sta s cally related, using whatever type of analysis is appropriate for the nature of the variables. X must come before Y in me (temporal precedence). We must be able to rule out all possible rival explana ons (variables other than X that might be the real causes of Y). In prac ce, this last condi on is difficult to achieve. Well- controlled experiments provide experimental control for rival explanatory variables (though methods such as holding variables constant or using random assignment of cases to group to try to achieve equivalence). Par al correla on and regression analyses provide forms of sta s cal control that can poten ally rule out some rival explanatory variables. However, there are limita ons to sta s cal control. We can sta s cally control only for variables that are measured and included in analyses, and there could always be addi onal variables that should have been considered as rival explana ons, about which we have no informa on. (At least in theory, experimental methods such as random assignment of cases to groups should control for all rival explanatory variables, whether they are explicitly iden fied or not). This chapter assumes familiarity with bivariate regression and correla on. Throughout this chapter, assump ons for the use of correla on and regression are assumed to be sa sfied. Except where noted otherwise, all variables are quan ta ve. None of the following analyses would make any sense if assump ons for correla on (par cularly the assump on of linearity) are violated. When we begin to examine sta s cal control, we need informa on about the nature of the associa on among all three pairs of variables: X1 with Y, X2 with Y, and X1 with X2. Subscripts are added to correla ons to make it clear which variables are involved. A few early examples in this chapter use a categorical X2 variable as the control variable; however, all examples used in this chapter assume that X1 and Y are quan ta ve.

 The bivariate correla on between X1 and Y is denoted r1Y (or rY1, because correla on is symmetrical; that is, you obtain the same value whether you correlate X1 with Y or Y with X1).

 The bivariate correla on between X2 and Y is denoted r2Y.  The bivariate correla on between X1 and X2 is denoted r12.

In this chapter, X1 denotes a predictor variable, Y denotes an outcome variable, and X2 denotes a third variable that may be involved in some manner in the X1, Y rela onship. The X2 variable can be called a control variable or a covariate. A control variable o en (but not always) represents a rival explanatory variable. This chapter describes two methods of sta s cal control for one covariate, X2, while examining the X1, Y associa on. The first method is separa ng data into groups, on the basis of scores on the X2 control variable, and then analyzing the X1, Y associa on. The second method is obtaining a par al correla on between X1 and Y controlling for X2. Use of these methods can help understand how sta s cal control works, and these can be useful as forms of preliminary data screening. However, mul ple regression and mul variate analyses are generally the way sta s cal control is done when data are analyzed and reported in journal ar cles.

You will see that when an X2 variable is sta s cally controlled, the correla on between X1 and Y can change in any way you can imagine. When the correla on between X1 and Y is substan ally different when we control for X2, we need to explain why the rela onship between X1 and Y is different when X2 is sta s cally controlled (than when X2 is not controlled). 3.2 FIRST RESEARCH EXAMPLE: CONTROLLING FOR A CATEGORICAL X2 VARIABLE Suppose that X1 is height, Y is vocabulary test score, and X2 is grade level (Grade 1, 5, or 9). In this example, X1 and Y are both quan ta ve variables. We assume that X1 and Y are linearly related. X2, grade level, is a convenient type of variable for the following examples because it can be treated as either a categorical variable that defines three groups (different grade levels) or as a quan ta ve variable that happens to have few different score values. The analysis in this sec on includes two simple steps.

1. Find the bivariate correla on between X1 and Y (ignoring X2). This answers the ques on, How do X1 and Y appear to be related when you do not control for X2? Obtain an X1, Y sca erplot as addi onal informa on about the rela onship. You may also want to add case markers for values of X2 to the plot, as discussed below.

2. Use the SPSS split file procedure to divide the data set into groups on the basis of the X2 control variable (first, fi h, and ninth grade groups). Within each grade-level group, obtain an X1, Y sca erplot and the X1, Y correla on, r1Y. You’ll have values of r1Y for the first grade group, the fi h grade group, and the ninth grade group. These r values within groups are sta s cally controlled to remove the effects of X2, grade, because the value of X2 is constant within each group.

Using these results, you can answer two ques ons:

 Do the values of r1Y within the groups (first grade, fi h grade, and ninth grade) differ from the overall value of r1Y obtained in Step 1? If so, how do they differ? Are they smaller or larger? (Unless you conduct sta s cal significance tests between correla ons, as discussed in Appendix 10C in Chapter 10 in Volume I [Warner, 2020], these comparisons are only qualita ve. Very large samples are required to have enough sta s cal power to judge differences between correla ons significant; do not overinterpret small differences.)

 Do the values of r1Y and the slopes in the sca erplots differ across these groups (i.e., between the first grade group, the fi h grade group, and the ninth grade group)? If these within-group correla ons differ, this is possible preliminary evidence of an interac on between X1 and X2 as predictors of Y. If there is an interac on between X1 and X2 as predictors of Y, par al correla on and regression results will be misleading (unless the regression analysis includes interac on terms).

In the first hypothe cal study, measures of height (X1) and vocabulary (Y) were obtained for groups of schoolchildren in Grades 1, 5, and 9 (grade is the categorical X2 control variable). Data for this example are in the file named heightvocabulary.sav. Before you start the analysis, you probably suspect that any correla on between height and vocabulary is silly or misleading

(another word for this is spurious; spuriousness is discussed later in the chapter). Using these data, the following analyses were done. First, before examining correla ons it is a good idea to look at sca erplots. To obtain the sca erplot, make the following SPSS menu selec ons: <Graphs> → <Legacy Dialogs> → <Sca er/Dot>. In the first dialog box, click Simple Sca er, then Define. The Simple Sca erplot dialog box appears in Figure 3.1. Move the name of the dependent variable into the “Y Axis” box, the predictor or X1 variable into the “X Axis” box, and the name of the control variable X2 into the “Set Markers by” box. (This yields reasonable results only if X2 has a small number of different values.) Then click OK. Figure 3.1 SPSS Simple Sca erplot Dialog Box Descrip on Figure 3.2 Sca erplot for Vocabulary With Height, With Fit Line at Total When the sca erplot appears, click on it twice to open it in the Chart Editor. Under the menu heading “Elements,” click “Fit Line at Total” (this requests the best regression line for the total sample). The resul ng sca erplot (with addi onal edi ng to improve appearance) appears in Figure 3.2. Case markers are used to iden fy group membership on the control variable (i.e., grade level). Scores for first graders appear as 1, scores for fi h graders appear as 5, and scores for ninth graders appear as 9 in this sca erplot. The three groups of scores show some separa on across grade levels. Both height and vocabulary increase across grade levels. If you examine the graph in Figure 3.2, you can see that the groups of scores for grade levels do not overlap very much. Figure 3.3 shows an exaggerated version of this sca erplot to make the pa ern more obvious; circles are added to highlight the three separate groups of scores. If you focus on just one group at a me, such as Grade 5 (circled), you can see that within each group, there is no associa on between height and vocabulary. To confirm this, we can run the correla on (and/or bivariate regression) analysis separately within each group. You should also be able to see that height increases across grade levels and that vocabulary increases across grade levels. Next obtain the bivariate correla on for the en re sample of N = 48 cases. From the top menu bar in SPSS, make the following menu selec ons: <Analyze> → <Correlate> → <Bivariate>, then move the names of the X1 and Y variables into the “Variables” pane in the main bivariate correla on dialog box, as shown in Figure 3.4. The zero-order correla on between height and vocabulary (not controlling for grade) that appears in Figure 3.5 is r(46) = .716, p < .01. The number in parentheses a er r is usually the df. The df for a bivariate correla on = N – 2, where N is the number of cases. There is a strong, posi ve, linear associa on between height and vocabulary when grade level is ignored. What happens when we control for grade level?

To examine the grade-level groups separately, use the SPSS split file procedure to divide the data into grade levels. Select <Data> → <Split File> to open the Split File dialog box; the menu selec ons and first dialog box appear in Figure 3.6. (Do not select the similarly named <Split into Files> command.) Descrip on Figure 3.3 Exaggerated Group Differences Across Grade Levels Descrip on Figure 3.4 Bivariate Correla ons Dialog Box Descrip on Figure 3.5 Correla on Between Height (X1) and Vocabulary (Y) in En re Sample In the Split File dialog box in Figure 3.6, select the radio bu on for “Organize output by groups,” then enter the name of the categorical control variable (grade) into the “Groups Based on” window, then click OK. All subsequent analysis will be reported separately for each grade level un l you go back into the split file dialog box and make the selec on “Analyze all cases, do not create groups.” Figure 3.6 Menu Selec ons and First Dialog Box for SPSS Split File Procedure Now obtain the correla ons between height (X1) and vocabulary (Y) again, using the bivariate correla on procedure. The results for each of the three grade-level groups appear in Figure 3.7. The within-group correla ons in Figure 3.7 (r = .067 for Grade 1, r = .031 for Grade 5, and r = – .141 for Grade 9) did not differ significantly from 0; these correla ons tell us how height and vocabulary are related when grade level is sta s cally controlled. Remember that the zero- order correla on between height and vocabulary, not controlling for grade level, was +.72 (p < .01) in Figure 3.5. Height and vocabulary appeared to be strongly, posi vely related when grade level was ignored; but when grade level was sta s cally controlled by looking at correla ons within groups, height and vocabulary were not related. If you look separately at the clusters of data points for each grade, within each grade, the correla ons between height and vocabulary do not differ significantly from zero. When we examine the height–vocabulary correla on separately with each grade-level group, we find out how height and vocabulary are related when height is held constant (by looking only within groups in which all members have the same grade-level score). In this situa on we can say that controlling for the X2 control variable “explained away” or accounted for the seemingly posi ve correla on between height and vocabulary. We can conclude that the zero-order correla on between height and vocabulary was a spurious correla on (i.e., it was misleading). Later you will see that there are other possible interpreta ons of situa ons in which controlling for an X2 variable makes a correla on between X1 and Y drop to zero.

Descrip on Figure 3.7 Correla ons Between Height and Vocabulary Separately Within Each Grade 3.3 ASSUMPTIONS FOR PARTIAL CORRELATION BETWEEN X1 AND Y, CONTROLLING FOR X2 Another way to evaluate the nature of the rela onship between X1 (height) and Y (vocabulary) while sta s cally controlling for X2 (grade) is to compute a par al correla on between X1 and Y, controlling for or par alling out X2. The par al correla on between X1 and Y controlling for X2 is denoted r1Y.2. The subscripts before the dot indicate which variables are being correlated. The subscripts a er the dot indicate which variable(s) are being controlled. In this case we read r1Y.2 as “the par al correla on between X1 and Y, controlling for X2.” For par al correla on to provide accurate informa on about the rela onship between variables, the following assump ons about scores on X1, X2, and Y must be reasonably well sa sfied. Detailed data screening procedures are not covered here; see Chapter 10 in Volume I (Warner, 2020) for review. Data screening should include the following:

1. Assess the types of variables. Par al correla on makes sense when X1, X2, and Y are all quan ta ve variables. (Under some circumstances, a dichotomous or dummy variable can be used in correla on analysis; for example, sex coded 1 = male and 2 = female can be correlated with height. However, you cannot use dichotomous variables as outcome or dependent variables in regression analysis.)

2. Ideally, scores on all variables should be approximately normally distributed. This can be assessed by examining histograms for all three variables. (The formal assump on is that scores are randomly sampled from normally distributed popula ons, and we have no way to test that assump on.)

3. There should not be extreme outliers or extreme bivariate outliers. Univariate outliers can be detected using boxplots (or other decision rules chosen prior to analysis). Bivariate outliers can be detected in sca erplots.

4. Examine sca erplots for all three pairs of variables. All three pairs of variables (X1 with Y, X2 with Y, and X1 with X2) must be linearly related. If they are not, use of Pearson correla on and par al correla on is not appropriate.

5. Other assump ons for use of Pearson’s r (such as homogeneity of variance of Y across values of X1) should be sa sfied. Unfortunately, small samples usually don’t provide enough informa on to evaluate these assump ons.

6. There must not be an interac on between X1 and X2 as predictors of Y (to say this another way, X2 must not moderate the associa on between X1 and Y). In the previous sec on, the SPSS split file procedure was used to divide the data set into groups on the basis of the categorical X2 control variable, grade level. If the correla ons or regression slopes for height and vocabulary had been different across groups, that would suggest a possible interac on between X1 and X2. Chapter 7, on modera on, explains how to test sta s cal significance for interac ons. If an interac on is present, but not included in the analysis, par al correla ons are misleading.

7. Factors that can ar factually influence the magnitudes of Pearson correla ons must be considered whenever we examine other sta s cs that are based on these correla ons.

These are discussed in Appendix 10D at the end of the Chapter 10, on correla on, in Volume I (Warner, 2020). For example, if X1 and Y both have low measurement reliability, the correla on between X1 and Y will be a enuated or reduced, and any par al correla on that is calculated using r1Y may also be inaccurate.

3.4 NOTATION FOR PARTIAL CORRELATION To obtain a par al correla on between X1 and Y controlling for X2, we need the three bivariate or zero-order correla ons among X1, X2, and Y. When we say that a correla on is “zero-order,” we mean that the answer to the ques on “How many other variables were sta s cally controlled or par alled out when calcula ng this correla on?” is zero or none. The following nota on is used. Subscripts for r indicate which variables are involved in the analysis.

For a first-order par al correla on between X1 and Y, controlling for X2, the term first-order tells us that only one variable (X2) was sta s cally controlled when assessing how X1 and Y are related. In a second-order par al correla on, the associa on between X1 and Y is assessed while sta s cally controlling for two variables; for example, rY1.23 would be read as “the par al correla on between Y and X1, sta s cally controlling for X2 and X3.” Variables that follow the period in the subscript are control variables. In a kth-order par al correla on, there are k control variables. This chapter examines first-order par al correla on in detail; the conceptual issues involved in the interpreta on of higher order par al correla ons are similar. The three zero-order correla ons listed above (r1Y, r2Y, and r12) provide informa on we can use to answer the ques on “When we control for, or take into account, a third variable called X2, how does that change our descrip on of the rela on between X1 and Y?” However, examina on of separate sca erplots that show how X1 and Y are related separately for each level of the X2 variable provides addi onal, important informa on. In the following examples, a dis nc on is made among three variables: an independent or predictor variable (denoted by X1), a dependent or outcome variable (Y), and a control variable (X2). The preliminary analyses in this chapter provide ways of exploring whether the nature of the rela onship between X1 and Y changes when you remove, par al out, or sta s cally control for the X2 variable. The following nota on is used to denote the par al correla on between Y and X1, controlling for X2: rY1.2.The subscript 1 in rY1.2 refers to the predictor variable X1, and the subscript 2 refers to the control variable X2. When the subscript is read, pay a en on to the posi on in which each variable is men oned rela ve to the period in the subscript. The period within the subscript divides the subscripted variables into two sets. The variable or variables to the right of the period in the subscript are used as predictors in a regression analysis; these are the

variables that are sta s cally controlled or par alled out. The variable or variables to the le of the period in the subscript are the variables for which the par al correla on is assessed while taking one or more control variables into account. Thus, in rY1.2, the subscript Y1.2 denotes the par al correla on between X1 and Y, controlling for X2. In the par al correla on, the order in which the variables to the le of the period in the subscript are listed does not signify any difference in the treatment of variables; we could read either rY1.2 or r1Y.2 as “the par al correla on between X1 and Y, controlling for X2.” However, changes in the posi on of variables (before vs. a er the period) do reflect a difference in their treatment. For example, we would read rY2.1 as “the par al correla on between X2 and Y, controlling for X1.” Another common nota on for par al correla on is pr1. The subscript 1 associated with pr1 tells us that the par al correla on is for the predictor variable X1. In this nota on, it is implicit that the dependent variable is Y and that other predictor variables, such as X2, are sta s cally controlled. Thus, pr1 is the par al correla on that describes the predic ve rela on of X1 to Y when one or more other variables are controlled. 3.5 UNDERSTANDING PARTIAL CORRELATION: USE OF BIVARIATE REGRESSIONS TO REMOVE VARIANCE PREDICTABLE BY X2 FROM BOTH X1 AND Y To understand the par al correla on between X1 and Y, controlling for X2, it is helpful to do the following series of simple analyses. First, use bivariate regression to obtain residuals for the predic on of Y from X2. This involves two steps. First, find the predicted value of Y (denoted Y′) from the following bivariate regression: Other

(3.1) By defini on, Y′ represents the part of the Y scores that is predictable from X2. Then, to find the part of Y that is not predictable from X2, we obtain the residual, that is, the difference between the original Y score and the predicted Y score. This residual is denoted Y*. Other

Similar analyses are carried out to find the part of the X1 score that is not predictable from X2: First, do a bivariate regression to predict X1 from X2. The value of actual minus predicted X1 scores, denoted X1*, is the part of the X1 scores that is not related to X2. The par al correla on between X1 and Y, controlling for or par alling out X2, can be obtained by finding the correla on between X1* and Y* (the parts of the X1 and Y scores that are not related to X2).

Consider this situa on as an example. You want to know the correla on between X1, everyday life stress, and Y, self-reported physical illness symptoms. However, you suspect that people high in the personality trait neuro cism (X2) complain a lot about their everyday lives and also complain a lot about their health. Suppose you want to remove the effects of this complaining tendency on both X1 and Y. To do that, you find the residuals from a regression that predicts Y, physical illness symptoms, from X2, neuro cism. Call the residuals Y*. You also find the residuals for predic on of X1, everyday life stress, from X2, neuro cism; these residuals are called X1*. When you find the correla on between X1* and Y*, you can assess the strength of associa on between these variables when X2 (neuro cism effects) has been completely removed from both variables. Two control variables o en used in personality research are neuro cism and social desirability response bias (social desirability measures assess a tendency to report answers that are more socially approved, instead of accurate answers). In ability measurement studies, a common control variable is verbal ability. (I received the Be y Crocker Homemaker of the Year award in high school because of my high score on a “homemaking skills” self-report test. However, scores on that test depended mostly on verbal and arithme c ability. If they had par alled out verbal ability, this could have yielded the part of the homemaking test scores related to actual homemaking skills, and someone more deserving might have won the award. Did I men on that I flunked peach pie making?) A par al correla on was obtained for the variables in the heightvocabulary.sav data examined in previous sec ons: height (X1), vocabulary score (Y), and grade (X2). Height increases with grade level; vocabulary increases with grade level. The fact that these scores both increase across grade levels may completely explain why they appear to be related. Figure 3.8 shows the SPSS Linear Regression dialog box to run the regression specified in Equa on 3.1 (to predict X1 from X2—in this example, height from grade). Figure 3.9 shows the SPSS Data View worksheet a er performing the regressions in Equa ons 3.1 (predic ng height from grade) and 3.2 (predic ng vocabulary score from grade). The residuals from these two separate regressions were saved as new variables and renamed. Descrip on Figure 3.8 Bivariate Regression to Predict Height (X1) From Grade in School (X2) Figure 3.9 SPSS Data View Worksheet That Shows Scores on RES_1 and RES_2 Note: RES_1 and RES_2 are the saved unstandardized residuals for the predic on of height from grade and the predic on of vocabulary from grade. In the text these are renamed Resid_Height and Resid_Voc. Res_1, renamed Resid_Height, refers to the part of the scores on the X1 variable, height, that was not predictable from or related to the control or X2 variable, grade. Res_2, renamed Resid_Voc, refers to the part of the scores on the Y variable, vocabulary, that was not

predictable from the control variable, grade. The effects of the control variable grade level (X2) have been removed from both height and vocabulary by obtaining these residuals. Finally, we use the bivariate Pearson correla on procedure (Figure 3.10) to obtain the correla on between these two new variables, Resid_Height and Resid_Voc. The correla on between these residuals, r = –.012 in Figure 3.11, corresponds to the value of the par al correla on between X1 and Y, controlling for or par alling out X2. Note that X2 is par alled out or removed from both variables (X1 and Y). This par al r = –.012 tells us that X1 (height) is not significantly correlated with Y (vocabulary) when variance that is predictable from grade level (X2) has been removed from or par alled out of both the X1 and the Y variables. Later you will learn about semipar al correla on, in which the control variable X2 is par alled out of only one variable. The value of par al r between height and vocabulary, controlling for grade (–.012) is approximately the average of the within-group correla ons between height and vocabulary that appeared in Figure 3.7 (.067 + .031 − .141)/3 = (–.043)/3 ≈ –.012. This correspondence is not close enough that we can use within-group correla ons to compute an overall par al r, but it illustrates how par al r can be interpreted. Par al r between X1 and Y is approximately the mean of the correla ons between X1 and Y for separate groups on the basis of scores for X2. Descrip on Figure 3.10 Correla on Between Residuals for Predic on of Height From Grade (Resid_Height) and Residuals for Predic on of Vocabulary From Grade (Resid_Voc) Descrip on Figure 3.11 Correla on Between Residuals for Height and Vocabulary Using Grade as Control Variable Note: The control variable grade (X2) was used to predict scores on the other variables (X1, height, and Y, vocabulary). The variable Resid_Height contains the residuals from the bivariate regression to predict height (X1) from grade (X2). The variable Resid_Voc contains the residuals from the bivariate regression to predict vocabulary (Y) from grade (X2). These residuals correspond to the parts of the X1 and Y scores that are not related to or not predictable from grade (X2). In the preceding example, X2 had only three score values, so we needed to examine only three groups. In prac ce, the X2 variable o en has many more score values (which makes looking at subgroups more tedious and less helpful; numbers of cases within groups can be very small). 3.6 PARTIAL CORRELATION MAKES NO SENSE IF THERE IS AN X1 × X2 INTERACTION The interpreta on for par al correla on (as the mean of within-group correla ons) does not make sense if assump ons for par al correla on are violated. If X1 and X2 interact as predictors of Y, par al correla on analysis will not help us understand the situa on. A graduate student

once brought me data that he didn’t understand. He was examining predictors of job sa sfac on for male and female MBA students in their first jobs. In his data, the control variable X2 corresponded to sex, coded 1 = male, 2 = female. X1 was a measure of need for power. Y was job sa sfac on evalua ons. Figure 3.12 is similar to the data he showed me. Back in the late 1970s, when the data were collected, women who tried to exercise power over employees got more nega ve reac ons than men who exercised power. This made management posi ons more difficult for women than for men. We can set up a sca erplot to show how evalua ons of job sa sfac on (Y) are related to MBA students’ need for power (X1). Case markers iden fy which scores belong to male managers (X2 = m) and which scores belong to female managers (X2 = f). Sex of manager was the X2 or “controlled for” variable in this example. If we look only at the scores for male managers (denoted by “m” in Figure 3.12), there was a posi ve correla on between need for power (X1) and job sa sfac on (Y). If we look only at the scores for female managers (denoted by “f” in Figure 3.12), there was a nega ve correla on between need for power (X1) and job sa sfac on (Y). In this example, we could say that sex and need for power interact as predictors of job sa sfac on evalua on; more specifically, for male managers, their job sa sfac on evalua ons increase as their need for power scores increase, whereas for female managers, their job sa sfac on evalua ons decrease as their need for power scores increase. We could also say that sex “moderates” the rela onship between need for power and job sa sfac on evalua on. Descrip on Figure 3.12 Interac on Between Sex (X2) and Need for Power (X1) as Predictors of Job Sa sfac on Evalua on (Y) Note: Within the male group, X1 and Y are posi vely correlated; within the female group, X1 and Y are nega vely correlated. In this example, the slopes for the subgroups (male and female) had opposite signs. Modera on or interac on effects do not have to be this extreme. We can say that sex moderates the effect of X1 on Y if the slopes to predict Y from X1 are significantly different for men and women. The slopes do not actually have to be opposite in sign for an interac on to be present, and the regression lines within the two groups do not have to cross. Another type of interac on would be no correla on between X1 and Y for women and a strong posi ve correla on between X1 and Y for men. Yet another kind of interac on is seen when the b slope coefficient to predict Y from X1 is posi ve for both women and men but is significantly larger in magnitude for men than for women. The student who had these data found that the overall correla on between X1 and Y was close to zero. He also found that the par al correla on between X1 and Y, controlling for sex, was close to zero. When he looked separately at male and female groups, as noted earlier, he found

a posi ve correla on between X1 and Y for men and a nega ve correla on between X1 and Y for women. (Robert Rosenthal described situa ons like this as “different slopes for different folks.”) If he had reported r1Y (the correla on between need for power and job sa sfac on, ignoring sex) near 0 and r1Y.2 (the par al correla on between need for power and job sa sfac on, controlling for sex) also near 0, this would not be an adequate descrip on of his results. A reader would have no way to know from these correla ons that there actually were correla ons between need for power and job sa sfac on but that the nature of the rela on differed for men and women. One way to provide this informa on would be to report the correla on and regression separately for each sex. (Later you will see be er ways to do this by including interac on terms in regression equa ons; see Chapter 7, on modera on.) 3.7 COMPUTATION OF PARTIAL R FROM BIVARIATE PEARSON CORRELATIONS There is a simpler direct method for the computa on of the par al r between X1 and Y, controlling for X2, on the basis of the values of the three bivariate correla ons:

The formula to calculate the par al r between X1 and Y, controlling for X2, directly from the Pearson correla ons is as follows: Other

(3.2) In the preceding example, where X1 is height, Y is vocabulary, and X2 is grade, the corresponding bivariate correla ons were r1Y = +.716, r2Y = +.787, and r12 = +.913. If these values are subs tuted into Equa on 3.2, the par al correla on rY1.2 is as follows: Other

Within rounding error, this value of –.010 agrees with the value that was obtained from the correla on of residuals from the two bivariate regressions reported in Figure 3.11. In prac ce, it is rarely necessary to calculate a par al correla on by hand. If you read an ar cle that reports

only zero-order correla ons, you could use Equa on 3.2 to calculate par al correla ons for addi onal informa on. The most convenient way to obtain a par al correla on, when you have access to the original data, is the par al correla ons procedure in SPSS. The SPSS menu selec ons <Analyze> → <Correlate> → <Par al>, shown in Figure 3.13, open the Par al Correla ons dialog box, which appears in Figure 3.14. The names of the predictor and outcome variables (height and vocabulary) are entered in the pane that is headed “Variables.” The name of the control variable, grade, is entered in the pane under the heading “Controlling for.” (Note that more than one variable can be placed in this pane; that is, we can include more than one control variable.) The output for this procedure appears in Figure 3.15, where the value of the par al correla on between height and vocabulary, controlling for grade, is given as r1Y.2 = –.012; this par al correla on is not significantly different from 0 (and is iden cal to the correla on between Resid_Height and Resid_Voc reported in Figure 3.11). Par al correla on is approximately (but not exactly) the mean of the X1, Y correla ons obtained by running an X1, Y correla on for each score on the X2 variable. In this example, where the X2 variable is grade level, each grade level contained numerous cases. In situa ons where X2 is a quan ta ve variable with many possible values, the same thing happens (essen ally), but it is more difficult to imagine because some values of X2 have few cases. Figure 3.13 SPSS Menu Selec ons for Par al Correla on Descrip on Figure 3.14 SPSS Dialog Box for the Par al Correla ons Procedure Descrip on Figure 3.15 Output From SPSS Par al Correla ons Procedure Note: First-order par al correla on between height (X1) and vocabulary (Y), controlling for grade level (X2). The par al correla on between height and vocabulary controlling for grade (r = –.012) is iden cal to the correla on between Resid_Height and Resid_Voc (r = –.012) that appeared in Figure 3.11. 3.8 SIGNIFICANCE TESTS, CONFIDENCE INTERVALS, AND STATISTICAL POWER FOR PARTIAL CORRELATIONS 3.8.1 Sta s cal Significance of Par al r The null hypothesis that a par al correla on equals 0 can be tested by se ng up a t ra o similar to the test for the sta s cal significance of an individual zero-order Pearson correla on. The SPSS par al correla ons procedure provides this sta s cal significance test; SPSS reports an exact p value for the sta s cal significance of par al r. The degrees of freedom (df) for a par al correla on are N – k, where k is the total number of variables that are involved in the par al correla on, and N is the number of cases or par cipants.

3.8.2 Confidence Intervals for Par al r Textbooks do not present detailed formulas for standard errors or confidence intervals for par al correla ons. Olkin and Finn (1995) and Graf and Alf (1999) provided formulas for computa on of the standard error for par al correla ons; however, the formulas are complicated and not easy to work by hand. SPSS does not provide standard errors or confidence interval es mates for par al correla ons. 3.8.3 Effect Size, Sta s cal Power, and Sample Size Guidelines for Par al r Like Pearson’s r (and r2), the par al correla on rY1.2 and squared par al correla on r2Y1.2 can be interpreted directly as informa on about effect size or strength of associa on between variables. Effect size labels for values of Pearson’s r can reasonably be used to describe effect sizes for par al correla ons (r = .10 is small, r = .30 is medium, and r = .50 is large). Algina and Olejnik (2003) provided sta s cal power tables for correla on analysis with discussion of applica ons in par al correla on and mul ple regression analysis. Later chapters discuss sta s cal power further in the situa on where it is more o en needed: mul ple regression. In general, no ma er what minimum sample sizes are given by sta s cal power tables, it is desirable to have large sample sizes for par al correla on, on the order of N = 100. 3.9 COMPARING OUTCOMES FOR RY1.2 AND RY1 When we compare the size and sign of the zero-order correla on between X1 and Y with the size and sign of the par al correla on between X1 and Y, controlling for X2, several different outcomes are possible. The value of r1Y, the zero-order correla on between X1 and Y, can range from –1 to +1. The value of r1Y.2, the par al correla on between X1 and Y, controlling for X2, can also poten ally range from –1 to +1 (although in prac ce its actual range may be limited by the correla ons of X1 and Y with X2). In principle, any combina on of values of r1Y and r1Y.2 can occur (although some outcomes are much more common than others). Here is a list of possible outcomes when r1Y is compared with rY1.2:

1. Both r1Y and rY1.2 are not significantly different from zero. 2. Par al correla on rY1.2 is approximately equal to r1Y. 3. Par al correla on rY1.2 is not significantly different from zero, even though r1Y differed

significantly from 0. 4. Par al correla on rY1.2 is smaller than r1Y in absolute value, but rY1.2 is significantly

greater than 0. 5. Par al correla on rY1.2 is larger than r1Y in absolute value, or opposite in sign from rY1.

Each of these is discussed further in later sec ons. Before examining possible interpreta ons for these five outcomes, we need to think about reasonable causal and noncausal hypotheses about associa ons among the variables X1, Y, and X2. Path models provide a way to represent these hypotheses. A er an introduc on to path models, we return to possible interpreta ons for the five outcomes listed above.

3.10 INTRODUCTION TO PATH MODELS Path models are diagrams that represent hypotheses about how variables are related. These are o en called “causal” models; that name is unfortunate because sta s cal analyses based on these models generally don’t provide evidence that can be used to make causal inferences. However, some paths in these models represent causal hypotheses. The arrows shown in Table 3.1 represent three different hypotheses about how a pair of variables, X1 and Y, may be related. Given an r1Y correla on from data, we can make only one dis nc on. If r1Y does not differ significantly from 0, we prefer the model in row 1 (no associa on). If r1Y does differ significantly from 0 (and whether it is posi ve or nega ve), we prefer one of the models in rows 2, 3, and 4. However, a significant correla on cannot tell us which of these three models is “true.” First, we need a theory that tells us what poten al causal connec ons make sense. Second, results from experiments in which the presumed causal variable is manipulated and other variables are controlled and observed provide stronger support for causal inferences. A noncausal associa on hypothesis should be the preferred interpreta on of Pearson’s r, unless we have other evidence for possible causality. Table 3.1 Four Possible Hypothesized Paths Between Two Variables (X1 and Y)

3.11 POSSIBLE PATHS AMONG X1, Y, AND X2 When we take a third variable (X2) into account, the number of possible models to represent rela onships among variables becomes much larger. In Figure 3.16, there are three pairs of variables (X1 and X2, X1 and Y, and X2 and Y). Each rectangle can be filled in with almost any1 of the four possible types of path (no rela on, noncausal associa on, or one of the two arrows that correspond to a causal hypothesis). The next few sec ons of this chapter describe examples of causal models that might be proposed as hypotheses for the rela onships among three variables. Conven onally, causal arrows point from le to right, or from top down. Loca ons of variables can be rearranged so that this can be done.

One of the most common (o en implicit) models corresponds to the path model regression that represents X1 and X2 as correlated predictors of Y. A major difference between experimental and nonexperimental research is that experimenters can usually arrange for manipulated independent variables to be uncorrelated with each other. In nonexperimental research, we o en work with predictors that are correlated, and analyses must take correla ons between predictors into account. For the analysis in Figure 3.17, the bidirec onal or noncausal path between X1 and X2 corresponds to the correla on r12. The unidirec onal arrows from X1 to Y and from X2 to Y can represent causal hypotheses, but when numerical results are obtained, it is usually be er to interpret them as informa on about strength of predic ve associa ons (and not make causal inferences). You will see later that regression with more than one predictor variable is used to obtain values (called path coefficients) that indicate the strength of associa on for each path. We haven’t yet covered these methods yet, and for now, the “causal” path coefficients are denoted by ques on marks, indica ng that you do not yet have methods to obtain these values. When standardized path coefficients (β coefficients) are reported, their values can be interpreted like correla ons. Values for these path coefficients will be es mated later using regression coefficients, which in turn depend on the set of correla ons among all the variables in the path diagram or model. Figure 3.16 Blank Template for All Possible Paths Among Three Variables Descrip on Figure 3.17 Path Model for Predic on of Y From Correlated Predictor (or Causal) Variables X1 and X2 Here is a hypothe cal situa on in which the model in Figure 3.17 might be used: Suppose we are interested in predic ng Y, number of likes people receive on social media. Number of likes might be influenced or caused by X1, quality of posts, and X2, number of posts. In addi on, X1 and X2 may be correlated or confounded; for instance, a person who posts frequently may also post higher quality material. Data for this situa on can be analyzed using regression with two correlated predictor variables (discussed in the next chapter). Some mes an X2 variable is added to an analysis because a researcher believes that X1 and X2 together can predict more variance in the Y outcome than X1 alone. In addi on, researchers want to evaluate how adding X2 to an analysis changes our understanding of the associa on between X1 and Y. There are numerous possible outcomes for a three-variable analysis. The correla on for path r12 can either be nonsignificant or significant with either a posi ve or nega ve sign. The coefficient for any path marked with a ques on mark can be either significant or nonsignificant, posi ve or nega ve. A significant path coefficient ?1 indicates that X1 is significantly related to Y, in the context of an analysis that also includes X2 and paths that relate X1 and Y to X2. Obtaining sta s cally significant paths for both the path X1 → Y and the path X2 → Y is not proof that X1 causes Y or that X2 causes Y. A model in which all paths are noncausal, as in Figure 3.18, would be equally consistent with sta s cally significant es mates for all path coefficients.

Path models are o en more interes ng and informa ve when at least one path is not sta s cally significant. Recall the example examining the associa on between height (X1) and vocabulary (Y), controlling for grade level (X2). When X2 was included in the analysis, the associa on between height and vocabulary dropped to 0. Figures 3.19 and 3.20 depict two reasonable corresponding path models. If r1Y.2 is close to 0, we do not need a direct path between X1 and Y in a model that includes X2; for example, because the correla on of height and vocabulary dropped to zero when grade level was taken into account, we can drop the direct path between height and vocabulary from the model. We can conclude that the only reason we find a correla on between height and vocabulary is that each of them is correlated with (or perhaps caused by) grade level. Unidirec onal arrows in these models represent causal hypotheses. We cannot prove or disprove any of these hypotheses using data from nonexperimental research. However, we can interpret some outcomes for correla on and par al correla on as consistent with, or not consistent with, different possible models. This makes it possible, some mes, to reduce the set of models that are considered plausible explana ons for the rela onships among variables. The next sec ons describe possible explana ons to consider on the basis of comparison of r1Y (not controlling for X2) and r1Y.2 (controlling for X2). One of the few things a par al correla on tells us is whether it is reasonable to drop one (or for more complex situa ons, more than one) of the paths from the model. Beginning in the next chapter (regression with two predictor variables) we’ll use coefficients from regression equa ons to make inferences about path model coefficients. This has advantages over the par al correla on approach. Descrip on Figure 3.18 Path Mode With Only Correla on Paths (No Causal Paths) Figure 3.19 Path Model: Height and Vocabulary Are Both Correlated With Grade Level but Are Not Directly Related to Each Other Figure 3.20 Height and Vocabulary Are Both Influenced or Caused by Matura on (Grade Level) 3.12 ONE POSSIBLE MODEL: X1 AND Y ARE NOT RELATED WHETHER YOU CONTROL FOR X2 OR NOT One possible hypothe cal model is that none of the three variables (X1, X2, and Y) is either causally or noncausally related to the others. This would correspond to a model that has no path (either causal or noncausal) between any pair of variables. If we obtain Pearson’s r values for r12, r1Y, and r2Y that are not significantly different from 0 (and all three correla ons are too small to be of any prac cal or theore cal importance), those correla ons are consistent with a model that has no paths among any of the three pairs of variables. The par al correla on between X1 and Y, controlling for X2, would also be 0 or very close to 0 in this situa on. A researcher who obtains values close to 0 for all the bivariate (and par al) correla ons would

probably conclude that none of these variables are related to the others either causally or noncausally. This is usually not considered an interes ng outcome. In path model terms, this would correspond to a model with no path between any pair of variables. 3.13 POSSIBLE MODEL: CORRELATION BETWEEN X1 AND Y IS THE SAME WHETHER X2 IS STATISTICALLY CONTROLLED OR NOT (X2 IS IRRELEVANT TO THE X1, Y RELATIONSHIP) If the par al correla on between X1 and Y, controlling for X2, is approximately equal to the zero-order correla on between X1 and Y, that is, r1Y.2 ≈ r1Y, we can say that the X2 variable is “irrelevant” to the X1, Y rela onship. If a researcher finds a correla on he or she “likes” between an X1 and a Y variable, and the researcher is asked to consider a rival explanatory variable X2 he or she does not like, this may be the outcome the researcher wants. This outcome could correspond to a model with only one path, between X1 and Y. The path could be either noncausal or causal; if causal, the path could be in the direc on X1 → Y or Y → X1). There would be no paths connec ng X2 with either X1 or Y. 3.14 WHEN YOU CONTROL FOR X2, CORRELATION BETWEEN X1 AND Y DROPS TO ZERO When r1Y.2 is close to zero (but r1Y is not close to zero), we can say that the X2 control variable completely accounts for or explains the X1, Y rela onship. In this situa on, the path model does not need a direct path from X1 to Y, because we can account for or explain their rela onship through associa ons with X2. There are several possible explana ons for this situa on. However, because these explana ons are all equally consistent with r1Y.2 = 0, informa on from data cannot determine which explana on is best. On the basis of theory, a data analyst may prefer one interpreta on over others, but analysts should always acknowledge that other interpreta ons or explana ons are possible (MacKinnon, Krull, & Lockwood, 2000). If r1Y differs significantly from 0, but r1Y.2 does not differ significantly from zero:

 X1 and X2 could be interpreted as strongly correlated or confounded predictors (or causes) of Y; when r1Y.2 = 0, all the predic ve informa on in X1 may also be included in X2. In that case, a er we use X2 to predict Y, we don’t gain anything by adding X1 as another predictor.

 The X1, Y associa on may be spurious; that is, the r1Y correla on may be nonzero only because X1 and Y are both correlated with, or both caused by, X2.

 The X1, Y associa on may be completely mediated by X2 (discussed below).  The X1, Y associa on might involve some form of suppression; that is, the absence of

direct associa on between X1 and Y may be clear only when X2 is sta s cally controlled. (There are other forms of suppression in which r1Y would not be close to 0.)

3.14.1 X1 and X2 Are Completely Redundant Predictors of Y In the next chapter you will see that when X1 and X2 are used together as regression predictors of Y, it is possible for the contribu on of X1 to the predic on of Y (indexed by the unstandardized regression slope b) can be nonsignificant 0, even in situa ons where r1Y is sta s cally significant. If all the predic ve informa on available in X1 is already available in X2, this outcome is consistent with the path model in Figure 3.21. For example, suppose a

researcher wants to predict college grades (Y) from X1 (verbal SAT score) and X2 (verbal SAT and math SAT scores). In this situa on, X1 could be completely redundant with X2 as a predictor. If the informa on in X1 that is predic ve of Y is also included in X2, then adding X1 as a predictor does not provide addi onal informa on that is useful to predict Y. Researchers try to avoid situa ons in which predictor variables are very highly correlated, because when this happens, any separate contribu ons of informa on from the predictor variables cannot be dis nguished. 3.14.2 X1, Y Correla on Is Spurious In the height, vocabulary, and grade level example, the associa on between height and vocabulary became not sta s cally significant when grade level was controlled. There is no evidence of a direct associa on between height and vocabulary, so there is not likely to be a causal connec on. Researchers are most likely to decide that a correla on is spurious when it is silly, or when there is no reasonable theory that would point to a direct associa on between X1 and Y. There are several possible path models. For example, X1 and Y might be correlated with each other only because they are both correlated with X2 (as in the path model in Figure 3.22), or X2 might be a common or shared cause of both X1 and Y (as in Figure 3.23). Models in which one of the causal paths in Figure 3.23 changed to a correla onal path are also possible. Alterna vely, we could propose that the matura on process that occurs from Grades 1 to 5 to 9 causes increases in both height and vocabulary. The hypothesis that X1 and Y have a shared cause (X2) corresponds to the path model in Figure 3.23. In this hypothe cal situa on, the only reason why height and vocabulary are correlated is that they share a common cause; when variance in height and vocabulary that can be explained by this shared cause is removed, these variables are not directly related. Examples of spurious correla on inten onally involve foolish or improbable variables. For example, ice cream sales may increase as temperatures rise; homicide rates may also increase as temperatures rise. If we control for temperature, the correla on between ice cream sales and homicide rates drops to 0, so we would conclude that there is no direct rela onship between ice cream sales and homicide but that the associa on of each of these variables with outdoor temperature (X2) creates the spurious or misleading appearance of a connec on between ice cream consump on and homicide. Figure 3.21 Outcome Consistent With Completely Redundant Predictor: All Predic ve Informa on in X1 Is Included in X2 Figure 3.22 Path Model for One Kind of Spurious Associa on Between Height and Vocabulary

Figure 3.23 Path Model in Which Height and Vocabulary Have a Shared or Common Cause (Matura on) 3.14.3 X1, Y Associa on Is Completely Mediated by X2 A media on model involves a two-step causal sequence hypothesis. First, X1 is hypothesized to cause X2; then, X2 is hypothesized to cause Y. If the X1, Y associa on drops to 0 when we control for X2, a direct path from X1 to Y is not needed. For example, we might hypothesize that increases in age (X1) cause increases in body weight (X2); and then, increases in body weight (X2) cause increases in blood pressure (Y). This is represented as a path model in Figure 3.24. If the X1, Y associa on drops to 0 when we control for X2, one possible inference is that the X1 associa on with Y is completely mediated by X2. Consider an example that involves the variables age (X1), body weight (X2), and systolic blood pressure (Y). It is conceivable that blood pressure increases as people age but that this influence is mediated by body weight (X2) and only occurs if weight changes with age and if blood pressure is increased by weight gain. The corresponding path model appears in Figure 3.24. The absence of a direct path from X1 to Y is important; the absence of a direct path is what leads us to say that the associa on between X1 and Y may be completely mediated. Methods for es ma on of path coefficients for this model are covered in Chapter 9, on media on. Figure 3.24 Complete Media on Model: Effects of Age (X1) on Systolic Blood Pressure (Y) Are Completely Mediated by Body Weight (X2) Note: Path diagrams for media on o en denote the media ng variable as M (instead of X2). 3.14.4 True Nature of the X1, Y Associa on (Their Lack of Associa on) Is “Suppressed” by X2 The “true” nature of the X1, Y associa on may be suppressed or disguised by X2. In the shared cause example (Figure 3.23), the true nature of the height, vocabulary associa on is hidden when we look at their bivariate correla on; the true nature of the associa on (i.e., that there is no direct associa on) is revealed when we control for X2 (grade level or matura on). 3.14.5 Empirical Results Cannot Determine Choice Among These Explana ons When we find a large absolute value for r1Y, and a nonsignificant value of rY1.2, this may suggest any one of these explana ons: spuriousness, completely redundant predictors, shared common cause, complete media on, or suppression. Theories and common sense can help us rule out some interpreta ons as nonsense (for instance, it’s conceivable that age might influence blood pressure; it would not make sense to suggest that blood pressure causes age). X1 cannot cause Y if X1 happens later in me than Y, and we would not hypothesize that X1 causes Y unless we can think of reasons why this would make sense. Even when theory and common sense are applied, data analysts are o en s ll in situa ons where it is not possible to decide among several explana ons.

3.15 WHEN YOU CONTROL FOR X2, THE CORRELATION BETWEEN X1 AND Y BECOMES SMALLER (BUT DOES NOT DROP TO ZERO OR CHANGE SIGN) This may be one of the most common outcomes when par al correla ons are compared with zero-order correla ons. The implica on of this outcome is that the associa on between X1 and Y can be only partly accounted for by a (causal or noncausal) path via X2. A direct path (either causal or noncausal) between X1 and Y is needed in the model, even when X2 is included in the analysis. We can consider the same poten al explana ons as for rY1.2 = 0 and add the word partly. The r1Y correla on might be partly spurious (this language is not common). X1 and X2 might be partly, but not completely, redundant predictors. X1 and X2 might share common causes, but the shared causes might not be sufficient to completely explain their associa on. The X1, Y associa on might be partly mediated by X2 (illustrated in Figure 3.25). There may be par al suppression by X2 of the true nature of the X1, Y associa on. Figure 3.25 Path Model for Par al Media on of Effects of X1 on Y by X2 As in previous situa ons: empirical values of r1Y and rY1.2, and of path coefficients from other analyses such as regression, cannot determine which among these explana ons is more likely to be correct. 3.16 SOME FORMS OF SUPPRESSION: WHEN YOU CONTROL FOR X2, R1Y.2 BECOMES LARGER THAN R1Y OR OPPOSITE IN SIGN TO R1Y The term suppression has been used to refer to different rela ons among variables. Cohen, Cohen, West, and Aiken (2013) suggested a broad defini on: X2 can be called a suppressor variable if it hides or suppresses the “true” associa on between X1 and Y. Under that broad defini on, any me r1Y.2 differs significantly from r1Y, we could say that X2 acted as a suppressor variable. Implicitly we assume that the “true” nature of the X1, Y associa on is seen only when we sta s cally control for X2. However, outcomes where rY1.2 is approximately equal to 0 or rY1.2 is less than r1Y, described in the preceding sec ons, are common outcomes that are not generally seen as surprising or difficult to explain. Many authors describe outcomes as suppression only if they are surprising or difficult to explain, for example, when r1Y.2 is larger than r1Y, or when rY1.2 is opposite in sign from r1Y. Paulhus, Robins, Trzesniewski, and Tracy (2004) explained different ways suppression has been defined and described three different types of suppression. Classical suppression occurs when an X2 variable that is not predic ve of Y makes X1 a stronger predictor of Y (see Sec on 3.16.1 for a hypothe cal example). These outcomes are not common in behavioral and social science research. However, it is useful to understand that they exist, par cularly if you happen to find one of these outcomes when you compare r1Y.2 to r1Y. The following sec ons describe three specific types of suppression.

3.16.1 Classical Suppression: Error Variance in Predictor Variable X1 Is “Removed” by Control Variable X2 “Classical” suppression occurs if an X2 variable that is not related to Y improves the ability of X1 to predict Y when it is included in the analysis. This can happen if X2 helps us remove irrelevant or error variance from X1 scores. Consider the following hypothe cal situa on. A researcher develops a wri en test of “mountain survival skills.” The score on this test is the X1 predictor variable. The researcher wants to demonstrate that scores on this test (X1) can predict performance in an actual mountain survival situa on (the score for this survival test is the Y outcome variable). The researcher knows that, to some extent, success on the wri en test depends on the level of verbal ability (X2). However, verbal ability is completely uncorrelated with success in the actual mountain survival situa on. The diagram in Figure 3.26 uses overlapping circles to represent shared variance for all pairs of variables (as discussed in Chapter 10, on correla on, in Volume I [Warner, 2020]). For this hypothe cal example, consider the problem of predic ng skill in a mountain survival situa on (Y) from scores on a wri en test of mountain survival skills (X1). That predic on may not be good, because verbal skills (X2) probably explain why some people do be er on wri en tests (X1). (I am guessing that a Daniel Boone–type hero might not have done well on such a test but would do very well outdoors.) It may be quite difficult to come up with ques ons that assess actual skill, independent of verbal ability. Verbal skills (X2) probably have li le or nothing to do with mountain survival (Y). From the point of view of a person who really wants to predict mountain survival skills, verbal ability is a nuisance or error variable. We might think of the wri en test score as being made up of two parts:

 a part of the score that is relevant to survival skill and  a part of the score that is related to verbal ability but is completely irrelevant to survival

skill. Figure 3.26 X2 (a Measure of Verbal Ability) Is a Suppressor of Error Variance in the X1 Predictor (a Wri en Test of Mountain Survival Skills); Y Is a Measure of Actual Mountain Survival

Table 3.2 has hypothe cal correla ons among the three variables. Using Equa on 3.2 or an online calculator, the par al correla on based on these three bivariate correla ons is r1Y.2 = .33. If verbal ability is not par alled out, the zero-order bivariate correla on between the wri en test survival skills, r1Y, is .25. When verbal ability is par alled out, the correla on between the wri en test and survival skills, r1Y.2, is .33. We can predict actual survival be er when we use only the part of the test scores that is not related to verbal ability. A path diagram is not helpful in understanding this situa on. Use of overlapping circles is more helpful. In an overlapping circle diagram, each circle represents the total variance of one variable. Shared or overlapping area corresponds to shared variance. For variables X1 and X2, with r = .65, the propor on of overlap would be r2 = .42. If two variables have a correla on of zero, their circles do not overlap. Figure 3.26 shows how par on of variance into explained and unexplained variance on the actual mountain survival task will work, given that X2 isn’t correlated at all with Y, but is highly correlated with X1. (This is an uncommon outcome. You will more o en see par on of variance that looks like the examples in the next chapter.) The overlapping circle diagrams that appear in Figure 3.26 can help us understand what might happen in this situa on. The top diagram shows that X1 is correlated with Y and X2 is correlated with X1; however, X2 is not correlated with Y (the circles that represent the variance of Y and the variance of X2 do not overlap). If we ignore the X2 variable, the squared correla on between X1 and Y (r21Y) corresponds to Area c in Figure 3.26. The total variance in X1 is given by the sum of Areas a + b + c. In these circle diagrams, the total area equals 1.00; therefore, the sum a + b + c = 1. The propor on of variance in X1 that is predic ve of Y (when we do not par al out the variance associated with X2) is equivalent to c/(a + b + c) = c/1 = c. When we sta s cally control for X2, we remove all the variance that is predictable from X2 from the X1 variable, as shown in the bo om diagram in Figure 3.26. The second diagram shows that a er the variance associated with X2 is removed, the remaining variance in X1 corresponds to the sum of Areas b + c. The variance in X1 that is predic ve of Y corresponds to Area c. The propor on of the variance in X1 that is predic ve of Y a er we par al out or remove the variance associated with X2 now corresponds to c/(b + c). Because (b + c) is less than 1, the propor on of variance in X1 that is associated with Y a er removal of the variance associated with X2 (i.e., r2Y1.2) is actually higher than the original propor on of variance in Y that was predictable from X1 when X2 was not controlled (i.e., r2Y1). In this situa on, the X2 control variable suppresses irrelevant or error variance in the X1 predictor variable. When we remove the verbal skills part of the wri en test scores by controlling for verbal ability, the part of the test score that is le becomes a be er predictor of actual survival. It is not common to find a suppressor variable that makes some other predictor variable a be er predictor of Y in actual research. However, some mes a researcher can iden fy a factor that influences scores on the X1 predictor and that is not related to or predic ve of the scores on the outcome variable Y. In this example, verbal ability was one factor that influenced scores on the wri en test, but it was almost completely unrelated to actual mountain survival skills. Controlling for verbal ability (i.e., removing the variance associated with verbal ability

from the scores on the wri en test) made the wri en test a be er predictor of mountain survival skills. If X2 has a nearly 0 correla on with Y, and X1 becomes a stronger predictor of Y when X2 is sta s cally controlled, classical suppression is a possible explana on. It is be er not to grasp at straws. If r1Y = .30 and r1Y.2 = .31, rY1.2 is (a li le) larger than r1Y; however, this difference between r1Y and rY1.2 may be too small to be sta s cally significant or to have a meaningful interpreta on. It may be desirable to find variables that can make your favorite X1 variable a stronger predictor of Y, but this does not happen o en in prac ce. 3.16.2 X1 and X2 Both Become Stronger Predictors of Y When Both Are Included in Analysis This outcome in which both X1 and X2 are more predic ve of Y when the other variable has been sta s cally controlled has been described as coopera ve, reciprocal, or mutual suppression. This can happen when X1 and X2 have opposite signs as predictors of Y, and X1 and X2 are posi vely correlated with each other. In an example provided by Paulhus et al. (2004), X1 (self-esteem) and X2 (narcissism) had rela onships with opposite signs for the outcome variable Y (an social behavior). In their Sample 1, Paulhus et al. reported an empirical example in which the correla on between self-esteem and narcissism was +.32. Self-esteem had a nega ve zero- order rela onship with an social behavior (–.27) that became more strongly nega ve when narcissism was sta s cally controlled (–.38). Narcissism had a posi ve associa on with an social behavior (.21) that became more strongly posi ve when self-esteem was sta s cally controlled (.33). In other words, each predictor had a stronger rela onship with the Y outcome variable when controlling for the other predictor. 3.16.3 Sign of X1 as a Predictor of Y Reverses When Controlling for X2 Another possible form of suppression occurs when the sign of rY1.2 is opposite to the sign of rY1. This has some mes been called nega ve suppression or net suppression; I prefer the term proposed by Paulhus et al. (2004), crossover suppression. In the following example, r1Y, the zero-order correla on between crowding (X1) and crime rate (Y2) across neighborhoods is large and posi ve. However, when you control for X2 (level of neighborhood socioeconomic status [SES]), the sign of the par al correla on between X1 and Y, controlling for X2, rY1.2, becomes nega ve. A hypothe cal situa on where this could occur is shown in Figure 3.27. In this hypothe cal example, the unit of analysis or case is “neighborhood”; for each neighborhood, X1 is a measure of crowding, Y is a measure of crime rate, and X2 is a categorical measure of income level (SES). X2 (SES) is coded as follows: 1 = upper class, 2 = middle class, 3 = lower class. The pa ern in this graph represents the following hypothe cal situa on. This example was suggested by correla ons reported by Freedman (1975), but it illustrates a much stronger form of suppression than Freedman found in his data. For the hypothe cal data in Figure 3.27, if you ignore SES and obtain the zero-order correla on between crowding and crime, you would obtain a large posi ve correla on, sugges ng that crowding predicts crime. However, there are two confounds present: Crowding tends to be greater in lower SES neighborhoods (3 = low SES), and the incidence of crime also tends to be greater in lower SES neighborhoods.

Once you look separately at the plot of crime versus crowding within each SES category, however, the rela onship becomes quite different. Within the lowest SES neighborhoods (SES code 3), crime is nega vely associated with crowding (i.e., more crime takes place in “deserted” areas than in areas where there are many poten al witnesses out on the streets). Freedman (1975) suggested that crowding, per se, does not “cause” crime; it just happens to be correlated with something else that is predic ve of crime, namely, poverty or low SES. In fact, within neighborhoods matched in SES, Freedman reported that higher popula on density was predic ve of lower crime rates. Descrip on Figure 3.27 Example: Crossover Suppression Note: On this graph, cases are marked by socioeconomic status (SES) level of the neighborhood (1 = high SES, 2 = medium SES, 3 = low SES). When SES is ignored, there is a large posi ve correla on between X1 (neighborhood crowding) and Y (neighborhood crime). When the X1, Y correla on is assessed separately within each level of SES, the rela onship between X1 and Y becomes nega ve. The X2 variable (SES) suppresses the true rela onship between X1 (crowding) and Y (crime). Crowding and crime appear to be posi vely correlated when we ignore SES; when we sta s cally control for SES, it becomes clear that within SES levels, crowding and crime appear to be nega vely related. 3.17 “NONE OF THE ABOVE” The foregoing sec ons describe some possible interpreta ons for comparisons of par al correla on outcomes with zero-order correla on outcomes. This does not exhaust all possibili es. Par al correla ons can be misleading or difficult to interpret. Do not strain to explain results that don’t make sense. Strange results may arise from sampling error, outliers, or problems with assump ons for correla ons. 3.18 RESULTS SECTION The first research example introduced early in the chapter examined whether height (X1) and vocabulary (Y) are related when grade level (X2) is sta s cally controlled. The results presented in Sec on 3.2 can be summarized briefly. Results The rela on between height and vocabulary score was assessed for N = 48 students in three different grades in school: Grade 1, Grade 5, and Grade 9. The zero-order Pearson’s r between height and vocabulary was sta s cally significant, r(46) = .72, p < .001, two tailed. A sca erplot of vocabulary scores by height (with individual points labeled by grade level) suggested that both vocabulary and height tended to increase with grade level. It seemed likely that the correla on between vocabulary and height was spurious, that is, a ributable en rely to the tendency of both these variables to increase with grade level.

To assess this possibility, the rela on between vocabulary and height was assessed controlling for grade. Grade was controlled for in two different ways. A first-order par al correla on was computed for vocabulary and height, controlling for grade. This par al r was not sta s cally significant, r(45) = –.01, p = .938. In addi on, the correla on between height and vocabulary was computed separately for each of the three grade levels. For Grade 1, r = .067; for Grade 5, r = .031; and for Grade 9, r = –.141. None of these correla ons was sta s cally significant, and the differences among these three correla ons were not large enough to suggest the presence of an interac on effect (i.e., there was no evidence that the nature of the rela onship between vocabulary and height differed substan ally across grades). When grade was controlled for, either by par al correla on or by compu ng Pearson’s r separately for each grade level, the correla on between vocabulary and height became very small and was not sta s cally significant. This is consistent with the explana on that the original correla on was spurious. Vocabulary and height are correlated only because both variables increase across grade levels (and not because of any direct causal or noncausal associa on between height and vocabulary). 3.19 SUMMARY Par al correla on can be used to provide preliminary exploratory informa on about rela ons among variables. When we take a third variable, X2, into account, our understanding of the nature and strength of the associa on between X1 and Y can change in several different ways. This chapter outlines two methods to evaluate how taking X2 into account as a control variable may modify our understanding of the way in which an X1 predictor variable is related to a Y outcome variable. The first method involved dividing the data set into separate groups, on the basis of scores on the X2 control variable (using the split file procedure in SPSS), and then examining sca erplots and correla ons between X1 and Y separately for each group. In the examples in this chapter, the X2 control variables had a small number of possible score values (e.g., when sex was used as a control variable, it had just two values, male and female; when grade level in school and SES were used as control variables, they had just three score values). The number of score values on X2 variables was kept small in these examples to make it easy to understand the examples. However, the methods outlined here are applicable in situa ons where the X2 variable has a larger number of possible score values, as long as the assump ons for Pearson correla on and par al correla on are reasonably well met. Note, however, that if the X2 variable has 40 possible different score values, and the total number of cases in a data set is only N = 50, it is quite likely that when any one score is selected (e.g., X2 = 33), there may be only one or two cases with that value of X2. When the n’s within groups based on the value of X2 become very small, it becomes impossible to evaluate assump ons such as linearity and normality within the subgroups, and es mates of the strength of associa on between X1 and Y that are based on extremely small groups are not likely to be very reliable. The minimum sample sizes that are suggested for Pearson correla on and bivariate regression are on the order of N = 100. Sample sizes should be even larger for studies where an X2 control variable is taken into account, par cularly in situa ons where the researcher suspects the presence of an

interac on or modera ng variable; in these situa ons, the researcher needs to es mate a different slope to predict Y from X1 for each score value of X2. We can use par al correla on to sta s cally control for an X2 variable that may be involved in the associa on between X1 and Y as a rival explanatory variable, a confound, a mediator, a suppressor, or in some other role. However, sta s cal control is generally a less effec ve method for dealing with extraneous variables than experimental control. Some methods of experimental control (such as random assignment of par cipants to treatment groups) are, at least in principle, able to make the groups equivalent with respect to hundreds of different par cipant characteris c variables. However, when we measure and sta s cally control for one specific X2 variable in a nonexperimental study, we have controlled for only one of many possible rival explanatory variables. In a nonexperimental study, there may be dozens or hundreds of other variables that are relevant to the research ques on and whose influence is not under the researcher’s control; when we use par al correla on and similar methods of sta s cal control, we are able to control sta s cally for only a few of these variables. In this chapter, many ques ons were presented in the context of a three-variable research situa on. For example, is X1 confounded with X2 as a predictor? When you control for X2, does the par al correla on between X1 and Y drop to 0? In mul variate analyses, we o en take several addi onal variables into account when we assess each X1, Y predic ve rela onship. However, the same issues that were introduced here in the context of three-variable research situa ons con nue to be relevant for studies that include more than three variables. A researcher may hope that adding a third variable (X2) to the analysis will increase the ability to predict Y. A researcher may hope that adding an X2 covariate will reduce or increase the strength of associa on between X1 and Y. Alterna vely, a researcher may hope that adding an X2 covariate does not change the strength of associa on between X1 and Y. The next chapter (regression with two predictors) shows how a regression to predict Y from both X1 and X2 provides more informa on. This regression will tell us how much variance in Y can be predicted from X1 and X2 as a set. It will also tell us how well X1 predicts Y when X2 is sta s cally controlled and how well X2 predicts Y when X1 is sta s cally controlled. COMPREHENSION QUESTIONS When we assess X1 as a predictor of Y, there are several ways in which we can add a third variable (X2) and several “stories” that may describe the rela ons among variables. Explain what informa on can be obtained from the following two analyses: Assess the X1, Y rela on separately for each group on the X2 variable. Obtain the par al correla on (par al r of Y with X1, controlling for X2). Which of these analyses (I or II) makes it possible to detect an interac on between X1 and X2? Which analysis assumes that there is no interac on? If there is an interac on between X1 and X2 as predictors of Y, what pa ern would you see in the sca erplots in Analysis I?

Discuss each of the following as a means of illustra ng the par al correla on between X1 and Y, controlling for X2. What can each analysis tell you about the strength and the nature of this rela onship? Sca erplots showing Y versus X1 (with X2 scores marked in the plot). Par al r as the correla on between the residuals when X1 and Y are predicted from X2. Explain how you might interpret the following outcomes for par al r: r1Y = .70 and r1Y.2 = .69 r1Y = .70 and r1Y.2 = .02 r1Y = .70 and r1Y.2 = –.54 r1Y = .70 and r1Y.2 = .48 What does the term par al mean when it is used in connec on with correla ons? NOTE 1Some issues with path models are omi ed from this simple introduc on. Analysis methods described here cannot handle path models with feedback loops, such as X1 → X2 → Y → X1 → X2 → Y, and so on, or paths for both X1 → X2 and X2 → X1. There are real-world situa ons where these models would be appropriate; however, different analy c methods would be required. DIGITAL RESOURCES Find free study tools to support your learning, including eFlashcards, data sets, and web resources, on the accompanying website at edge.sagepub.com/warner3e. Descrip ons of Images and Figures Back to Figure The dialog box is divided into two ver cal panes. The first pane on the le side displays a list of variables with the nominal icon. The list reads as follows: “Resid_Height[Res_1]” and “Resid_ Voc[Res_1]”. The second pane is subdivided into two horizontal panes. The first horizontal pane has the following data fields.

 Y- Axis 1: Vocabulary.  X- Axis 1: Height.  Set markers by: grade.  Label cases by: empty.

The second horizontal pane is labelled, “Panel by” and has empty data fields for “Rows” and “columns”. Below each of these empty data fields, there is a disabled check box that reads, “Nest Variables (no empty columns)”. Another pane labelled the, “Template” is given at the bo om of the dialog box. It has a check box for, “Use Chart specifica ons from…” and a disabled tab for “File”.

At the bo om in the dialog box are the following tabs, “Reset”, “Paste”, “Cancel”, “OK”, and “Help”. The tab of “OK” is selected. Back to Figure The second horizontal pane is labelled, “Panel by” and has empty data fields for “Rows” and “columns”. Below each of these empty data fields, there is a disabled check box that reads, “Nest Variables (no empty columns)”. Another pane labelled the, “Template” is given at the bo om of the dialog box. It has a check box for, “Use Chart specifica ons from…” and a disabled tab for “File”. At the bo om in the dialog box are the following tabs, “Reset”, “Paste”, “Cancel”, “OK”, and “Help”. The tab of “OK” is selected. Back to Figure The horizontal axis represents height and ranges from 45 to 70, in increments of 5. The ver cal axis represents vocabulary and ranges from 20 to 80, in increments of 20. A sca er plot for grades 1, 5, and 9 is plo ed in the graph. A best fit line is seen going upwards from an approximate value of 36 on the ver cal axis. An equa on y equals nega ve 53.26 plus1.99 mul plied by X is seen displayed on the best fit line. For the graph r equals .72 and r square equals .521. The details of the sca er plot are approximately summarized below:

 The data points for grade 1 are seen sca ered between the values 45 and 55 on the horizontal axis and between 20 to 60 on the ver cal axis.

 The data points for grade 5 are seen sca ered between the values 53 and 60 on the horizontal axis and between 40 to 80 on the ver cal axis.

 The data points for grade 9 are seen sca ered between the values 57 and 68 on the horizontal axis and between 50 to 80 on the ver cal axis.

Back to Figure The horizontal axis represents height and is labeled X1. The ver cal axis represents vocabulary and is labeled Y. A sca er plot for grades 1, 5, and 9 is plo ed on the graph. A best fit line is seen going upwards from the bo om le of the graph to the top right. The data points for grade 1 are seen plo ed on the bo om le of the graph around the best fit line. The data points for grade 5 are seen plo ed at the center of the graph around the best fit line. The data points for grade 9 are seen plo ed on the top right of the graph around the best fit line. Back to Figure

The dialog box is tled bivariate correla ons and is divided into two ver cal panes. In the first pane a variable labeled grade is listed. In the second pane tled “Variables” the variables of height and vocabulary are listed. There is a bu on with a “Return Key” between the two panes. There are two tabs to the right side of the second pane: op ons and style. There are two more horizontal panes below the variables pane. The first pane is labelled, “Correla on Coefficients” and has three checkboxes: Pearson Kendall’s tau-b Spearman The check box of Pearson is selected. The second pane is labelled, “Test of Significance” and has the following radio bu ons. Two –tailed One-tailed. The radio bu on of “Two-tailed” is selected. Under these panes, a check box with the following text is given and is selected, “Flag significant correla ons”. At the bo om in the dialog box are the following tabs, “Reset”, “Paste”, “Cancel”, and “OK” are visible.

From the menu bar in the SPSS window the menu data is selected. The pull down menu shows various op ons like define value proper es, iden fy duplicate cases, sort cases, split file, split into file, and so on. From this expanded view the op on split file is elected. A dialog box for split file is displayed on the right hand side of this SPSS window. The dialog box is labelled “Split File” and is divided into two ver cal panes. In the first pane variables such as height and vocabulary are listed. The second pane has the following radio bu ons:

 Analyze all cases, do not create groups  Compare groups  Organize output by groups  Sort the file by grouping the variables  File is already sorted

The radio bu ons of “Compare Groups” and “Sort the file by grouping the variables” are selected. Below the data field of “Organize output by groups” a cell labelled, “Groups based on” is visible and has the text “grade” in the cell. At the bo om of the dialog box the text reads, Current Status: Analysis by groups is off. There are tabs for “Reset”, “Paste”, “Cancel”, and “OK”. The tab of “OK” is selected in this screenshot. Back to Figure The details of the output table are as follows: For grade 1:

The dialog box is tled “Linear Regression” and is divided into two panes. In the first pane variables grade and vocabulary are listed. The second pane has a data field labeled, “Dependent” the text reads height. A horizontal pane below these data fields is labeled, “Block 1 of 1” with two tabs “Previous” and “Next”, the la er is in an enabled state. It has a data field and is labeled “Independents” and the text reads, “grade”. Below this is a data field labeled Method with a “Enter” wri en in a dropdown cell. There are three data fields with empty cells below this pane and are listed below. Selec on Variable with a disabled tab labeled rule. Case Labels. WLS Weight. There are tabs to the right side of this pane that read as follows: Sta s cs, plots, save, op ons, style, and bootstrap. At the bo om in the dialog box are the following tabs, “Reset”, “Paste”, “Cancel”, and “OK”. The tabs of “OK” is selected. Back to Figure The dialog box is tled “Bivariate Correla ons” and is divided into two ver cal panes. In the first pane variables are listed in alphabe cal order and with a nominal and measurement icons. The list reads as follows: grade, gender, height, and vocabulary. The second pane is tled “Variables” and lists the following variables: “Resid_Height [Res 1]” and “Resid_Voc [Res 2]”. There is a bu on with a “Return Key” between the two panes and an op ons tab to the right side of the second pane. There are two more horizontal panes below in the dialog box. The first pane is labelled, “Correla on Coefficients” and has three checkboxes:

Pearson Kendall’s tau-b Spearman The check box of Pearson is selected. The second pane is labelled, “Test of Significance” and has the following radio bu ons. Two –tailed One-tailed. The radio bu on of “Two-tailed” is selected. Under these panes, a check box with the following text is given and is selected, “Flag significant correla ons”. At the bo om in the dialog box are the following tabs, “Reset”, “Paste”, “Cancel”, and “OK” are visible and the “OK” tab is selected. Back to Figure The details of the table are as follows:

The horizontal axis represents the variable need for power and the ver cal axis represents job performance evalua on. In males the variables are posi vely correlated. The data points are seen going upwards from bo om right to the top le . In females the variables seem nega vely correlated. The data points are seen coming downwards from top le to bo om right. Back to Figure The dialog box is tled “Par al Correla ons” and is divided into four panes. In the first pane variables are listed in alphabe cal order and with a nominal and measurement icons. The list reads as follows: gender, Resid_Height[Res 1], and Resid_Voc [Res 2].

The second pane is tled “Variables” and lists the variables height and vocabulary. The variable of vocabulary is highlighted. The third pane labeled, “Controlling for” has the Variable grade and is selected. There is a bu on with a “Return Key” between the two panes and an op ons tab to the right side of the second pane. Another horizontal pane labeled, “Test of Significance” has the following radio bu ons. Two –tailed One-tailed. The radio bu on of “Two-tailed” is selected. Under these panes, a check box with the following text is given and is selected, “Display actual significance level”. At the bo om in the dialog box are the following tabs, “Reset”, “Paste”, “Cancel”, and “OK” are visible and the “OK” tab is selected. Back to Figure The details of the table are as below:

All the variables are arranged in the form of a triad. On the top in the illustra on are variables X1 and X2 and at the bo om is the variable Y. Variables X1 and X2 are connected through a bidirec onal curved arrow. The variable X1 is connected to Y and is labeled ques on mark 1. The Variable X2 is labeled ques on mark 2 is also connected to Y. Both the variables are connected to Y through separate unidirec onal arrows. Back to Figure All the variables are arranged in the form of a triad. On the top in the illustra on are variables X1 and X2 and at the bo om is the variable Y.

Variables X1 and X2 are connected through a bidirec onal curved arrow. The variable X1 is connected to Y and is labeled ques on mark 1. The Variable X2 is labeled ques on mark 2 is also connected to Y. Both the variables are connected to Y through separate bidirec onal arrows. Back to Figure In the first diagram three overlapping circles are arranged ver cally. The intersec ng area between the first circle and the second circle is labeled c, the intersec ng area between the second and the third circle is labeled a. The le over area in the second circle is labeled b. The second and the third circles are labeled X1 and X2 respec vely. Text on the right reads: Total variance in X1: a plus b plus c equals 1. Variance in X1 that is predic ve of Y: c. Propor on of total variance in X1 that is predic ve of Y when X2 is ignored: c divided by a plus b plus c equals c divided by 1 equals c. In the second diagram a complete circle labeled Y and a par al circle labeled X1 is shown. The area of intersec on between Y and X1 is labeled c and the le over area in the second circle is labeled b. Text on the right reads: Total variance that remains in X1 a er X2 is “par alled out” is b plus c. Variance in X1 that is predic ve of Y corresponds to area c. Propor on of variance in X1 that is predic ve of Y when all variance in X1 that is related to X2 has been removed is: c divided by b plus c. Back to Figure The horizontal axis represents crowding and the ver cal axis represents crime. The sca er plot marks the data points on the basis of socioeconomic status (SES) level of the neighborhood. In this graph: 1 is high SES, 2 is medium SES, and 3 is low SES. The crowding and crime correla on is assessed separately within each level of SES.

 The data points for SES level 1 are seen clustered at the level where both crime and crowding is minimum.

 The data points for SES level 2 are seen clustered at the level where both crime and crowding is medium

 The data points for SES level 3 are seen clustered at the level where both crime and crowding is maximum.