1Simple Linear Regression, Scatterplots, and Bivariate CorrelationThis section covers procedures for testing the association between two continuousvariables using the SPSS Regression and Correlate analyses. Specifically, we demonstrateprocedures for running Simple Linear Regression, Producing Scatterplots, and running BivariateCorrelations, which report the Pearson’s r statistic. These analyses will allow us to identify thestrength (Pearson’s r) and direction (the sign of r and b) of the relationship between ourvariables of interest and make predictions regarding the value of the outcome variable (Y) forknown values of a predictor (X). For the following examples, wehave recreated the data set based oncartoon 8.1 (Santa Deals with the reindeer)found in Table 8.1. Again, theIndependent variable is the number oftimes in a month that the reindeercomplain to Santa. The dependent variableis the size of the herd for a given month.2Setting Up the DataFigure 8.1 presents thevariable view of the SPSS data editorwhere we have defined two variables(both continuous). The first variablerepresents the frequency with whichthe reindeer complain during themonths we sampled. We have givenit the variable name complain andgiven it the variable label “Numberof Complaints Received perMonth.” The second variablerepresents the herd size duringeach month that we sampled. Wehave given it the variable nameherdsize and the variable label“Current Size of the Herd.”Figure 8.2 presents the dataview of the SPSS data editor.Here, we have entered thecomplaint and herd size data for3the 12 months we have sampled. Remember that the columns represent each of the differentvariables and the rows represent each observation, which in this case is each cow. For example,during the first month, 2 reindeer complained and the herd size was 25. Similarly, during the 12thmonth, 14 reindeer complained and herd size consisted of 9 reindeer.Simple Linear RegressionSimple Linear Regression allows us to determine the direction and of the associationbetween two variables and to identify the least squares regression line that best fits the data. Inconjunction with the regression equation (Y = a + bX), this information can be used to makepredictions about the value of Y for known values of X. For example, we can make predictionsabout the number of reindeer that will be turned into venison if they complain a certain numberof times. Further, the SPSS simple regression analysis will tell us whether a significant amountof the variance in one variable is accounted for (predicted) by another variable. That is, theseanalyses will tell us wether the relationship between reindeer complaints and the size of the herdis a significant relationship (not likely to have occurred by chance alone).Running the AnalysesSimple Linear Regression (See Figure 8.3): From the Analyze (1) pull down menu, select Regression (2), then select Linear... (3) from the side menu. In the Linear Regressiondialogue box, enter the variable herdsize in the Dependent: field by left-clicking on the variable4and left-clicking on theboxed arrow (4) pointing tothe Dependent: field. Next,enter the variable complainin the Independent(s):field by left-clicking on thevariable and left-clickingon the boxed arrow (5)pointing to theIndependent(s): field.Finally, double check yourvariables and either selectOK (6) to run, or Paste tocreate syntax to run at alater time.If you selected the paste option from the procedure above, you should have generated thefollowing syntax:REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT herdsize /METHOD=ENTER complain .To run the analyses using the syntax, while in the Syntax Editor, select All from the Run pull-down menu.5Reading the Simple Linear Regression OutputThe Linear Regression Output is presented in Figure 8.4. This output consists of fourparts: Variables Entered/Removed, Model Summary, Anova, and Coefficients. For our purposeswe really only need to concern ourselves with the Coefficients output. Interpretation of the otherparts of the output is more fully described in Chapter 12 on Multiple Regression. The first row ofthis output, labeled (constant), reports the Y-intercept (a) in the first column, labeled B. Thesecond column of this row, provides us with a Standard Error (SE) of the Y-intercept. Like thestandard error of the mean found in Chapter 8 (confidence intervals and 9 (t-tests), this SE is anestimate of how much the Y-intercept for sample potentially differs from the Y-intercept foundin the population. The last two columns of this row report a t value and the associatedsignificance level, which tests whether the Y intercept is significantly different from zero. The tvalue is obtained by dividing the Y-intercept by the Standard Error of the Y-intercept. In thiscase, the Y-intercept for our data is 26.551, the SE of the Y-intercept is .537, and is significantlydifferent from zero (t = 49.406) at a significance level of less than .001. The second row of this output, labeled with variable label. presents the slope of theregression line (b) in first column, labeled B. In our example, b is -1.258. That is for every6complaint that the reindeer lodge, the herd size decreases by 1.258 reindeer. The second columnof this row presents the Standard Error of b (often referred to as the SE of the Regression Coefficient). Again, this SE is an estimate of how much the true b found in thepopulation potentially differs from b found in our sample. Skipping now to the last two columnof the second row, SPSS reports a t statistic that allows us to determine whether relationshipbetween X and Y is significant. Specifically, this t-test tells us whether the slope of theregression equation is significantly different from zero and is obtained by dividing b by the SE ofb. In this example b is -1.258, SE of b is .075, and t is -16.696, which is significant at least at the.001 level. Thus, we can conclude that the number of complaints reindeer make is significantlyassociated with a decrease in the number of reindeer in the herd.The third column of this output reports a statistic that we have not previously discussed, the Standardized Regression Coefficient Beta. This coefficient is the slope of the regression linefor X and Y after both variables have been converted to Z scores. For all practical purposes, insimple regression the Standardized Beta is the same as the Pearson’s r