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UT Knoxville STAT 201 - 1) bivariate_cor_reg

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SAS OUTPUT FOR CORRELATIONActual and Predicted SalaryObs salary Value ResidualBRIEF REVIEW OF BASIC STATISTICS AND NOTATIONA DATA SETGRAPHIC REPRESENTATION OF BIVARIATE RELATIONSCourse: Multiple Regression Topic: Bivariate Correlation & Regression 1 BIVARIATE CORRELATION & REGRESSIONThis semester we will discuss multiple regression/correlation analysis (MRC). MRC is a flexible analysis strategy that enables us to examine the association among a continuous (and at times categorical) dependent variable and multiple continuous and/or categorical independent variables. Indeed, analysis of variance, when examined in terms of the general linear model, can be considered a special case of MRC. When we specify the models used in MRC we assume that the independent variable(s) (IV) cause (i.e., produce changes in) the dependent variable (DV). Keep in mind, however, that MRC only indicates whether there is an association among the variables. The extent to which we can infer that the direction of the association flows from the IV to the DV and is causal in nature is strictly limited by the methodology used to collect the data. We can more comfortably infer causal direction to the extent to which data were collected using an experimental method in which the (a) independent variables were manipulated, (b) participants (or observations) were randomly assigned to levels of the independent variable, and (c) extraneous (i.e., confounding variables) were controlled. To the extent which the above criteria (a, b, and c) were not satisfied the less comfortably we can infer causation and can simply talk about associations among variables. Furthermore, we must keep in mind that the statistical analyses are performed typicallyon sample data in an attempt to make inferences about population level associations. Consequently, issues of sampling distributions and hypothesis testing continue to be relevant.Today we will discuss the simple situation in which we examine the relationship between two variables. (It’s likely that you covered much of today’s material in an introductory statistics class – in which case this will serve as a review) As we will discuss in future classes, bivariate associations can be very misleading when the data are not collected using a strict experimental method because a portion of the relationship between the independent variable (or predictor) and dependent variable (or criterion) is usually shared with other predictor variables. Procedures for handling such complications must wait another day.BRIEF REVIEW OF BASIC STATISTICS AND NOTATIONBefore venturing into the world of bivariate associations, it might be fruitful to quickly review standard deviation and z-scores. Such statistics are frequently used in MRC. Recall that there are separate (yet related) formulas for determining the standard deviation (i.e, variation) of a population () and sample (s).NX2)( 1)(2nXXsThe sample formula is typically used when we wish to estimate the population standard deviationbased on sample data. The major difference between formulas is that the sample uses n-1 in the denominator, as opposed to N, to adjust for the tendency of samples to underestimate population variability. The numerator of the formulas, which sum the squared deviations of the scores of a distribution from the mean of the distribution, is often abbreviated SS. So standard deviation can also be expressed as NSS and 1nSSs. Variance is simply the standard deviation squared.Course: Multiple Regression Topic: Bivariate Correlation & Regression 2 The authors of our textbook, Cohen and Cohen, use slightly different notation to represent standard deviation. They use sd to represent the population standard deviation and ds~to represent the sample standard deviation (i.e., when estimating from the sample to the population). Furthermore, the authors represent deviations scores (i.e., XX ) with a lower caseletter (e.g., x). Consequently, SS is represented as x.When formulas require that standard deviation of variable Y be divided by the standard deviation of variable X, it need not matter whether we use the sample or population formulas when samples sizes are equal because the denominators of each formula cancel and we essentially divide by SS.xyxxyyxxyySSSSSSNNSSNSSNSS *and xyxxyyxxyySSSSSSnnSSnSSnSS1*111 When comparing variables that are measured on different scales (e.g., Celsius and miles per hour), it is often useful to transform the variables into a Z-score metric, which indicates the number of standard deviations by which a score deviates from the mean of the distribution:XzIf all of the scores in a distribution are transformed into z-scores the transformed z-distribution will have a mean =0, standard deviation = 1, and the shape of the transformed distribution will have the same shape as the original distribution. Now let’s proceed to our discussion of bivariate associations.A DATA SETFor the next few classes, we will use the bogus data set on academic salary provided by Cohen and Cohen (1983, p. 99). The data set is reproduced in the following table.Subject Salary PhD Pubs Sex Citations1 18000 1 2 0 12 19961 2 4 0 03 19828 5 5 1 14 17030 7 12 1 05 19925 10 5 0 06 19041 4 9 0 17 27132 3 3 1 08 27268 8 1 0 19 32483 4 8 0 010 27029 16 12 1 411 25362 15 9 0 012 28463 19 4 0 313 32931 8 8 0 514 28270 14 11 0 015 38362 28 21 0 3PhD = years since PhD, Pubs = number of publications, Sex (0 = male, 1 = female), Citations = number of times a publication was cited. Data are from Cohen & Cohen (1983). Applied multiple regression/correlation analysis for the behavioral sciences (p. 99). Hillsdale: New Jersey.Course: Multiple Regression Topic: Bivariate Correlation & Regression 3 GRAPHIC REPRESENTATION OF BIVARIATE RELATIONSA useful and relatively easy way of assessing the nature of a bivariate relationship is to visually examine the relationship in a scatter plot. For example, we can visualize the relationship between salary and number of publications by plotting a given individuals salary by his/her number of publications. The following tables creates a SAS data set and uses the proc plot procedure to plot the salary by publications data:Such a procedure plots salary on the Y-axis and publications on the X-axis. Simply reverse the order of variables to reverse the axis on which the variables are plotted. Additional


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UT Knoxville STAT 201 - 1) bivariate_cor_reg

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