Understanding Bivariate Linear RegressionFixed-Effects Model Assumptions for Bivariate Linear RegressionRandom-Effects Model Assumptions for Bivariate Linear RegressionEffect Size Statistics for Bivariate Linear RegressionConducting a Bivariate Linear Regression AnalysisSelected SPSS Output for Bivariate Linear RegressionUsing SPSS Graphs to Display the ResultsSIMPLE LINEAR REGRESSION: PREDICTIONFor the bivariate linear regression problem, data are collected on an independent or predictorvariable (X) and a dependent or criterion variable (Y) for each individual. Bivariate linearregression computes an equation that relates predicted Y scores (Ŷ) to X scores. The regressionequation includes a slope weight for the independent variable, Bslope (b), and an additive constant,Bconstant (a):Ŷ = Bslope X + Bconstant(or)Ŷ = bX + aIndices are computed to assess how accurately the Y scores are predicted by the linear equation.We will focus on applications in which both the predictor and the criterion are quantitative(continuous – interval/ratio data) variables. However, bivariate regression analysis may be usedin other applications. For example, a predictor could have two levels like gender and be scored 0for females and 1 for males. A criterion may also have two levels like pass-fail performance,scored 0 for fail and 1 for pass.Linear regression can be used to analyze data from experimental or non-experimental designs. Ifthe data are collected using experimental methods (e.g., a tightly controlled study in whichparticipants have been randomly assigned to different treatment groups), the X and Y variablesmay be referred to appropriately as the independent and the dependent variables, respectively.SPSS uses these terms. However, if the data are collected using non-experimental methods (e.g.,a study in which subjects are measured on a variety of variables), the X and Y variables are moreappropriately referred to as the predictor and the criterion, respectively.UNDERSTANDING BIVARIATE LINEAR REGRESSIONA significance test can be conducted to evaluate whether X is useful in predicting Y. This test canbe conceptualized as evaluating either of the following null hypotheses: the population slopeweight is equal to zero or the population correlation coefficient is equal to zero.The significance test can be derived under two alternative sets of assumptions, assumptions for afixed-effects model and those for a random-effects model. The fixed-effects model is probablymore appropriate for experimental studies, while the random-effects model seems moreappropriate for non-experimental studies. If the fixed-effects assumptions hold, linear or non-linear relationships can exist between the predictor and criterion. On the other hand, if therandom-effects assumptions hold, the only type of statistical relationship that can exist betweentwo variables is a linear one.Regardless of the choice of assumptions, it is important to examine a bivariate scatterplot of thepredictor and the criterion variables prior to conducting a regression analysis to assess if a non-linear relationship exists between X and Y and to detect outliers. If the relationship appears to benon-linear based on the scatterplot, you should not conduct a simple bivariate regression analysisbut should evaluate the inclusion of higher-order terms (variables that are squared, cubed, and soon) in your regression equation. Outliers should be checked to ensure that they were notincorrectly entered in the data set and, if correctly entered, to determine their effect on the resultsof the regression analysis.FIXED-EFFECTS MODEL ASSUMPTIONS FOR BIVARIATE LINEAR REGRESSIONAssumption 1: The Dependent Variable is Normally Distributed in the Population for EachLevel of the Independent VariableIn many applications with a moderate or larger sample size, the test of the slope mayyield reasonably accurate p values even when the normality assumption is violated. Tothe extent that population distributions are not normal and sample sizes are small, the pvalues may be invalid. In addition, the power of this test may be reduced if the populationdistributions are non-normal.Assumption 2: The Population Variances of the Dependent Variable are the same for AllLevels of the Independent VariableTo the extent that this assumption is violated and the sample sizes differ among the levelsof the independent variables, the resulting p value for the overall F test is not trustworthy.Assumption 3: The Cases Represent a Random Sample from the Population, and the Scoresare Independent of Each Other from One Individual to the NextThe significance test for regression analysis will yield inaccurate p values if theindependence assumption is violated.RANDOM-EFFECTS MODEL ASSUMPTIONS FOR BIVARIATE LINEAR REGRESSIONAssumption 1: The X and Y Variables are Bivariately Normally Distributed in the PopulationIf the variables are bivariately normally distributed, each variable is normally distributedignoring the other variable and each variable is normally distributed at every level of theother variable. The significance test for bivariate regression yields, in most cases,relatively valid results in terms of Type I errors when the sample is moderate to large insize. If X and Y are bivariately normally distributed, the only type of relationship thatexists between these variables is linear.Assumption 2: The Cases Represent a Random Sample from the Population, and the Scoreson Each Variable are Independent of Other Scores on the Same VariableThe significance test for regression analysis will yield inaccurate p values if theindependence assumption is violated.REGRESSIONPAGE - 2EFFECT SIZE STATISTICS FOR BIVARIATE LINEAR REGRESSIONLinear regression is a more general procedure that assesses how well one or more independentvariables predict a dependent variable. Consequently, SPSS reports strength-of-relationshipstatistics that are useful for regression analyses with multiple predictors. Four correlationalindices are presented in the output for the Linear Regression procedure: the Pearson product-moment correlation coefficient (r), the multiple correlation coefficient (R), its squared value (R2),and the adjusted R2. However, there is considerable redundancy among these statistics for