REPRESENTING NON LINEAR ASSOCIATIONS Linear Association Direction of relation b w X Y is constant across X Non Linear Association Direction of relation b w X Y is changes across X Quadratic Cubic Quartic 1 change in direction 1 bend 2 changes in direction 2 bends 3 changes in direction 3 bends How Many Non Linear Patterns Can Be Tested There are an infinite number of non linear patterns of measured X values limits of testable patterns e g need at least 3 values of X to test a quadratic pattern Can t test if direction of relation changes with only 2 values of X Can test G 1 changes in direction of association between X Y where G of measured values of X If measure X 1 2 3 4 5 can test linear quadratic cubic and quartic G 1 Values of X are Necessary But Not Sufficient Need to sample X values that span the range of X across which the direction change s occurs E g Imagine in the population Y increases across X until X 6 and then decreases as X increases beyond 6 If we sample X values between 1 and 5 we will only be able to detect the linear increase and will miss the direction change that occurs when X 6 How Many Non Linear Patterns Should Be Tested Test as many non linear patterns as is dictated by theory Goal of science is to parsimoniously explain behavior Identifying 29 changes in direction is not the same as explaining why there are 29 changes in direction Theory should dictate the regression model Specifying Non Linear Associations with a Linear Model A model is linear if the parameters are limited to the first power and not multiplied or divided by other parameters Y Bo B1 X 1 B2 X 2 A model is non linear if the parameters are raised to a power other than 1 or multiplied or divided by other parameters Y Bo B12 X 1 B 32 X 2 In this class we examine only linear models Specifying Non Linear Associations with a Linear Model The trick to testing a non linear pattern with a linear model is to transform the X values Retain linearity in parameters and incorporate non linearity with transformations in the measured variables Two ways of transforming X orthogonal polynomial contrasts power polynomials Orthogonal Polynomial Contrasts We discussed this last semester when testing non linear trends in ANOVA Divide the X variable into g 1 predictors i e each value of X represents a level of X Contrast code the g 1 predictors and use the orthogonal polynomial contrast weightings Power Polynomials Transform X with powers X2 X3 X4 Each power indicates the of discernable slopes relating Y to X or 1 more than the number of direction changes E g Linear and Quadratic is specified as Y Bo B1 X 1 B2 X 12 Partialling is essential for B2 to reflect quadratic pattern A Bogus Data Set Pretend we are interested in the effect of perceived groupness on persuasion this is all imaginary but plausible Theory of numbers suggests that increasing the number of agents of influence increases persuasion e g a person will be more strongly persuaded to adopt a belief if confronted by 3 vs 2 people Past research indicates that persuasion increases linearly from 1 to 2 to 3 agents A Bogus Data Set However Group vs Individual level perception at times others are perceived as a group rather than distinct persons When simultaneously confronted by multiple persons perceiver may chunk persons into one group in which case the group is perceived as the unit and may be less influential than if individuals were the perceptual unit Persuasion may have a linear and quadratic relation with number of others as increases rate of persuasion might diminish A Bogus Data Set Past research was limited to 1 2 or 3 agents We increase of agents to provide a better test of linear quadratic pattern 34 Participants are confronted with either 1 2 3 4 5 6 7 8 9 10 persons arguing for a tuition increase Participant then rates attitude toward raising tuition on a 9point scale 1 strongly disagree to 9 strongly agree A Bogus Data Set Forming Quadratic Polynomial in SAS data nonlin input num attitude cards 1 1 1 1 1 2 1 2 2 2 2 3 3 2 3 4 enter all data data temp set nonlin numsqr num num Non Essential Ill Conditioning Power Polynomials Power polynomials are formed by multiplying a variable by itself e g num3 num num num Variables representing the polynomials will be highly correlated Non Essential Ill Conditioning Power Polynomials High correlations introduce multicollinearity in model Multicollinearity inflates standard error lower order betas e g Attitude B0 B1num B2num2 B3num3 B4num4 B5num5 SE for B1 B2 B3 and B4 will be inflated Non Essential Ill Conditioning Power Polynomials Such ill conditioning of model is non essential because collinearity is an artifact of computation num num Two Strategies to Deal with Non Essential Collinearity Hierarchical Regression Simultaneous Regression on Centered Data Hierarchical Regression high correlation between num and num2 poses a problem for our hypothesis test Avoid problem with hierarchical procedure Model 1 Attitude B0 Model 2 Attitude B0 B1num Model 3 Attitude B0 B1num B2num2 Model 2 vs 1 is test of linear component Model 3 vs 2 is test of quadratic component 2 RFull RRe2 stricted F 2 1 RFull k F k R n k F 1 Testing in SAS proc reg model attitude num model attitude num numsqr run Recall F test provided by SAS compares current model with a model with 0 predictors so F for first model is test of num Model Comparisons Linear Component Attitude B0 Attitude B0 B1num F 1 32 31 35 p 0001 4949 0 F 1 4949 1 0 31 35 34 1 1 Linear component is significant and accounts for 49 of variation in attitudes i e persuasion Quadratic component Attitude B0 B1num Attitude B0 B1num B2num2 F 1 31 4 51 p 05 5591 4949 F 1 5591 2 1 4 51 34 2 1 Quadratic component is significant and accounts for 6 42 of variation in attitudes i e persuasion beyond the linear component Summarizing Results in a Table Note increased SE for num in full model 08 vs 33 poses problem for test of num in full model so use model comparison Obtaining the Regression Equation from Hierarchical Model Use fullest model as the regression equation linking persuasion to linear and quadratic effect of number of others disregard significance test of lower order parameter use model comparison for the test Attitude 0 50 1 14 num 0 06 numsqr Plotting the Regression Line Obtain predicted value for each value of X and plot predicted value against X e g predicted value of attitude when num 2 numsqr 4 Attitude 0 50 1 14 2 0 06 4 2 54 repeat with each value of num 1 10 and
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