REPRESENTING NON-LINEAR ASSOCIATIONSLinear Association-Direction of relation b/w X&Y is constant across XNon-Linear Association-Direction of relation b/w X&Y is changes across XQuadratic1 change in direction(1 bend)Cubic2 changes in direction(2 bends)Quartic3 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 patternse.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 quarticG-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) occursE.g.,Imagine in the population Y increases across X until X = 6, and then decreases as X increases beyond 6If 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 > 6How 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 modelSpecifying 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 parameters2211XBXBBYo-A model is non-linear if the parameters are raised to a powerother than 1 or multiplied (or divided) by other parameters231221XBXBBYo-In this class we examine only linear modelsSpecifying Non-Linear Associationswith 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 polynomialsOrthogonal 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 weightingsPower 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:21211XBXBBYo-Partialling is essential for B2 to reflect quadratic patternA 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 agentsA 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 diminishA 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 9-point scale (1=strongly disagree to 9 = strongly agree)A Bogus Data SetForming Quadratic Polynomial in SASdata nonlin;input num attitude;cards;1 11 1 1 21 22 22 33 23 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 correlatedNon-Essential Ill-Conditioning & Power Polynomials-High correlations introduce multicollinearity in model-Multicollinearity inflates standard error lower order betase.g., Attitude = B0 + B1num + B2num2 + B3num3 + B4num4 + B5num5 SE for B1 B2 B3 and B4 will be inflatedNon-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 DataHierarchical Regression-high correlation between num and num2 poses a problem forour hypothesis test.-Avoid problem with hierarchical procedureModel 1: Attitude = B0Model 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)1()1()()(22Re2FFullRFstrictedFullknRkkRRFTesting in SASproc 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 ComponentAttitude = B0Attitude = B0 + B1num F(1,32)=31.35, p =.0001Linear component is significant and accounts for 49% of variation in attitudes (i.e., persuasion)-Quadratic componentAttitude = B0 + B1num Attitude = B0 + B1num + B2num2 F(1,31)=4.51, p < .05Quadratic component is significant and accounts for 6.42% of variation in attitudes (i.e., persuasion) beyond the linear component35.31)1134()4949.1()01()04949(.F51.4)1234()5591.1()12()4949.5591(.FSummarizing 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 fromHierarchical 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
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