Modeling discontinuous and nonlinear change (29 pages)

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Modeling discontinuous and nonlinear change



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Modeling discontinuous and nonlinear change ALDA Chapter Six Things have changed Bob Dylan Judith D Singer John B Willett Harvard Graduate School of Education Chapter Chapter 6 6 Modeling Modeling discontinuous discontinuous and and nonlinear nonlinear change change General idea All our examples so far have assumed that individual growth is smooth and linear But the multilevel model for change is much more flexible Discontinuous Discontinuousindividual individualchange change 6 1 especially 6 1 especiallyuseful usefulwhen whendiscrete discreteshocks shocksor or time limited treatments affect the life course time limited treatments affect the life course Using Usingtransformations transformationstotomodel modelnon linear non linearchange change 6 2 perhaps 6 2 perhapsthe theeasiest easiest way of fitting non linear change models way of fitting non linear change models Can Cantransform transformeither eitherthe theoutcome outcomeororTIME TIME We Wealready alreadydid didthis thiswith withALCUSE ALCUSE which whichwas wasaasquare squareroot rootofofaasum sumofof44items items Using Usingpolynomials polynomialsofofTIME TIMEtotorepresent representnon linear non linearchange change 6 3 6 3 While Whileadmittedly admittedlyatheoretical atheoretical it s it svery veryeasy easytotodo do Probably Probablythe themost mostpopular popularapproach approachininpractice practice Truly Trulynon linear non lineartrajectories trajectories 6 4 6 4 Logistic Logistic exponential exponential and andnegative negativeexponential exponentialmodels models for forexample example AAworld worldofofpossibilities possibilitieslimited limitedonly onlyby byyour yourtheory theory and andthe thequality qualityand andamount amountofofdata data Singer Willett page 2 Example Example for for discontinuous discontinuous individual individual change change Wage Wage trajectories trajectories the the GED GED Data source Murnane Boudett and Willett 1999 Evaluation Review Sample Sample the thesame same888 888male malehigh highschool school dropouts dropouts from frombefore before Research Researchdesign design Each Eachwas wasinterviewed interviewedbetween between11and and13 13times times after dropping out after dropping out 34 6 34 6 n 307 n 307 earned earnedaaGED GEDatatsome somepoint point during data collection during data collection OLD OLDresearch researchquestions questions How Howdo dolog WAGES log WAGES change changeover overtime time Do the wage trajectories differ by ethnicity Do the wage trajectories differ by ethnicityand and highest grade completed highest grade completed Additional AdditionalNEW NEWresearch researchquestions questions What Whatisisthe the effect of GED attainment Does earning a effect of GED attainment Does earning a GED GED affect affectthe thewage wagetrajectory s trajectory selevation elevation affect the wage trajectory s slope affect the wage trajectory s slope create createaadiscontinuity discontinuityininthe thewage wagetrajectory trajectory ALDA Section 6 1 1 pp 190 193 Singer Willett page 3 First First steps steps Think Think about about how how GED GED receipt receipt might might affect affect an an individual s individual s wage wage trajectory trajectory Let s start by considering four plausible effects of GED receipt by imagining what the wage trajectory might look like for someone who got a GED 3 years after labor force entry post dropout 2 5 F Immediate shifts in both elevation rate of change LNW D An immediate shift in rate of change no difference in elevation GED B An immediate shift in elevation no difference in rate of change 2 0 A No effect of GED whatsoever 1 5 How Howdo dowe wemodel modeltrajectories trajectorieslike like these within the context of a linear these within the context of a linear growth growthmodel model 0 2 4 6 8 10 EXPER ALDA Figure 6 1 p 193 Singer Willett page 4 Including Including aa discontinuity discontinuity in in elevation elevation not not slope slope Trajectory Trajectory B B Key Keyidea idea It s It seasy easy simply simplyinclude includeGED GEDas asaatime varying time varyingeffect effectatatlevel 1 level 1 2 4 Yij 0i 1i EXPERij 2i GEDij ij LNW Common rate of change Pre Post GED 1i 2 2 Post GED GED 1 Yij 0i 2i 1i EXPERij ij 2 0 Pre GED GED 0 Elevation differential on GED receipt i 1 8 1 6 Yij 0i 1i EXPERij ij LNW at labor force entry 0i 0 2 4 6 8 10 EXPER ALDA Section 6 1 1 pp 194 195 Singer Willett page 5 Including Including aa discontinuity discontinuity in in slope slope not not elevation elevation Trajectory Trajectory D D Using Usingan anadditional additionaltemporal temporalpredictor predictorto tocapture capturethe the extra extraslope slope post GED post GEDreceipt receipt Yij 0i 1i EXPER ij 3i POSTEXPij ij Post GED POSTEXP clocked in same cadence as EXPER Yij 0i 1i EXPERij 3i POSTEXP ij Yij 0i 1i 3i EXPER ij 2 4 LNW Slope differential Pre Post GED 3i 2 2 POSTEXPij 0 prior to GED POSTEXPij Post GED experience a new TV predictor that clocks TIME since GED receipt in the same cadence as EXPER 2 0 Rate of change Pre GED 1i 1 6 ALDA Section 6 1 1 pp 195 198 Pre GED POSTEXP 0 1 8 Yij 0i 1i EXPERij ij LNW at labor force entry 0i 0 2 4 6 8 10 EXPER Singer Willett page 6 Including Including aa discontinuities discontinuities in in both both elevation elevation and and slope slope Trajectory Trajectory F F Simple Simpleidea idea Combine Combinethe thetwo twoprevious previousapproaches approaches Yij 0i 1i EXPERij 2i GED 3i POSTEXPij ij 2 4 LNW Post GED Yij 0i 2i 1i 3i EXPER ij Slope differential Pre Post GED 3i 2 2 2 0 Rate of change Pre GED 1i Constant elevation differential on GED receipt i 1 8 1 6 LNW at labor force entry 0i 0 2 4 6 8 Pre GED Yij 0i 1i EXPERij ij 10 EXPER ALDA Section 6 1 1 pp 195 198 Singer Willett page 7 Many Many other other types types of of discontinuous discontinuous individual individual change change trajectories trajectories are are possible possible Just like a regular regression model the multilevel model for change can include discontinuities nonlinearities and other nonstandard terms Generally more limited by data theory or both than by the ability to specify the model Extra terms in the level 1 model translate into extra parameters to estimate What kinds of other complex trajectories could be used Effects on elevation and slope can depend upon timing of GED receipt ALDA pp 199 201 You might have non linear changes before or after the transition point The effect of GED receipt might be instantaneous but not


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