Discontinuous & Nonlinear Change (ALDA, Chapter 6)

Transcription

1 What wll we cover? Dscontnuous & Nonlnear Change (ALDA, Chapter 6) Dscontnuous Indvdual Change Usng Transformatons to Model Nonlnear Indvdual Change Usng Polynomals of TIME to Represent Indvdual Change Truly Nonlnear Trajectores p. 19 p. 8 p. 13 p. 3 John Wllett & Judy Snger Harvard Unversty Graduate School of Educaton May, 3 1

2 Dscontnuous Indvdual Change Data Example: Wage Trajectores & the GED Research Queston: Are the wage trajectores of hgh-school dropouts dsrupted by ther acquston of the GED? Ctaton: Murnane, Boudett & Wllett (1999). Sample: 888 male hgh-school dropouts. Research Desgn: Unbalanced longtudnal desgn, dropouts aged between 14 and 17 at frst wave. Intervews wth dropouts conducted at dfferent tmes durng the calendar year, approxmately annually, from the frst day of work. Work career lengths dffered from dropout to dropout. Some dropouts obtaned the GED durng ther work careers, at tmes of ther own choosng. Dscontnuous Indvdual Change Excerpt from the Person-Perod Dataset Wage Trajectores & the GED (ALDA, Table 6.1, p. 19) Dropout #6 never receved the GED Dropout #36 receved the GED after years of work experence Dropout #4384 receved the GED after 1.76 years of work experence Log of Hourly Wage Years of work experence GED, a tme-varyng dchotomous predctor that ndcated whether dropout has the GED on occason j.

3 Dscontnuous Indvdual Change How Mght Recept of GED Affect the Wage Trajectory? Wage Trajectores & the GED (ALDA, Fgure 6.1, p. 193). LNW D Dscontnuous Indvdual Change How Do You Model Ths Knd of Thng? Wage Trajectores & the GED (ALDA, Fg. 6., p. 196) Incorporatng a dscontnuty n elevaton s straghtforward, and treats predctor GED as a tme-varyng man effect at level-1: Y π EXPER GED + ε = + π1 + π C. GED B A Pre-GED (GED=): Y π EXPER + ε = + π1.4 LNW Post-GED (GED=1): Y π EXPER + ε = ( + π ) + π1. Common rate of change Pre-Post GED, π EXPER Perhaps there s A: No effect of GED? B: An mmedate shft n elevaton? C: An mmedate shft n rate of change? D: Immedate shfts n both elevaton and slope. 1.8 Elevaton dfferental on GED recept, π 1.6 LNW at labor force entry, π EXPER 3

4 Dscontnuous Indvdual Change How Do You Model Ths Knd of Thng? Wage Trajectores & the GED (ALDA, Fg. 6., p. 196) To create a dscontnuty n slope, you must add another predctor to the person-perod dataset: Dscontnuous Indvdual Change How Do You Model Ths Knd of Thng? Wage Trajectores & the GED (ALDA, Fg. 6., p. 196) Incorporatng a dscontnuty n elevaton s straghtforward, and treats predctor GED as a tme-varyng man effect at level-1: Y π EXPER POSTEXP + ε = + π1 + π 3 Pre-GED (POSTEXP=): Y π EXPER + ε = + π1 Post-GED: Y = π + π + π ) EXPER + ε ( 1 3 POSTEXP s set to zero pror to GED..4 LNW. Slope dfferental Pre-Post GED, π3 POSTEXP, for Post- GED Experence, s a new contnuous tmevaryng predctor that clocks experence from the recept of GED Rate of change Pre GED, π1 1.6 LNW at labor force entry, π EXPER 4

5 Dscontnuous Indvdual Change How Do You Model Ths Knd of Thng? Wage Trajectores & the GED (ALDA, Fg. 6., p. 196) Incorporatng a smultaneous dscontnuty n elevaton and n slope s acheved by combnng the two prevous approaches: Y π EXPER GED POSTEXP + ε = + π1 + π + π 3.4 LNW. Slope dfferental Pre-Post GED, π3 Dscontnuous Indvdual Change Wage Trajectores & the GED It s only the tp of the ceberg Effects on elevaton and slope could depend on the passage of tme (see ALDA, pp )? Possbltes of non-lnear change before, and after, the transton pont? An nstantaneous, rather than an endurng, mpact of transton? A delayed mpact of transton? Multple transton ponts?. (wrte your effect here?). Rate of change Pre GED, π1 1.8 Constant elevaton dfferental on GED recept, π LNW at labor force entry, π EXPER

6 Dscontnuous Indvdual Change Taxonomy of Ftted Multlevel Models for Change Wage Trajectores & the GED (ALDA, Equ. 6.6 & Table 6.) Hypotheszed Baselne Model (A): Y 1 = π π = γ π = γ 1 π = γ + π EXPER 1 + γ ( HGC 1 + γ ( BLACK 11 ε ~ N(, σ ) ε and + π ( UERATE 9) + ζ 9) + ζ 1 ζ σ ~ N, ζ 1 σ1 7) + ε σ 1 σ 1 Dscontnuous Indvdual Change Wage Trajectores & the GED (ALDA, Fgure 6.3, p. 6) Soluton for Fxed Effects Standard Effect Estmate Error DF t Value Pr > t Intercept <.1 EXPER <.1 HGC_ <.1 EXPER*BLACK <.1 UE_ <.1 GED POSTEXP

7 ALCUSE 1 PEER Usng Transformatons to Model Nonlnear Change (ALDA, Fgure 6.4, p. 9) Hgh Low COA = 1 COA = Earler, we modeled lnear change n ALCUSE, an outcome that we formed from the square root of the researchers orgnal adolescent alcohol use measurement. Usng Transformatons to Model Nonlnear Change Generalzng the Approach wth the Ladder of Powers (ALDA, Fgure 6., p.11) Hgh PEER Low AGE We can de-transform the earler fndngs, and return to the orgnal scale, by reversng the orgnal transformaton,.e., by squarng the predcted values of ALCUSE and re-plottng. By transformng the outcome before analyss, we have modeled nonlnear change over tme. 1 1 IQ TIME IQ (.3) 3,,, 1, 1, 1 1,, IQ TIME TIME (1/.3) 7

8 Usng Polynomals n TIME to Represent Change You Can Model Even More Complex Trajectores (ALDA, Table 6.4, 14) Usng Polynomals n TIME to Represent Change Data Example: Chld Externalzng Behavor Research Queston: Do grls and boys trajectores of externalzng behavor dffer, durng the elementary school years? Ctaton: Part of the larger study reported by Keley, Bates, Dodge & Pettt (). Sample: 4 chldren Research Desgn: Sx waves of annual data, obtaned at the end of frst through sxth grade. Outcome: Externalzng behavor -- ratng of the extent to whch the chld dsplays aggressve, dsruptve or delnquent behavor: Measured wth Achenbach s (1991) Chld Behavor Checklst. 34 tems, each w/ 3-pont scale (=rarely/never, 1=sometmes, =often). Total ratng out of 68. Prncpal Queston Predctor: FEMALE, tme-nvarant respondent gender (=male, 1=female). 8

9 4 Usng a Polynomals n TIME to Represent Change EDA, for 8 cases from the Externalzng Behavor Dataset (ALDA, Fgure 6.7, 18) 6 EXTERNAL Raw data ponts OLS ft, polynomal, order optmzed for ndvdual chld OLS ft, quartc, order common across chldren A EXTERNAL 4 B EXTERNAL 4 C EXTERNAL 4 D Usng a Polynomals n TIME to Represent Change Usng Model Comparsons to Select the Polynomal Order (ALDA, Table 6., 1) Process: Create polynomal functons of TIME (TIME, TIME 3, TIME 4, and so on) as predctors n the person-perod dataset. Ft successve uncondtonal MMC growth models. Compare goodness-of-ft across models. 6 EXTERNAL 4 E EXTERNAL 4 F EXTERNAL 4 G EXTERNAL 4 H Concluson: Consderable potental complexty n the temporal functonal form of the ndvdual growth trajectores across chldren. Hard to judge whch s most approprate, because of need to use sample (observed) data to draw conclusons about the underlyng (true) trajectores. A B H : γ 1 = ; σ1 = ; σ 1 = B C C D H : γ H : γ 3 = ; σ = ; σ = ; σ = ; σ = ; 3 3 σ = ; σ 13 1 = ; σ = 3 = χ (3)=7.8 χ (4)=9. χ ()=11.1 Rej Rej Not Rej 9

10 Truly Nonlnear Change Data Example: Learnng to Play Fox n Geese Truly Nonlnear Change Learnng Fox n Geese (ALDA, Fgure 6.8, p. 7) Research Queston: Does a chld s ablty to learn a novel cogntve task depend on hs/her pror cogntve ablty? Ctaton: Tvnan (198). Sample: 17 1 st and nd graders. Research Desgn: Three week expermental perod n whch each chld played Fox n Geese wth the expermenter, learnng from experence. Unbalanced desgn, wth up to 7 waves of data for each chld. Each wave conssts of one game. Outcome: NMOVES, the number of moves made by the chld before catastrophc error: Ranges between 1 and. Prncpal Queston Predctor: READ, score on standardzed readng test. NMOVES NMOVES NMOVES NMOVES NMOVES ID #1 ID #4 ID #6 ID #7 1 1 NMOVES NMOVES NMOVES ID #8 ID #11 ID #1 ID # Evdence of: Lower asymptote? Upper asymptote? Dfferental rates of change between asymptotes? 1

11 Truly Nonlnear Change Learnng Fox n Geese (ALDA, Equ. 6.8 & Fg. 6.9, p. 9) Truly Nonlnear Change Learnng Fox n Geese (ALDA, Table 6.6, p. 31) Truly Non- Lnear Change Change that s non-lnear n the ndvdual growth parameters For the Fox n Geese task, the hypotheszed ndvdual growth model s the logstc curve: Y 1 19 = + + π TIME 1+ π e 1 ε π s a knd of ntercept. π 1 s a knd of slope NMOVES NMOVES NMOVES π1 =. π1 =. 1 π1 =. 1 π1 =.3 1 π1 =.3 π1 =.1 NMOVES NMOVES 1 π1 =.3 1 π1 =.1 1 Hgh READ (1.8) π1 = π = 1 π = 1 π = 1. Can be ftted n pretty much the usual way, wth SAS PROC NLMIXED Low READ (-1.8) 11

12 Truly Nonlnear Change Selected Truly Non-lnear Trajectores (ALDA, Table 6.7, p. 3, & Fgure 6.11, p. 3) Truly Nonlnear Change (ALDA, Table 6.7, p. 3, & Fgure 6.11, p. 3) Hyperbola E(Y) α = 1 1 π1 =.1 E(Y) 1 Inverse polynomal α = 1 π =.1 π = E(Y) 1 Exponental π1 =.3 E(Y) 1 Negatve exponental α = 1 π1 =.3 π1 =. π1 =. 7 π1 =.1 7 π = π1 =.1 π1 = TIME π1 =. (for all curves) TIME π1 =.1 π = (for all curves) TIME π = (for all curves) TIME 1

13 Manpulate Your Data Manpulate Your Data 13