Introduction to Growth Curves Using Stata

Transcription

1 Multilevel/Mixed Models and Longitudinal Analysis Using Stata Alan C. Acock University Distinguished Professor of Family Studies & Knudson Chair for Family Research & Policy Oregon State University College of Health and Human Sciences Summer Workshop Series July 2010 Introduction to Growth Curves Using Stata 1

2 What s in a name 2 What s in a name: Cross Sectional When measured at one time Repeated measures on a case The case might be a family Repeated measures might be Dad s happiness, Mom s happiness, Oldest kid s happiness, Next oldest kid s happiness, etc. Idea is that the measurements are nested (repeated) in the case. We have 2+ measurements in each family 3 2

3 What s in a name: Longitudinal When measured longitudinally in a panel The case might be a an individual Repeated measures might be his/her happiness at wave 1, wave 2, wave 3, wave 4, etc. Idea is that the measurements are nested (repeated) in the case. We have 2+ measurements in each family 4 What s this about levels Cross Sectional? Cross-sectional has individuals nested in families. Level 1 is the individual s score (mom, dad, kid) Level 2 is the family Level 1 scores within a family more homogeneous than scores for random individuals Level 3 might be neighborhood 5 3

4 What s this about levels?--variables Can have different predictor variables at each level Level 1 variables might be personality, IQ, attitude Level 2 variables might be household income, days/ week family eats dinner together Level 3 might be neighborhood %white, median home value Key All this is interdependent because the levels are nested 6 What s this about levels--longitudinal? Longitudinal models have scores at each wave nested in individuals Level 1 is the score at wave 1, wave 2, etc. Level 2 is the individual Level 1 scores of individual at each wave are more homogeneous than scores for random individuals 7 4

5 Graphing the Interdependence Sophia Rabe-Hesketh & Anders Skrondal, Multilevel and Longitudinal Modeling Using Stata. College Station, TX: Stata Press. I change the labels of variables from what they use 8 Graphing the Interdependence Generate a mean for the husband twoway (scatter husband1 couple, msymbol(circle)) ///! (scatter husband2 couple, sort msymbol(circle_hollow)), ///! xtitle(couple) ytitle(husband's measure Stability) ///! legend(order(1 "Time 1" 2 "Time 2"))! 9 5

6 Graphing the Interdependence Husband's measure Stability Couple Time 1 Time 2 10 Wide to Long Format 11 6

7 Wide to Long Format 12 Variance Components Intraclass Correlation We run a regression with no predictors and tell Stata what is the id variable 13 7

8 The command--xtreg! Stata has many commands for multilevel models, all start with xt!. xtreg husband, i(couple) mle! Just enter the level 1 variable (repeated variable) in variable list In our data, each husband from 1 to 17 is identified by the variable couple. The i means whatever variable is in parentheses is the identification variable. This might be called id, case, etc. Here it happens to be called couple 14 The command--xtreg! The mle means we are asking for a maximum likelihood estimator The default is restricted maximum likelihood, reml! But reml makes it harder to compare models This command requires the data to be in the long format 15 8

9 The xtreg Result 16 The xtreg Result We have 34 level 1 observations (two measures) for each of our 17 level 2 cases (called groups since the level 1 values are grouped in the 17 level 2 husbands We have no missing values: min, avg, max all = 2. Stata automatically uses all available data, e.g., with families and mom, dad, kids some families (level 2) might have 1 kid, some might have 2 kids, etc. The chi-square test with no predictors is meaningless (df = 0) The maximized log likelihoods value is

10 The xtreg Result The _cons (constant/intercept) with no predictors, , is the overall mean (best guess in absence of predictors) The /sigma_u is that standard deviation (other programs report variance (option in Stata) Between(husbands). We expect this to be large The /sigma_e is the standard deviation Within (husbands). We expect this to be small. Rho ( ) is the intraclass correlation (ICC) Var(Between) ICC = Var(Between) + Var(Within = SD(Between) 2 SD(Between) 2 + SD(Within) = = The xtreg Result The _cons (constant/intercept) with no predictors, , is the overall mean (best guess in absence of predictors) The /sigma_u is that standard deviation (other programs report variance (option in Stata) Between(husbands). We expect this to be large The /sigma_e is the standard deviation Within(husbands). We expect this to be small. Rho ( ) is the intraclass correlation (ICC) is.967 Below table chi-square(1) = 46.27; p <.001 is the significance of the ICC 19 10

11 The Intraclass Correlation ICC = = Var(Between) Var(Between) + Var(Within = SD(Between) 2 SD(Between) 2 + SD(Within) =.967 Using the standard deviations is more easily interpretable than using the variances. About 95% of the husbands will be within 2*(107.05) of the mean of That is, the mean plus or minus Roughly between 250 and 650. About 95% of the two measures for each husband will be within 2*19.91 of the husband s mean. Roughly his mean plus or minus 40. Husbands are relatively stable. Most variance is between husbands rather than within husbands. 20 The xtmixed command The xtmixed command is much more general. xtmixed does not report the ICC. xtmixed husband couple:, mle! After the two vertical bars we have the identification variable followed by a colon After the comma we ask from a maximum likelihood estimator 21 11

12 The xtmixed result 22 The xtmixed result This has all of the same numbers as the xtreg! The variance components are shown in the bottom table labeled random-effects parameters The standard deviation between individuals is the standard deviation around the overall mean, This appears as the sd(_cons) and is The standard deviation within each husband, across his repeated measures is the sd(residual) and is The ICC is computed using the simple formula shown before 23 12

13 The xtmixed result Conf. Intervals Both results show standard errors for the estimated standard deviations and 95% confidence intervals These are somewhat problematic. The boundary space for a variance or standard deviation has a lower limit of zero A similar problem occurs putting a confidence interval around a correlation coefficient since it can t be below 0. Stata adjusts for this by reporting an asymmetric confidence interval. A symmetrical C.I. for sd(residual) would be Graphic Representation of Variance Components H 1 ε 11 Husband j's mean (true score) ε 21 H 2 ζ 1 Distance of husband j from overall mean M =

14 Graphic Representation of Variance Components Husband j s mean is j above the overall mean a happy guy At time 1, he is 1j point is above his average score At time 2, he is 2j point is above his average score The variance of his mean around the overall mean ( j ) is the between variance (should be big) The variance of his two scores around his own mean ( ij ) is the within variance (should be small) 26 Applications of Variance Components Often just a first step to get the ICC to show that the data is not independent and a multilevel analysis is needed If ICC is small some say you do not need to run multilevel analysis Counter argument If the design is multilevel then you need to run a multilevel analysis 27 14

15 Applications of Variance Components You don t change the test you planned to do to get a significant result If you set up a nonparametric test and it was not significant, but then you noticed a few outliers, what would you do? Change to a t-test that is sensitive to outliers and might be significant Stay the course with your research design FDA expects drug companies to indicate what tests they will run before they collect the data and does not allow them to try different tests till they find one is significant If you set up a test using a two-tail assumption, can you change it to one-tail after seeing the result? This is equivalent to not running a multilevel analysis after you see the ICC is small 28 Applications of Variance Components Can use ICC and graph to see who is most similar Are wives more consistent than husbands? Are identical twins more similar than other twins? Are students in all female math classes more similar than mixed math classes? Just compare the ICCs and possibly do a graph 29 15

16 How many 2nd level groups are needed? Here these are husbands Could be families, organizations, classrooms, etc In a very real sense, these are your cases. 30 to 50 seems reasonable It is possible to do a power analysis If you had 5 classes, it would be like having 5 observations a pretty small sample size 30 How many level 1 scores are needed? Here we only had 2, more would be very helpful Could be scores on members of a group students in a class (25-30), members of a family (3-6) Issue is getting a mean of these values to represent some sense of a true score. Husband s mean is his reference point Mean of 25 students in a class is the classes reference point 31 16

17 Do-file * intraclass.do! clear! cd "/Volumes/acock/1flash/1presentations/OSU 2010 Workshop/ data"! use intraclass.dta! list couple hus*! egen husband_mean = rowmean(husband1 husband2)! summarize husband_mean! * Use menu system to generate this graph! twoway (scatter husband1 couple, msymbol(circle)) ///! (scatter husband2 couple, sort msymbol(circle_hollow)), ///! xtitle(couple) ytitle(husband's measure Stability) ///! legend(order(1 "Time 1" 2 "Time 2"))! list! * Reshaping the data from wide to long! reshape long wife husband, i(couple) j(occassion)! list couple occassion husband husband_mean if couple < 5! * Variance Components models! xtreg husband, i(couple) mle! xtmixed husband couple:, mle! xtmixed husband couple:, mle nolog! 32 Do-file * Comparison table! quietly xtreg wife, i(couple) mle! estimates store her! quietly xtreg husband, i(couple) mle! estimates store him! estimates table her him! list in 1/10! gen id = _n! list in 1/10! rename wife pw! rename husband ph! list in 1/10! reshape long p, i(id) j(partner) string! list in 1/10! encode partner, gen(spouse)! list in 1/10, nolabel! recode spouse 2 = 0! list in 1/10, nolabel! 33 17

18 Do-file xtmixed p couple:, mle! estimates store model1! xtmixed p spouse couple:, mle! estimates store model2! twoway (scatter p couple if spouse==0, msymbol (circle)) ///!! (scatter p couple if spouse==1, msymbol (circle_hollow)), ///!! xtitle(couple) ytitle(marital Satisfaction) ///!! legend(order(0 "Wife" 1 "Husband")) xlabel (1/17)! * Three Way, measures nested in spouses who are nested in couples! xtmixed p spouse couple: spouse:, mle! estimates store model3! lrtest model2 model3! 34 Sometimes a Simple Example Helps Farmer Brown has 48 brand new pigs and his daughter, Emma, weighs each pig once a week for 9 weeks Farmer Brown wants to know what the weight trajectory Stata uses this data, but I ve added a catch. Emma is not reliable. In fact, she only records 294 of the 432 (9*48) possible weights so that we have 30% missing values. This means only 3 pigs got weighed all 9 weeks (listwise) The result for the first 2 pigs (in Long Format) appears on the next slide 35 18

19 Data for first two pigs 36 Graph for 10 pigs twoway connected weight week if id<=10, connect(line)! weight week 37 19

20 How about a fixed effects model? Brown really doesn t care much for individual differences and really just want to see how fast the pigs are growing overall To adjust for the lack of independence (9 weights nested in each pig), Brown does a fixed effects model using xtreg! 38 Fixed Effects Model 39 20

21 Making a graph of the fixed effect predict weightfe! twoway (line weightfe week)! Linear prediction week 40 Random Intercept model There are now two error terms, one for the variance around the intercept and one for the rest of the unexplained variance weight ij = β 0 + β 1 week ij + µ i + ε ij Pig i at week j now has i This error will be positive if the pig weighs more than the average initially It will be negative if weights less than average initially Intercept will be β 0 + µ i There is also an error, ij, for each pig at each wave. A pig might have been sick one week and lost weight that week

22 Estimating the random intercept model. xtmixed weight week id:, mle!. estimates store weightri! weight week part Response variable weight has a fixed portion depending on the week id: specifies a random effect by the grouping variable id. This gives us the random intercept. The mle uses a maximum likelihood estimator The estimates store weightri stores the results using the name weightri! 42 Random Intercept Model: Results 43 22

23 Random Intercept Model: Interpretation We have 294 cases where we have a weight for a pig (not 3 as would be the case with listwise deletion and not 432 The first estimation table reports the fixed effects We estimate B 0 = and B 1 = 6.21! Weight = week + error is our fixed effect part Second table is variance components. The 3.89 is the standard deviation of the constant/intercept and its standard error, 0.41 is quite small The sd(residual) = 2.10 is the standard deviation of the error (standard error) The chi-square(1) = , p < tells us we needed to use a multilevel model 44 A Random Slope Now let s try a random coefficient/slope weight ij = β 0 + β 1 week ij + µ 0i + µ 1i week ij + ε ij The 0i is the variance around the intercept The 1i week ij is the variance weekly variance around the slope Random intercept: (β 0 + µ 0i ) Random slope: (β 1 + µ 1i )week ij 45 23

24 Covariance of Intercept & Slope Need to decide on the covariance of the intercept and the slope The default assumes the covariance of the intercept variance and slope variance are uncorrelated, an identity matrix 46 A Random slope: cov(unstruct) Now let s try a random slope weight ij = β 0 + β 1 week ij + µ 0i + µ 1i week ij + ε ij The 0i is the variance around the intercept The 1i week ij is the variance weekly variance around the slope Unstructured covariance assumes the covariance of the intercept variance and slope variance are correlated: 47 24

25 Random Coefficients Model. xtmixed weight week id: week, nolog mle cov(unstruct) var!. estimates store weightrc!. lrtest weightri weightrc! The id: is the part of the command that gives us the random intercept Any variable after the colon will have a random coefficient The variable week is allowed to have a different slope for each pig since some grow faster than others The cov(unstruct) allows the random intercept and random slope to be correlated Notice the var at the end means we are estimating variances

26 School Engagement Example Data from Day and others of children and their parents from Seattle. They have 3 waves. Kids were 10, 11, 12, or 13 the first wave, 11, 12, 13, or 14 the second wave, and 12, 13, 14, or 15 the third year Reorganized data by age at birth (MCAR) birthyr wave1 wave2 wave3 wave4 wave5 wave6! ! ! ! ! ! ! Total ! ! 50 Correlation of Intercept and Slope We can see if the intercept and slope are correlated We need to do 494 separate regressions of school engagement on year for each child and save the 494 intercepts and slopes statsby inter=_b[_cons] slope = _b[yr], ///! by(id) saving(ols): regress sch yr! 51 26

27 Correlation of Intercept and Slope We merge the saved dataset with our active dataset Then we do the graph using twoway (scatter slope inter) (lfit slope /// inter), xtitle(intercept) ytitle(slope)! 52 Intercept and slope are correlated Slope r = Intercept 53 27

28 How do the means fit? We expect there to be a steady decline in school engagement 54 Using xtreg to estimate the ICC 55 28

29 Compare random intercept & random coefficient models. xtmixed sch female mom_ed nev_mar ///! div_sep other yr id:, mle ///! cov(unstructured)!. estimates store ri!. xtmixed sch female mom_ed nev_mar /// div_sep other yr id:, mle!. estimates store ri!. lrtest ri rc! 56 Telling a story We will run the model using random slopes (even though in this case they were not needed) We will create a graph comparing a male whose mother has low education and has never married to a female whose mother has a college degree and is married We think of these as ideal types. xtmixed sch female mom_ed nev_mar div_sep ///! other yr id: yr, mle cov(unstructured)!. predict sch_score!. twoway (connected sch_score yr if female==0 ///! & mom_ed==2 & nev_mar==1, sort)(connected ///! sch_score yr if female ==1 & mom_ed==4 & ///! mom_ed <. & nev_mar==0 & div_sep==0 & other==0)! 57 29

30 Telling a Story Linear prediction, fixed portion yr Male, Mom never married, low ed Female, Mom married, B.A. 58 *mkdaygrow! clear! cd "/Volumes/acock/1daygrow/data"! use "wave1-3_final_combinedsite_8.dta! destring family_id, gen(id)! keep if site == 1! fre p1_21b_1 p1_21c_1! gen birthyr = p1_21b_1! tab birthyr p1_21b_1! replace birthyr = 1995 if birthyr == ! drop if birthyr == 1993 birthyr == 1998! tab birthyr p1_21b_1! gen age1 = 13 if birthyr == 1994! gen age2 = 14 if birthyr == 1994! gen age3 = 15 if birthyr == 1994! replace age1 = 12 if birthyr == 1995! replace age2 = 13 if birthyr == 1995! replace age3 = 14 if birthyr == 1995! replace age1 = 11 if birthyr == 1996! replace age2 = 12 if birthyr == 1996! replace age3 = 13 if birthyr == 1996! replace age1 = 10 if birthyr == 1997! replace age2 = 11 if birthyr == 1997! replace age3 = 12 if birthyr == 1997! gen wave1 = 0 if age1 == 10! gen wave2 = 1 if age2 == 11! replace wave2 = 1 if age1 == 11! gen wave3 = 2 if age3 == 12! replace wave3 = 2 if age2 == 12! replace wave3 = 2 if age1 == 12! 59 30

31 factor c_scheng1_1 - c_scheng3_1 c_scheng5_1 c_scheng7_1 - c_scheng8_1 ///!!c_scheng15_1, pcf! factor c_scheng1_2 - c_scheng3_2 c_scheng5_2 c_scheng7_2 - c_scheng9_2, pcf! factor c_scheng1_3 - c_scheng3_3 c_scheng5_3 c_scheng7_3 - c_scheng9_3, pcf! alpha c_scheng1_1 - c_scheng3_1 c_scheng5_1 c_scheng7_1 - c_scheng8_1 ///!!c_scheng15_1, asis item! alpha c_scheng1_2 - c_scheng3_2 c_scheng5_2 c_scheng7_2 - c_scheng9_2, ///! asis item! alpha c_scheng1_3 - c_scheng3_3 c_scheng5_3 c_scheng7_3 - c_scheng9_3, ///! asis item! egen schengage1 = rowmean(c_scheng1_1 - c_scheng3_1 c_scheng5_1 ///! c_scheng7_1 - c_scheng8_1 c_scheng15_1)! egen schengage2 = rowmean(c_scheng1_2 - c_scheng3_2 c_scheng5_2 ///!! c_scheng7_2 - c_scheng9_2)! egen schengage3 = rowmean(c_scheng1_3 - c_scheng3_3 c_scheng5_3 ///!! c_scheng7_3 - c_scheng9_3)! pwcorr schengage1-schengage3, obs! /* make six wave for school engatement! */! 60 gen sch1 = schengage1 if birthyr == 1997! gen sch2 = schengage2 if birthyr == 1997! gen sch3 = schengage3 if birthyr == 1997! replace sch2 = schengage1 if birthyr == 1996! replace sch3 = schengage2 if birthyr == 1996! gen sch4 = schengage3 if birthyr == 1996! replace sch3 = schengage1 if birthyr == 1995! replace sch4 = schengage2 if birthyr == 1995! gen sch5 = schengage3 if birthyr == 1995! replace sch4 = schengage1 if birthyr == 1994! replace sch5 = schengage2 if birthyr == 1994! gen sch6 = schengage3 if birthyr == 1994! list id sch* birthyr in 1/50! tabstat sch1-sch6, statistics( count mean ) by(birthyr) columns(variables)! gen wave4 = 3 if age3 == 13! replace wave4 = 3 if age2 == 13! replace wave4 = 3 if age1 == 13! gen wave5 = 4 if age3 == 14! replace wave5 = 4 if age2 == 14! gen wave6 = 5 if age3 == 15! tabstat wave*, statistics( count ) by(birthyr) columns(variables)! /*! Summary statistics: N! by categories of: birthyr! birthyr wave1 wave2 wave3 wave4 wave5 wave6! ! ! ! ! ! ! Total ! ! */! /* School Engagement! Wave 2 and 3 had 9 items, wave 1 had 15. Droped items 4 and 6 as negqtively! worded. Kept 7 items that are in common! c_scheng1_1 - c_scheng3_1 c_scheng5_1 c_scheng7_1 - c_scheng9_1 c_scheng15_1! c_scheng1_2 - c_scheng3_2 c_scheng5_2 c_scheng7_2 - c_scheng9_2! c_scheng1_3 - c_scheng3_3 c_scheng5_3 c_scheng7_3 - c_scheng9_3! alphas are.80,.83, and.83 for waves 1, 2, and 3.! */! 61 31

32 factor c_scheng1_1 - c_scheng3_1 c_scheng5_1 c_scheng7_1 - c_scheng8_1 ///!!c_scheng15_1, pcf! factor c_scheng1_2 - c_scheng3_2 c_scheng5_2 c_scheng7_2 - c_scheng9_2, pcf! factor c_scheng1_3 - c_scheng3_3 c_scheng5_3 c_scheng7_3 - c_scheng9_3, pcf! alpha c_scheng1_1 - c_scheng3_1 c_scheng5_1 c_scheng7_1 - c_scheng8_1 ///!!c_scheng15_1, asis item! alpha c_scheng1_2 - c_scheng3_2 c_scheng5_2 c_scheng7_2 - c_scheng9_2, ///! asis item! alpha c_scheng1_3 - c_scheng3_3 c_scheng5_3 c_scheng7_3 - c_scheng9_3, ///! asis item! egen schengage1 = rowmean(c_scheng1_1 - c_scheng3_1 c_scheng5_1 ///! c_scheng7_1 - c_scheng8_1 c_scheng15_1)! egen schengage2 = rowmean(c_scheng1_2 - c_scheng3_2 c_scheng5_2 ///!! c_scheng7_2 - c_scheng9_2)! egen schengage3 = rowmean(c_scheng1_3 - c_scheng3_3 c_scheng5_3 ///!! c_scheng7_3 - c_scheng9_3)! pwcorr schengage1-schengage3, obs! /* make six wave for school engatement! */! gen sch1 = schengage1 if birthyr == 1997! gen sch2 = schengage2 if birthyr == 1997! gen sch3 = schengage3 if birthyr == 1997! replace sch2 = schengage1 if birthyr == 1996! replace sch3 = schengage2 if birthyr == 1996! gen sch4 = schengage3 if birthyr == 1996! replace sch3 = schengage1 if birthyr == 1995! replace sch4 = schengage2 if birthyr == 1995! gen sch5 = schengage3 if birthyr == 1995! replace sch4 = schengage1 if birthyr == 1994! replace sch5 = schengage2 if birthyr == 1994! gen sch6 = schengage3 if birthyr == 1994! list id sch* birthyr in 1/50! tabstat sch1-sch6, statistics( count mean ) by(birthyr) columns(variables)! 62 /* Generating covariates! gender! mom's education! marital status===redo! */! gen nev_mar = 1 if famstruct2_1 == 4! replace nev_mar = 0 if famstruct2_1 ~= 4 & famstruct2_1 <.! gen div_sep = 1 if famstruct2_1 == 1 famstruct2_1 == 5! replace div_sep = 0 if famstruct2_1 ~= 1 & famstruct2_1 ~= 5 & famstruct2_1 <.! gen married = 1 if famstruct2_1 == 2! replace married = 0 if famstruct2_1 ~= 2 & famstruct2_1 <.! gen other = 1 if famstruct2_1 == 3 famstruct2_1 == 6! replace other = 0 if famstruct2_1 ~= 3 & famstruct2_1 ~= 6 & famstruct2_1 <.! fre famstruct2_1 nev_mar div_sep married other! gen female = p1_21a_1-1! fre female! clonevar mom_ed = p1_4_1! reshape long sch, i(id) j(w)! keep id sch w female mom_ed nev_mar div_sep married other! list id sch w in 1/30! gen yr = w -1! /*!!We want to know if the means for school engagement go down/up in a linear!!fashion. We can make a table of the mean for each of the six years, year!!0 to year 5! */! tabstat sch, statistics(mean count) by(yr) columns(variables)! xtreg sch, i(id) mle! xtmixed sch yr female mom_ed nev_mar div_sep other id:! xtmixed sch yr female mom_ed id:yr! regress sch yr if id == ! 63 32

33 /*! Correlation of intercept and Slope! This section calculates the intercept and the slope when you regress sch on! yr for each case, then it creates a graph showing the link of school and year.! */! statsby inter=_b[_cons] slope = _b[yr], by(id) saving(ols): regress sch yr! sort id! merge id using ols! drop _merge! twoway (scatter slope inter) (lfit slope inter), xtitle(intercept) ytitle(slope)! corr inter slope! corr inter slope, cov! xtdescribe if yr <., i(id) t(yr)! xtsum sch female mom_ed nev_mar div_sep married other yr, i(id)! regress sch female mom_ed nev_mar div_sep married other yr! predict res, residuals! /* Correlation of residuals */! preserve! keep id res yr! reshape wide res, i(id) j(yr)! tabstat res*, statistics(count variance)! pwcorr res*,obs! restore! /* Fixed effects model! These effects are the within subject estimates effects of the time! varying covariates. We have none. The time invariant covariates have! no within subject variance and hence cannot be estimated (are dropped).! The estimates for time variant covariance are not biased because of! omitted time invariant covariates. Each subject serves as his/her own! control. We could add time varying family processes, for example.! */! 64 xtreg sch female mom_ed nev_mar div_sep other yr, i(id) fe! /* Random Intercept Model! */! xtmixed sch yr id:, ml cov(unstructured)! estimates store riyronly! xtmixed sch female mom_ed nev_mar div_sep other yr id:, mle! estimates store ri! /*! Mixed-effects ML regression Number of obs = 1386! ! sch Coef. Std. Err. z P> z [95% Conf. Interval]! ! female ! mom_ed ! nev_mar ! div_sep ! other ! yr ! _cons ! ! ! Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]! ! id: Identity! sd(_cons) ! ! sd(residual) ! ! LR test vs. linear regression: chibar2(01) = Prob >= chibar2 = ! ICC =.443^2/(.443^ ^2) = =.528! */! 65 33

34 66 Random Coefficients Model! */! xtmixed sch yr id: yr, mle cov(unstructured)! estimates store rcyronly! xtmixed sch female mom_ed nev_mar div_sep other yr id: yr, mle cov(unstructured)! estimates store rc! /*! Mixed-effects ML regression Number of obs = 1386! Group variable: id Number of groups = 483! Obs per group: min = 1! avg = 2.9! max = 3! Wald chi2(6) = ! Log likelihood = Prob > chi2 = ! ! sch Coef. Std. Err. z P> z [95% Conf. Interval]! ! female ! mom_ed ! nev_mar ! div_sep ! other ! yr ! _cons ! ! Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]! ! id: Unstructured! sd(yr) ! sd(_cons) ! corr(yr,_cons) ! ! sd(residual) ! ! LR test vs. linear regression: chi2(3) = Prob > chi2 = ! lrtest riyronly rcyronly! lrtest ri rc! /*!. estimates store riyronly!. lrtest riyronly rcyronly! Likelihood-ratio test LR chi2(2) = 2.62! (Assumption: riyronly nested in rcyronly) Prob > chi2 = ! Note: The reported degrees of freedom assumes the null hypothesis is not on the boundaryof! the parameter space. If this is not true, then the reported test is conservative.! lrtest ri rc! Likelihood-ratio test LR chi2(2) = 2.35! (Assumption: ri nested in rc) Prob > chi2 = !!DIVIDE THE P VALUE BY TWO BECAUSE THIS IS INHERENTLY A ONE TAIL TEST --CAN'T!!BE NEGATIVE! xtmixed sch female mom_ed nev_mar div_sep other yr id: yr, mle cov(unstructured)! predict sch_score! twoway (connected sch_score yr if female==0 & mom_ed==2 & nev_mar==1, sort) ///! (connected sch_score yr if female ==1 & mom_ed==4 & mom_ed <. & nev_mar==0 ///! & div_sep==0 & other==0)! gen yrxfemale = yr * female! gen yrxmom_ed = yr * mom_ed! gen yrxnev_mar = yr*nev_mar! gen yrxdiv_sep = yr*div_sep! xtmixed sch female mom_ed nev_mar div_sep other yr yrxfemale ///! id: yr, mle cov(unstructured)! xtmixed sch female mom_ed nev_mar div_sep other yr yrxmom_ed ///! id: yr, mle cov(unstructured)! xtmixed sch female mom_ed nev_mar div_sep other yr yrxnev_mar ///! id: yr, mle cov(unstructured)! xtmixed sch female mom_ed nev_mar div_sep other yr yrxdiv_sep ///! id: yr, mle cov(unstructured)! 67 34