Longitudinal Data Analysis via Linear Mixed Model

Transcription

1 Longitudinal Data Analysis via Linear Mixed Model Edps/Psych/Stat 587 Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2017

2 Outline Introduction Approaches to Longitudinal Data Analysis Longitudinal HLM by Example The Riesby Data Exploratory Analysis Model Selection Models for Serial Correlation C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

3 Reading and References Reading: Snijders & Bosker, chapter 12 Additional References: Diggle, P.J., Liang, K.L., & Zeger, S.L. (2002). Analysis of Longitudinal Data, 2nd Edition. London: Oxford Science. Notes by Donald Hedeker. Available from his web-site hedeker. Hedeker, D., & Gibbons, R.D. (2006). Longitudinal Data Analysis. Wiley. Singer, J.D. & Willett, J.B. (2003). Applied Longitudinal Data Analysis. Oxford. Verbeke, G, & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

4 Introduction Purpose: Study change and the factors that effect change. Data: Longitudinal data consist of repeated measurements on the same unit over time. Models: Hierarchical Linear Models (linear mixed models) with extensions for possible serial correlation and non-linear pattern of change. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

5 Purpose: Study Nature of Change Goal: Study change and the factors that effect intra- and inter-individual change. Differences found in cross-sectional data often explained as reflecting change in individuals. A model for cross-sectional data Y i1 = β 0 +β cs x i1 +ǫ i1 where i = 1,...,N (individuals) and x i1 is some time measure (e.g., age). Interpretation: β cs = difference in Y between 2 individuals that differ by 1 unit of time (x). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

6 Cross-Sectional Data Ignoring longitudinal structure: ( reading) i = (age) i C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

7 Cross-Sectional Data (continued) Occasion 1: ( reading) i1 = (age) i1 Occasion 2: ( reading) i2 = (age) i2 C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

8 A Model for Longitudinal Data or repeated observations. Y it = β 0 +β cs x i1 +β l (x it x i1 )+ǫ it When t = 1, the model is the same as the cross-sectional model. β l = the expected change in Y over time per unit change in the time measure x (within individual differences). β cs still reflects differences between individuals. β cs and β l reflect different processes. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

9 A Model for Longitudinal Data ( reading) it = (age) i [(age) it (age) i1 ] C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

10 Advantages: Longitudinal Data More Powerful. Inference regarding β cs is a comparison of individuals with the same value of x. Inference regarding β l is a comparison of an individual s response at two times = Assuming y changes systematically with time and retains it s meaning. Each individual is their own control group. Often there is much more of variability between individuals than within individuals and the between variability is consistent over time. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

11 Advantages: Longitudinal Data (continued) Distinguish Among Sources of Variation. Variation in Y may be due Between individuals differences. Within individuals: Measurement error & unobserved covariates. Serial correlation. A step toward showing causality. Causal relativity (i.e., effect of cause relative to another). Causal manipulation. Cause precedes effect (i.e. temporal ordering). Rule out all other possibilities. See Schneider, Carnoy, Kilpatrick & Shavelson (2010). Estimating Causal Effects Using Experimental and Observational Designs:A think Thank White Paper. The Governing Board of the AERA Grant Program. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

12 Studying Change Longitudinal data is required to study the pattern of change and the factors that effect it, both within and between individuals. Level 1: How does the outcome change over time? (descriptive) Level 2: Can we predict differences between individuals in terms of how they change? (relational). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

13 Time Time is a level 1 (micro level) predictor. The number of time points/occasions needed. Measure of time should be Reliable Valid Makes sense for outcome and research questions. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

14 Metric for Time Example from Singer & Willett: If you want to study the longevity of automobiles. Change in appearance of cars Age. Tire wear Miles. Wear of ignition system Trips (# of starts). Engine wear Oil changes. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

15 Metric & Clock for Time Example from Nicole Allen et al. Study the change in arrest rates following passage of law in 1994 requiring coordinated responses to cases of domestic violence. Daily data from all municipalities in Illinois (excluding those in Cook) from 1996 to Zero point? 1996? When council (coordinated response) began? Others Metric? Level? (Daily, Weekly, Monthly, Quarterly, Yearly?) (Municipality? County? Circuit?) C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

16 Three Major Approaches To analyzing longitudinal data. Classic reference: Diggle, Liang & Zeger Marginal Analysis: Only interested in average response. Transition Models: Focus on how Y it depends on past values of Y and other variables (i.e., a conditional model). Random Effects Models: Focus on how regression coefficients vary over individuals. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

17 Marginal Analysis Focus on average of the response variable: Ȳ +t = 1 N N i=1 and how the mean changes over time. Y it In HLM terms, only interested in the fixed effects, E(Y it ) = X i Γ. Observations are correlated, so need to make adjustments to variance estimates, i.e., var(y i ) = V i (θ) where θ are parameters. Sandwich estimator or Robust estimation. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

18 Transition Models Focus on how Y it depends on pervious values of Y (i.e., Y i,(t 1), Y i,(t 2),...) and other variables. Model the Conditional Distribution of Y it, E(Y it Y i,(t 1),...,Y i,1,x) = p k=1 Such models include assumptions about Dependence of Yit on x it s. Correlation between repeated measures. (t 1) β k x itk + α k Y i,k k=1 C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

19 Transition Models (continued) We have focused on continuous/numerical Y s, but when Y is categorical, Stage sequential models (e.g., must master addition and subtraction before can master multiplication). The gateway hypothesis of drug use. Digression: When an event occurs is another type discrete outcome variable, but we re not considering such discrete variables in this class. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

20 Random Effects Models Observations are correlated because repeated measurements are made on the same individual. Regression coefficients vary over individuals, i.e., E(Y it β i1,...,β ip ) = p k=1 β ikx ikt One individual s data does not contain enough information to estimate β ik s ; therefore, we assume a distribution for β ik s, where U i N(0,T) i.i.d.. β i = X i Γ+Z i U i C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

21 Advantages of HLM for Longitudinal Data Explicitly model individual change over time. Simultaneously and explicitly model between- and within-individual variation. Explanatory variables can be time-invariant or time-varying. Flexible modeling of covariance structure of the repeated measures. Many non-linear patterns can be represented by linear models (e.g., polynomial, spline). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

22 Advantages of HLM Flexible treatment of time Time can be treated as a continuous variable or as a set of fixed points. Can have a different numbers of repeated observations. (implication: can handle missing data). Can extend HLM models to higher level structures (e.g., repeated measurements on students within classes, etc). Generalizations exist for non-linear data. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

23 HLM for Longitudinal Data Uses everything we ve learned about HLM s, but requires a slight change in terminology and notation: Level 1 units are occasions of measurement and indexed by t (t for time where t = 1,...,r). Level 2 units are individuals. Y it = measurement of response/dependent variable for individual i at time t. The level 1 model: within individual model. The level 2 model: between individuals model. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

24 HLM for Longitudinal Data One major change: May need a more complex model for the level 1 (within individual) residuals; that is, R i N(0,Σ i ) where Σ i = σ 2 I (constant and uncorrelated) may be too simple. One s that we ll explicitly cover are lag 1: Auto-correlated errors, AR(1). Moving average, MA(1). Auto-correlated, moving average ARMA(1,1). TOEP(#). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

25 Longitudinal HLM by Example The Riesby Data, from Hedeker s web-site (and used in Hedeker & Gibbons book, 2006). Drug Plasma Levels and Clinical Response. Risby and associates (Riesby, et al, 1977) examined the relationship between Imipramine (IMI) and Desipramine (DMI) plasma levels and clinical response in 66 depressed inpatients (37 endogenous and 29 non-endogenous). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

26 The Riesby Data Outcome variable: Hamilton Depression Score (HD). Independent variables: Gender. D where = 1 for endogenous and = 0 of non-endogenous. IMI (impipramine) drug-plasma levels (µg/1). Antidepressant given 225 mg/day, weeks 3-6. DMI (desipramine) drug-plasma levels (µg/1). Metabolite of imipramine. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

27 The Design Drug-Washout day 0 day 7 day 14 day 21 day 28 day 35 wk 0 wk 1 wk 2 wk 3 wk 4 wk 5 Hamilton Y it Depression HD 1 HD 2 HD 3 HD 4 HD 5 HD 6 Level 2 Gender Diagnosis G D Level 1 IMI IMI 3 IMI 4 IMI 5 IMI 6 DMI DMI 3 DMI 4 DMI 5 DMI 6 N Note: n = 6 and N = 66. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

28 More Information About the Topic From a Psychiatrist friend: Everyone uses Hamilton Depression Score Good that both IMI and DMI are used. In the psychiatric literature, the sum is usually reported. Impipramine is an older drug, which has many undesirable side effects, but it works. Distinction between diagnosis with respect to drug not done (relevant to practice). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

29 Exploring Individual Structures C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

30 Exploring Individual Structures C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

31 Exploring Individual Structures C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

32 Overlay Individual Regressions C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

33 Overlay Individual Regressions C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

34 Overlay Individual Regressions C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

35 Exploring Mean Structure General linear decline and increasing variance. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

36 Exploring Mean Structure (continued) C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

37 Exploring Mean Structure (continued) C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

38 Exploring Individual Specific Models Based on the figures, a plausible for level 1 Y it = β 0i +β i1 (week) it +ǫ it Using OLS, fit this model to each person s data and compute: R 2 i = SSMOD i /SSTO i. R 2 meta = i SSMOD i/ i SSTO i. And if there are two possible (nested) models, Compute an F-statistic. Try to see who improves. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

39 Linear Model: R 2 i and R 2 meta Fit statistics for the model Y it = β 0i +β i1 (week) it +ǫ it C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

40 Quadratic: R 2 i and R 2 meta Y it = β 0i +β i1 (week) it +β i2 (week) 2 it +ǫ it C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

41 Comparison C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

42 Who Improved? C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

43 Who Improved? (continued) C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

44 F Test for Quadratic Term Reduced Model: Y it = β o +β 1 (week) it +ǫ it p = 2. Full Model: Y it = β o +β 1 (week) it +β 2 (week) 2 it +ǫ it p = 1. F-statistic: F = ( i SSE(R) i SSE(F) i )/ i p i i SSE(F) i/ i (n i p p ) = / /177 = 1.55 Comparing F = 1.55 to the F distribution with df num = 66 and df den = 177, the p-value =.01. To be used with a grain of sand (assumptions violated) C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

45 Preliminary HLM Level 1: Y it = β oi +β 1i (week) it +β 2i (week) 2 it +R it Level 2: β oi = γ 00 +γ 01 (endog) i β 1i = γ 10 β 2i = γ 20 Preliminary Mixed Linear Model: Y it = γ 00 +γ 01 (endog) i +γ 10 (week) it +γ 20 (week) 2 it +R it C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

46 Exploring Random Effects Fitting this model to each individual s data using ordinary least squares regression we look at Raw residuals, Squared residuals, ˆR 2 it. ˆR it = (Y it Ŷit) = Z i U i +R i. Correlations between residuals to look for serial correlation (i.e., need model for Σ i?) corr(r it,r it ). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

47 Raw Residuals by Week C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

48 Raw Residuals by Week 2 C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

49 Raw Residuals with Endog C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

50 Mean Raw Residuals C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

51 Squared Raw Residuals C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

52 Mean Squared Raw Residuals C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

53 Variance Function Given Preliminary Mixed Linear Model: Y it = γ 00 +γ 01 (endog) i +γ 10 (week) it +γ 20 (week) 2 it +R it Assuming R it N(0,σ 2 I) and random intercept and slopes, i.e., The variance of Y it (U 0i,U 1i,U 2i ) N(0,T) var(y it ) = τ 2 0 +τ2 1 week2 it +τ2 2 week4 it +2τ 01week it +2τ 02 week 2 it +2τ 12week 3 it +σ2 C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

54 Variance Function: 2 Random Effects Assuming R it N(0,σ 2 I) and random intercept and slope for week, i.e., The variance of Y it (U 0i,U 1i ) N(0,T) var(y it ) = τ 2 0 +τ2 1 week2 it +2τ 01week it +σ 2 How many random effects? Need to also consider possible serial correlation. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

55 Correlation Between Time Points Entries in the table are correlation(ˆr it, ˆR it ). week0 week1 week2 week3 week4 week5 week week week week week week Note: 46 n 66 due to individuals with missing observations. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

56 (R) Plot of Correlations rwide.week0 rwide.week1 rwide.week2 rwide.week3 rwide.week4 rwide.week5 rwide.week0 rwide.week1 rwide.week2 rwide.week3 rwide.week4 rwide.week5 C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

57 Plot of Correlations C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

58 Plot of Correlations: Lags C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

59 Mini-Outline (Next Steps) Before covering possible models for the level one, fit some HLM models to Riesby data (nothing new here). Consider some models for level 1 residuals. Simulation of data with different error structures. Analyze Riesby data using alternative error structures for level 1. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

60 Model for Riesby Data Within Individual (level 1) (HamD) it = β 0i +β 1i (week) it +β 2i (week) 2 it +R it where R it N(0,σ 2 ). Between Individuals (level 2) β 0i β 1i β 2i = γ 00 +γ 01 (endog) i +U 0i = γ 10 +γ 11 (endog) i +U 1i = γ 20 +U 2i where U i N(0,T). C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

61 Linear Mixed Model Scalar form (HamD) it = γ 00 +γ 10 (week) it +γ 20 (week) 2 it +γ 01 (endog) i +γ 11 (endog) i (week) it +U 0i +U 1i (week) it +U 2i (week) 2 it +R it In matrix form, Y i = X i Γ+Z i U i +R i C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

62 Linear Mixed Model Y i = X i Γ+Z i U i +R (HamD) i1 (HamD) i2. (HamD) ir = 1 (week) i1 (week) 2 i1 D i D i (week) i1 1 (week) i2 (week) 2 i2 D i D i (week) i (week) ir (week) 2 ir D i D i (week) ir 1 (week) i1 (week) 2 i1 1 (week) i2 (week) 2 i2 + U R i1 0i R i2 U 1i +... U 2i. 1 (week) ir (week) 2 ir R ir γ 00 γ 10 γ 20 γ 01 γ 11 where D i = { 0 Non-endogenous 1 endogenous C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

63 Marginal Model (HamD) it N(X i Γ,(Z i TZ i +σ2 I)) The covariance matrix (Z i TZ i +σ2 I) The (k,t) element of Z i = {z itk } for k = 0,(q 1) and t = 0,...r. The covariance matrix for U i : T = τ 00 τ 10 τ 20 τ 10 τ 11 τ 12 τ 20 τ 12 τ 22 (t,t ) element of Z i TZ i = q (q) k=0 l=k τ klz itk z it l C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

64 Marginal Model: Our Example (t,t) element of Z i TZ i +σ2 I var(y it ) = τ 2 0 +τ 2 1(week) 2 it +τ2 2(week) 4 it +2τ 01(week) it +2τ 02 (week) 2 it +2τ 12(week) 3 it +σ2 (t,t ) element of Z i TZ i +σ 2 I cov(y it,y it ) = τ 00 +τ 10 (week it +week it )+τ 20 (week 2 it +week2 it ) cov(y it,y i t) = cov(y it,y i t ) = 0. +τ 11 (week it )(week it )+τ 22 (week 2 it)(week 2 it ) +τ 12 (week 2 it )(week it )+τ 12(week it )(week 2 it ) +σ 2 C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

65 Covariance Parameter Estimates Empty/Null Preliminary HLM Cov Std Std Parm Estimate Error Estimate Error UN(1,1) UN(2,1) UN(2,2) UN(3,1) UN(3,2) UN(3,3) Residual ˆρ = 13.62/( ) =.26 C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

66 Solution for fixed Effects Empty/Null Preliminary HLM Std Std Effect Estimate Error Estimate Error Intercept Week Week*Week Endog = Endog = 1 0. Week*Endog = Week*Endog = 1 0. hline C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

67 Global Fit Statistics Model 2LnLike AIC BIC Empty/Null Preliminary HLM Preliminary HLM w/ cubic week C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

68 Model Reduction: Random Effects (i.e., Covariance Structure) Test whether need random term for (week) 2 it, H 0 : τ 2 2 = τ 02 = τ 12 = 0 versus H a : not H 0 The Reduced Model, (HamD) it = γ 00 +γ 10 (week) it +γ 20 (week) 2 it +γ 01 (endog) i +γ 11 (endog) i (week) it +U 0i +U 1i (week) it +R it, has 2lnLike = C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

69 Model Reduction: Random Effects Test statistic: difference between 2lnLike of full and reduced models, = 10.5 Sampling distribution is a mixture of χ 2 3 and χ2 2, p-value =.5(.015) +.5(.005) =.01 Conclusion: Reject H 0. AIC favors the model without U 2i while BIC favors the model with U 2i. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

70 Model Reduction: Fitted Effects Do not remove (week) it or (week) 2 it, because of non-zero τ2 1 and τ 2 2. Possible Reductions: endog and endog*week. t-tests indicate don t need these ; however, Likelihood ratio test statistic for week*endog (i.e., H o : γ 11 = 0 versus H a : γ 11 0), = =.002 df = 1, p-value large...retain H o (i.e. drop the interaction). After removing week*endog, we do a likelihood ratio test for endog C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

71 Reduced Model Covariance Parameters Preliminary HLM No interaction Cov Std Std Parm Estimate Error Estimate Error UN(1,1) UN(2,1) UN(2,2) UN(3,1) UN(3,2) UN(3,3) Residual C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

72 Reduced Model fixed Effects Preliminary HLM No interaction Cov Std Std Parm Estimate Error Estimate Error Intercept Week Week*Week Endog = Endog = Week*Endog = Week*Endog = 1 0. Estimates are pretty similar. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

73 Do we need endog? H o : γ 01 = 0 versus H a : γ 01 0 t-test using the estimates from the model without the cross-level interaction, t = = 1.94, df = 65.7, p value =.056 Likelihood ratio test statistic, = Comparing this to χ 2 1, p-value=.056. Conclusion: maybe/undecided about endog, keep it for now. C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

74 Global Fit Statistics Model 2LnLike AIC BIC Empty/Null Preliminary HLM No endog*week No endog*week and no endog We ll go with this model: Y it = β 0i +β 1i (week) it +β 2i (week) 2 it +R it β 0i = γ 00 +endog i +U 0i β 1i = γ 10 +U 1i β 2i = γ 20 +U 2i What other analyses should we do to adequacy of this model? C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

75 Interpretation (ĤamD) it = (week) it +.05(week) 2 it 1.79D i C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

76 Estimated Model Individuals (ĤamD) it = (week) it +.05(week) 2 it 1.79D i +Û 0i +Û 1i (week) it +Û 2i (week) 2 it C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77

77 Summary Just an HLM To study the nature of change & requires systematic change. Level 1: Time is a level 1 variable (should have at least 3 time points). Need to choose metric of time. time variant variables are level 1 variables. Level 2: Study differences between individuals. time invariant variables are level 2 variables. What about covariance matrix for R it?... C.J. Anderson (Illinois) Longitudinal Data Analysis via Linear Mixed Model Fall / 77