1. Missing Data Michael C. Neale International Workshop on Methodology for Genetic Studies of Twins and Families Boulder CO 2006 Virginia Institute for Psychiatric and Behavioral Genetics Virginia Commonwealth University Vrije Universiteit Amsterdam

2. Various forms of missing data  Twins volunteer to participate or not  Twins are obtained from hospital records  Attrition during longitudinal studies  Equipment failure:  Systematic (if < value then scored zero)  Random  Structured interviews  Don’t bother asking question 2 if response to Q1 is no  Designs to increase statistical power

3. Plan of Talk 1. Review likelihood & Mx facilities 2. Conceptual view of ascertainment 1. Ordinal 2. Continuous 3. Ascertainment in twin studies 1. Ordinal 2. Continuous 4. Missing data 1. Systematic (if < value then scored zero) 2. Random 5. Structured interviews 1. Don’t bother asking question 2 if response to Q1 is no

4. Likelihood approach  Usual nice properties of ML remain  Flexible  Simple principle  Consideration of possible outcomes  Re-normalization  May be difficult to compute Advantages & Disadvantages

5. Maximum Likelihood Estimates  Asymptotically unbiased  Minimum variance of all asymptotically unbiased estimators  Invariant to transformations  a 2 = (a) 2 Have nice properties ^ ^

6. Example: Two Coin Toss 3 outcomes HHHT/THTT Outcome 0 0.5 1 1.5 2 2.5 Frequency Probability i = freq i / sum (freqs)

7. Example: Two Coin Toss 2 outcomes HHHT/THTT Outcome 0 0.5 1 1.5 2 2.5 Frequency Probability i = freq i / sum (freqs)

8. Non-random ascertainment  Probability of observing TT globally  1 outcome from 4 = 1/4  Probability of observing TT if HH is not ascertained  1 outcome from 3 = 1/3  or 1/4 divided by 'ascertainment correction' of 3/4 = 1/3 Example

9. Correcting for ascertainment Univariate continuous case; only subjects > t ascertained 01234-2-3-4 0 0.1 0.2 0.3 0.4 0.5 : G xixi N likelihood t

10. Correcting for ascertainment Univariate continuous case; only subjects > t ascertained 01234-2-3-4 0 0.1 0.2 0.3 0.4 0.5 : G xixi N likelihood t

11. Correcting for ascertainment  Without ascertainment, we compute  With ascertainment, the correction is Dividing by the realm of possibilities Does likelihood increase or decrease after correction? pdf, N(: ij, G ij ), at observed value x i divided by: I -4 N(: ij, G ij ) dx = 1 4 t I -4 N(: ij, G ij ) dx < 1

12. Ascertainment in Practice BG Studies  Studies of patients and controls  Patients and relatives  Twin pairs with at least one affected  Single ascertainment pi 6 0  Complete ascertainment pi = 1  Incomplete0 < pi <1  Linkage studies  Affected sib pairs, DSP etc  Multiple affected families pi = probability that someone is ascertained given that they are affected

13. Correction depends on model  1 Correction independent of model parameters: "sample weights"  2 Correction depends on model parameters: weights vary during optimization  In twin data almost always case 2  continuous data  binary/ordinal data

14. High correlation Concordant>t txtx 0 1 I tx I ty N(x,y) dy dx 4 4 1010 tyty

15. Medium correlation txtx tyty 0 1 I tx I ty N(x,y) dy dx 4 4 1010

16. Low correlation tyty 0 1 I tx I ty N(x,y) dy dx 4 4 1010 txtx

17. Two approaches for twin data in Mx  Contingency table approach  Automatic  Limited to two variable case  Raw data approach  Manual  Multivariate  Moderator / Covariates

18. Contingency Table Case  Feed program contingency table as usual  Use -1 for frequency for non-ascertained cells  Correction for ascertainment handled automatically Binary data

19. At least one twin affected txtx tyty                                  1010 0 1 1-I -4 I -4 N(x,y) dy dx txty Ascertainment Correction  

20. Ascertain iff twin 1 > t txtx tyty                                  1010 0 1 I ty I -4 N(x,y) dx dy 4 4 I ty N(y) dy = 4 Twin 1 Twin 2

21. Contingency Tables  Use -1 for cells not ascertained  Can be used for ordinal case  Need to start thinking about thresholds  Supply estimated population values  Estimate them jointly with model

22. Mx Syntax Classical Twin Study: Contingency Table ftp://views.vcu.edu/pub/mx/examples/ncbook2/categor.mx G1: Model parameters Data Calc NGroups=4 Begin Matrices; X Lower 1 1 Free Y Lower 1 1 Free Z Lower 1 1 Free W Lower 1 1 End Matrices; ! parameters are fixed by default, unless declared free Begin Algebra; A= X*X'; C= Y*Y'; E= Z*Z'; D= W*W'; End Algebra: End

23. Mx Syntax Group 2 G2: young female MZ twin pairs Data Ninput=2 CTable 2 2 329 83 95 83 Begin Matrices= Group 1 T full 2 1 Free End Matrices; Covariances A+C+D+E | A+C+D _ A+C+D | A+C+D+E ; Thresholds T ; Options RSidual End

24. Mx Syntax Group 3 G3: young female DZ twin pairs Data Ninput=2 CTable 2 2 201 94 82 63 Begin Matrices= Group 1 H Full 1 1 Q Full 1 1 T Full 2 1 Free End Matrices; Matrix H.5 Matrix Q.25 Start.6 All Covariances A+C+D+E | H@A+C+Q@D _ H@A+C+Q@D | A+C+D+E / Thresholds T ; Options RSidual NDecimals=4 End

25. Mx Syntax Group 4 Group 4: constrain variance to 1 Constraint NI=1 Begin Matrices = Group 1 ; I unit 1 1 End Matrices; Constraint I = A+C+E+D ; Option Multiple End Specify 2 t 8 8 Specify 3 t 8 8 End

26. Raw data approach  Correction not always necessary  ML MCAR/MAR  Prediction of missingness  Correct through weight formula

27. What can we do with Mx? Normal theory likelihood function for raw data in Mx j=1 ln L i = f i ln [ 3 w j g(x i,: ij, G ij )] m x i - vector of observed scores on n subjects :ij - vector of predicted means Gij - matrix of predicted covariances - functions of parameters

28. Likelihood Function Itself  Example: Normal pdf The guts of it j=1 ln L i = f i ln [ 3 w ij g(x i,: ij, G ij )] m g(x i,: ij, G ij ) - likelihood function

29. Normal distribution N(: ij, G ij ) Likelihood is height of the curve 01234-2-3-4 0 0.1 0.2 0.3 0.4 0.5 : G xixi N likelihood

30. Weighted mixture of models Finite mixture distribution j=1 m j = 1....m models w ij Weight for subject i model j e.g., Segregation analysis ln L i = f i ln [ 3 w ij g(x i,: ij, G ij )]

31. Mixture of Normal Distributions Two normals, propotions w1 & w2, different means But Likelihood Ratio not Chi-Squared - what is it? 0123-2-3-4 0 0.1 0.2 0.3 0.4 0.5 :1:1 xixi g :2:2 w 1 x l 1 w 2 x l 2

32. General Likelihood Function  Sample frequencies binary data  Sometimes 'sample weights'  Might also vary over model j Finally the frequencies f i - frequency of case i j=1 ln L i = f i ln [ 3 w j g(x i,: ij, G ij )] m

33. General Likelihood Function  Model for Means can differ  Model for Covariances can differ  Weights can differ  Frequencies can differ Things that may differ over subjects i = 1....n subjects (families) j=1 ln L i = f i ln [ 3 w ij g(x i,: ij, G ij )] m

34. Raw Ordinal Data Syntax  Read in ordinal file  May use frequency command to save space  Weight uses \mnor function  \mnor(R_M_U_L_K)  R - covariance matrix (p x p)  M - mean vector (1xp)  U - upper threshold (1xp)  L - lower threshold (1xp)  K - indicator for type of integration in each dimension (1xp)  0: L=-4; 1: U=+4  2: Iu; 3: L=-4 U=4 L

35. Mx Syntax G1: Model parameters Data Calc NGroups=4 Begin Matrices; X Lower 1 1 Free Y Lower 1 1 Free Z Lower 1 1 Free W Lower 1 1 End Matrices; ! parameters are fixed by default, unless declared free Begin Algebra; A= X*X'; C= Y*Y'; E= Z*Z'; D= W*W'; End Algebra: End

36. Mx Syntax G2: MZ twin pairs Data Ninput=3 Ordinal File=mz.frq Labels T1 T2 Freq Definition Freq ; Begin Matrices= Group 1 T full 2 1 Free F full 1 1 ! Frequency End Matrices; Specify F Freq Covariances A+C+D+E | A+C+D _ A+C+D | A+C+D+E ; Thresholds T ; Frequency F; Options RSidual End

37. Mx Syntax G3: DZ twin pairs Data Ninput=3 Labels T1 T2 Freq Ordinal File=dz.frq Definition Freq ; Begin Matrices= Group 1 H Full 1 1 Q Full 1 1 T Full 2 1 Free F full 1 1 ! Frequency End Matrices; Specify F Freq Matrix H.5 Matrix Q.25 Start.6 All Covariances A+C+D+E | H@A+C+Q@D _ H@A+C+Q@D | A+C+D+E / Thresholds T ; Frequency F ; Options RSidual NDecimals=4 End

38. Mx Syntax Group 4: constrain variance to 1 Constraint NI=1 Begin Matrices = Group 1 ; I unit 1 1 End Matrices; Constraint I = A+C+E+D ; Option Multiple End Specify 2 t 8 9 Specify 3 t 8 9 End

39. Ascertainment additional commands Why inverse of J and K? Begin Algebra; M=(A+C+E|A+C_A+C|A+C+E); N=(A+C+E|h@A+C_h@A+C|A+C+E); J=I-\mnor(M_Z_T_T_Z); ! Z = [0 0] K=I-\mnor(N_Z_T_T_Z); ! DZ case End Algebra; Weight J~; ! for MZ group Weight K~; ! DZ group

40. Correcting for ascertainment  Multivariate selection: multiple integrals  double integral for ASP  four double integrals for EDAC  Use (or extend) weight formula  Precompute in a calculation group  unless they vary by subject Linkage studies

41. When ascertainment is NOT necessary  Various flavors of missing data mechanisms  MCAR: Missing completely at random  MAR: Missing at random  NMAR: Not missing at random Little & Rubin Terminology

42. Simulation: 3 types of missing data  Selrand: MCAR  missingness function of independent random variable  Selonx: MAR  missingness predicted by other measured variable in analysis + MCAR  Selony: NMAR  missingness mechanism related to ''residual" variance in dependent variable

43. Method  Simulate bivariate normal data X,Y  Sigma = 1.5 .5 1  Mu = 0, 0  Make some variables missing  Generate independent random normal variable, Z, if Z>0 then Y missing  If X>0 then Y missing  If Y>0 then Y missing  Estimate elements of Sigma & Mu  Constrain elements to population values 1,.5, 0 etc  Compare fit  Ideally, repeat multiple times and see if expected 'null' distribution emerges

44. Method  Check to see if estimates ‘look right’  Quantify ‘look right’  Test model with parameters fixed to population values  Look at chi-squared  Repeat!  Examine distribution of chi-squareds

45. R simulation 'model'

46. R simulation script: selrand.R r<-0.5 #Correlation (r) N<-1000 #Number of pairs of scores outfile<-"selrand.rec"#output file for the simulation vals<-matrix(rnorm((N*3)),,3);#3 random normals per pair of observed scores rootr<-sqrt(r);#Square root of r root1mr<-sqrt(1-r); #Square root of (1-r) #Data generation follows: pairs<-matrix(cbind(rootr*vals[,1]+root1mr*vals[,2],rootr*vals[,1] +root1mr*vals[,3]),,2) # pairs now contains pairs of scores drawn from population # with unit variances and correlation r

47. # Now to make some of the scores missing, using a loop for (i in 1:N) { x<-rnorm(1,0,1) # x is an independent random number from N(0,1) if (x>0) pairs[i,2] <- NA } write.table (pairs,file=outfile,row.names=FALSE,na=".",col.names=FALSE) R simulation script: selrand.R

48. R simulation script: selonx.R # Now to make some of the scores missing, using a loop for (i in 1:N) { if (pairs[i,1]>0) pairs[i,2] <- NA } write.table(pairs,file=outfile,row.names=FALSE,na=".", col.names=FALSE) # note how missingness of 2 nd column of pairs depends on # value of 1 st column

49. R simulation script: selony.R # Now to make some of the scores missing, using a loop for (i in 1:N) { if (pairs[i,2]>0) pairs[i,2] <- NA } write.table(pairs,file=outfile,row.names=FALSE,na=".", col.names=FALSE) # note how missingness of 2 nd column of pairs depends on # value of 2 nd column itself

50. Mx Script Rather basic, like Monday morning Estimate pop cov matrix of X&Y, with Y observed iff X>0 Data ngroups=1 ninput=2 Rectangular file=selonx.rec Begin Matrices; a sy 2 2 free ! covariance of x,y m fu 1 2 free ! mean of x,y End Matrices; Means M / Covariance A / matrix a 1.3 1 bound.1 2 a 1 1 a 2 2 option rs mu issat end fix all matrix 1 a 1.5 1 matrix 1 m 0 end

51. Mx Scripts & Data  Check output:  Summary statistics (obs means)  Estimated means & covariance matrices  Difference in fit between estimated values and population values  Interpretation? F:\mcn\2006\sel

52. ML estimation under different missingness mechanisms Missingnessmean xmean yvar xcov xyvar y LR Chisq MCAR (rand) MLE MAR (on x) MLE NMAR (on y) MLE

53. ML estimation under different missingness mechanisms Missingnessmean xmean yvar xcov xyvar y LR Chisq MCAR (rand) MLE -0.0116-0.11.05050.49980.87696.492 sample-0.0116-0.09191.05050.8839 MAR (on x) MLE 0.00480.09981.00840.44811.10255.768 sample0.00140.44371.00840.9762 NMAR (on y) MLE -0.02040.68050.99960.13560.2894227.262 sample0.04480.73730.99960.2851

54. BG Studies: Screen + Examination  Bivariate analysis of screen & exam  No ascertainment correction required  Example: all pairs where at least one screens positive are examined  Works for continuous & ordinal  Undersampling: some proportion of pairs concordant negative for screen are also examined  Ascertainment correction required  Different correction for screen -- vs +-/-+/++ Only a subset, selected on basis of screen, are examined

55. Other examples of necessarily missing data  Consider:  Measures of abuse/dependence in non-users  Age at onset in the healthy  Symptoms of schizophrenia in non- schizophrenics  How do we know whether, e.g.,  Drug use is related to drug abuse  Age at onset predicts severity

56. Two binary variables X and Y  Both X and Y observed  XY Outcomes: 00, 01, 10, 11  But Y observed only if X=1  XY Outcomes 0?, 10, 11  Only three cell frequencies “0/1/2”  No use / Use with no abuse / Use and abuse  No variance in use when abuse is observed Amnesiac researcher, forgot to get data from relatives

57. Two binary variables in twins Twin 1 No Yes No Yes No Yes No Yes No Yes No Yes Twin 2 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 0 0 1 1 0 1 Enlightened researcher, good memory, has data from relatives

58. Censored case Y iff X=1 Twin 1 No Yes No Yes No Yes No Yes No Yes No Yes Twin 2 0 ? 0 ? 1 0 0 ? 1 1 1 0 0 ?1 1 0 ? 1 0 1 0 1 1 1 1 1 0 1 1

59. Can estimate parameters! Correlation for Initiation Correlation for Dependence Correlation Twin 1 initiation with Twin 2 Dependence (“proxy”) Indirectly assess correlation between initiation and dependence A Twin 1 A Twin 2 B Twin 1 B Twin 2 b b VA VB* VAVB* rB*rA

60. Multivariate case: Stem and probe Stem = Yes STEM P2P3 P4 P5 P6 F1 l6l6 l1l1

61. Stem and probe items Stem = No Probes Missing STEM P2P3 P4 P5 P6 F1 l6l6 l1l1

62. Proxy information from cotwin identifies the model STEM P2P3 P4 P5 P6 F1 l6l6 l1l1 STEM P2P3 P4 P5 P6 F2 l6l6 l1l1 R > 0

63. Conclusion  Be careful when designing studies with non-random ascertainment  Usually possible to correct  In principle, heritability should not change  In practice, it might