MDM for univariate sampling Missing values of Y are missing completely at random (MCAR) if R ⊥ ⊥ Y , or equivalently if f ( r | y ) = f ( r ) For univariate samples, this is also classified as missing at random (MAR). More on this distinction later. Hogan (MDEpiNet) Missing Data October 22, 2018 28 / 160
MDM for univariate sampling Missing values of Y are missing not at random (MNAR) if there exists at least one value of y such that f ( r | y ) � = f ( r ) Or in words, if the probability of response is systematically higher/lower for particular values of y . Hogan (MDEpiNet) Missing Data October 22, 2018 29 / 160
MDM for univariate sampling Under MAR, methods applied to the observed data only will generally yield valid inferences about the population. ◮ Estimates will be consistent ◮ Standard errors may be larger than if you had the full data Under MNAR, methods applied to observed data only generally will not yield valid inferences Hogan (MDEpiNet) Missing Data October 22, 2018 30 / 160
MDM for multivariate sampling – regression Consider the setting where we are interested in the regression of Y on X . Let µ ( X ) = E ( Y | X ). Assume there are no other covariates available The model we have in mind is g { µ ( X ) } = β 0 + β 1 X Here, the function g tells you the type of regression model you are fitting (linear, logistic, etc.) The full data are ( Y 1 , X 1 , R 1 ) , ( Y 2 , X 2 , R 2 ) , . . . , ( Y n , X n , R n ) Hogan (MDEpiNet) Missing Data October 22, 2018 31 / 160
MDM for multivariate sampling – regression In cases like this, we can define missing data mechanisms relative to the objective of inference. The Y ’s are missing at random if Y ⊥ ⊥ R | X In words, the MDM is a random deletion mechanism within distinct levels of X Another way to write this: f ( r | y , x ) = f ( r | x ) The deletion mechanism depends on X , but within levels of X it does not depend on Y Hogan (MDEpiNet) Missing Data October 22, 2018 32 / 160
Examples of MAR in regression Let Y denote blood pressure, X denote gender (1 = F, 0 = M). The regression model of interest is E ( Y | X ) = β 0 + β 1 X , so that β 0 = mean BP among men, β 1 = mean difference. Let’s assume men have higher BP on average. Randomly delete BP for 20% of men and 40% of women. R does depend on Y , but only through X ◮ Men have higher BP ◮ Men less likely to be deleted ◮ ⇒ those with higher BP less likely to be deleted Within levels of X , deletion mechanism is completely random. Hogan (MDEpiNet) Missing Data October 22, 2018 33 / 160
MAR in regression – some practical issues Revisit the MAR condition. If R ⊥ ⊥ Y | X , this also means f ( y | x , r ) = f ( y | x ) , or f ( y | x , r = 1) = f ( y | x , r = 0) The relationship between Y and X is the same whether R = 1 or R = 0. Consequence is that a regression fit to those with R = 1 gives valid estimates of regression parameters. (Standard errors will be higher relative to having all the data.) Hogan (MDEpiNet) Missing Data October 22, 2018 34 / 160
MAR in regression – some practical issues Under MAR, the inferences are still valid even if ◮ The X distribution is different between those with missing and observed Y ’s Question you have to ask to (subjectively) assess MAR: Is the missing data mechanism a random deletion of Y ’s among people who have the same X values? Equivalent formulation of this question: Is the relationship between X and Y the same among those with missing and observed Y values? Hogan (MDEpiNet) Missing Data October 22, 2018 35 / 160
MAR for regression when auxiliary variables are available In some cases we have information on more than just the X variables In a clinical trial we may be interested in E ( Y | X ) when X is a treatment group, but we have collected lots of baseline covariates V . In a longitudinal study, we may be interested in the mean outcome at the last measurement time, but we have accumulated information on the outcome at previous measurement times. When auxiliary information is available, we can use it in some cases to make MAR more plausible. Here MAR has a slightly different formulation. Hogan (MDEpiNet) Missing Data October 22, 2018 36 / 160
MAR with auxiliary covariates The relationship of interest is g { µ ( X ) } = β 0 + β 1 X The full data are ( Y 1 , X 1 , V 1 , R 1 ) , ( Y 2 , X 2 , V 2 , R 2 ) , . . . , ( Y n , X n , V n , R n ) where Y is observed when R = 1 and is missing when R = 0. Hogan (MDEpiNet) Missing Data October 22, 2018 37 / 160
MAR with auxiliary covariates Values of Y are missing at random (MAR) if Y ⊥ ⊥ R | ( X , V ) Two equivalent ways to write this are: f ( r | x , v , y ) = f ( r | x , v ) f ( y | x , v , r = 1) = f ( y | x , v , r = 0) The first says that within distinct levels defined by ( X , V ), missingness in Y is a random deletion mechanism The second says that the relationship between Y and ( X , V ) is the same whether Y is missing or not Hogan (MDEpiNet) Missing Data October 22, 2018 38 / 160
MAR with auxiliaries – example Return to our BP example, but now assume V denotes income level. Recall, we are interested in the coefficient β 1 from E ( Y | X ) = β 0 + β 1 X and not the coefficient α 1 from E ( Y | X , V ) = α 0 + α 1 X + α 2 V Hogan (MDEpiNet) Missing Data October 22, 2018 39 / 160
Missing data mechanisms for longitudinal data Need to define some notation for longitudinal data Y j = value of Y at time j R j = 1 if Y j observed, 0 otherwise = ( Y 1 , Y 2 , . . . , Y j ) Y j = outcome history up to time j = covariate history up to time j X j H j = ( X j , Y j − 1 ) Allows us to define MCAR, MAR, MNAR for longitudinal data Hogan (MDEpiNet) Missing Data October 22, 2018 40 / 160
Missing data mechanisms for longitudinal data MAR Missing at random (MAR) If interest is in marginal means such as E ( Y j ), MAR means R j ⊥ ⊥ Y j | ( R j − 1 = 1 , H j ) Interpretation: ◮ Among those in follow up at time j, missingness is independent of the outcome Y j conditional on the previously observed Y ’s . ◮ Missingness does not depend on present or future Y ’s, given the past. Hogan (MDEpiNet) Missing Data October 22, 2018 41 / 160
Missing data mechanisms for longitudinal data MAR Implications 1 Selection mechanism [ R j | R j − 1 = 1 , H J ] = [ R j | R j − 1 = 1 , H j ] ◮ Can model selection probability as a function of observed past 2 Imputation mechanism [ Y j | R j = 0 , R j − 1 = 1 , H j ] = [ Y j | R j = 1 , R j − 1 = 1 , H j ] ◮ Can impute missing Y j using a model of the observed Y j ◮ Critical: Must correctly specify observed-data model Hogan (MDEpiNet) Missing Data October 22, 2018 42 / 160
LOCF We can characterize LOCF in this framework It is an imputation mechanism ◮ Missing Y j is equal to the most recently observed value of Y ◮ Missing value filled in with probability one (no variance) Formally Y j ∗ with probability one , [ Y j | R j = 0 , H j ] = where j ∗ = max k < j { R k = 1 } Not an MAR mechanism in general ◮ Conditional distribution of missing Y j not equal to that for observed Y j Hogan (MDEpiNet) Missing Data October 22, 2018 43 / 160
Random effects and parametric models Assume a joint distribution for the repeated measures Model applies to the full data , hence cannot be checked Example: Multivariate normal ( Y 1 , . . . , Y J ) T ∼ N ( µ, Σ) , where µ J × 1 = E ( Y ) and Σ J × J = var( Y ) Special case: Random effects model ◮ Particular way of structuring mean and variance Hogan (MDEpiNet) Missing Data October 22, 2018 44 / 160
Random effects and parametric models When do these models yield valid inference? Most parametric models valid if ◮ MAR holds ◮ All parts of the model are correctly specified These models have an implied distribution for the conditionals [ Y j | Y 1 , . . . , Y j − 1 ] Under MAR, the implied distribution applies to those with complete and incomplete data, but .... Parametric assumptions cannot be checked empirically Hogan (MDEpiNet) Missing Data October 22, 2018 45 / 160
GEE I Assume a mean and variance structure for the repeated measures ◮ Not necessarily a full parametric model Assumed variance structure is the ‘working covariance’ With complete data ◮ Inferences are most efficient when covariance correctly specified ◮ Correct inference about time-specific means even if covariance mis-specified ◮ Reason: all information about time-specific means is already observed Hogan (MDEpiNet) Missing Data October 22, 2018 46 / 160
GEE II With incomplete data ◮ Information about time-specific means relies on ‘imputation’ of missing observations ◮ These imputations come from the conditional distribution [ Y j | Y 1 , . . . , Y j − 1 ] ◮ The form of the conditional distribution depends on the working covariance Implication: Correct inference about time-specific means only when both mean and covariance are correctly specified ◮ Can get different treatment effects with different working covariances Hogan (MDEpiNet) Missing Data October 22, 2018 47 / 160
Dependence of estimates on working covariance From Hogan et al., 2004 Statistics in Medicine Hogan (MDEpiNet) Missing Data October 22, 2018 48 / 160
Structure of case studies 1 Introduce modeling approach 2 Relate modeling approach to missing data hierarchy 3 Illustrate on simple cases 4 Include a treatment comparison 5 Discussion of key points from case study Hogan (MDEpiNet) Missing Data October 22, 2018 49 / 160
CASE STUDY I: MIXTURE MODEL ANALYSIS OF GROWTH HORMONE TRIAL Hogan (MDEpiNet) Missing Data October 22, 2018 50 / 160
Outline of analysis Objective: Compare EG to EP at month 12 ◮ Variable: Y 3 Estimation of E ( Y 3 ) for EG arm only ◮ Ignoring baseline covariates ◮ Using information from baseline covariate Y 1 ◮ MAR and MNAR (sensitivity analysis) Treatment comparisons ◮ Expand to longitudinal case ◮ MAR – using regression imputation ◮ MNAR – sensitivity analysis Hogan (MDEpiNet) Missing Data October 22, 2018 51 / 160
Estimate E ( Y ) from univariate sample Y3 R [1,] NA 0 [2,] 82.6 1 [3,] 118.7 1 [4,] 99.6 1 [5,] NA 0 [6,] 57.5 1 [7,] NA 0 [8,] NA 0 [9,] 87.5 1 [10,] 97.4 1 [11,] 21.4 1 [12,] 47.2 1 [13,] NA 0 [14,] 68.6 1 Hogan (MDEpiNet) Missing Data October 22, 2018 52 / 160
Estimating E ( Y ) from univariate sample Model: E ( Y | R = 1) = µ 1 E ( Y | R = 0) = µ 0 (not identifiable) Target of estimation E ( Y ) = µ 1 P ( R = 1) + µ 0 P ( R = 0) Question: what to assume about µ 0 ? In a sense, we are going to impute a value for µ 0 , or impute values of the missing Y ’s that will lead to an estimate of µ 0 . Hogan (MDEpiNet) Missing Data October 22, 2018 53 / 160
Parameterizing departures from MAR Target of estimation: E µ 0 ( Y ) = µ 1 P ( R = 1) + µ 0 P ( R = 0) = µ 1 + ( µ 0 − µ 1 ) P ( R = 0) Suggests sensitivity parameter ∆( µ 0 ) = µ 0 − µ 1 . Leads to E ∆ ( Y ) = µ 1 + ∆ P ( R = 0) Features of this format: Centered at MAR (∆ = 0) ∆ cannot be estimated from observed data Can vary ∆ for sensitivity analysis Allows Bayesian approach by placing prior on ∆ Hogan (MDEpiNet) Missing Data October 22, 2018 54 / 160
Estimation under MNAR Recall model E ∆ ( Y ) = µ 1 + ∆ P ( R = 0) Estimate known quantities � n 1 = i R i � n 0 = i (1 − R i ) � P ( R = 0) = n 0 / ( n 1 + n 0 ) (1 / n 1 ) � µ 1 � = i Y i Have one unknown quantity ∆ = µ 0 − � µ 1 Hogan (MDEpiNet) Missing Data October 22, 2018 55 / 160
Estimation under MAR Plug into model � � E ∆ ( Y ) = � µ 1 + ( µ 0 − � µ 1 ) P ( R = 0) � �� � � = � µ 1 + ∆ P ( R = 0) Interpretation Under MAR (∆ = 0), ◮ � E ∆ ( Y ) = � µ 1 ◮ Estimator is the observed-data mean Under MNAR (∆ � = 0), ◮ Shift observed-dta mean by ∆ P ( R = 0) ◮ Shift is proportional to fraction of missing observations Hogan (MDEpiNet) Missing Data October 22, 2018 56 / 160
105 100 95 y3.delta 90 85 80 75 70 -10 -5 0 5 10 delta Y [ R =1] = 88 . 3 � P ( R = 0) = 0 . 42 Hogan (MDEpiNet) Missing Data October 22, 2018 57 / 160
Using information from baseline covariates Y1 Y3 R [1,] 35.0 NA 0 [2,] 74.5 82.6 1 [3,] 120.2 118.7 1 [4,] 84.8 99.6 1 [5,] 68.6 NA 0 [6,] 47.9 57.5 1 [7,] 39.0 NA 0 [8,] 52.0 NA 0 [9,] 92.9 87.5 1 [10,] 98.8 97.4 1 [11,] 48.6 21.4 1 [12,] 45.7 47.2 1 [13,] 63.4 NA 0 [14,] 64.2 68.6 1 Hogan (MDEpiNet) Missing Data October 22, 2018 58 / 160
General model for Y Objective: Inference for E ( Y 3 ) Model – General form [ Y 3 | Y 1 , R = 1] ∼ F 1 ( y 3 | y 1 ) [ Y 3 | Y 1 , R = 0] ∼ F 0 ( y 3 | y 1 ) The general form encompasses all possible models that can be assumed for the observed and missing values of Y 3 The model F 0 cannot be estimated from data Hogan (MDEpiNet) Missing Data October 22, 2018 59 / 160
The model under MAR and MNAR Under MAR, F 0 = F 1 . Suggests following strategy ◮ Fit model for F 1 using observed data; call it � F 1 ◮ Use this model to impute missing values of Y 3 Under MNAR, F 0 � = F 1 . Suggests following strategy ◮ Parameterize a model so that F 0 is related to F 1 through a sensitivity parameter ∆ ◮ Generically write this as F 0 = F ∆ 1 ◮ Use fitted version of F ∆ 1 to impute missing Y 3 Hogan (MDEpiNet) Missing Data October 22, 2018 60 / 160
Regression parameterization of F 1 and F 0 Take the case of MAR first MAR implies [ Y 3 | Y 1 , R = 1] = [ Y 3 | Y 1 , R = 0] Assume regression model for [ Y | X , R = 1] E ( Y 3 | Y 1 , R = 1) = α 1 + β 1 Y 1 Assume a model for [ Y | X , R = 0] of similar form E ( Y 3 | Y 1 , R = 0) = α 0 + β 0 Y 1 Cannot estimate parameters from observed data Hogan (MDEpiNet) Missing Data October 22, 2018 61 / 160
Regression parameterization of F 1 and F 0 Recall model E ( Y 3 | Y 1 , R = 1) = α 1 + β 1 Y 1 E ( Y 3 | Y 1 , R = 0) = α 0 + β 0 Y 1 Link the models. One way to do this: β 0 = β 1 + ∆ β α 0 = α 1 + ∆ α Under MAR: ∆ α = ∆ β = 0 Hogan (MDEpiNet) Missing Data October 22, 2018 62 / 160
Caveats to using this (or any!) approach Recall model E ( Y 3 | Y 1 , R = 1) = α 1 + β 1 Y 1 E ( Y 3 | Y 1 , R = 0) = α 0 + β 0 Y 1 More general version of missing-data model E ( Y 3 | Y 1 , R = 0) = g ( Y 1 ; θ ) Do we know the form of g ? Do we know the value of θ ? Do we know that Y 1 is sufficient to predict Y 3 ? We are assuming we know all of these things. Hogan (MDEpiNet) Missing Data October 22, 2018 63 / 160
Estimation of E ( Y 3 ) under MAR 1 Fit the model E ( Y 3 | Y 1 , R = 1) = α 1 + β 1 Y 1 α 1 , � ⇒ obtain � β 1 . 2 For those with R = 0, impute predicted value via � � Y 3 i = E ( Y 3 | Y 1 i , R i = 0) α 1 + � = � β 1 Y 1 i 3 Estimate overall mean as the mixture � � R i Y 3 i + (1 − R i ) � E ( Y 3 ) = (1 / n ) Y 3 i i Hogan (MDEpiNet) Missing Data October 22, 2018 64 / 160
Regression imputation under MAR 140 + 120 100 + y3[r == 1] + + 80 + + + + + + 60 + + + + 40 20 60 80 100 120 y1[r == 1] Hogan (MDEpiNet) Missing Data October 22, 2018 65 / 160
Sample means and imputed means under MAR n r Y 1 Y 3 R = 0 16 57 67 R = 1 22 78 88 MAR 38 79 Hogan (MDEpiNet) Missing Data October 22, 2018 66 / 160
Some intuition behind this (simple) estimator Could base estimate purely on regression model E ( Y 3 ) = E Y 1 , R { E ( Y 3 | Y 1 , R ) } � � = E R E Y 1 | R { E ( Y 3 | Y 1 , R ) } = E R [ α 1 + β 1 E ( Y 1 | R )] = α 1 + β 1 E ( Y 1 ) Plug in the estimators for each term May be more efficient when regression model is correct Hogan (MDEpiNet) Missing Data October 22, 2018 67 / 160
Some more details ... If we don’t want to use the regression model for Y 3 , can write E ( Y 3 ) = E ( Y 3 | R = 1) P ( R = 1) + E ( Y 3 | R = 0) P ( R = 0) , where the term in red is E ( Y 3 | R = 0) = E Y 1 | R =0 { E ( Y 3 | Y 1 , R = 0) } = E Y 1 | R =0 ( α 1 + β 1 Y 1 | R = 0) = α 1 + β 1 E ( Y 1 | R = 0) Hence � � [ R =1] [ R =0] � ˆ α 1 + � ˆ E ( Y 3 ) = Y P ( R = 1) + � β 1 Y P ( R = 0) 3 1 Hogan (MDEpiNet) Missing Data October 22, 2018 68 / 160
Inference and treatment comparisons SE and CI: bootstrap ◮ Draw bootstrap sample ◮ Carry out imputation procedure ◮ Repeat for lots of bootstrap samples (say B ) ◮ Base SE and CI on the B bootstrapped estimators Why not multiple imputation? ◮ Estimators are linear ◮ Bootstrap takes care of missing data uncertainty here Treatment comparisons – coming later Hogan (MDEpiNet) Missing Data October 22, 2018 69 / 160
Estimation of E ( Y 3 ) under MNAR Recall model E ( Y 3 | Y 1 , R = 1) = α 1 + β 1 Y 1 E ( Y 3 | Y 1 , R = 0) = α 0 + β 0 Y 1 More general version of model E ( Y 3 | Y 1 , R = 1) = g 1 ( Y 1 ; θ 1 ) E ( Y 3 | Y 1 , R = 0) = g 0 ( Y 1 ; θ 1 , ∆) = h { g 1 ( Y 1 ; θ 1 ) , ∆ } The h function relates missing-data and observed-data models. User needs to specify form of h The parameter ∆ should not be estimable from observed data Vary ∆ in a sensitivity analysis Hogan (MDEpiNet) Missing Data October 22, 2018 70 / 160
Regression-based specification under MNAR Specify observed-data model E ( Y 3 | Y 1 , R = 1) = g 1 ( Y 1 , θ 1 ) = α 1 + β 1 Y 1 Specify missing-data model E ( Y 3 | Y 1 , R = 0) = h { g 1 ( Y 1 , θ 1 ) , ∆ } = ∆ + g 1 ( Y 1 , θ 1 ) = ∆ + ( α 1 + β 1 Y 1 ) Many other choices are possible Here, add a constant to the MAR imputation Have MAR when ∆ = 0, and MNAR otherwise Hogan (MDEpiNet) Missing Data October 22, 2018 71 / 160
Estimation of E ( Y 3 ) under MNAR 1 Fit the model E ( Y 3 | Y 1 , R = 1) = α 1 + β 1 Y 1 α 1 , � ⇒ obtain � β 1 . 2 For those with R = 0, impute predicted value via � � Y 3 i = E ( Y 3 | Y 1 i , R i = 0) α 1 + � = ∆ + � β 1 Y 1 i 3 Estimate overall mean as the mixture � � R i Y 3 i + (1 − R i ) � E ( Y 3 ) = (1 / n ) Y 3 i i � � [ R =1] [ R =0] ˆ α 1 + � ˆ = Y P ( R = 1) + ∆ + � β 1 Y P ( R = 0) 3 1 Hogan (MDEpiNet) Missing Data October 22, 2018 72 / 160
Sensitivity analysis based on varying ∆ What should be the ‘anchor point’? ◮ Usually appropriate to anchor analysis at MAR ◮ Examine effect of MNAR by varying ∆ away from 0 How to select a range for ∆? ◮ Will always be specific to application. ◮ Ensure that range is appropriate to context (see upcoming example) ◮ Can use data-driven range for ∆, e.g. based on SD Reporting final inferences ◮ ‘Stress test’ approach ◮ Inverted sensitivity analysis — find values of ∆ that would change substantive conclusions ◮ Average over plausible ∆ values Hogan (MDEpiNet) Missing Data October 22, 2018 73 / 160
Calibrating ∆ How should the range and scale of ∆ be chosen? Direction ◮ ∆ > 0 ⇒ dropouts have higher mean ◮ ∆ < 0 ⇒ dropouts have lower mean Range and scale ◮ Residual variation in outcome quantified by SD of regression error [ Y 3 | Y 1 , R = 1] = α 1 + β 1 Y 1 + e σ 2 var( e ) = ◮ Suggests scaling ∆ in units of σ ◮ Will illustrate in longitudinal case Hogan (MDEpiNet) Missing Data October 22, 2018 74 / 160
Moving to longitudinal setting Set-up for single treatment arm Illustrate ideas with analysis of GH data ◮ Compare treatments ◮ Illustrate sensitivity analysis under MNAR ◮ Discuss how to report results Hogan (MDEpiNet) Missing Data October 22, 2018 75 / 160
Longitudinal case: notation Assume missing data pattern is monotone K = dropout time � = R j j E k ( Y j ) = E ( Y j | K = k ) When j > k , cannot estimate E k ( Y j ) from data Hogan (MDEpiNet) Missing Data October 22, 2018 76 / 160
Longitudinal model with J = 3: Set up E k ( Y 1 ) = E ( Y 1 | K = k ) identified for k = 1 , 2 , 3 For other means, we have j = 2 j = 3 K = 1 E 1 ( Y 2 | Y 1 ) E 1 ( Y 3 | Y 1 , Y 2 ) K = 2 E 2 ( Y 2 | Y 1 ) E 2 ( Y 3 | Y 1 , Y 2 ) K = 3 E 3 ( Y 2 | Y 1 ) E 3 ( Y 3 | Y 1 , Y 2 ) Components in red cannot be estimated. Need assumptions Hogan (MDEpiNet) Missing Data October 22, 2018 77 / 160
Longitudinal model with J = 3: MAR j = 2 j = 3 E 3 ( Y 3 | Y 1 , � K = 1 ω E 2 ( Y 2 | Y 1 ) + (1 − ω ) E 3 ( Y 2 | Y 1 ) Y 2 ) K = 2 E 2 ( Y 2 | Y 1 ) E 3 ( Y 3 | Y 1 , Y 2 ) K = 3 E 3 ( Y 2 | Y 1 ) E 3 ( Y 3 | Y 1 , Y 2 ) Here, ω is a weight such that 0 ≤ ω ≤ 1 Hogan (MDEpiNet) Missing Data October 22, 2018 78 / 160
Longitudinal model with J = 3: MNAR j = 2 j = 3 E 3 ( Y 3 | Y 1 , � K = 1 ω E 2 ( Y 2 | Y 1 ) + (1 − ω ) E 3 ( Y 2 | Y 1 ) + ∆ 1 Y 2 ) + ∆ 2 K = 2 E 2 ( Y 2 | Y 1 ) E 3 ( Y 3 | Y 1 , Y 2 ) + ∆ 3 K = 3 E 3 ( Y 2 | Y 1 ) E 3 ( Y 3 | Y 1 , Y 2 ) Hogan (MDEpiNet) Missing Data October 22, 2018 79 / 160
Procedure with several longitudinal measures Start by imputing those with missing data at j = 2 1 Fit model for E ( Y 2 | Y 1 , R 2 = 1) α (2) + β (2) E ( Y 2 | Y 1 , R 2 = 1) = 1 Y 1 α (2) , � β (2) ⇒ obtain � 1 This is a model that combines those with K = 2 and K = 3 2 Impute missing Y 2 as before α (2) + � � β (2) Y 2 i = ∆ + � 1 Y 1 i Hogan (MDEpiNet) Missing Data October 22, 2018 80 / 160
Procedure with several longitudinal measures Now impute those with missing data at j = 3 1 Fit model for E ( Y 3 | Y 1 , Y 2 , R 3 = 1) α (3) + β (3) 1 Y 1 + β (3) E ( Y 3 | Y 1 , Y 2 , R 3 = 1) = 2 Y 2 α (3) , � β (3) 1 , � β (3) ⇒ obtain � 2 2 Impute missing Y 3 as follows: ◮ For those with Y 1 , Y 2 observed, α (3) + � β (3) β (3) � 1 Y 1 i + � Y 3 i = ∆ + � 2 Y 2 i ◮ For those with only Y 1 observed, α (3) + � β (3) β (3) � 1 Y 1 i + � � = ∆ + � Y 3 i Y 2 i 2 Hogan (MDEpiNet) Missing Data October 22, 2018 81 / 160
Side note Recall imputation for those with only Y 1 observed: α (3) + � � β (3) 1 Y 1 i + � β (3) � Y 3 i = ∆ + � Y 2 i 2 This is really just using information from the observed Y 1 because α (2) + � � β (2) Y 2 i = ∆ + � 1 Y 1 i Hence the imputation is from the (linear) model of E ( Y 3 | Y 1 ) that is implied by the other imputation models. Hogan (MDEpiNet) Missing Data October 22, 2018 82 / 160
Calibration of ∆ At each time point j , ∆ is actually a multiplier of the residual SD for the observed data regression [ Y j | Y 1 , . . . , Y j − 1 , R = 1] For example, the imputation model at j = 3 is actually α (3) + � � β (3) 1 Y 1 i + � β (3) Y 3 i = ∆ � σ 3 + � 2 Y 2 i where σ 2 3 = var( Y 3 | Y 1 , Y 2 , R 3 = 1) Generally will suppress this for clarity Hogan (MDEpiNet) Missing Data October 22, 2018 83 / 160
Procedure with several longitudinal measures Final step: compute estimate of E ( Y 3 ) � � R 3 i Y 3 i + (1 − R 3 i ) � E ∆ ( Y 3 ) = (1 / n ) Y 3 i (∆) i [ K =3] [ K =2] Based on the imputations, this turns out to be a weighted average of Y , Y , 3 2 [ K =1] and Y . 1 Weights depend on dropout rates at each time coefficients in imputation models sensitivity parameter(s) Hogan (MDEpiNet) Missing Data October 22, 2018 84 / 160
Analysis components Fitted regression models at each time point ◮ Can check validity of imputation Contour plots: vary ∆ separately by treatment ◮ Treatment effect estimates ◮ p -values Summary table ◮ Treatment effect, SE, p-value ◮ These are computed using bootstrap Hogan (MDEpiNet) Missing Data October 22, 2018 85 / 160
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Case Study 2: Chronic Schizophrenia Major breakthroughs have been made in the treatment of patients with psychotic symptoms. However, side effects associated with some medications have limited their usefulness. RIS-INT-3 (Marder and Meibach, 1994, Chouinard et al. , 1993) was a multi-center study designed to assess the effectiveness and adverse experiences of four fixed doses of risperidone compared to haliperidol and placebo in the treatment of chronic schizophrenia. Hogan (MDEpiNet) Missing Data October 22, 2018 90 / 160
RIS-INT-3 Patients were required to have a PANSS (Positive and Negative Syndrome Scale) score between 60 and 120. Prior to randomization, one-week washout phase (all anti-psychotic medications discontinued). If acute psychotic symptoms occurred, patients randomized to a double-blind treatment phase, schedule to last 8 weeks. Patients randomized to one of 6 treatment groups: risperidone 2, 6, 10 or 16 mg, haliperidol 20 mg, or placebo. Dose titration occurred during the first week of the double-blind phase. Hogan (MDEpiNet) Missing Data October 22, 2018 91 / 160
RSIP-INT-3 Patients scheduled for 5 post-baseline assessments at weeks 1,2,4,6, and 8 of the double-blind phase. Primary efficacy variable: PANSS score Patients who did not respond to treatment and discontinued therapy or those who completed the study were eligible to receive risperidone in an open-label extension study. 521 patients randomized to receive placebo ( n = 88), haliperidol 20 mg ( n = 87), risperidone 2mg ( n = 87), risperidone 6mg ( n = 86), risperidone 10 mg ( n = 86), or risperidone 16 mg ( n = 87). Hogan (MDEpiNet) Missing Data October 22, 2018 92 / 160
Dropout and withdrawal Only 49% of patients completed the 8 week treatment period. The most common reason for discontinuation was “insufficient response.” Other main reasons included: adverse events, uncooperativeness, and withdrawal of consent. Hogan (MDEpiNet) Missing Data October 22, 2018 93 / 160
Dropout and Withdrawal Placebo Haliperidol Risp 2mg Risp 6mg Risp 10mg Risp 16 mg ( n = 88) ( n = 87) ( n = 87) ( n = 86) ( n = 86) ( n = 87) Completed 27 31% 36 41% 36 41% 53 62% 48 56% 54 62% Withdrawn 61 69% 51 59% 51 59% 33 38% 38 44% 33 38% Lack of Efficacy 51 58% 36 41% 41 47% 12 14% 25 29% 18 21% Other 10 11% 15 17% 10 11% 21 24% 13 15% 15 17% Hogan (MDEpiNet) Missing Data October 22, 2018 94 / 160
Central Question What is the difference in the mean PANSS scores at week 8 between risperidone at a specified dose level vs. placebo in the counterfactual world in which all patients were followed to that week? Hogan (MDEpiNet) Missing Data October 22, 2018 95 / 160
Sample means and imputed means under MAR N INS HOST EPS Y 0 µ R � Placebo R = 0 61 3.9 10.5 3.3 94 ?? R = 1 27 3.7 8.1 3.2 89 78 Risperidone R = 0 51 3.8 10.9 3.5 98 ?? R = 1 36 3.8 8.1 2.8 87 71 Hogan (MDEpiNet) Missing Data October 22, 2018 96 / 160
Sample means and imputed means under MAR N INS HOST EPS Y 0 µ R � Placebo R = 0 61 3.9 10.5 3.3 94 79 R = 1 27 3.7 8.1 3.2 89 78 Risperidone R = 0 51 3.8 10.9 3.5 98 74 R = 1 36 3.8 8.1 2.8 87 71 Hogan (MDEpiNet) Missing Data October 22, 2018 97 / 160
Regression imputation under MAR Placebo Hogan (MDEpiNet) Missing Data October 22, 2018 98 / 160
Regression imputation under MAR Risperidone Hogan (MDEpiNet) Missing Data October 22, 2018 99 / 160
Sample means by dropout time: Aggregated data 120 100 PANSS 80 60 1 2 3 4 5 6 Visit Hogan (MDEpiNet) Missing Data October 22, 2018 100 / 160
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