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Term 4, 2006BIO656--Multilevel Models 1 Missing Data Measurement Error Part 13

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Term 4, 2006BIO656--Multilevel Models 2 PROJECTS ARE DUE By midnight, Friday, May 19 th Electronic submission only to Please name the file: [myname]-project.[filetype] or [name1_name2]-project.[filetype]

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Term 4, 2006BIO656--Multilevel Models 3 Overview Missing data are inevitable Some missing data are “inherent” Prevention is better than statistical “cures” Too much missing information invalidates a study There are many methods for accommodating missing data –Their validity depends on the missing data mechanism and the analytic approach Issues can be subtle A little data on the missingness process can be helpful

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Term 4, 2006BIO656--Multilevel Models 4 Common types of missing data Survey non-response Missing dependent variables Missing covariates Dropouts Censoring –administrative, due to competing events or due to loss to follow-up Non-reporting or delayed reporting Noncompliance Measurement error

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Term 4, 2006BIO656--Multilevel Models 5 Implications of missing data Missing data produces/induces Unbalanced data Loss of information and reduced efficiency Extent of information loss depends on –Amount of missingness –Missingness pattern –Association between the missing and observed data –Parameters of interest –Method of analysis Care is needed to avoid biased inferences, inferences that target a reference population other than that intended e.g., those who stay in the study

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Term 4, 2006BIO656--Multilevel Models 6 Inherent missingness Right-censoring We know only that the event has yet to occur –Issue: “No news is no news” versus “no news is good news” Latent disease state Disease Free/Latent Disease/Clinical Disease –Screen and discover latent disease –Only known that transition DF LD occurred before the screening time and that LD CD has yet to occur

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Term 4, 2006BIO656--Multilevel Models 7 Missing Data Mechanisms Missing Data Mechanisms Little RJA, Rubin D. Statistical analysis with missing data. Chichester, NY: John Wiley & Sons; 2002 Missing Completely at random (MCAR) Pr(missing) is unrelated to process under study Missing at Random (MAR) Pr(missing) depends only on observed data Not Missing at Random (NMAR) Pr(missing) depends on both observed and unobserved data These distinctions are important because validity of an analysis depends on the missing data mechanism

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Term 4, 2006BIO656--Multilevel Models 8 Notation Notation (for a missing dependent variable in a longitudinal study) i indexes participant (unit), i = 1,…,n j indexes measurement (sub-unit), j = 1,…,J Potential response vector Y i = (Y i1, Y i2, …, Y iJ ) Response Indicators R i = (R i1, R i2, …, R iJ ) R ij = 1 if Y ij is observed and R ij = 0 if Y ij is missing Given R i, Y i can be partitioned into two components: Y i O observed responses Y i M missing responses

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Term 4, 2006BIO656--Multilevel Models 9 Schematic Representation of Response vector and Response indicators Response vectorResponse indicators PatientY1Y1 Y2Y2 Y3Y3 …YJYJ R1R1 R2R2 R3R3 …RJRJ 1y 11 y 12 y 13 …y 1J 111…1 2y 21 *y 23 …y 2J 101…1 3y 31 y 32 *…y 3J 110…1 … ………………………… ny n1 **…*100…0 Eg:Y 2 = (Y 21, Y 22, Y 23, …, Y 2J )R 2 = (1, 0, 1, …, 1) Y 2 O = (Y 21, Y 23, …, Y 2J )Y 2 M = (Y 22 )

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Term 4, 2006BIO656--Multilevel Models 10 More general missing data A similar notation can be used for missing regressors (X ij ) and for missing components of an even more general data structure Using “ Y ” to denote all of the potential data (regressors, dependent variable, etc.), the foregoing notation applies in general

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Term 4, 2006BIO656--Multilevel Models 11 Missing Data Mechanisms Some mechanisms are relatively benign and do not complicate or bias an analysis Others are not benign and can induce bias Example Goal is to predict weight from gender and height Use information from Bio656 students Possible reasons for missing data –Absence from class –Gender-associated, non-response –Weight-associated, non-response How would each of the above reasons affect results ?

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Term 4, 2006BIO656--Multilevel Models 12 Missing Completely at Random (MCAR) Missingness is a chance mechanism that does not depend on observed or unobserved responses –R i is independent of both Y i O and Y i M Pr(R i | Y i O, Y i M ) = Pr(R i ) In the weight survey example, missingness due to absence from class is unlikely to be related to the relation between weight, height and gender The dataset can be regarded as a random sample from the target population (the full class, Bio620 over the years,....) A complete-case analysis is appropriate, albeit with a drop in efficiency relative to obtaining more data

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Term 4, 2006BIO656--Multilevel Models 13 Missing Completely at Random (MCAR) The probability of having a missing value for variable Y is unrelated to the value of Y or to any other variables in the data set A complete-case analysis is appropriate Height (cm)

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Term 4, 2006BIO656--Multilevel Models 14 Missing at random (MAR) Missingness depends on the observed responses, but does not depend on what would have been measured, but was not collected Pr(R i |Y i O,Y i M ) = Pr(R i |Y i O ) The observed data are not a random sample from the full population –In the weight survey example, data are MAR if Pr(missing weight) depends on gender or height but not on weight Even though not a random sample, the distribution of Y i M conditional on Y i O is the same as that in the reference population (the full class) Therefore, Y i M can be validly predicted using Y i O –Of course, validity depends on having a correct model for the mean and dependency structure for the observed data But, we don ’ t need to do these predictions to get a valid inferences

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Term 4, 2006BIO656--Multilevel Models 15 Height (cm) Missing at random (MAR) The probability of missing data on Y is unrelated to the value of Y, after controlling for other variables in the analysis Analysis using the wrong model is not valid –e.g., uncorrelated regression, when correlation is needed A complete case analysis gives a valid slope, when selection is on the predictors, BUT correlation will be biased.

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Term 4, 2006BIO656--Multilevel Models 16 When the mechanism is MAR Complete-case methods and standard regression methods based on all the available data can produce biased estimates of mean response or trends If the statistical model for the observed data is correct, likelihood- based methods using only the observed data are valid Requires that the joint distribution of the observed Y i s is correctly specified, –when the mean and covariance are correct –when using a correct GEE working model –when using correct random effects Ignorability With a correct model for the observeds, under MAR the details of the missing data mechanism are not needed; the mechanism is ignorable –Ignorability is not an inherent property of the mechanism –It depends on the mechanism and on the analytic model

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Term 4, 2006BIO656--Multilevel Models 17 Not missing at random (NMAR) Not missing at random (NMAR) Missingness depends on the responses that could have been observed Pr(R i |Y i O,Y i M ) does depend on Y i M The observed data cannot be viewed as a random sample of the complete data The distribution of Y i M conditional on Y i O is not the same as that in the reference population (the full class) Y i M depends on Y i O and on Pr(R i |Y i O,Y i M ) and on Pr(Y) In the weight survey example, data are NMAR if missingness depends on weight

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Term 4, 2006BIO656--Multilevel Models 18 Missing Data Mechanisms: Not missing at random (NMAR) Also known as –Non-ignorable missing The probability of missing data on Y is related to the value of Y even if we control for other variables in the analysis. A complete-case analysis is NOT valid Any analysis that does not take dependence on Y into account is not valid Inferences are highly model dependent Height (cm)

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Term 4, 2006BIO656--Multilevel Models 19 MAR for Y vs X NMAR for cor(X,Y)

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Term 4, 2006BIO656--Multilevel Models 20 When the mechanism is NMAR Almost all standard methods of analysis are invalid –Valid inferences require joint modeling of the response and the missing data mechanism Pr(R i |Y i O,Y i M ) Importantly, assumptions about Pr(R i |Y i O,Y i M ) cannot be empirically verified using the data at hand Sensitivity analyses can be conducted (Dan Scharfstein ’ s research focus) Obtaining values from some missing Ys can inform on the missing data mechanism

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Term 4, 2006BIO656--Multilevel Models 21 Dropouts (if missing, missing thereafter) Dropout Completely at Random Dropout at each occasion is independent of all past, current, and future outcomes –Is assumed for Kaplan-Meier estimator and Cox PHM Dropout at Random Dropout depends on the previously observed outcomes up to, but not including, the current occasion –i.e., given the observed outcomes, dropout is independent of the current and future unobserved outcomes Dropout Not at Random, “informative dropout” Dropout depends on current and future unobserved outcomes

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Term 4, 2006BIO656--Multilevel Models 22 Probability of a follow-up lung function measurement depends on smoking status and current lung function Is the mechanism MAR? We don’t know!

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Term 4, 2006BIO656--Multilevel Models 23 LUNG FUNCTION DECLINE IN ADULTS

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Term 4, 2006BIO656--Multilevel Models 24 Longitudinal dropout example Longitudinal dropout example Repeated measurements Y it i indexes people, i=1,…,n t indexes time, t=1,…,5 Y it = μ it = 0 + 1 t + e it cor = cov(e is, e it ) = |s-t| ; 0 0 = 5, 1 = 0.25, = 1, = 0.7

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Term 4, 2006BIO656--Multilevel Models 25 Longitudinal dropout example Longitudinal dropout example the dropout mechanism Dropout indicator, D i D i = k if person i drops out between the (k-1) st and k th occasion Assume that Dropout is MCARif 2 = 3 = 0 Dropout is MAR if 3 = 0 Dropout is NMARif 3 ≠ 0

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Term 4, 2006BIO656--Multilevel Models 26 Population Regression Line vs. Observed Data Means = -0.5, 2 = 3 = 0) MCAR ( 1 = -0.5, 2 = 3 = 0) = -0.5, 2 =0.5, 3 = 0) MAR ( 1 = -0.5, 2 =0.5, 3 = 0) = -0.5, 2 =0, 3 = 0.5) NMAR ( 1 = -0.5, 2 =0, 3 = 0.5) YY Y T T T

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Term 4, 2006BIO656--Multilevel Models 27 Analysis results Analysis results The true regression parameters are intercept = 5.0 and slope = 0.25, = 0.7 ML (se) GEE/OLS (se) Dropout Mechanism ParameterEstimate MCAR Intercept5.015 (0.031) (0.032) Slope0.257 (0.016) (0.018) MAR Intercept5.003 (0.041) (0.043) Slope0.261 (0.016) (0.018) NMAR Intercept5.058 (0.040) (0.043) Slope0.201 (0.016) (0.018)

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Term 4, 2006BIO656--Multilevel Models 28 Misspecified GEE Misspecified GEE (when the truth is random intercepts and slopes) Complete Data (GEE)Partial Missing Data (GEE) Time YY

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Term 4, 2006BIO656--Multilevel Models 29 Correctly specified Random Effects Correctly specified Random Effects (when the truth is random intercepts and slopes) Complete Data (REM)Partial Missing Data (REM) Time Y Y

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Term 4, 2006BIO656--Multilevel Models 30 The probability of dropping out depends on the observed history

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Term 4, 2006BIO656--Multilevel Models 31 One step at a time

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Term 4, 2006BIO656--Multilevel Models 32 The OLS analysis has regressors 0, 1, 2 and dependent variables 0, , 2 There are 5 different “trajectories” with relative weights The Indep. Increments analysis has a constant regressor “1” and so is just estimating the mean. The dependent variable is either + or -

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Term 4, 2006BIO656--Multilevel Models 33 If the missing data process is MAR and if we use the correct model for the observed data, the missing data mechanism is “ignorable” In the foregoing example, computing first differences (current value – previous value) and averaging them differences is an unbiased estimate (of 0) no matter how complicated the MAR missing data process We don’t have to know the details of the dropout process (it can be very complicated), as long as the probabilities depend only on what has been observed and not on what would have been observed Ignorability depends on using the correct model for the observed data (mean and dependency structure) If the errors were independent (rather than the first differences), then standard OLS would be unbiased

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Term 4, 2006BIO656--Multilevel Models 34 Analytic Approaches Complete Case Analysis Global complete case analysis Individual model complete case analysis Augment with missing data indicators –primarily for missing Xs Weighting Imputation –Single –Multiple Likelihood-based (model-based) methods

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Term 4, 2006BIO656--Multilevel Models 35 Analytic Approaches Global complete-case Analysis (use only data for people with fully complete data) Biased, unless the dropout is MCAR Even if MCAR is true, can be immensely inefficient Analyze Available Data (use data for people with complete data on the regressors in the current model) More efficient than complete-case methods, because uses maximal data Biased unless the dropout is MCAR Can produce floating datasets, producing “illogical” conclusions –R 2 relations are not monotone Use Missing data indicators (e.g., create new covariates)

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Term 4, 2006BIO656--Multilevel Models 36 Weighting Stratify samples into J weighting classes –Zip codes – propensity score classes Weight the observed data inversely according to the response rate of the stratum –Lower response rate higher weight Unbiased if observed data are a random sample in a weighting class (a special form of the MAR assumption) Biased, if respondents differ from non-respondents in the class Difficult to estimate the appropriate standard error because weights are estimated from the response rates

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Term 4, 2006BIO656--Multilevel Models 37 Simple example of weighting adjustment Estimate the average height of villagers in two villages Surveys sent to 10% of the population in both villages Direct, unweighted: 1.7*(2/3) + 1.4*(1/3) = 1.60m Weighted: 100*1.7* *1.4*0.01 = 1.55m (= 1.7* *.5) Village AVillage B # villagers1000 # survey sent 100 # providing data Avg height1.7m1.4m 2 x Weight

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Term 4, 2006BIO656--Multilevel Models 38 Single Imputation Fill in missing values with imputed values Once a filled-in dataset has been constructed, standard methods for complete data can be applied Problem Fails to account for the uncertainty inherent in the imputation of the missing data Don’t use it!

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Term 4, 2006BIO656--Multilevel Models 39 Multiple Imputation Multiple Imputation Rubin 1987, Little & Rubin 2002 Multiply impute “m” pseudo-complete data sets –Typically, a small number of imputations (e.g., 5 ≤ m ≤10) is sufficient Combine the inferences from each of the m data sets Acknowledges the uncertainty inherent in the imputation process Equivalently, the uncertainty induced by the missing data mechanism Rubin DB. Multiple Imputation for Nonresponse in Surveys, Wiley, New York, 1987 Little RJA, Rubin D. Statistical analysis with missing data. Chichester, NY: John Wiley & Sons; 2002

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Term 4, 2006BIO656--Multilevel Models 40 Multiple Imputation

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Term 4, 2006BIO656--Multilevel Models 41 Multiple Imputation: Combining Inferences Combine m sets of parameter estimates to provide a single estimate of the parameter of interest Combine uncertainties to obtain valid SEs In the following, “ k ” indexes imputation Within-imputation variance Between-imputation variance

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Term 4, 2006BIO656--Multilevel Models 42 Multiple Imputation: Combining Inferences Combine m sets of parameter estimates to provide a single estimate of the parameter of interest Combine uncertainties to obtain valid SEs In the following, “ k ” indexes imputation Within-imputation covariance Between-imputation covariance

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Term 4, 2006BIO656--Multilevel Models 43 Producing the Imputed Values Last value carried forward (LVCF) Single Imputation (never changes) Assumes the responses following dropout remain constant at the last observed value prior to dropout Unrealistic unless, say, due to recovery or cure Underestimates SEs Hot deck Randomly choose a fill-in from outcomes of “similar” units Distorts distribution less than imputing the mean or LVCF Underestimates SEs

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Term 4, 2006BIO656--Multilevel Models 44 Valid Imputation Build a model relating observed outcomes Means and covariances and random effects,... Goal is prediction, so be liberal in including predictors Don’t use P-values; don’t use step-wise Do use multiple R 2, predictions sums of squares, cross-validation,...

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Term 4, 2006BIO656--Multilevel Models 45 Producing Imputed Values Sample values of Y i M from pr(Y i M |Y i O, X i ) Can be straightforward or difficult Monotone case: draw values of Y i M from pr(Y i M |Y i O,X i ) in a sequential manner Valid when dropouts are MAR or MCAR Propensity Score Method Imputed values are obtained from observations on people who are equally likely to drop out as those lost to follow up at a given occasion Requires a model for the propensity (probability) of dropping out, e.g.,

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Term 4, 2006BIO656--Multilevel Models 46 Producing Imputed Values Producing Imputed Values Recall that “Y” is all of the data, not just the dependent variable Predictive Mean Matching (build a regression model!) A series of regression models for Y ik, given Y i1, …,Y ik-1, are fit using the observed data on those who have not dropped out by the k th occasion. For example, E(Y ik ) = 1 + 2 Y i1 + … + k Y i(k-1) V(Y ik ) = Yields and 1.Parameters * and 2* are then drawn from the distribution of the estimated parameters (to account for the uncertainty in the estimated regression) 2.Missing values can then be predicted from 1 * + 2 * Y i1 + … + k * Y ik-1 + * e i, where e i is simulated from a standard normal distribution 3.Repeat 1 and 2

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Term 4, 2006BIO656--Multilevel Models 47 Missing, presumed at random Estimate the difference in cost between transurethral resection (TURP) and contact-laser vaporization of the prostate (Laser) 100 patients were randomized to one of the two treatments –TURP: n = 53; Laser: n = categories of medical resource usage were measured –e.g., GP visit, transfusion, outpatient consultation, etc. Cost-analysis with incomplete data* * Briggs A et al. Health Economics. 2003; 12,

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Term 4, 2006BIO656--Multilevel Models 48 Missing data TURP n = 53Laser n = 47Total n = 100 Patients with no missing resource counts 34 (59%)21 (51%)55 (55%) Observed resource counts 570 (90%)510 (90%)1080 (90%) Complete-case analysis uses only half of the patients in the study even though 90% of resource usage data were available

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Term 4, 2006BIO656--Multilevel Models 49 Comparison of inferences Note that mean imputation understates uncertainty.

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Term 4, 2006BIO656--Multilevel Models 50 Multiple Imputation versus likelihood analysis when data are MAR Both multiple imputation or used of a valid statistical model for the observed data (likelihood analysis) are valid –The model-based analysis will be more efficient, but more complicated Validity of each depends on correct modeling to produce/induce ignorability

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Term 4, 2006BIO656--Multilevel Models 51 What if you doubt the MAR assumption What if you doubt the MAR assumption (you should always doubt it!) You can never empirically rule out NMAR Methods for NMAR exist, but they require information and assumptions on pr(Missing | observed, unobserved) Methods depend on unverifiable assumptions Sensitivity analysis can assess the stability of findings under various scenarios –Set bounds on the form and strength of the dependence –Evaluate conclusions within these bounds

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Term 4, 2006BIO656--Multilevel Models 52 MEASUREMENT ERROR If a covariate (X) is measured with error, what is the implication for regression of Y on X? See also “Air” and “Cervix” in volume II of the BUGS examples

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Term 4, 2006BIO656--Multilevel Models 53 Measurement Error Measurement Error Another type of missing data Measurement error is a special case of missing data because we do not get to “observe the true value” of the response or covariates Depending on the measurement error mechanism and on the analysis, inferences can be –inefficient (relative to no measurement error) –biased

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Term 4, 2006BIO656--Multilevel Models 54 Differential attenuation across studies complicates “exporting” and synthesizing

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Term 4, 2006BIO656--Multilevel Models 55

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Term 4, 2006BIO656--Multilevel Models 56 The two “Pure Forms” The two “Pure Forms” relating X t & X o Classical: X o = X t + , (0, 2 ) What you see is a random deviation from the truth Measured & true blood pressure Measured and true social attitudes Berkson: X t = X o + The truth is a random deviation from what you see Individual SES measured by ZIP-code SES Personal air pollution measured by centrally monitored value Actual temperature & thermostat setting

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Term 4, 2006BIO656--Multilevel Models 57 Hybrids are possible X t and X o have a general joint distribution

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Term 4, 2006BIO656--Multilevel Models 58 Measurement error’s effect on a simple regression coefficient Classical The regression coefficient on X o is attenuated towards 0 relative to the “true” regression coefficient on X t Because, the spread of X o is greater than that for X t Berkson No effect on the expected regression coefficient Variance inflation

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Term 4, 2006BIO656--Multilevel Models 59 Berkson X t = X 0 + , (0, 2 ) true: Y = int + X t + resid = int + (X 0 + ) + resid observed: Y = int + * X 0 + resid Var(X 0 ) = 0 2 No attenuation * = because E(X t | X 0 ) = X 0

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Term 4, 2006BIO656--Multilevel Models 60 Classical X o = X t + , (0, 2 ) true: Y = int + X t + resid observed: Y = int + * X 0 + resid = int + * (X t + ) + resid Var(X 0 ) = t 2 + 2 (X 0 is stretched out) Attenuation (attenuation factor ) * = = t 2 /( t 2 + 2 ) slope = cov(Y, X)/Var(X), but E(X t | X 0 ) = X 0

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Term 4, 2006BIO656--Multilevel Models 61 Y versus X t

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Term 4, 2006BIO656--Multilevel Models 62 Y versus X 0

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Term 4, 2006BIO656--Multilevel Models 63 An illustration An illustration Back to the basic example W = Weight (lb) H = Height (cm) Analysis: simple linear regression W i = 0 + 1 H i + i where i ~ N(0, Assume the true model to be: W i = H i + i where i ~ N(0, 8 2 Measurement error 1.Error in W: observe W * = W + i * where i ~ N(0, 4 2 2.Error in H : observe H * = H + i * where i * ~ N(0, 10 2

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Term 4, 2006BIO656--Multilevel Models 64 Scenario 1: Measurement Error in Response Standard regression estimate for 1 is unbiased, but less efficient The larger is the measurement error, the greater the loss in efficiency Results: 1 = 1.16 SE( 1 )= 0.15 1 = 1.08 SE( 1 ) = 0.18

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Term 4, 2006BIO656--Multilevel Models 65 Scenario 2: measurement error in H Standard regression estimate for 1 is biased (attenuated) The larger is the measurement error, the greater the attenuation Results: 1 = 1.16 SE( 1 )= 0.15 1 = 0.69 SE( 1 )= 0.21

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Term 4, 2006BIO656--Multilevel Models 66 Multivariate Measurement Error X o = X t + , (0, )

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Term 4, 2006BIO656--Multilevel Models 67

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Term 4, 2006BIO656--Multilevel Models 68 The Multiple Imputation Algorithm in SAS The MIANALYZE Procedure –Combines the m different sets of the parameter and variance estimates from the m imputations –Generates valid inferences about the parameters of interest PROC MIANALYZE ; BY variables; VAR variables;

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Term 4, 2006BIO656--Multilevel Models 69 Multiple Imputation Algorithm in SAS PROC MI ; BY variables; FREQvariable; MULTINORMAL ; VAR variables; Available options in PROC MI include: NIMPU=number (default=5) Available options in MULTINORMAL statement: METHOD=REGRESSION METHOD=PROPENSITY METHOD=MCMC The default is METHOD=MCMC

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