Missing data measurement error
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Missing Data & Measurement Error. Welcome to Rachel Whitaker. 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

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Missing data measurement error

Missing Data& Measurement Error

Welcome to Rachel Whitaker

Bio753—Advanced Methods

in Biostatistics, III


Overview

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

Bio753—Advanced Methods

in Biostatistics, III


Common types of missing data

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

Bio753—Advanced Methods

in Biostatistics, III


Implications of missing data

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

Bio753—Advanced Methods

in Biostatistics, III


Inherent missingness

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

Bio753—Advanced Methods

in Biostatistics, III


Missing data measurement error

Missing Data MechanismsLittle 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

Bio753—Advanced Methods

in Biostatistics, III


Notation for a missing dependent variable in a longitudinal study

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

    Yi = (Yi1, Yi2, …, YiJ)

  • Response Indicators

    Ri = (Ri1, Ri2, …, RiJ)

    Rij = 1 if Yij is observed and Rij = 0 if Yij is missing

  • Given Ri, Yi can be partitioned into two components:

    YiO observed responses

    YiM missing responses

Bio753—Advanced Methods

in Biostatistics, III


Schematic representation of response vector and response indicators

Schematic Representation of Response vector and Response indicators

  • Eg:Y2 = (Y21, Y22, Y23, … , Y2J)R2 = (1, 0, 1, … , 1)

    • Y2O = (Y21, Y23, …, Y2J)Y2M = (Y22)

Bio753—Advanced Methods

in Biostatistics, III


More general missing data

More general missing data

  • A similar notation can be used for missing regressors (Xij) 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

Bio753—Advanced Methods

in Biostatistics, III


Missing data mechanisms

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?

Bio753—Advanced Methods

in Biostatistics, III


Missing completely at random mcar

Missing Completely at Random (MCAR)

  • Missingness is a chance mechanism that does not depend on observed or unobserved responses

    • Ri is independent of both YiO and YiM

      Pr(Ri | YiO , YiM ) = Pr(Ri)

  • 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

Bio753—Advanced Methods

in Biostatistics, III


Missing completely at random mcar1

Height (cm)

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

Bio753—Advanced Methods

in Biostatistics, III


Missing at random mar

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(Ri|YiO,YiM) = Pr(Ri|YiO)

  • 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 YiM conditional on YiO is the same as that in the reference population (the full class)

  • Therefore, YiM can be validly predicted using YiO

    • 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

Bio753—Advanced Methods

in Biostatistics, III


Missing at random mar1

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.

Bio753—Advanced Methods

in Biostatistics, III


When the mechanism is mar

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 Yis 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

Bio753—Advanced Methods

in Biostatistics, III


Not missing at random nmar

Not missing at random (NMAR)

  • Missingness depends on the responses that could have been observed

    Pr(Ri|YiO,YiM)does depend on YiM

  • The observed data cannot be viewed as a random sample of the complete data

  • The distribution of YiM conditional on YiO is not the same as that in the reference population (the full class)

  • YiM depends on YiOand on Pr(Ri|YiO,YiM) and on Pr(Y)

  • In the weight survey example, data are NMAR if missingness depends on weight

Bio753—Advanced Methods

in Biostatistics, III


Missing data mechanisms not missing at random nmar

Height (cm)

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

Bio753—Advanced Methods

in Biostatistics, III


Mar for y vs x y x nmar for cor x y

MAR for Y vs X [Y | X] NMAR for cor(X,Y)

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When the mechanism is nmar

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(Ri|YiO,YiM)

  • Importantly, assumptions about Pr(Ri|YiO,YiM) 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

Bio753—Advanced Methods

in Biostatistics, III


Dropouts if missing missing thereafter

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

Bio753—Advanced Methods

in Biostatistics, III


Missing data measurement error

Probability of a follow-up lung function measurement depends on smoking status and current lung function

Is the mechanism MAR?

We don’t know!

Bio753—Advanced Methods

in Biostatistics, III


Lung function decline in adults

LUNG FUNCTION DECLINE IN ADULTS

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Longitudinal dropout example

Longitudinal dropout example

  • Repeated measurements Yit

    i indexes people, i=1,…,n

    t indexes time, t=1,…,5

    Yit = μit = 0 + 1t + eit

    cor = cov(eis, eit) = |s-t|;  0

  • 0 = 5, 1 = 0.25,  = 1,  = 0.7

Bio753—Advanced Methods

in Biostatistics, III


Longitudinal dropout example the dropout mechanism

Longitudinal dropout examplethe dropout mechanism

  • Dropout indicator, Di

  • Di = k if person i drops out between the (k-1)st and kth occasion

  • Assume that

  • Dropout is MCARif q2 = q3 = 0

  • Dropout is MAR if q3 = 0

  • Dropout is NMARif q3 ≠ 0

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Population regression line vs observed data means

Population Regression Line vs. Observed Data Means

MCAR (q1= -0.5,q2=q3 = 0)

MAR (q1= -0.5, q2=0.5,q3 = 0)

Y

Y

6.5

6.5

6

6

5.5

5.5

5

5

T

T

1

2

3

4

5

1

2

3

4

5

NMAR (q1= -0.5, q2=0,q3 = 0.5)

Y

6.5

6

5.5

5

T

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1

2

3

4

5


Analysis results the true regression parameters are intercept 5 0 and slope 0 25 0 7

Analysis resultsThe true regression parameters are intercept = 5.0 and slope = 0.25,  = 0.7

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Misspecified gee when the truth is random intercepts and slopes

Misspecified GEE(when the truth is random intercepts and slopes)

CompleteData (GEE)

PartialMissing Data (GEE)

Y

Y

Time

Time

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Correctly specified random effects when the truth is random intercepts and slopes

Correctly specified Random Effects(when the truth is random intercepts and slopes)

Complete Data (REM)

Partial Missing Data (REM)

Y

Y

Time

Time

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The probability of dropping out depends on the observed history

The probabilityof dropping out depends on theobserved history

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One step at a time

One step at a time

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Missing data measurement error

There are 5 different “trajectories”

with relative weights 2 2 1 1 2

The OLS analysis has regressors 0, 1, 2 and dependent variables

0, ,2

The Indep. Increments analysis has a constant regressor “1” and so is just estimating the mean. The dependent variable is either + or -

Bio753—Advanced Methods

in Biostatistics, III


Missing data measurement error

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

Bio753—Advanced Methods

in Biostatistics, III


Analytic approaches

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

Bio753—Advanced Methods

in Biostatistics, III


Analytic approaches1

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

    • R2 relations are not monotone

      Use Missing data indicators (e.g., create new covariates)

Bio753—Advanced Methods

in Biostatistics, III


Weighting

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

Bio753—Advanced Methods

in Biostatistics, III


Simple example of weighting adjustment

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*0.005 + 50*1.4*0.01 = 1.55m (= 1.7*.5 + 1.4*.5)

2 x Weight

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Single imputation

Single Imputation

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!

Bio753—Advanced Methods

in Biostatistics, III


Multiple imputation rubin 1987 little rubin 2002

Multiple ImputationRubin 1987, Little & Rubin 2002

  • Multiply impute “m” pseudo-complete data sets

    • Typically, a small number of imputations (e.g., 5 ≤ m ≤10) is generally 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

Bio753—Advanced Methods

in Biostatistics, III


Multiple imputation

Multiple Imputation

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Multiple imputation combining inferences

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

This computation is correct

for fully efficient estimators.

Within-imputation variance

Between-imputation variance

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Multiple imputation combining inferences1

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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Producing the imputed values

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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Valid imputation

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 R2, predictions sums of squares, cross-validation, ...

Bio753—Advanced Methods

in Biostatistics, III


Producing imputed values

Producing Imputed Values

Sample values of YiM from pr(YiM|YiO, Xi)

  • Can be straightforward or difficult

  • Monotone case: draw values of YiM from pr(YiM|YiO,Xi) 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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Producing imputed values recall that y is all of the data not just the dependent variable

Producing Imputed ValuesRecall 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 Yik, given Yi1, …,Yik-1, are fit using the observed data on those who have not dropped out by the kth occasion. For example,

    E(Yik) = 1 + 2Yi1 +…+ kYi(k-1)

    V(Yik) =

    Yields and

  • Parameters * and 2* are then drawn from the distribution of the estimated parameters (to account for the uncertainty in the estimated regression)

  • Missing values can then be predicted from

    1*+ 2*Yi1+…+ k*Yik-1+ *ei,

    where ei is simulated from a standard normal distribution

  • Repeat 1 and 2

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Missing presumed at random

Missing, presumed at random

Cost-analysis with incomplete data*

  • 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 = 47

  • 12 categories of medical resource usage were measured

    • e.g., GP visit, transfusion, outpatient consultation, etc.

* Briggs A et al. Health Economics. 2003; 12, 377-392

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Missing data

Missing data

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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Comparison of inferences

Comparison of inferences

Note that mean imputation understates uncertainty.

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Multiple imputation versus likelihood analysis when data are mar

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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What if you doubt the mar assumption you should always doubt it

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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Measurement error

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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Measurement error another type of missing data

Measurement ErrorAnother 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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Missing data measurement error

  • Differential attenuation across

  • studies complicates “exporting”

  • and synthesizing

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Missing data measurement error

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The two pure forms relating x t x o

The two “Pure Forms”relating Xt & Xo

Classical: Xo = Xt + , (0, 2)

What you see is a random deviation from the truth

  • Measured & true blood pressure

  • Measured and true social attitudes

    Berkson: Xt = Xo + 

    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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Hybrids are possible

Hybrids are possible

Xt and Xo have a general joint distribution

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Measurement error s effect on a simple regression coefficient

Measurement error’s effecton a simple regression coefficient

Classical

  • The regression coefficient on Xo is attenuated towards 0 relative to the “true” regression coefficient on Xt

  • Because, the spread of Xo is greater than that for Xt

    Berkson

  • No effect on the expected regression coefficient

  • Variance inflation

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Berkson

Berkson

Xt = X0 + , (0, 2)

true: Y = int + Xt + resid

= int + (X0 + ) + resid

observed: Y = int + * X0 + resid

Var(X0) = 02

No attenuation * = 

because E(Xt | X0) = X0

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Classical

Classical

Xo = Xt + , (0, 2)

true: Y = int + Xt + resid

observed: Y = int + *X0 + resid

= int + *(Xt + ) + resid

 Var(X0) = t2+ 2 (X0 is stretched out)

Attenuation (attenuation factor )

* = 

 = t2 /(t2 + 2)

slope = cov(Y, X)/Var(X), but E(Xt | X0) =  X0

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Y versus x t

Y versus Xt

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Y versus x 0

Y versus X0

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An illustration

An illustration

Back to the basic example

  • W = Weight (lb)

  • H = Height (cm)

  • Analysis: simple linear regression

    Wi = b0 + b1 Hi+ ei where ei ~ N(0, s2)

    Assume the true model to be:

    Wi = 3 + 1.0Hi+ ei whereei ~ N(0, 82)

    Measurement error

  • Error in W: observe W* = W + ei* where ei ~ N(0, 42)

  • Error in H : observe H* = H + i* where i*~ N(0, 102)

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Scenario 1 measurement error in response

Results:

b1 = 1.16

SE(b1)= 0.15

b1 = 1.08

SE(b1) = 0.18

Scenario 1: Measurement Error in Response

  • Standard regression estimate for b1 is unbiased, but less efficient

  • The larger is the measurement error, the greater the loss in efficiency

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Scenario 2 measurement error in h

Results:

b1 = 1.16

SE(b1)= 0.15

b1 = 0.69

SE(b1)= 0.21

Scenario 2: measurement error in H

  • Standard regression estimate for b1 is biased (attenuated)

  • The larger is the measurement error, the greater the attenuation

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Missing data measurement error

Multivariate Measurement Error

Xo = Xt + , (0, )

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Missing data measurement error

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The multiple imputation algorithm in sas

The Multiple Imputation Algorithm in SAS

The MIANALYZEProcedure

  • 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 <options>;

    BY variables;

    VAR variables;

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Multiple imputation algorithm in sas

Multiple Imputation Algorithm in SAS

  • PROC MI <options>;

    BY variables;

    FREQvariable;

    MULTINORMAL <options>;

    VAR variables;

  • Available options in PROC MI include: NIMPU=number (default=5)

  • Available options in MULTINORMAL statement:

    METHOD=REGRESSION

    METHOD=PROPENSITY<(NGROUPS=number)>

    METHOD=MCMC<(options)>

    The default is METHOD=MCMC

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Missing data measurement error

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