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Missing Data & Measurement Error

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Missing Data& Measurement Error

Welcome to Rachel Whitaker

Bio753—Advanced Methods

in Biostatistics, III

- 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

- 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

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

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

- Issue: “No news is no news” versus
- Disease Free/Latent Disease/Clinical Disease
- Screen and discover latent disease
- Only known that transition DFLD occurred before the screening time and that LDCD has yet to occur

Bio753—Advanced Methods

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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

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

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- Eg:Y2 = (Y21, Y22, Y23, … , Y2J)R2 = (1, 0, 1, … , 1)
- Y2O = (Y21, Y23, …, Y2J)Y2M = (Y22)

Bio753—Advanced Methods

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- 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

- 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

- 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)

- Ri is independent of both YiO and YiM
- 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

Height (cm)

- 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

- 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

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Height (cm)

- 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

- 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

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- 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

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Height (cm)

- 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

Bio753—Advanced Methods

in Biostatistics, III

- 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

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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

- Is assumed for Kaplan-Meier estimator and Cox PHM
- 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”

- i.e., given the observed outcomes, dropout is independent of the current and future unobserved outcomes
- Dropout depends on current and future unobserved outcomes

Bio753—Advanced Methods

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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

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LUNG FUNCTION DECLINE IN ADULTS

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- 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

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- 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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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

Bio753—Advanced Methods

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1

2

3

4

5

Bio753—Advanced Methods

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CompleteData (GEE)

PartialMissing Data (GEE)

Y

Y

Time

Time

Bio753—Advanced Methods

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Complete Data (REM)

Partial Missing Data (REM)

Y

Y

Time

Time

Bio753—Advanced Methods

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The probabilityof dropping out depends on theobserved history

Bio753—Advanced Methods

in Biostatistics, III

Bio753—Advanced Methods

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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

- 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

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

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)

- R2 relations are not monotone

Bio753—Advanced Methods

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- 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

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- 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

- 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

- 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

Bio753—Advanced Methods

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- 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

Bio753—Advanced Methods

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- 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

Bio753—Advanced Methods

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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

Bio753—Advanced Methods

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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, ...

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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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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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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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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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Note that mean imputation understates uncertainty.

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- 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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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

Bio753—Advanced Methods

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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

Bio753—Advanced Methods

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- 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

Bio753—Advanced Methods

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- Differential attenuation across
- studies complicates “exporting”
- and synthesizing

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Bio753—Advanced Methods

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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

Bio753—Advanced Methods

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Xt and Xo have a general joint distribution

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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

Bio753—Advanced Methods

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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

Bio753—Advanced Methods

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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 Xt

Bio753—Advanced Methods

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Y versus X0

Bio753—Advanced Methods

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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)

Bio753—Advanced Methods

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Results:

b1 = 1.16

SE(b1)= 0.15

b1 = 1.08

SE(b1) = 0.18

- Standard regression estimate for b1 is unbiased, but less efficient
- The larger is the measurement error, the greater the loss in efficiency

Bio753—Advanced Methods

in Biostatistics, III

Results:

b1 = 1.16

SE(b1)= 0.15

b1 = 0.69

SE(b1)= 0.21

- Standard regression estimate for b1 is biased (attenuated)
- The larger is the measurement error, the greater the attenuation

Bio753—Advanced Methods

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Multivariate Measurement Error

Xo = Xt + , (0, )

Bio753—Advanced Methods

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Bio753—Advanced Methods

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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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- 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

Bio753—Advanced Methods

in Biostatistics, III

Bio753—Advanced Methods

in Biostatistics, III