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Part 13 Missing Data Measurement Error Term 4, 2006 BIO656--Multilevel Models 1 PROJECTS ARE DUE • By midnight, Friday, May 19th • Electronic submission only to [email protected] • Please name the file: [myname]-project.[filetype] or [name1_name2]-project.[filetype] Term 4, 2006 BIO656--Multilevel Models 2 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 Term 4, 2006 BIO656--Multilevel Models 3 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 Term 4, 2006 BIO656--Multilevel Models 4 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 Term 4, 2006 BIO656--Multilevel Models 5 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 DFLD occurred before the screening time and that LDCD has yet to occur Term 4, 2006 BIO656--Multilevel Models 6 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 Term 4, 2006 BIO656--Multilevel Models 7 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 Term 4, 2006 BIO656--Multilevel Models 8 Schematic Representation of Response vector and Response indicators Response vector Response indicators Patient Y1 Y2 Y3 … YJ R1 R2 R3 … RJ 1 y11 y12 y13 … y1J 1 1 1 … 1 2 y21 * y23 … y2J 1 0 1 … 1 3 y31 y32 * … y3J 1 1 0 … 1 … … … … … … … Eg: … yn1 … … … n * * … * 1 0 0 … 0 Y2 = (Y21, Y22, Y23, … , Y2J) Y2O = (Y21, Y23, …, Y2J) Term 4, 2006 R2 = (1, 0, 1, … , 1) Y2M = (Y22) BIO656--Multilevel Models 9 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 Term 4, 2006 BIO656--Multilevel Models 10 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? Term 4, 2006 BIO656--Multilevel Models 11 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 Term 4, 2006 BIO656--Multilevel Models 12 Missing Completely at Random (MCAR) Scatterplot: Weight vs Height by Gender 90 80 70 60 H[G == 0 & R == 1] 70 80 90 100 FEMALE Observed Missing 60 • A complete-case analysis is appropriate Weight (lb) • 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 Weight (lb) 100 MALE Observed Missing 50 60 70 80 90 100 Height (cm) Height (cm) Term 4, 2006 BIO656--Multilevel Models 13 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 Term 4, 2006 BIO656--Multilevel Models 14 Missing at random (MAR) Term 4, 2006 60 70 80 90 100 MALE Observed Missing H[G == 0 & R == 1] 70 80 90 100 FEMALE Observed Missing 60 A complete case analysis gives a valid slope, when selection is on the predictors, BUT correlation will be biased. Weight (lb) • Analysis using the wrong model is not valid – e.g., uncorrelated regression, when correlation is needed Scatterplot: Weight vs Height by Gender Weight (lb) • The probability of missing data on Y is unrelated to the value of Y, after controlling for other variables in the analysis 50 BIO656--Multilevel Models 60 70 80 90 100 (cm) H[GHeight == 1 & R == 1] 15 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, likelihoodbased 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 Term 4, 2006 BIO656--Multilevel Models 16 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 YiO and on Pr(Ri|YiO,YiM) and on Pr(Y) • In the weight survey example, data are NMAR if missingness depends on weight Term 4, 2006 BIO656--Multilevel Models 17 Missing Data Mechanisms: Not missing at random (NMAR) 70 80 90 100 MALE Observed Missing 60 Weight (lb) H[G == 0 & R == 1] 70 80 90 100 FEMALE Observed Missing 60 Weight (lb) • 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 Scatterplot: Weight vs Height by Gender 50 60 70 80 90 100 (cm) H[GHeight == 1 & R == 1] Term 4, 2006 BIO656--Multilevel Models 18 MAR for Y vs X NMAR for cor(X,Y) Scatterplot: Weight vs Height with fitted line analysis with missing Weight (lb) 80 40 60 Weight (lb) 100 120 initial analysis 50 60 70 80 Height (cm) Term 4, 2006 90 100 50 60 70 80 90 100 Height (cm) BIO656--Multilevel Models 19 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 Term 4, 2006 BIO656--Multilevel Models 20 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 Term 4, 2006 BIO656--Multilevel Models 21 Probability of a follow-up lung function measurement depends on smoking status and current lung function Term 4, 2006 Is the mechanism MAR? BIO656--Multilevel Models We don’t know! 22 LUNG FUNCTION DECLINE IN ADULTS Term 4, 2006 BIO656--Multilevel Models 23 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 Term 4, 2006 BIO656--Multilevel Models 24 Longitudinal dropout example the dropout mechanism • Dropout indicator, Di • Di = k if person i drops out between the (k-1)st and kth occasion • Assume that Pr( Di k | Di k , Yi1 ,..., Yik ) log q1 q2 Yik 1 q3Yik Pr( Di k | Di k , Yi1 ,..., Yik ) • Dropout is MCAR if q2 = q3 = 0 • Dropout is MAR if q3 = 0 • Dropout is NMAR if q3 ≠ 0 Term 4, 2006 BIO656--Multilevel Models 25 Population Regression Line vs. Observed Data Means MCAR (q1= -0.5, q2= q3 = 0) Y MAR (q1= -0.5, q2=0.5, q3 = 0) Y 6.5 6.5 6 6 5.5 5.5 5 1 2 3 Y 4 5 T 5 1 2 3 4 T 5 NMAR (q1= -0.5, q2=0, q3 = 0.5) 6.5 6 5.5 Term 4, 2006 5 BIO656--Multilevel Models 1 2 3 4 5 T 26 Analysis results The true regression parameters are intercept = 5.0 and slope = 0.25, = 0.7 ML(se) GEE/OLS(se) Estimate Estimate Dropout Mechanism Parameter MCAR Intercept 5.015(0.031) 5.022(0.032) Slope 0.257(0.016) 0.253(0.018) Intercept 5.003(0.041) 5.062(0.043) Slope 0.261(0.016) 0.182(0.018) Intercept 5.058(0.040) 5.071(0.043) Slope 0.201(0.016) 0.162(0.018) MAR NMAR Term 4, 2006 BIO656--Multilevel Models 27 Misspecified GEE (when the truth is random intercepts and slopes) Complete Data (GEE) Partial Missing Data (GEE) Y Y Time Term 4, 2006 BIO656--Multilevel Models Time 28 Correctly specified Random Effects (when the truth is random intercepts and slopes) Complete Data (REM) Partial Missing Data (REM) Y Y Time Term 4, 2006 BIO656--Multilevel Models Time 29 The probability of dropping out depends on the observed history Term 4, 2006 BIO656--Multilevel Models 30 One step at a time Term 4, 2006 BIO656--Multilevel Models 31 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 - Term 4, 2006 BIO656--Multilevel Models 32 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 Term 4, 2006 BIO656--Multilevel Models 33 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 Term 4, 2006 BIO656--Multilevel Models 34 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) Term 4, 2006 BIO656--Multilevel Models 35 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 Term 4, 2006 BIO656--Multilevel Models 36 Simple example of weighting adjustment • Estimate the average height of villagers in two villages • Surveys sent to 10% of the population in both villages Village A Village B # villagers 1000 1000 # survey sent 100 100 # providing data 100 50 Avg height 1.7m 1.4m • 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 Term 4, 2006 BIO656--Multilevel Models 37 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! Term 4, 2006 BIO656--Multilevel Models 38 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 Term 4, 2006 BIO656--Multilevel Models 39 Multiple Imputation Term 4, 2006 BIO656--Multilevel Models 40 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 1 m ˆ (k) β β m k 1 m 1 m 1 1 ˆ (k) β Var( β ) Var(βˆ (k) ) 1 β m k 1 m m 1 k 1 Within-imputation variance Term 4, 2006 2 Between-imputation variance BIO656--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 1 m ˆ (k) β β m k 1 m 1 m 1 1 ˆ (k) β βˆ (k) β Cov( β ) Cov( βˆ (k) ) 1 β m k 1 m m 1 k 1 Within-imputation covariance Term 4, 2006 Between-imputation covariance BIO656--Multilevel Models 42 ' 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 Term 4, 2006 BIO656--Multilevel Models 43 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, ... Term 4, 2006 BIO656--Multilevel Models 44 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., Pr(Di k | Di k, Yi1,, Yik ) log θ1 θ2 Yik 1 Pr(Di k | Di k, Yi1,, Yik ) Term 4, 2006 BIO656--Multilevel Models 45 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 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) = ˆ 2 Yields ̂ and ̂ 2 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*Yi1+…+ k*Yik-1+ *ei, where ei is simulated from a standard normal distribution 3. Repeat 1 and 2 Term 4, 2006 BIO656--Multilevel Models 46 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 Term 4, 2006 BIO656--Multilevel Models 47 Missing data TURP n = 53 Laser n = 47 Total 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 Term 4, 2006 BIO656--Multilevel Models 48 Comparison of inferences Note that mean imputation understates uncertainty. Term 4, 2006 BIO656--Multilevel Models 49 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 Term 4, 2006 BIO656--Multilevel Models 50 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 Term 4, 2006 BIO656--Multilevel Models 51 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 Term 4, 2006 BIO656--Multilevel Models 52 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 Term 4, 2006 BIO656--Multilevel Models 53 • Differential attenuation across studies complicates “exporting” and synthesizing Term 4, 2006 BIO656--Multilevel Models 54 Term 4, 2006 BIO656--Multilevel Models 55 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 Term 4, 2006 BIO656--Multilevel Models 56 Hybrids are possible Xt and Xo have a general joint distribution Term 4, 2006 BIO656--Multilevel Models 57 Measurement error’s effect on 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 Term 4, 2006 BIO656--Multilevel Models 58 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 Term 4, 2006 BIO656--Multilevel Models 59 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 Term 4, 2006 BIO656--Multilevel Models 60 Y versus Xt Term 4, 2006 BIO656--Multilevel Models 61 Y versus X0 Term 4, 2006 BIO656--Multilevel Models 62 An illustration Back to the basic example • W = Weight (lb) • H = Height (cm) • Analysis: simple linear regression Wi = 0 + 1 Hi+ ei where ei ~ N(0, 2 Assume the true model to be: Wi = 3 + 1.0Hi+ ei where ei ~ N(0, 82 Measurement error 1. Error in W: observe W* = W + ei* where ei ~ N(0, 42 2. Error in H : observe H* = H + i* where i* ~ N(0, 102 Term 4, 2006 BIO656--Multilevel Models 63 Scenario 1: Measurement Error in Response 100 Scatterplot: Weight vs Height Results: 90 1 = 1.16 80 SE(1)= 0.15 70 Weight (lb) No error With error 60 1 = 1.08 50 SE(1) = 0.18 60 70 80 90 Height (cm) • Standard regression estimate for 1 is unbiased, but less efficient • The larger is the measurement error, the greater the loss in efficiency Term 4, 2006 BIO656--Multilevel Models 64 Scenario 2: measurement error in H 100 Scatterplot: Weight vs Height Results: 90 80 SE(1)= 0.15 70 1 = 1.16 60 1 = 0.69 SE(1)= 0.21 50 Weight (lb) No error With error 50 60 70 80 90 100 Height (cm) • Standard regression estimate for 1 is biased (attenuated) • The larger is the measurement error, the greater the attenuation Term 4, 2006 BIO656--Multilevel Models 65 Multivariate Measurement Error Xo = Xt + , (0, ) Term 4, 2006 BIO656--Multilevel Models 66 Term 4, 2006 BIO656--Multilevel Models 67 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 <options>; BY variables; VAR variables; Term 4, 2006 BIO656--Multilevel Models 68 Multiple Imputation Algorithm in SAS • • • PROC MI <options>; BY variables; FREQ variable; 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 Term 4, 2006 BIO656--Multilevel Models 69 Term 4, 2006 BIO656--Multilevel Models 70