1 MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group
2 ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could arise when these techniques are used 3. Implementation of SAS Proc MI procedure Assuming MVN Assuming FCS 4. Imputation Diagnostics
3 GOALS OF STATISTICAL ANALYSIS WITH MISSING DATA Minimize bias Maximize use of available information Obtain appropriate estimates of uncertainty
4 THE MISSING DATA MECHANISM DESCRIBES THE PROCESS THAT IS BELIEVED TO HAVE GENERATED THE MISSING VALUES. 1. Missing completely at random (MCAR) Neither the unobserved values of the variable with missing nor the other variables in the dataset predict whether a value will be missing. Example: Planned missingness 2. Missing at random (MAR) Other variables (but not the variable with missing itself) in the dataset can be used to predict missingness. Example: Men may be more likely to decline to answer some questions than women 3. Missing not at random (MNAR) The unobserved value of the variable with missing predicts missingness. Example: Individuals with very high incomes are more likely to decline to answer questions about their own income
5 OUR DATA High School and Beyond N= Variables Student Demographics and Achievement including test scores
6 ANALYSIS OF FULL DATA
7 COMMON TECHNIQUES FOR DEALING WITH MISSING DATA 1. Complete case analysis (listwise deletion) 2. Mean Imputation 3. Single Imputation 4. Stochastic Imputation
8 Method: Drop cases with missing data on any variable of interest Appeal: Nothing to implement default method Drawbacks: COMPLETE CASE ANALYSIS (LISTWISE DELETION) Loss of cases/data Biased estimates unless MCAR
9 COMPLETE CASE ANALYSIS (LISTWISE DELETION) proc means data = ats.hsb_mar nmiss N min max mean std; var _numeric_ ; run;
10 LISTWISE DELETION ANALYSIS DROPS OBSERVATIONS WITH MISSING VALUES
11 UNCONDITIONAL MEAN IMPUTATION Method: Replace missing values for a variable with its overall estimated mean Appeal: Simple and easily implemented Drawbacks: Artificial reduction in variability b/c imputing values at the mean. Changes the magnitude of correlations between the imputed variables and other variables.
12 MEAN, SD AND CORRELATION MATRIX OF 5 VARIABLES BEFORE & AFTER MEAN IMPUTATION
13 SINGLE OR DETERMINISTIC (REGRESSION) IMPUTATION Method: Replace missing values with predicted scores from a regression equation. Appeal: Uses complete information to impute values. Drawback: All predicted values fall directly on the regression line, decreasing variability. Also known as regression imputation
14 SINGLE OR DETERMINISTIC (REGRESSION) IMPUTATION p.46, Applied Missing Data Analysis, Craig Enders (2010)
15 SINGLE OR DETERMINISTIC (REGRESSION) IMPUTATION Imputing values directly on the regression line: Underestimates uncertainty Inflates associations between variables because it imputes perfectly correlated values Upwardly biases R-squared statistics, even under the assumption of MCAR
16 STOCHASTIC IMPUTATION Stochastic imputation addresses these problems with regression imputation by incorporating or "adding back" lost variability. Method: Add randomly drawn residual to imputed value from regression imputation. Distribution of residuals based on residual variance from regression model.
18 STOCHASTIC IMPUTATION Appeals: Restores some lost variability. Superior to the previous methods as it will produce unbiased coefficient estimates under MAR. Drawback: SE s produced during stochastic estimation, while less biased, will still be attenuated.
19 WHAT IS MULTIPLE IMPUTATION? Iterative form of stochastic imputation. Multiple values are imputed rather than a single value to reflect the uncertainty around the true value. Each imputed value includes a random component whose magnitude reflects the extent to which other variables in the model cannot predict it's true value Common misconception: imputed values should represent "real" values. Purpose: To correctly reproduce the full data variance/covariance matrix
20 ISN'T MULTIPLE IMPUTATION JUST MAKING UP DATA? No. This is argument applies to single imputation methods MI analysis methods account for the uncertainty/error associated with the imputed values. Estimated parameters never depend on a single value.
21 THREE PHASES 1. Imputation or Fill-in Phase: Missing values are imputed, forming a complete data set. This process is repeated m times. 2. Analysis Phase: Each of the m complete data sets is then analyzed using a statistical model (e.g linear regression). 3. Pooling Phase: The parameter estimates (e.g. coefficients and standard errors) obtained from each analyzed data set are then combined for inference.
22 THE IMPORTANCE OF BEING COMPATIBLE The imputation model should be "congenial to or consistent with your analytic model: Includes, at the very least, the same variables as the analytic model. Includes any transformations to variables in the analytic model E.g. logarithmic and squaring transformations, interaction terms All relationships between variables should be represented and estimated simultaneously. Otherwise, you are imputing values assuming they are uncorrelated with the variables you did not include.
23 PREPARING FOR MULTIPLE IMPUTATION 1. Examine the number and proportion of missing values among your variables of interest. 2. Examine Missing Data Patterns among your variables of interest. 3. If necessary, identify potential auxiliary variables 4. Determine imputation method
24 EXAMINE MISSING VALUES: PROC MEANS NMISS OPTION
25 EXAMINE MISSING VALUES: NOTE VARIABLE(S ) WITH HIGH PROPORTION OF MISSING -- THEY WILL IMPACT MODEL CONVERGENCE THE MOST
26 EXAMINE MISSING DATA PATTERNS: SYNTAX proc mi data=hsb_mar nimpute=0 ; var write read female math prog ; ods select misspattern; run;
27 EXAMINE MISSING DATA PATTERNS
28 IDENTIFY POTENTIAL AUXILIARY VARIABLES Characteristics: Correlated with missing variable (rule of thumb: r> 0.4) Predictor of missingness Not of analytic interest, so only used in imputation model Why? Including auxiliary variables in the imputation model can: Improve the quality of imputed values Increase power, especially with high fraction of missing information (FMI >25%) Be especially important when imputing DV Increase plausibility of MAR
29 HOW DO YOU IDENTIFY AUXILIARY VARIABLES? A priori knowledge Previous literature Identify associations in data
30 AUXILIARY VARIABLES ARE CORRELATED WITH MISSING VARIABLE
31 AUXILIARY VARIABLES ARE PREDICTORS OF MISSINGNESS
32 IMPUTATION MODEL EXAMPLE 1: MI USING MULTIVARIATE NORMAL DISTRIBUTION (MVN)
33 ASSUMING A JOINT MULTIVARIATE NORMAL DISTRIBUTION Probably the most common parametric approach for multiple imputation. Assumes variables are individually and jointly normally distributed Assuming a MVN distribution is robust to violations of normality given a large enough N. Uses the data augmentation (DA) algorithm to impute. Biased estimates may result when N is relatively small and the fraction of missing information is high.
34 IMPUTATION PHASE proc mi data= new nimpute=10 out=mi_mvn seed=54321; var socst science write read female math progcat1 progcat2 ; run;
35 MULTIPLY IMPUTED DATASET
36 ANALYSIS PHASE: ESTIMATE MODEL FOR EACH IMPUTED DATASET proc glm data = mi_mvn ; model read = write female math progcat1 progcat2; by _imputation_; ods output ParameterEstimates=a_mvn; run; quit;
37 PARAMETER ESTIMATE DATASET
38 proc mianalyze parms=a_mvn; modeleffects intercept write female math run; POOLING PHASE- COMBINING PARAMETER ESTIMATES ACROSS DATASETS progcat1 progcat2;
40 COMPARE MIANALYZE ESTIMATES TO ANALYSIS WITH FULL DATA FULL DATA ANALYSIS MIANALYZE OUPUT
41 HOW DOES PROC MIANALYZE WORK PROC MIANALYZE combines results across imputations Regression coefficients are averaged across imputations Standard errors incorporate uncertainty from 2 sources: "within imputation" - variability in estimate expected with no missing data The usual uncertainty regarding a regression coefficient "between imputation" - variability due to missing information. The uncertainty surrounding missing values
42 OUTPUT FROM MIANALYZE
43 VARIANCE WITHIN Sampling variability expected with no missing data. Average of variability of coefficients within an imputation Equal to arithmetic mean of sampling variances (SE 2 ) Example: Add together 10 estimated SE 2 for write and divide by 10 V w =
44 VARIANCE BETWEEN Variability in estimates across imputations i.e. the variance of the m coefficients Estimates the additional variation (uncertainty) that results from missing data. Example: take all 10 of the parameter estimates (β) for write and calculate the variance V B =
45 TOTAL VARIANCE The total variance is sum of 3 sources of variance. Within, Between Additional source of sampling variance. What is the sampling variance? Variance Between divided by number of imputations Represents sampling error associated with the overall coefficient estimates. Serves as a correction factor for using a specific number of imputations.
46 DEGREES OF FREEDOM DF for combined results are determined by the number of imputations. *By default DF = infinity, typically not a problem with large N but can be with smaller samples The standard formula to estimate DF can yield estimates that are fractional or that far exceed the DF for complete data. Correction to adjust for the problem of inflated DF has been implemented Use the EDF option on the proc mianalyze line to indicate to SAS what is the proper adjusted DF.
47 RELATIVE INCREASES IN VARIANCE (RVI) Proportional increase in total sampling variance due to missing information [V B + V B /m] V w For example, the RVI for write coefficient is 0.048, meaning that the sampling variance is 4.8% larger than it would have been with complete data.
48 FRACTION OF MISSING INFORMATION (FMI) Directly related to RVI. Proportion of total sampling variance that is due to missing data [V B + V B /m] V T For a given variable, FMI based on percentage missing and correlations with other imputation model variables. Interpretation similar to an R 2. Example: FMI=.046 for write means that 4.6% of sampling variance is attributable to missing data.
49 RELATIVE EFFICIENCY: IS 5 IMPUTATIONS ENOUGH? Captures how well true population parameters are estimated Related to both the FMI and m Low FMI + few m = high efficiency As FMI increase so should m: Better statistical power and more stable estimate
50 DIAGN0STICS: HOW DO I KNOW IF IT WORKED? Compare means and frequencies of observed and imputed values. Use boxplots to compare distributions Look at Variance Information tables from the proc mianalyze output Plots - Assess convergence of DA algorithm
51 TRACE PLOTS: DID MY IMPUTATION MODEL CONVERGE? Convergence for each imputed variable can be assessed using trace plots. Examine for each imputed variables Special attention to variables with a high FMI proc mi data= ats.hsb_mar nimpute=10 out=mi_mvn; mcmc plots=trace plots=acf ; var socst write read female math; run;
53 EXAMPLE OF A POOR TRACE PLOT
54 AUTOCORRELATION PLOTS: DID MY IMPUTATION MODEL CONVERGE? Assess possible auto correlation of parameter values between iterations. Assess the magnitude of the observed dependency of imputed values across iterations. proc mi data= ats.hsb_mar nimpute=10 out=mi_mvn; mcmc plots=trace plots=acf ; var socst write read female math; run;
56 IMPUTATION MODEL EXAMPLE 2: MI USING FULLY CONDITIONAL SPECIFICATION (FCS)
57 WHAT IF I DON T WANT TO ASSUME A MULTIVARIATE NORMAL DISTRIBUTION? Alternative method for imputation is Fully Conditional Method (FCS) FCS does not assume a joint distribution and allows the use of different distributions across variables. Each variable with missing is allowed its own type of regression (linear, logistic, etc) for imputation Example uses: Logistic model for binary outcome Poisson model for count variable Other bounded values
58 AVAILABLE DISTRIBUTIONS FCS methods available: Discriminant function or logistic regression for binary/categorical variables Linear regression and predictive mean matching for continuous variables. Properties to Note: 1. Discriminant function only continuous vars as covariates (default). 2. Logistic regression assumes ordering of class variables if more then two levels (default). 3. Regression is default imputation method for continuous vars. 4. PMM will provide plausible values. 1. For an observation missing on X, finds cases in data with similar values on other covariates, then randomly selects an X value from those cases
59 IMPUTATION PHASE proc mi data= ats.hsb_mar nimpute=20 out=mi_fcs ; class female prog; fcs plots=trace(mean std); var socst write read female math science prog; fcs discrim(female prog /classeffects=include) nbiter =100 ; run;
60 ALTERNATE EXAMPLE proc mi data= ats.hsb_mar nimpute=20 out=mi_new1; class female prog; var socst write read female math science prog; fcs logistic (female= socst science) ; fcs logistic (prog =math socst /link=glogit) regpmm(math read write); run;
61 ANALYSIS PHASE: ESTIMATE GLM MODEL USING EACH IMPUTED DATASET proc genmod data=mi_fcs; class female prog; model read=write female math prog/dist=normal; by _imputation_; ods output ParameterEstimates=gm_fcs; run;
63 POOLING PHASE- COMBINING PARAMETER ESTIMATES ACROSS DATASETS PROC MIANALYZE parms(classvar=level)=gm_fcs; class female prog; MODELEFFECTS INTERCEPT write female math prog; RUN;
64 TRACE PLOTS: DID MY IMPUTATION MODEL CONVERGE?
65 FCS HAS SEVERAL PROPERTIES THAT MAKE IT AN ATTRACTIVE ALTERNATIVE 1. FCS allows each variable to be imputed using its own conditional distribution 2. Different imputation models can be specified for different variables. However, this can also cause estimation problems. Beware: Convergence Issues such as complete and quasi-complete separation (e.g. zero cells) when imputing categorical variables.
66 BOTTOM LINE MI improves over single imputation methods because: Single value never used Appropriate estimates of uncertainty The nature of your data and model will determine if you choose MVN or FCS Both are state of the art methods for handling missing data Produce unbiased estimates assuming MAR
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