practical imputation of missing data

Dealing with Missing Survey Data

A Short Course in Practical Applications of Imputation

Mansour Fahimi, Ph.D.VP, Statistical Research Services

American Association for Public Opinion Research (AAPOR)Phoenix, Arizona

May 11, 2011

Darryl CreelResearch Statistician

Course Outline

Sources of Missing Data

Why Imputation?

Non-stochastic Methods

Stochastic Methods

Special Topics and Software Packages:

Weighted Sequential Hot-Deck Imputation

Multiple Imputation

Multivariate (Vector) Imputation

Imputation Diagnostics2

Course Structure

Introduction & Background:

2:30 – 3:30

Non-Stochastic and Stochastic Methods:

3:45 – 4:45

Special Topics and Software Packages:

5:00 – 6:00

3

Introduction & Background

We are all here for a spell;get all the good laughs you can.

Will Rogers

4

Sources of Missing Survey Data

Item Nonresponse:

Refusal

Don’t know

Inconsistent data due to respondent or processing errors

Partial Nonresponse:

Excessive item nonresponse

Improper application of skips

Nonresponse at a later level in a multi-stage design

Unit (Total) Nonresponse:

Refusal

Failure to contact or cooperate

Undercoverage5

Handling Missing Data

Ignore the problem and live with its consequences:

Concerns for technical and time resources

Concerns for “messing up” survey data

Use standard methods to handle missing data:

Imputation:

Item nonresponse

Partial nonresponse

Weighting:

Total nonresponse

Undercoverage6

Pros and Cons of Imputation

Pros:

Simplifies presentation

Eliminates database attrition

Reduces bias due to differential item nonresponse

Removes analytical inconsistencies across software

Allows more effective weighting options

Cons:

Can generate inconsistent data

Can distort relationship among items

Can artificially reduce variance of survey estimates

7

Ignoring the ProblemAwkward Presentations

7%8%

9%

12%

25%

18%

6%

15%

0K to 10K 11K to 19K 20K to 29K 30K to 49K 50K to 80K 80K to 120K Over 120K Missing

Distribution of Household Income

8

Ignoring the ProblemCan Bias Results

9

ID Will Buy Gender Income ID Will Buy Gender Income ID Will Buy Gender Income ID Will Buy Gender Income

1 No Female 61 26 Yes Female 65 51 Yes Male 31 76 No Female 32

2 Yes Female 32 27 No Male 19 52 Yes Female 61 77 No Male 56

3 No Male 38 28 Yes Female 74 53 No Female 31 78 No Male 55

4 No Male 42 29 No Male 36 54 Yes Female 41 79 No Female 34

5 No Male 37 30 No Male 74 55 Yes Female 48 80 No Male 50

6 Yes Female 39 31 No Male 55 56 Yes Female 34 81 Yes Female 25


8 Yes Female 54 33 No Male 50 58 No Female 44 83 Yes Female 49

9 Yes Female 61 34 No Male 38 59 No Male 35 84 No Female 67

10 Yes Female 55 35 No Male 52 60 No Female 67 85 No Male 58


12 No Male 44 37 Yes Female 91 62 No Male 45 87 Yes Male 95

13 Yes Female 50 38 No Male 20 63 No Female 76 88 No Male 80

14 No Male 24 39 Yes Male 45 64 Yes Female 99 89 No Male 90

15 Yes Female 99 40 No Female 39 65 Yes Male 57 90 Yes Female 83

16 No Male 45 41 Yes Female 60 66 No Female 34 91 No Female 45

17 Yes Female 11 42 No Female 29 67 No Female 99 92 No Female 19

18 Yes Male 53 43 Yes Female 55 68 No Male 21 93 No Male 44

19 Yes Female 26 44 No Female 35 69 No Male 79 94 No Female 26

20 No Male 38 45 Yes Male 73 70 No Male 60 95 Yes Female 65

21 No Male 86 46 Yes Male 70 71 No Male 98 96 No Female 37

22 No Male 82 47 Yes Male 85 72 No Male 65 97 No Female 68

23 No Male 73 48 Yes Male 24 73 Yes Female 82 98 No Female 24

24 No Male 49 49 Yes Male 40 74 Yes Female 81 99 Yes Male 77

25 No Male 45 50 Yes Female 27 75 No Male 62 100 No Female 75


The LOGISTIC Procedure

Model Information

Number of Observations Read 100

Number of Observations Used 100

Response Profile

Ordered Total

Value Will Buy Frequency

1 Yes 41

2 No 59

Class Level Information

Design

Class Value Variables

Gender Female 1

Male -1

10


Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -1.3692 0.6147 4.9618 0.0259

Gender Female 1 0.8757 0.2303 14.4644 0.0001

Income 1 0.0173 0.0104 2.7590 0.0967

Odds Ratio Estimates

Point 95% Wald

Effect Estimate Confidence Limits

Gender Female vs. Male 5.763 2.337 14.211

Income 1.017 0.997 1.038

Association of Predicted Probabilities and Observed Responses

_________________________________________________

Percent Concordant 73.8 Somers' D 0.482

Percent Discordant 25.6 Gamma 0.484

Percent Tied 0.6 Tau-a 0.235

Pairs 2419 c 0.741

11



Number of Observations Read 100

Number of Observations Used 86

Response Profile

Ordered Total

Value Will Buy Frequency

1 Yes 32

2 No 54

NOTE: 14 observations were deleted due to missing values for the response

or explanatory variables.


Design

Class Value Variables

Gender Female 1

Male -1

12



Standard Wald


Intercept 1 -1.5947 0.6523 5.9766 0.0145

Gender Female 1 0.6280 0.2407 6.8083 0.0091

Income 1 0.0191 0.0108 3.1273 0.0770


Point 95% Wald



Income 1.019 0.998 1.041

Association of Predicted Probabilities and Observed Responses

_________________________________________________

Percent Concordant 69.2 Somers' D 0.391

Percent Discordant 30.2 Gamma 0.393

Percent Tied 0.6 Tau-a 0.185

Pairs 1728 c 0.695

13

ImputationWhen

Apply imputation only after:

Data cleaning is complete

Edit and range checks have been applied

Skip patterns have been observed

Missing rates are reasonably low

Covariates are available with observed values

When items are needed for analysis and weighting

14

ImputationHow

Non-stochastic imputation:

Logical

Cold-deck

Stochastic (statistical) imputation:

Hot-deck

Regression-based

Multiple

Bootstrap

Multivariate (Vector)15

Non-stochastic ImputationRetrieval of True Value Without Estimation

Logical imputation – Deducing true values based on:

Other observed values

Questionnaire logic and skip patterns

Cold-deck imputation – Obtaining “true” values via:

Administrative records

Prior surveys

16

Stochastic ImputationEstimation/Guestimation of Missing Values

Hot-deck imputation:

Cell Mode/Mean

Sequential

Random

Proportional

Regression-based imputation:

Predicted Value Substitution

Predicted Value Matching

Random Residuals

Multiple and multivariate imputation17

Stochastic ImputationCell Mode/Mean

GenderCups of Coffee per Day Income

Before After Before After

Male

1 1 45 45

2 2 40 40

. 1 38 38

1 1 54 54

1 1 . 44.25

Female

2 2 60 60

3 3 72 72

1 1 58 58

2 2 . 67.5

. 2 80 80

18

Stochastic ImputationSequential Imputation

GenderCups of Coffee per Day Income


Male

1 1 45 45

2 2 40 40

. 2 38 38

1 1 54 54

1 1 . 54

Female

2 2 60 60

3 3 72 72

1 1 58 58

2 2 . 58

. 2 80 80

19

Stochastic ImputationRandom Imputation

Gender RandomCups of Coffee per Day Income


Male

4 1 1 45 45

2 2 2 40 40

1 . 1 38 38

2 1 1 54 54

2 1 1 . 40

Female

3 2 2 60 60

4 3 3 72 72

3 1 1 58 58

1 2 2 . 60

2 . 3 80 80

20

Stochastic ImputationProportional Imputation

Gender Cups of Coffee Frequency Distribution CDF

Male1 3 75% 75%

2 1 25% 100%

Female

1 1 25% 25%

2 2 50% 75%

3 1 25% 100%

Gender RandomCups of Coffee per Day

Before After

Male

37.2% 1 1

79.4% 2 1

73.4% . 1

34.5% 1 2

60.9% 1 1

Female

31.1% 2 2

99.6% 3 3

70.5% 1 1

74.7% 2 2

47.4% . 2

21

Stochastic ImputationRegression-Based Imputation

ID Age Income

1 26 45

2 28 48

3 32 .

4 35 54

5 38 .

6 41 60

7 44 72

8 47 78

9 49 .

10 52 84

Regression Statistics

Number of Observations 10

Observations Used 7

R Square 0.95

Adjusted R Square 0.95

Standard Error 3.54

ANOVA DF SS MS F P-Value

Regression 1 1323.4 1323.4 105.7 0.000

Residual 5 62.6 12.5

Total 6 1386

Parameter Estimate t Stat P-value LCB UCB

Intercept 3.47 0.58 0.58 -11.81 18.75

Age 1.53 10.28 0.00 1.15 1.91

Income = 3.47 + 1.53 Age

22

Stochastic ImputationRegression-Based Imputation

23

Regression-Based ImputationPredicted Value Substitution

Age Income Imputed Income

26 45 45

28 48 48

32 . 3.47 + 1.53 32 = 52.31

35 54 54

38 . 3.47 + 1.53 38 = 61.61

41 60 60

44 72 72

47 78 78

49 . 3.47 + 1.53 49 = 78.44

52 84 84

24

Regression-Based ImputationPredicted Value Matching

Age Income Predicted Income Imputed Income

26 45 3.47 + 1.53 26 = 43.25 45

28 48 46.31 48

32 . 52.43 54

35 54 57.02 54

38 . 61.61 54

41 60 66.2 60

44 72 70.79 60

47 78 75.38 78

49 . 78.44 78

52 84 83.03 84

25

Regression-Based ImputationRandom Residual

Age Income Predicted Income Random Imputed Income

26 45 3.47+1.53 26 = 43.25 45

28 48 46.31 48

32 . 52.43 1.3 52.43+1.3 = 53.73

35 54 57.02 54

38 . 61.61 -1.7 61.61-1.7 = 59.91

41 60 66.2 60

44 72 70.79 72

47 78 75.38 78

49 . 78.44 0.05 78.44+0.05 = 78.49

52 84 83.03 84

26

Stochastic ImputationRegression-Based Imputation for Discrete Variables

Binary Variables: Yes/No

Logistic Regression

Nominal Variables: White/Black/Other

Generalized Logit Model

Ordinal Variable: Low/Medium/High

Proportional Odds Model

27



Probability modeled is Gender='Male'


Class Value Design Variables

Will_Buy No 1

Yes -1



Intercept 1 -0.0580 0.233 0.0617 0.8038

Will_Buy No 1 0.5886 0.233 6.3585 0.0117

28


If P(Male) 0.5 Then Gender = Male

If P(Male) < 0.5 Then Gender = Female

Observed

Predicted

Male Female

Male 45 45

Female 41 41

Missing 14 5 9

Total 86 50 50

29

Stochastic ImputationRegression-Tree Imputation for Discrete Variables

WILL_BUY$ = (Yes)

Class = Female

Class Cases %

Female 21 65.6

Male 11 34.4

N = 32

INCOME <= 36

Class = Female

Class Cases %

Female 9 56.3

Male 7 43.8

N = 16

INCOME > 36

Class = Male

Class Cases %

Female 11 28.9

Male 27 71.1

N = 38

WILL_BUY$ = (No)

Class Cases %

Female 20 37.0

Male 34 63.0

Class Cases %

Female 41 47.7

Male 45 52.3

30

Stochastic ImputationRegression-Tree Imputation for Discrete Variables

Will Buy

66%

Income

34%

Relative Importance of Predictors

Prediction Success

Observed

Predicted

Male Female Correct

Male 45 27 18 60.0%

Female 41 11 30 73.2%

Total 86 38 48 66.3%

31

Stochastic Cold-Deck

District School

Students Before Students After

Male Female Total Male Female Total

A

1 137 59 196 137 59 196

2 82 62 144 82 62 144

3 . . . 98.5 55.5 154

4 111 60 171 111 60 171

5 64 41 105 64 41 105

Subtotal 394 222 616 492.5 277.5 770

B

1 84 74 158 84 74 158

2 121 64 185 121 64 185

3 . 54 54 91 54 145

4 67 . 67 67 64 131

Subtotal 272 192 464 363 256 619

32

Special Topics and Software Packages

Everything should be made as simple as possible, but not simpler.

Einstein

33

Special Topics and Software Packages

Weighted Sequential Hot-Deck

Multiple Imputation:

SAS: Proc MI Proc MIANALYZE

R: MICE POOL

Approximate Bayesian Bootstrap

Multivariate (Vector) Imputation

34

Types of Missing Data

Missing Completely at Random (MCAR): outcome variable is random subsample of the sampled units

Missing at Random (MAR): Missingness for the outcome variable depends on covariates

Oftentimes in practice:

Cannot assume MCAR at the overall level

May assume MAR within subgroups created from covariates

This assumes MCAR within subgroups (imputation cells)

35

Patterns of MissingnessMonotone Missing Data Pattern

Var1 Var2 Var3 Var4

X X X X

X X X X

X X X .

X X . .

X X . .

X X . .

X . . .

x . . .

36

Patterns of MissingnessSwiss Cheese Missing Data Pattern

Var1 Var2 Var3 Var4

X . X X

. X X X

X . X .

. X . X

X X X X

X X . .

X . X .

X . . X

37

Population StatisticsFull Data Without Missing Values


Point 95% Wald



Income 1.017 0.997 1.038

---------------------------------------------------------------

Cumulative Cumulative

Gender Frequency Percent Frequency Percent

Female 50 50.00 50 50.00

Male 50 50.00 100 100.00

38

ScenarioFemale vs. Male

(OR)Female vs. Male

(CI OR)Income

(OR)Income (CI OR)

Population 5.76 N/A 1.017 N/A

Complete Case Analysis


Point 95% Wald



Income 1.019 0.998 1.041

39

GenderImputed

Missing Male Female Total

Known

Male 9 41 0 50

Female 5 0 45 50

Total 14 41 45 100

Complete Case Analysis


(OR)Female vs. Male

(CI OR)Income

(OR)Income (CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

40

Most ProbableMissing Gender Set to Male

41

GenderImputed

Male Female Total

Known

Male 50 0 50

Female 9 41 50

Total 59 41 100


Point 95% Wald


genderMiss_male Female vs. Male 2.091 0.915 4.778

Income 1.014 0.995 1.033

Most ProbableMissing Gender Set to Male


(OR)Female vs. Male

(CI OR)Income

(OR)Income (CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

42

ProportionalMissing Gender Set to Mode

43

GenderImputed

Male Female Total

Known

Male 44 6 50

Female 3 47 50

Total 47 53 100


Point 95% Wald


genderMiss_prop Female vs. Male 2.382 1.037 5.472

Income 1.016 0.996 1.035

ProportionalMissing Gender Set to Mode


(OR)Female vs. Male

(CI OR)Income

(OR)Income (CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

44

Weighted Sequential Hot-Deck ImputationProc HOTDECK in SUDAAN

Weighted Sequential Hot Deck (WSHD)

Imputation:

Weighted: Weights used in selecting donors

Sequential: Order matters in selecting donors

Hot Deck: Uses observe values as donors

Weighted Sequential Hot Deck references - Cox

(1980), Iannacchione (1982), and SUDAAN (2009)

45

Proc HOTDECK in SUDAAN

What is the WSHD algorithm doing?

nr is the number of item respondents (5)

wh is the sample weight for the hth respondent

nm is the number of item nonrespondents (3)

si is the scaled weight for the ith nonrespondent

46

w1 w

2 w3 w

5w

4

s1 s

2s3

Proc HOTDECK in SUDAANGeneral Steps

Identify:

Imputation variables (IMPVAR)

Imputation classes (IMPBY)

Item respondents (donors)

Item nonrespondents (recipients)

Imputation revised output variable names (IMPNAME)

Performs:

Selection algorithm to pick donor within imputation class

Assign value from donor to recipient

Create output dataset with imputed values (OUTPUT)47

Proc HOTDECK in SUDAANSyntax

Proc Hotdeck data = withMissing;

weight _one_;

impvar genderMiss_num;

impby _one_;

impid id;

output / impute = all

filename = wshd_ran_imp

replace;

run;

48

Proc HOTDECK in SUDAANRandom

49

GenderImputed

Male Female Total

Known

Male 45 5 50

Female 2 48 50

Total 47 53 100


Point 95% Wald


genderMiss_wshd_ran 1 vs. 2 2.678 1.167 6.146

Income 1.014 0.995 1.034

Proc HOTDECK in SUDAANRandom


(OR)Female vs. Male

(CI OR)Income

(OR)Income (CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

50

Proc HOTDECK in SUDAANSorted by Income

51

GenderImputed

Male Female Total

Known

Male 45 5 50

Female 2 48 50

Total 47 53 100


Point 95% Wald


genderMiss_wshd_inc 1 vs. 2 2.783 1.205 6.426

Income 1.015 0.996 1.035

Proc HOTDECK in SUDAANSorted by Income


(OR)Female vs. Male

(CI OR)Income

(OR)Income (CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

WSHD Income 2.78 (1.205, 6.426) 1.015 (0.996, 1.035)

52

Multiple Imputation (MI)

Single imputation may not reflect the uncertainty due to the

imputation but MI can incorporate the imputation

uncertainty in standard errors by:

Repeating the imputation process several times

Estimating the variance inflation from multiple imputations

Why is MI not used more widely?

Lack of knowledge

Lack of software

Lack of infrastructure

Time versus quality

Reference for multiple imputation (Rubin 1987)53

Multiple Imputation Process with SAS and RThree Rounds of Imputations

54

Multiple Imputation Estimates

The MI point estimate is the average of the

individual point estimates:

55


The MI variance estimate has two components

within and between components:

The within component is the average of the individual

variance estimates from each imputation

The between component is the variance between the

individual point estimates from each imputation

The two components are combined to produce the

total MI variance

56


57


Within and Between Variance Component Calculations

ImputationPoint

Estimate

Squared

Deviation

Variance

Estimates

1 0.799 0.0372 0.052

2 0.710 0.0108 0.049

3 0.499 0.0114 0.045

4 0.596 0.0001 0.048

5 0.426 0.0324 0.045

Total 3.030 0.0920 0.239

Average 0.606 0.0230 0.0478

58


59

SAS Proc MISyntax

Proc MI is the SAS procedure (SAS 2009) for

multiple imputation:

Proc MI data = withMissing seed = 92358792 out =

genderMiss_sasmi;

class genderMiss will_buy;

monotone logistic ( genderMiss = will_buy

income /details );

var will_buy income genderMiss;

run;

Creates a stacked data set (indexed by _imputation_)

60

SAS Proc MISelected Output

Model Information

Data Set WORK.WITHMISSING

Method Monotone

Number of Imputations 5

Seed for random number generator 92358792

Monotone Model Specification

Imputed

Method Variables

Logistic Regression genderMiss

Missing Data Patterns

gender -Group Means-

Group Will_Buy Income Miss Freq Percent Income

1 X X X 86 86.00 54.232558

2 X X . 14 14.00 50.214286

61

SAS Proc LOGISTIC

Proc Logistic performs the logistic regression by

the separate imputation data sets:

Proc Logistic Data = genderMiss_sasmi descending

covout outest = sasmi noPrint;

class GenderMiss;

model Will_Buy = GenderMiss Income;

by _imputation_;

run;

Creates the parameter estimates for each imputed

dataset

62

SAS Proc Logistic

_Imputation_ _TYPE_ _NAME_ Intercept Female Income

1 PARMS Will_Buy -1.428 0.799 0.019

1 COV Intercept 0.376 -0.036 -0.006

1 COV genderMissFemale -0.036 0.052 0.001

1 COV Income -0.006 0.001 0.000

2 PARMS Will_Buy -1.258 0.710 0.016

2 COV Intercept 0.353 -0.022 -0.006


2 COV Income -0.006 0.000 0.000

3 PARMS Will_Buy -1.138 0.499 0.014

3 COV Intercept 0.331 -0.011 -0.005


3 COV Income -0.005 0.000 0.000

4 PARMS Will_Buy -1.290 0.596 0.017

4 COV Intercept 0.346 -0.025 -0.005


4 COV Income -0.005 0.000 0.000

5 PARMS Will_Buy -1.172 0.426 0.015

5 COV Intercept 0.328 -0.014 -0.005


5 COV Income -0.005 0.000 0.000

63

SAS Proc MIANALYZESyntax

Proc MIANALYZE combines the information from

different imputations to produce the MI estimates:

Proc mianalyze data = sasmi;

modeleffects intercept gendermissFemale income;

run;

Creates the MI estimates, i.e., the combined

estimates

64

SAS Proc MIANALYZESelected Output

The MIANALYZE Procedure

Model Information

Data Set WORK.SASMI


Variance Information

Relative Fraction

-----------------Variance-------------- Increase Missing Relative

Parameter Between Within Total DF in Variance Information Efficiency

intercept 0.012920 0.346675 0.362179 2182.8 0.044722 0.043683 0.991339

gendermissFemale 0.023019 0.047655 0.075278 29.7 0.579652 0.405660 0.924956

income 0.000003 0.000010 0.000103 4567.8 0.030494 0.030017 0.994032

Parameter Estimates

t for H0:

Parameter Estimate Std Error 95% Confidence Limits DF Parameter=Theta0 Pr > |t|

intercept -1.257118 0.601813 -2.43730 -0.07693 2182.8 -2.09 0.0368

gendermissFemale 0.605981* 0.274369 0.04541 1.16655 29.7 2.21 0.0351

income 0.016126 0.010140 -0.00375 0.03601 4567.8 1.59 0.1118

*Note because of the cell coding this needs to be multiplied by 2 before it is transformed. OR = 3.36

65

SAS Proc MI


(OR)Female vs. Male

(CI OR)Income

(OR)Income(CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

WSHD Income 2.78 (1.205, 6.426) 1.015 (0.996, 1.035)

SAS Proc MI 3.36 (2.480, 10.310) 1.033 (0.993, 1.075)

66

Multiple Imputation Process with SAS and RThree Rounds of Imputations

67

Multiple Imputation in RMICE Syntax

Multivariate Imputation by Chained Equations

(MICE) is the R package (R Development Core

Team 2011)

The actual function call is mice():

imp <- mice ( mice.dat, method = "logreg" )

summary ( imp )

Creates the imputed data sets

68

Multiple Imputation in RMICE Output

Multiply imputed data set

Call: mice(data = mice.dat, method = "logreg")

Number of multiple imputations: 5

Missing cells per column: Will_Buy Income genderMissNA

0 0 14

Imputation methods:

Will_Buy Income genderMissNA

"logreg" "logreg" "logreg"

VisitSequence:

genderMissNA

3

PredictorMatrix:

Will_Buy Income genderMissNA

Will_Buy 0 0 0

Income 0 0 0

genderMissNA 1 1 0

Random generator seed value: NA

69

Multiple Imputation in RWITH Syntax

The function call with() evaluates an expression in

multiply imputed data sets. Here the expression is a

logistic regression:

fit <- with ( data = imp, exp = glm ( Will_Buy ~

genderMissNA+Income, family = binomial ))

summary ( fit )

Creates the parameter estimates for each imputation

70

Multiple Imputation in RWITH Output

Summary of imputation 1:

Call:

glm(formula = Will_Buy ~ genderMissNA + Income, family = binomial)

Deviance Residuals: Min 1Q Median 3Q Max

-1.4366 -1.0399 -0.8137 1.1971 1.6758

Coefficients: Estimate Std. Error z value Pr(>|z|)

(Intercept) -0.78055 0.58719 -1.329 0.184

genderMissNAMale -0.67439 0.41739 -1.616 0.106

Income 0.01386 0.00958 1.446 0.148

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 135.37 on 99 degrees of freedom

Residual deviance: 130.70 on 97 degrees of freedom

AIC: 136.7

Number of Fisher Scoring iterations: 4

71

Multiple Imputation in RPOOL Syntax

The function call pool() combines the results of the

multiple imputations:

miInfo <- pool ( fit )

summary ( miInfo )

Creates the MI estimates, i.e., combined estimates

72

Multiple Imputation in RPOOL Output

Estimate SE t Pr(>|t|)

(Intercept) -0.61304767 0.60336291 -1.016051 90.17776 0.31232328

genderMissNAMale*-1.20913640 0.56869594 -2.126156 16.39488 0.04900864

Income 0.01548992 0.01001359 1.546890 93.07291 0.12528334

Low 95 High 95 nmis fmi Lambda

(Intercept) -1.811701210 0.585605860 NA 0.05153238 0.03072758

genderMissNAMale*-2.412362324 -0.005910468 NA 0.47515975 0.41481552

Income -0.004394874 0.035374719 0 0.03593733 0.01544126

*Note that the base category has changed to male. The point estimate and

confidence interval estimates need to be multiplied by -1 to be consistent

with the other examples. The odds ratio for the point estimate is 3.35.

73

Multiple Imputation in R


(OR)Female vs. Male

(CI OR)Income

(OR)Income(CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

WSHD Income 2.78 (1.205, 6.426) 1.015 (0.996, 1.035)

SAS Proc MI 3.36 (2.480, 10.310) 1.033 (0.993, 1.075)

R MICE 3.35 (1.006, 11.160) 1.016 (0.996, 1.036)

74

Approximate Bayesian Bootstrap (ABB)

Multiple Imputation without explicit distributional

assumptions (Rubin and Schenker 1986)

Respondents: nr

Nonrespondents: nm

Total sample: n = nr + nm

For one round of imputation:

Select a sample WR of size nr from the respondents to create a donor pool

Select a sample WR of size nm as donors

For better results can create classes within which donors are

independently selected75

ABB

Data Set WORK.ABBMI



Relative Fraction

-----------------Variance----------------- Increase Missing Relative


intercept 0.030344 0.381395 0.417808 526.61 0.095474 0.090600 0.982202

gendermissFemale 0.000278 0.051936 0.052270 97972 0.006431 0.006410 0.998720

income 0.000004 0.000105 0.000110 1789.8 0.049620 0.048337 0.990425

Parameter Estimates

t for H0:

Parameter Estimate Std Error 95% Confidence Limits DF Parameter=Theta0 Pr > |t|

intercept -1.810986 0.646381 -3.08079 -0.54118 526.61 -2.80 0.0053

gendermissFemale 0.637589 0.228627 0.18948 1.08569 97972 2.79 0.0053

income 0.022036 0.010494 0.00145 0.04262 1789.8 2.10 0.0359

76

ABB


(OR)Female vs. Male

(CI OR)Income

(OR)Income(CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

WSHD Income 2.78 (1.205, 6.426) 1.015 (0.996, 1.035)

SAS Proc MI 3.36 (2.480, 10.310) 1.033 (0.993, 1.075)

R MICE 3.35 (1.006, 11.160) 1.016 (0.996, 1.036)

ABB 3.58 (1.461, 8.770) 1.045 (1.000, 1.089)

77

Tree, WSHD, and MI

Use the terminal nodes from the tree as imputation

classes, or define them deterministically

Make sure the imputation classes are not too small,

i.e., they have enough donors

Apply WSHD or ABB within the imputation classes

Make sure data set is appropriately configured for

the software to do the analysis

78

Tree, WSHD, and MI

WILL_BUY$ = (Yes)

Class = Female

Class Cases %

Female 21 65.6

Male 11 34.4

N = 32

INCOME <= 36

Class = Female

Class Cases %

Female 9 56.3

Male 7 43.8

N = 16

INCOME > 36

Class = Male

Class Cases %

Female 11 28.9

Male 27 71.1

N = 38

WILL_BUY$ = (No)

Class Cases %

Female 20 37.0

Male 34 63.0

Class Cases %

Female 41 47.7

Male 45 52.3

79

Tree, WSHD, and MI

Could use ABB but here we used WSHD as an example

Proc Hotdeck data = withMissing seed =646388;

weight _one_;

impvar genderMiss_num / multimp = 5;

impby mTreeImpClass;

impid id;

output / impute = all filename = wshd_mi replace;

Run;

80

Tree, WSHD, and MI

Alphabetic List of Variables and Attributes

# Variable Type Len Format Label

9 DONORID1 Num 8 8. Donor ID #1





2 ID Num 8 8. ID

4 IMPUTEVAL1_1 Num 8 Imputation #1 for Imputed variable #1





3 ORIGVAL Num 8 Orig Variable

1 PROCNUM Num 8 2. Procedure Number

15 _ONE_ Num 8 SUDAAN Reserved Variable One

14 mTreeImpClass Num 8 MTREEIMPCLASS

Imputed variables are added to the file. If you want to use PROC MI or some other

software, you need to configure the data set appropriately, e.g., stack the data set.

81

Tree, WSHD, and MI

Data Set WORK.WSHDMI



Relative Fraction

-----------------Variance----------------- Increase Missing Relative


intercept 0.001779 0.339011 0.341146 102153 0.006297 0.006277 0.998746

gendermiss_wshd_mi1 0.011362 0.046557 0.060192 77.956 0.292857 0.245628 0.953175

income 0.0000005 0.000098 0.000099 114624 0.005942 0.005925 0.998816

Parameter Estimates

t for H0:

Parameter Estimate Std Error 95% Confidence Limits DF Parameter=Theta=0 Pr > |t|

intercept -1.206797 0.584077 -2.35158 -0.06201 102153 -2.07 0.0388

gendermiss_wshd_mi1 0.570888 0.245340 0.08245 1.05933 77.956 2.33 0.0226

income 0.015578 0.009928 -0.00388 0.03504 114624 1.57 0.1166

82

Tree, WSHD, and MI


(OR)Female vs. Male

(CI OR)Income

(OR)Income(CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

WSHD Income 2.78 (1.205, 6.426) 1.015 (0.996, 1.035)

SAS Proc MI 3.36 (2.480, 10.310) 1.033 (0.993, 1.075)

R MICE 3.35 (1.006, 11.160) 1.016 (0.996, 1.036)

ABB 3.58 (1.461, 8.770) 1.045 (1.000, 1.089)

Tree/WSHD/MI 3.13 (1.179, 8.320) 1.032 (0.992, 1.073)

83

WSHD Vector Imputation

84

Obs ID Gender genderMiss Income

2 2 Female 32

8 8 Female 54

11 11 Female 41

16 16 Male 45

17 17 Female 11

24 24 Male 49

26 26 Female 65

31 31 Male 55

43 43 Female 55

55 55 Female 48

62 62 Male 45

64 64 Female 99

78 78 Male 55

83 83 Female 49


85

Obs ID Gender genderMiss Income incomeMiss

2 2 Female 32 32

8 8 Female 54 .

11 11 Female 41 41

16 16 Male 45 .

17 17 Female 11 11

24 24 Male 49 49

26 26 Female 65 65

31 31 Male 55 55

43 43 Female 55 55

55 55 Female 48 .

62 62 Male 45 .

64 64 Female 99 99

78 78 Male 55 55

83 83 Female 49 49


Proc Hotdeck data = vectorMissing;

weight _one_;

impvar genderMiss_num incomeMiss;

impby _one_;

impid id;

output / impute = all filename = wshd_vec_imp

replace;

run;

86


87

Obs ID ORIGVAL1 ORIGVAL2 IMPUTEVAL1 IMPUTEVAL2 DONORID

1 58 1 44 1 44 .

2 43 . 55 2 55 21

3 59 2 35 2 35 .

4 21 2 86 2 86 .

5 80 2 50 2 50 .

13 95 1 65 1 65 .

14 74 1 81 1 81 .

15 64 . 99 1 99 50

51 55 . . 2 44 12

52 4 2 42 2 42 .

53 99 2 77 2 77 .

54 12 2 44 2 44 .

55 27 2 19 2 19 .

56 16 . . 1 61 9

57 85 2 58 2 58 .



Point 95% Wald


genderMiss_wshd_vec 1 vs 2 2.802 1.196 6.566

incomeMiss 1.014 0.995 1.034

88

Tree, WSHD, and MI


(OR)Female vs. Male

(CI OR)Income

(OR)Income(CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

WSHD Income 2.78 (1.205, 6.426) 1.015 (0.996, 1.035)

SAS Proc MI 3.36 (2.480, 10.310) 1.033 (0.993, 1.075)

R MICE 3.35 (1.006, 11.160) 1.016 (0.996, 1.036)

ABB 3.58 (1.461, 8.770) 1.045 (1.000, 1.089)

Tree/WSHD/MI 3.13 (1.179, 8.320) 1.032 (0.992, 1.073)

WSHD Vector 2.80 (1.196, 6.566) 1.014 (0.995, 1.034)

89

Imputation Diagnostics

90

Imputation DiagnosticsConditional Plot

91

Imputation DiagnosticsMosaic Plot

92

Imputation DiagnosticsBar Chart

93

Another Classification AlternativeTree in R (Development Core Team 2o11)

94

R Classification TreeTerminal Nodes

95

R Classification TreeImputed Values


Point 95% Wald


genderMiss_tree 1 vs. 2 5.763 2.337 14.211

Income 1.017 0.997 1.038

96

Multiple Imputation with IVEware

IVEware (Raghuhatan et alia 2002) is a suite of SAS macros for conducting multiple imputation

Two-step process

Impute

Creates a stacked data set (indexed by _mult_)

Need to create separate data sets from the stacked data set

Analyze

Creates individual estimates

Combines individual estimates to create MI estimate

97

Multiple Imputation with IVEwareSyntax

Creating stacked dataset indexed my _mult_

%impute ( name=mysetup, dir=“Yours", setup=new )

datain forIve;

dataout imp_ive all;

default continuous;

categorical will_buy genderMiss;

transfer id stratum psu weight;

iterations 5;

multiples 5;

seed 728728;

run;

98

Multiple Imputation with IVEwareAnalyze

Producing individual estimates and combing them to produce MI estimates

%regress ( setup = new, name = regsetup, dir = "Yours" )

datain temp1 temp2 temp3 temp4 temp5;

estout est;

repout rep;

weight weight;

stratum stratum;

cluster psu;

mdata skip;

link logistic;

categorical will_buy genderMiss;

id id;

dependent will_buy;

predictor genderMiss income;

print details;

run;

99

Multiple Imputation with IVEwareEstimation

Variable Odds 95% Confidence Interval

Ratio Lower Upper

Intercept

genderMiss* 0.2783180 0.1121444 0.6907245

income 0.9840339 0.9639162 1.0045716

* Note that the base category has changed to male. The point estimate

and confidence interval estimates need to be inverted to be

consistent with the other examples. The odds ratio for the point

estimate is 3.59.

100

Multiple Imputation with IVEware


(OR)Female vs. Male

(CI OR)Income

(OR)Income(CI OR)


Complete Case 3.51 (1.367, 9.021) 1.019 (0.998, 1.041)

Most Probable 2.09 (0.915, 4.778) 1.014 (0.995, 1.033)

Proportional 2.38 (1.037, 5.472) 1.016 (0.996, 1.035)

WSHD Random 2.68 (1.167, 6.146) 1.014 (0.995, 1.034)

WSHD Income 2.78 (1.205, 6.426) 1.015 (0.996, 1.035)

SAS Proc MI 3.36 (2.480, 10.310) 1.033 (0.993, 1.075)

R MICE 3.35 (1.006, 11.160) 1.016 (0.996, 1.036)

IVEware 3.59 (1.448, 8.917) 1.016 (0.995, 1.037)

101

practical imputation of missing data

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