Wish You Were Here!Strategies for Handling

Missing Data

Overview

Types of Missing Data

Strategies for Handling Missing Data

Software Applications and Examples

Agenda

Sources of Missing Data

◦ Item non-response Missing value for any given item

◦ Scale non-response Missing value for any given scale Often a result of item non-response

◦ Attrition Missing value (item and/or scale) for any given time point

◦ Data entry error Observed value not included

Overview

So I have missing data…what’s the big deal?

◦ Missing data, no matter how minimal, can (and probably do) result in biased results

◦ Statistical power

◦ Validity

Overview

How much missing data is “problematic”? Depends on who you ask… Answer #1

ANY

Answer #2 Its never “too much” Optimal methods can easily accommodate 50% missing data

Answer #3 >5% (Schafer, 1999) >10% (Bennett, 2001) >20% (Peng, et al., 2006)

Answer #4 (Widaman, 2006) 1%-2% (Negligible) 5%-10% (Minor) 10%-25% (Moderate) 25%-50% (High) >50% (Excessive)

Overview

Missing Completely at Random (MCAR)

Missing at Random (MAR)

Not Missing at Random (NMAR)

Types of Missing Data

Missing Completely at Random (MCAR)

◦ Missing values on Y are unrelated to any other variable in the analysis

◦ Cases with missing data can be treated as a random subset of the entire sample

◦ Best case scenario; difficult to ascertain

Types of Missing Data

Missing at Random (MAR)

◦ Missing values on Y are related to X but not to Y

◦ Missing values on Y arerandom (random effect)after

controlling for X (systematic effect

◦ Can test systematic effect but not random effect

Types of Missing Data

Not Missing at Random (NMAR)

◦Missing values on Y are related to Y itself

◦Missing data are “non-ignorable”

◦Difficult to ascertain; difficult to manage

Types of Missing Data

Testing for MCAR

◦ Little’s Test of MCAR Omnibus χ2 test of all specified variables

If significant, data are not MCAR May be MAR or MNAR

If not significant, can assume MCAR

Available in SPSS under “Missing Value Analysis” and as a SAS Macro

Determining Type of Missing Data

Testing for MAR

◦ Create a “dummy” variable for not missing/missing on the variable of interest

◦ Conduct statistical tests to see if other relevant variables are associated with values of the new variable Binomial logistic regression χ2 test of independence t-tests

◦ If significant relationships are found, then have MAR; these variables need to be included in any analyses

◦ If no significant relationships found, then you have more work to do

Determining Type of Missing Data

If not MCAR or MAR, does that mean it is MNAR?

◦ Not necessarily…

Might still be MAR but you haven’t found the right indicator variable

◦ Consider other potentially relevant variables and test against the missing data “dummy” variable

Determining Type of Missing Data

Patterns of missing data◦ Monotone pattern

Variables v1-vj can be ordered so that if data are missing on v1, they are missing on all successive variables

VERY common with longitudinal data

Determining Type of Missing Data

Patterns of missing data◦ Non-monotone pattern

Patterns of missing data are arbitrary

Determining Type of Missing Data

Deletion Methods◦Remove cases with missing values

Non-Stochastic Methods◦Replace missing values with “known” values

Stochastic Methods◦Replace missing values with estimated values

Methods for Handling Missing Data

List-Wise Deletion◦ Mechanism

Deletes cases from analysis with missing data on any variable (even if that variable isn’t part of the analysis)

Only uses “complete cases”

◦ Pros Easy to implement Works for any kind of statistical analysis If data are MCAR, does not introduce any bias in parameter estimates Standard error estimates are appropriate

◦ Cons May delete a large proportion of cases, resulting in loss of statistical power May introduce bias if MAR but not MCAR

Deletion Methods

Pair-Wise Deletion◦ Mechanism

Deletes cases when missing data on a specific variable involved in parameter estimation

Uses all available information for each estimation, independent of information available for other estimations

◦ Pros Approximately unbiased if MCAR Uses all available information

◦ Cons Standard errors are incorrect

Deletion Methods

Deletion Methods

Mean Imputation◦ Mechanism

All missing values on a given variable are replaced by the sample mean for that variable

◦ Pros Leaves sample mean of non-missing values unchanged

◦ Cons Often leads to biased parameter estimates (e.g., variances) Usually leads to standard error estimates that are biased downward

Treats imputed data as real data, ignores inherent uncertainty in imputed values.

Non-Stochastic Methods

Individual Mean Imputation◦ Mechanism

Scale scores are computed by taking the mean of non-missing values Ex: Respondent answered 8 of 10 questions on Miller Anxiety Scale –

Compute Scale score by taking mean of available cases

◦ Pros All available information for a given individual is used in the estimation of

missing values

◦ Cons Assumes the items with missing values are similar in difficulty or extremity to

items with non-missing data May lead to biased scores

Non-Stochastic Methods

Regression◦ Mechanism

Missing values are replaced by “predicted” values derived from MR using all relevant variables

◦ Pros Predicted values maintain relationships among variables

◦ Cons Predicted values are “perfect” and lead to positively

biased estimates

Non-Stochastic Methods

Non-Stochastic Methods

Stochastic Regression (aka “Simple Imputation”)◦ Mechanism

Similar to non-stochastic regression in the available data are used to predict missing values

Adds a random value to the predicted value by sampling from a normal distribution with a mean of zero and variance equal to the residual variance of the regression equation

◦ Pros Improvement over Non-Stochastic methods Provides unbiased variance estimates

◦ Cons Only uses a single estimation step and may produce inaccurate or

unusual values

Stochastic Methods

Stochastic Methods (Regression)

Expectation Maximization (EM)◦ Mechanism

2-step iterative process Step 1: Expectation

Use parameter values (initially based on complete-case data) to estimate values for missing data

Step 2: Maximization Use complete-case data and estimated values for missing data to estimate new

model parameters Repeat until results converge (Successive iterations will not yield different parameters)

◦ Pros Minimizes bias in parameter estimates (larger samples yield less bias) Ideal for exploratory and reliability analyses

◦ Cons Initial estimates based on list-wise deletion (doesn’t use all available data) Biased standard errors (minimized with larger samples) Less efficient than FIML for hypothesis testing

Stochastic Methods

Stochastic Methods (EM)

Full Information Maximum Likelihood (FIML)◦ Mechanism

Directly estimates parameters using all observed data for every case

◦ Pros Only requires a single step for imputation and analysis Uses all available data even if some cases are missing data Unbiased standard errors Can be used with smaller samples (N<100)

◦ Cons All variables related to missing data need to be included in the analysis

Stochastic Methods

Stochastic Methods (FIML)

Multiple Imputation (MI)◦ Mechanism

Creates multiple data set using stochastic regression Minimum of 3-5 recommended, but no limit on maximum (Schafer, 1997)

Each data set will be slightly different because of the random component Parameters are estimated for each data set and then averaged

◦ Pros Produces unbiased parameter estimates Produces unbiased standard errors Easy to include auxiliary variables

◦ Cons Labor intensive Can be difficult to integrate multiple data sets

Stochastic Methods

Stochastic Methods (MI)

Comparison of Stochastic Methods

Stochastic Methods

Good Better Best• Stochastic Regression •Expectation-Maximization • Multiple Imputation

• Full Information Maximum Likelihood

Software ApplicationsSPSS/PASW SAS AMOS/MPLUS/

LISRELDeletion

Non-Stochastic Replacement

Simple Imputation

EM

FIML

MI

Modeling problematic child behavior outcomes

Predictors◦ Positive Parenting◦ Social Skills◦ Interpartner Violence◦ Child Sex

N=181

Original data set missing 4 observations (<.5%)

New data set created for purpose of demonstration

Example

◦Little’s Test of MCAR can be obtained as part of PASW “Missing Values Analysis”

Little's MCAR test: Chi-Square = 36.014, DF = 18, Sig. = .007

Conclude that data are not MCAR (not surprising given that I did not delete values in a random manner)

Testing for Type of Missing Data

Test of MAR can be conducted by creating new dichotomous variable for “Not Missing/Missing” and using it as the outcome variable in a logistic regression model Most interested in missing data on outcome variable in this example, but method is

not limited to that Conclude that pattern of missing data is related to Gender

Little's MCAR test for Boys: Chi-Square = 8.338, DF = 14, Sig. = .871* Little's MCAR test for Girls: Chi-Square = 13.026, DF = 18, Sig. = .790*

*We can conclude that data are MCAR within each group. Gender must be included in any missing data analysis to minimize bias.

Testing for Type of Missing Data

Variables in the Equation

B S.E. Wald df Sig. Exp(B)

Step 1a Gender 3.091 1.046 8.726 1 .003 22.003

Parenting .074 .087 .718 1 .397 1.076

Skills .010 .023 .195 1 .658 1.010

Aggression -.003 .022 .024 1 .877 .997

Constant -9.058 2.936 9.516 1 .002 .000

a. Variable(s) entered on step 1: Gender, Parenting, Skills, Aggression.

Patterns of missing data can be obtained using “Analyze Patterns” option available under “Multiple Imputation”

Results of pattern analysis

Variable Summarya,b

Missing

Valid N Mean Std. Deviation N Percent

Behavior 59 32.6% 122 55.75 10.333

Positive Parenting 44 24.3% 137 18.4293 3.04990

Interpartner Violence 36 19.9% 145 12.77 12.229

Social Skills 27 14.9% 154 51.75 11.501

a. Maximum number of variables shown: 25

b. Minimum percentage of missing values for variable to be included: 10.0%

Results of pattern analysis

Although the pattern is not monotone, these cases only make up a very small %

PASW provides several options for handling missing data The add-on module for “Missing Values Analysis” allows

you to implement several different strategies simultaneously◦ In addition to saving time, comparison output is provided for

means, SDs, and correlation/covariance matrices Available options:

◦ List-wise deletion◦ Pair-wise deletion◦ Stochastic regression◦ EM

Missing Values Analysis in PASW

Missing Values Analysis in PASW

Enter continuous and categorical variables

Choose strategies

Additional options

The “Multiple Imputation” option is part of the basic PASW package◦ Provides numerous options

Choose # of iterations Choose estimation method

(monotone vs. non-monotone patterns) Create new data sets

Multiple Imputation in PASW

Multiple Imputation in PASW

Enter all variables to use in imputation (model + auxiliary)

Choose # of iterations

Create a new data set with imputed data

Note: PASW allows you to run analysis on all imputed sets simultaneously

Multiple Imputation in PASW

“Automatic” is the default

Can manually select method based on pattern of missing data

If your data include interactions, so should your imputation model

Missing Data and LISRELMultiple Imputation available in PreLIS under “Statistics”

I have included both model and auxiliary variables

Select estimation methodEM -> monotoneMCMC -> non-monotone

Decide how to handle cases when all data are missing

Output is a “complete” data set for analysis

Missing Data and LISREL

An alternative to MI is to use FIML estimation with the original data set containing missing values

LISREL will default to this option if there is missing data

Comparing Results

Complete List-Wise Pair-Wise

BStd.

Error Sig. BStd.

Error Sig. BStd.

Error Sig.(Constant) 83.71 5.29 .000 91.47 6.57 .000 91.34 7.01 .000

Child's Sex -.75 1.38 .586 -.64 1.72 .709 -.58 1.79 .748

Positive Parenting -1.03 .22 .000 -1.27 .27 .000 -1.34 .28 .000

Social Skills -.20 .06 .001 -.26 .08 .001 -.21 .08 .000

Interpartner Violence .14 .06 .024 .10 .07 .136 .07 .07 .006

Comparing Results

Complete Mean Substitution Simple Imputation

BStd.

Error Sig. BStd.

Error Sig. BStd.

Error Sig.(Constant) 83.71 5.29 .000 85.37 5.21 .000 80.87 6.01 .000

Child's Sex -.75 1.38 .586 -.42 1.19 .709 -.18 1.48 .904

Positive Parenting -1.03 .22 .000 -1.17 .22 .000 -1.06 .24 .000

Social Skills -.20 .06 .001 -.16 .05 .001 -.12 .06 .049

Interpartner Violence .14 .06 .024 .07 .05 .136 .05 .06 .390

Comparing Results

Complete EM-PASW MCMC-LISREL FIML-LISREL

BStd.

Error Sig. BStd.

Error Sig. BStd.

Error Sig. BStd.

Error Sig.(Constant) 83.71 5.29 .000 91.52 4.99 .000 92.96 5.32 .000 88.83 5.86 .000

Child's Sex -.75 1.38 .586 -.35 1.16 .761 -.18 1.59 .359 -.23 .79 .799

Positive Parenting

-1.03 .22 .000 -1.36 .21 .000 -1.24 .26 .000 -1.19 .26 .000

Social Skills -.20 .06 .001 -.22 .05 .000 -.23 .06 .000 -.25 .07 .000

Interpartner Violence

.14 .06 .024 .09 .05 .073 .11 .06 .051 .11 .06 .076

The goal of handling missing data is to find values close to the “real” (but absent) values. (T or F)◦ FALSE – the goal is to estimate unbiased standard errors and parameter

estimates

Which is more important – amount of missing data or type of missing data?◦ Both are important, but type is more important than amount

List-wise deletion is a good strategy for handling missing data? (T or F)◦ TRUE – if data are MCAR; if not MCAR, then there are better alternatives

There are no “good” strategies for handling data that are NMAR. (T or F)◦ TRUE – but FIML is considered to yield the least biased results

Test

Deletion is the only strategy for handling missing categorical data.(T or F)◦ FALSE – can use both non-stochastic and stochastic methods

If using multiple imputation, it is best to include all available variables. (T or F)◦ FALSE – only include variables related to those with missing data

Values such as “not applicable”, “not sure”, “I don’t know”, etc. should be treated as missing data. (T or F)◦ FALSE – if you included these as possible response categories, then they constitute valid

responses (i.e., they are not missing)

List-wise deletion is better than non-stochastic imputation. (T or F)◦ TRUE – if data are MCAR and/or unless using a small sample with minimal power

Test

Missing data should only be imputed for predictor variables and never for outcome variables. (T or F)◦ DEPENDS – if you have good auxiliary variables for the outcome variable, then you

should impute on the outcome variable; otherwise you should not impute.

Values such as “not applicable”, “not sure”, “I don’t know”, etc. can be treated as missing data. (T or F)◦ TRUE – IF you have a strong theoretical argument that a different response would

have been obtained under different circumstances

The most important factor in choosing a strategy is the type of missing data. (T or F) TRUE

Analyses should always be conducted and reported using data with and without missing values. (T or F)◦ TRUE

Test

Causes (actual and/or hypothesized) of missing data should be discussed

The amount of missing data and the strategy used to handle it should be reported

Results of analyses with and without missing data should be discussed

The most appropriate strategy should be used

Summary

Strategy Type of Missing DataMCAR MAR NMAR

List-wise DeletionPair-wise DeletionNon-stochastic ReplacementSimple ImputationEMFIMLMultiple Imputation

Summary

Allison, P. D. (2001). Missing data. Thousand Oaks, CA: Sage Publications.Bennett, D.A. (2001). How can I deal with missing data in my study? Australian and New

Zealand Journal of Public Health, 25, 464-469.Little, R.J.A. (1988). A test of missing completely at random for multivariate data with missing

values. Journal of the American Statistical Association , 83, 1198-1202. Little, R. J. A., & Rubin, D.B. (1987). Statistical analysis with missing data. John Wiley & Sons,

New York.Peng, C.Y., Harwell, M., Liou, S.M., & Ehman, L.H. (2006). Advances in missing data methods

and implications for educational research. In S Sawilowsky (Ed.), Real data analysis (pp.31-78), Greenwich, CT: Information Age.

Schafer, J.L. (1997). Analysis of incomplete multivariate data. Thousand Oaks, CA: Sage.Schafer, J.L. (1999). Multiple imputation: A primer. Statistical Methods in Medical Research. 8:

3-15. Schlomer, G.L., Bauman, S., & Card, N.A. (2010). Best practices for missing data management

in counseling psychology. Journal of Counseling Psychology, 57(1), 1-10.Widaman, K.F. (2006). Missing data: What to do with or without them. Monographs of the

Society for Research in Child Development, 71(3), 42-64.

References

Questions?