# Some birds, a cool cat and a wolf

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Some birds, a cool cat and a wolf
Tricks of the trade Some birds, a cool cat and a wolf Dick Wiggins, City University, London Gopal Netuveli, Imperial College, University of London RSS Official Statistics/Statistical Computing Section 18th May 2005

Acknowledgments Economic and Social Research Council
Human Capability and Resilience Network

Missing data is a pervasive fact of life.

Sample dataset 100 records randomly selected from British Household Panel Survey with the condition that all cases had complete information on age, sex and socio-economic position. The data contains variables selected from wave 1 and wave11.

Terminology Unit nonresponse: complete absence of any information from a sampled individual or case. Item nonresponse: an individual who cooperates but for some reason has missing values for certain items. Attrition: In longitudinal data, attrition is the cumulative rate of unit nonresponse across waves.

Levels of measurement Nominal Ordinal Interval Ratio
Values are just names e.g. 1 = male 2 = female Ordinal Inherent ranking, but intervals are not equal e.g. RG’s social class Interval Numerical, intervals are meaningful, but no zero e.g. temperature scales Celsius and Farenheit Ratio Numerical, meaningful intervals, zero defined e.g. height, income

The distribution of the measure is important and needs to be specified.

Pattern of missingness -monotone
Percentage of missingness (Lambda) = number of missing values/number of values *100 Pattern of missingness -monotone Lambda for both monotone and non-monotone missingness = 820/3500 = 23.4

Process of missingness
Missing completely at random (MCAR) assumes that missing values are a simple random sample of all data values. Missing at random (MAR) assumes that missing values are a simple random sample of all data values with in subclasses defined by observed data. Missing not at random (MNAR)

MCAR, MAR, MNAR Let Y represent the data which actually consists of Yobs (observed data) and Ymis (missing data) Let the missingness be described by a binary variable R R = 1 if data is missing, 0 otherwise Then a simple way of describing the pattern of missingness will be by evaluating the probability P(R=1) using the data Y. P(R=1|Y) In MCAR we can not evaluate that probability using Y In MAR we assume we can evaluate the probability using Yobs, Ymis is not needed In MNAR, we need both Yobs & Ymis to evaluate the probability

Dick’s menagerie The Ostrich The Hawk The Cuckoo The Owl The Pussycat
The Wolf

The Ostrich aka Listwise Deletion
Ignores missingness i.e. assumes MCAR and drops all cases with missing values. The Hawk aka ad hoc methods Ad hoc methods used are pairwise deletion, mean substituition, last value carry forward

The Cuckoo aka hot decking
Like the cuckoo, hot decking ‘steals’ from other complete records to replace missing records The choice of the complete record is based on a set of observed variables so that the complete and the missing records are as much similar as possible Substituting from an adjacent record is a very simple application of this principle on the assumption that adjacent records will be very similar

The Owl aka Multiple imputation
Works with standard complete-data analysis methods One set of imputations may be used for many analyses Can be highly efficient

Efficiency= 1/(1+(proportion missing/No. of imputations))

Rubin’s rule for combining estimate
Point estimate: Average of point estimates from each imputed sample Variance estimate: Average of within imputation variance + between imputation variance inflated by a factor equal to (1+(1/number of imputations))

The Pussy Cat – Modelling (Heckman 2 step procedure)
What is modelled? The probability of having a missing value based on fully observed characteristics (e.g. age, sex, socio-economic status) AND The model of interest (e.g. predictors of casp19)

Equations P(R=1) = f (age, sex, ses) Step 1
CASP-19= f (age, sex, financial situation, social network, P(R=1)) Step 2

Strengths and weaknesses
Strength: Useful for sensitivity analysis. If the error terms in step 1 and step 2 are significantly correlated then MNAR should be considered. Weakness: Full information needed on variables in step 1

Setting up the illustration in STATA
Listwise: default Hotdeck single imputation Multiple imputation m=5 Heckman ML

Comparison of results from different methods used to manage missingness
Significant coefficients are emboldened Hot deck stratification by agegr & sex Heckman sample equation = agegr+0.06 sex ses Rho (correlation of errors terms in selection and sustantive equations) significantly different from 0. (p <0.0001). MNAR to be considered.

Advice Don’t be an Ostrich Ignore the Hawk
Be the Cuckoo if Lambda is small Otherwise, use the Owl Always stroke the Pussy Cat Await the Wolf