Critique of Traditional Tests

1.7 Critique of Traditional Tests

To avoid missing data biases, we need to analyze missing data bearing in mind the mechanism type because invalid results occur when analyzing arbitrarily without thinking of the right mechanism to suit the missing data on hand. For instance, the missing data mechanism is ignorable for likelihood inference if the data are MAR, also for pairwise and listwise deletion will be biased if the data are MAR and MNAR and it fist only for MCAR data with the allowable missing percentage. MAR data is usually unbiased only if mixed-effects models (data that consists of two parts fixed and random) are identifiable.  MNAR data can’t   be easily tested and need more advanced analysis methods [12].

Although Little’s test is widely used to test MCAR data, it might get some drawback if not identifying the specific variable that violates this assumption, also assumes that missing data patterns share a common covariates matrix, and the possibility of having propensity to produce type II errors.

For MAR tests proposed by Diggle, Kolmogorov, and Fisher, they suffer from the p-values whether these values are uniformly distributed under the null hypotheses, then Fisher [] proposed several tests of combining p-values of these tests by the use of

C= – 2                             (13)

As overall test criterion, where    is a random sample from a uniform distribution.

Fairclough [ 13 ] outlined an approach for MNAR using logistic regression where the dataset is restricted to responders and regarded the reminder-response as missing data. The drawback of the approach is that the prerequisite assumes all responder data must be used and the true value of the data regarded as missing must be known which make it impossible to identify MNAR.

New Approaches to Differentiate Missing Data type

Equation (3) Pr = ( |  ,  ,  ) = Pr ( | ,   ) MAR written as

Pr ( |Y) =   Pr ( | ) means that missingness happen at certain rate and the rate of missing can be explained if we now other factors.

Based on this definition, let  be the linear predictors, we have missing data indicator R=1 for observed so, we fit logistic regression model as

P(R=1 | ) = log (  ).                (14)

A statistically significant association between missingness indicator and observed data reveals evidence for MAR data.

• Identify MAR

Identifying Missing Type

Incomplete data Y= Yo+Ym

Not MAR Maybe MNAR

Significant

Test for MAR Pr ( |Y) = Pr ( )

MAR

Insignificant

P(R=1 | ) = log ( )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure (5) A flowchart for identifying MAR data.

 

Besides this approach calculate Likelihood test ratio for the data. Also, some quick check for this MAR mechanism is to record a column with one (1) if the data is missing and with zero (0) if the data is observed, regress other variables onto it using logistic regression if we get significant p-value that will indicates MAR. Repeating for all other columns with missing values. Notably, some p-value may be significant by chance, so we are adjusting cutoff for significance based on the number of regressions.

 

Equation (5)  Pr = ( |  ,  ,   ) = Pr ( |  ) MCAR written as  Pr ( |Y)=   Pr ( ) means that the data is completely goes at consistent rate no matter what and not based or depend on anything at all. It is allowable to deal with it with deletion for each value 5% to10% pairwise deletion or use multiple imputation (MICE) approaches.

Based on the definition R=1 or R= 0 and assessing mean and covariance differences between the observed and missing and test if  :  =  and  : otherwise;  :      and    otherwise. Accepting of  Would mean no significant difference between the group of both terms of means and covariances, so confirming that an observation is MCAR.

 

Identifying Missing Type

inIncomplete data Y= Yo+Ym

:  =

:

Not MCAR Maybe MAR or MNAR

MCAR

Test for MCAR Pr ( |Y) = Pr ( )

Accept

Reject

Accept

Reject

– Identify MCAR

 

 

 

 

 

 

 

 

 

 

 

Figure (6) A flowchart for Identifying MCAR data

 

Both approaches of mean and covariances differences(above) and Little’s test for MCAR are to be used in the KDDsubset simulation to make sure that we are using the right assumption for the data analysis.

Equation (8a)   Pr ( |  ,  ,  ) is MNAR written as Pr ( |Y)=   Pr ( | ) means that it is fundamentally different from the other two assumptions, a kind of missingness that is neither MCAR nor MAR,  the missing of certain values depend on the true value itself. The probability of missingness depends on the unobserved values. And expressed mathematically as P (R, Y), this joint distribution can be factored into two components: P (R, Y) = P (R |Y)P(Y).

The two parts model combine the substantive regression model, with an additional regression equation that predicts response probabilities.

R serves as a clear indicator of , such that the cases that score above some threshold on have complete data R= 1, and cases below have missing values R= 0.

Hence the model expresses the predicted probability of a complete response as:

The propensity of missing data on the outcome variable

P(P=1 | )=   )                                                  (15)

Phi  is the cumulative normal distribution function [14]

Identifying Missing Type IIIIII

iincomplete data Y= Yo+Ym

Re- examine the data.

MNAR

Test for MNAR Pr ( |Y) =  Pr ( )

P(R=1 | )=   )

Significant

Insignificant

– Identify MNAR 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure (7) A flowchart for identifying MANR data.[15]