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.
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]
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