1. CH2. Cleaning and Transforming Data
Graphical methods
Missing data processes
Impact of missing data
Approaches available
Outliers
Underlying assumptions
Data transformation
Nonmetric data as metric data
Fall, 2010
Multivariate Analysis Lec 2

2. Graphical Examination
The shape of distribution
Histogram; stem and leaf diagram
The relationships among variables
Bivariate relationships: scatterplot (matrix)
Examining groups difference
Boxplot
Multivariate profiles: rarely used
Fall, 2010
Multivariate Analysis Lec 2

3. Multivariate Analysis Lec 2
Deviate from normal distribution
Fall, 2010
Multivariate Analysis Lec 2

4. Multivariate Analysis Lec 2
correlation
Fall, 2010
Multivariate Analysis Lec 2

5. Multivariate Analysis Lec 2
Outlier
Three groups are essentially equal
Three groups have substantial difference
Fall, 2010
Multivariate Analysis Lec 2

6. Multivariate Analysis Lec 2
Missing Data
A fact of life
Missing data that affect the generalizability of the results
Determining the reasons underlying missing data -> select the appropriate course of action
A missing data process; unknown so ask:
Randomly scattered throughout?
How prevalent?
Action is similar to non-respondents
Potential biases and sample size available
Few guidelines and remedies to the problem
Fall, 2010
Multivariate Analysis Lec 2

7. The Reasons leading to Missing Data
Explicitly expected: ignorable missing data
MV analysis itself is a way to overcome missing data
Nonsampled observations; probability sampling (the missing data process is random)
When data are censored; e.g., causes of death
Other types:
Procedural factors
Inapplicable responses
Refusal to respond …
Fall, 2010
Multivariate Analysis Lec 2

8. Patterns of Missing Data
First, ascertain the degree of randomness presented in the missing data
Missing at Random (MAR): if missing values of Y depend on X, but not on Y (not necessarily represent a truly random sample of all Y values
Missing Completely at Random (MCAR): the observed values of Y are truly a random sample of all Y values
Fall, 2010
Multivariate Analysis Lec 2

9. Diagnosing the Randomness
Assess the missing data process of a single variable Y by forming two groups
With missing data and with valid data
Dichotomized correlations
Statistical significance tests of the correlations provide a conservative estimate of the degree of randomness
Nonsignificant: MCAR; otherwise, MAR
Overall test: compare the pattern of missing data with the pattern expected for a random missing data process
Fall, 2010
Multivariate Analysis Lec 2

10. Approaches for Dealing with Missing Data
Complete Case Approach: Use of observations with complete data only
Only for MCAR
Delete Variables
When a nonrandom pattern of missing data is present
No firm guidelines, should based on empirical and theoretical considerations
Fall, 2010
Multivariate Analysis Lec 2

11. Multivariate Analysis Lec 2
Imputation Methods
The process of estimating the missing value based on valid values of other variables and/or cases in the sample
The idea is both seductive and dangerous
Two approaches
Use of all the available information from a subset of cases to generalize to the entire sample
Methods of estimating replacement values for the missing data
Fall, 2010
Multivariate Analysis Lec 2

12. All-available Approach
Estimate correlations and maximize the pairwise information available in the sample
Use correlation to represent and only for MCAR
Problems
Correlations may be “out of range” and inconsistent with other correlations
The eigenvalue in the correlation matrix can become negative
Fall, 2010
Multivariate Analysis Lec 2

13. Replacement of Missing Data
Case substitution (by nonsampled observation)
Mean substituion (based on all valid responses)
Understate the true variance; actual distribution is distorted; depress the observed correlation
Deck imputation: external source
Regression imputation: based on regression
Reinforce the relationship already in the data; understate the variance; assume substantial correlations with the other variables; not constrained in the estimates it makes
Multiple imputation: a combination of several methods
Model-based procedures
Fall, 2010
Multivariate Analysis Lec 2

14. An Illustration of Missing Data Diagnosis
Number of cases without missing data
V1 and V3: the two most likely variables for deletion
V1-V9: metric variables
Total number of cases = 70
Fall, 2010
Multivariate Analysis Lec 2

15. An Illustration of Missing Data Diagnosis
Delete just V1
= 32
Fall, 2010
Multivariate Analysis Lec 2

16. An Illustration of Missing Data Diagnosis
Reduced sample: 64 cases
(70 – 6 = 64)
Fall, 2010
Multivariate Analysis Lec 2

17. Multivariate Analysis Lec 2
Rules of Thumb 2–1
How Much Missing Data Is Too Much?
Missing data under 10% for an individual case or observation can generally be ignored, except when the missing data occurs in a specific nonrandom fashion (e.g., concentration in a specific set of questions, attrition at the end of the questionnaire, etc.).
The number of cases with no missing data must be sufficient for the selected analysis technique if replacement values will not be substituted (imputed) for the missing data.
Fall, 2010
Multivariate Analysis Lec 2

18. Multivariate Analysis Lec 2
Rules of Thumb 2–3
Under 10% – Any of the imputation methods can be applied when missing data is this low, although the complete case method has been shown to be the least preferred.
10 to 20% – The increased presence of missing data makes the all available, hot deck case substitution and regression methods most preferred for MCAR data, while model-based methods are necessary with MAR missing data processes
Over 20% – If it is necessary to impute missing data when the level is over 20%, the preferred methods are:
the regression method for MCAR situations, and
model-based methods when MAR missing data occurs.
Fall, 2010
Multivariate Analysis Lec 2

19. Multivariate Analysis Lec 2
Outliers
Observations with a unique combination of characteristics identifiable as distinctly different from the other observations
Beneficial and problematic outliers
Four classes
Procedure error
A result of an extraordinary event
Extraordinary observations of no explanation
Unique in combination of values across the variables
Fall, 2010
Multivariate Analysis Lec 2

20. Multivariate Analysis Lec 2
Fall outside the ellipse more than 2 times
Fall, 2010
Multivariate Analysis Lec 2

21. Multivariate Analysis Lec 2
Fall, 2010
Multivariate Analysis Lec 2

22. Testing the Assumptions of Multivariate Analysis
Why need it
The complexity of the relationships
The complexity of the analyses and of the results may mask the “signs” of assumption violations apparent in the simpler univariate analysis
Tested twice
First for separate variables
Second for the multivariate model variate
Fall, 2010
Multivariate Analysis Lec 2

23. Multivariate Analysis Lec 2
Normality
Normal distribution
The benchmark for statistical methods
Required to use the F and t statistics
Univariate normal, does not imply multivariate normality
Multivaraite normal, imply univariate normal; depend on the technique
Fall, 2010
Multivariate Analysis Lec 2

24. Multivariate Analysis Lec 2
Analysis and Test
Graphical analysis: normal probability plot
Statistical tests
Skewness
Kurtosis
Remedies
Data transformations
Fall, 2010
Multivariate Analysis Lec 2

25. Multivariate Analysis Lec 2
Fall, 2010
Multivariate Analysis Lec 2

26. Multivariate Analysis Lec 2
Other Problems
Homoscedasticity: equal variance dispersion
Levene test; Box’s M test
Fall, 2010
Multivariate Analysis Lec 2

27. Multivariate Analysis Lec 2
Linearity, problems for correlation-based analysis
Residual analysis
Absence of correlated errors
Often due to data collection process
Data transformation
Typical, squared, square root, logarithm, inverse, depending on data shapes
Fall, 2010
Multivariate Analysis Lec 2

28. Multivariate Analysis Lec 2
For 2-9 (a) either variable can be squared to achieve linearity
Fall, 2010
Multivariate Analysis Lec 2

29. Multivariate Analysis Lec 2
Fall, 2010
Multivariate Analysis Lec 2

30. Multivariate Analysis Lec 2
Fall, 2010
Multivariate Analysis Lec 2

31. Multivariate Analysis Lec 2
Fall, 2010
Multivariate Analysis Lec 2

32. Multivariate Analysis Lec 2
Fall, 2010
Multivariate Analysis Lec 2

33. Nonmetric Data with Dummy Variables
Many techniques require metric data
Using dummy (dichotomous) variables as replacement variables
With k categories, need k – 1 dummy variables
Indicator coding and effects coding: same result with different interpretation of the dummy-variable coefficients
Fall, 2010
Multivariate Analysis Lec 2

34. Multivariate Analysis Lec 2
Fall, 2010
Multivariate Analysis Lec 2