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Home , Multivariate statistics

Multivariate

Wan Nor Arifin

Unit of and Research Methodology, Universiti Sains Malaysia.

email: [email protected]

December 1, 2018

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 1 / 28 1 Multivariate

2 Screening of for accuracy

3 Normality, linearity and

4 Data transformation

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 2 / 28 Multivariate

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 3 / 28 Introduction

Multivariate? Strictly speaking:

variate = outcome/dependent variable (DV) = one DV bivariate = two DVs multivariate = > two DVs

→ regardless of the number of independent variables (IVs)/predictors In general, involving > 2 variables = multivariate analysis.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 4 / 28 Introduction

Why bother?

most studies and research involve many variables. consider many predictors and many outcomes at the same time. computer!

I availability of software I processing power

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 5 / 28 Screening of data for accuracy

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 6 / 28 Methods

proofreading – compare form with dataset. exploratory data analysis:

I . I graphical exploration.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 7 / 28 Descriptive statistics

numerical variables

I , I SD, IQR, MAD I minimum, maximum categorical variables

I n, %

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 8 / 28 Practical

multivariate.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 9 / 28 Graphical exploration

numerical variables

I , box-and-whisker plot, Q-Q plot. I more details in normality. categorical variables

I bar charts, pie charts etc. I descriptive statistics are more informative.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 10 / 28 Practical

multivariate.R

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 11 / 28 Normality, linearity and homoscedasticity

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 12 / 28 Normality, linearity and homoscedasticity

All are concerned with numerical variables

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 13 / 28 Normality

Normal distribution of data of DV and IV.

graphical

I Univariate: histogram, box-and-whisker plot I Bivariate: I Multivariate: 1 F Q-Q plot (multivariate) – Mahalanobis distance vs expected values. 2 F χ vs Mahalanobis distance plot (Arifin, 2015).

1The distance of a case from the centroid, where centroid is the intersection of the of all variables (Tabachnick & Fidell, 2007). Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 14 / 28 Practical

multivariate.R

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 15 / 28 Normality

statistical

I – symmetry

F < 2-3 times of its SE:

r 6 SE = N

I – peakness/flatness

F < 2-3 times of its SE:

r24 SE = N

I statistical tests – Shapiro-Wilk test. I Multivariate – Mardia’s skewness and kurtosis.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 16 / 28 Practical

multivariate.R

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 17 / 28 Linearity

Linear relationship between two variables.

graphical

I Bivariate: scatter plot. statistical

I , correlations.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 18 / 28 Practical

multivariate.R

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 19 / 28 Homoscedasticity

Equality/homogeneity of :

across groups (categorical IV). for each levels of IV (numerical IV). graphical

I Univariate per group: Compare and box-and-whisker plots. I Bivariate: scatter plot. statistical

I Tests of equality of .

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 20 / 28 Practical

multivariate.R

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 21 / 28 Data transformation

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 22 / 28 Data transformation

whenever numerical data are not normally distributed. to turn these data into normally distributed data.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 23 / 28 Common data transformation

√ square root – X natural log – lnX log 10 – log10X 1 reciprocal – X power of k – X k , e.g. X 2, X 3

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 24 / 28 Suitable data transformation

Depending on the tail of the skewness, we may try suitable transformations2:

Table 1: Skewness tail and suitable transformations.

Tail Transformation (R format) Purpose Right sqrt(x), log(x), log10(x), 1/x Make larger values smaller Left xˆk Make smaller values larger

2More details can be referred to Kutner, Nachtsheim, Neter, & Li (2005), Hair, Black, Babin, & Anderson (2010) and Tabachnick & Fidell (2007), i.e. transformation of Y/X/both to handle normality, and normality. Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 25 / 28 Practical

multivariate.R

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 26 / 28 Other issues3 for self-study:

Missing data Multivariate outliers Multicollinearity and singularity

3Not covered in your syllabus. Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 27 / 28 References

Arifin, W. N. (2015). The graphical assessment of multivariate normality using SPSS. Education in Medicine Journal, 7(2), e71–e75. Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2010). Multivariate data analysis. New Jersey: Prentice Hall. Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2005). Applied linear (5th ed.). Singapore: McGraw-Hill Education (Asia). Tabachnick, B., & Fidell, L. (2007). Using multivariate statistics (5th ed.). USA: Pearson.

Wan Nor Arifin (USM) Multivariate statistics December 1, 2018 28 / 28