Multivariate Analysis
Overview
Introduction
y Multivariate thinking
◦ Body of thought processes that illuminate the interrelatedness between and within sets of variables.
y The essence of multivariate thinking is to expose the inherent structure and
meaning revealed within these sets of variables through application and
interpretation of various statistical
methods
Why the multivariate approach?
y
Big idea- multiple response outcomes
y
With univariate analyses we have just one dependent variable of interest
y
Although any analysis of data involving more than one variable could be seen as ‘multivariate’, we typically reserve the term for multiple dependent variables
y
So MV analysis is an extension of UV ones, or
conversely, many of the UV analyses are special cases of
MV ones
Why MV over the univariate approach?
y Complexity
◦ The subject/data studied may be more
complex than what univariate methods can offer in terms of analysis
y Reality
◦ In some cases it would be inappropriate to conduct univariate analysis as the
data/research demand a multivariate analysis
Why MV over the univariate approach?
y
Experimental data
◦ Although experimental research can be and often is
multivariate, typically subjects are assigned to groups and the manipulations regard corresponding changes to a
single outcome
x Different doses of caffeine Æ test performance x Causality is more easily deduced
y
Non-experimental data
◦ Likewise survey/inventory data might be analyzed in univariate fashion, but typically it will require the
multivariate approach to solve the questions stemming from it
x Correlational
Why not MV?
y In the past the computations were
overwhelming even with smaller datasets, and so MV analyses were typically avoided
y Now this is not a problem but there are
still reasons to not do a MV analysis
Why not MV?
y
Ambiguity
◦ MV analysis may result in a less clear understanding of the data
x E.g. group differences on a linear combination of DVs (Manova)
x Differences are easily interpreted in a univariate sense
◦ Ambiguity because of ignorance of the technique is not a valid reason however
y
Unnecessary complexity
◦ Just because SEM looks neat/is popular doesn’t mean you have to do one, or that it is the best way to answer your research question
y
No free lunch
◦ MV analyses come with their own rules and assumptions that may make analysis difficult or not as strong
Multivariate Pros and Cons Summary
y
Advantages of using a multivariate statistic
◦ Richer realistic design
◦ Looks at phenomena in an overarching way (provides multiple levels of analysis)
◦ Each method differs in amount or type of Independent Variables (IVs) and DVs
◦ Can help control for Type I Error y
Disadvantages
◦ Larger Ns are often required
◦ More difficult to interpret
◦ Less known about the robustness of assumptions
Primary purposes of MV analysis
y Prediction and explanation
y Determining structure
Prediction
y
The goal in most research situations is to be able to predict outcomes based on prior
information
◦ E.g. given a person’s gender and region, what will their attitude be on some social issue?
◦ Given a number of variables how well can we predict group membership?
y
Explanation
◦ Which variables are most important in the prediction of some outcome?
◦ In many cases this is end goal of an analysis, though a very problematic one
A caveat regarding ‘explanation’
y
Determining variable importance can be a suspect endeavor
y
Something that might be deemed a statistically significant variable may not make the cut had the study been conducted again
y
Depending on a number of factors, results may be sample specific
◦ i.e. you may not see the same ordering next time
Structure
y A different goal in MV analysis is to determine the structure of the data
◦ Is there an underlying dimension that can describe the data in a simpler fashion?
y Methods involve classification and/or data reduction
y Latent variables (constructs)
◦ Example:
x Observed variables Giddiness, Silliness, Irrationality, Possessiveness and Misunderstanding reduced to the underlying construct of ‘Love’
y Interest may be in reducing variables (Factor analysis),
emphasis on group membership (Cluster analysis), stimulus structure (MDS) etc.
Prediction and Structure
y Both prediction and structure may be the goal of analysis
◦ SEM and path analysis
y How well does the model fit the data?
Multivariate Themes
Multiplicity Theme
Multiple considerations at all levels of focus, with greater multiplicity generally leading to greater reliability, validity, and generalization:
Multiple Theories and
Hypotheses Multiple Empirical
Studies Multiple Measures
Multiple Time Points
Multiple Controls
Multiple Samples
Practical Implications
Multiple Statistical
Methods
Multivariate Themes
Central Themes All multivariate methods focus on these central themes:
Variance
Systematic Random
Covariance
Ratio of Variances
Linear Combinations (e.g., Components,
Factors)
Interpretation Themes
Big picture and Specific levels
Macro-Assessment (e.g., Significance Test
& Effect Size)
Micro-Assessment (e.g., Examining Means or Weights)
Things to consider
y
Initial variable choice
y
Comes down to:
◦ Familiarity with previous research
◦ Instrument used
◦ Expertise with field of study
◦ Common sense
y
Much of the ‘hard work’ consists of developing a
plan of attack and deciding on how to study the
problem
Initial Examination of Data
y Preliminary analysis
◦ A thorough initial examination of the data is not only required but also necessary for a full
understanding of any research
◦ Such initial analyses provide a better grasp of
what is happening in the data and may inform the MV analysis to a certain extent
y However, in the MV case, if the actual goal is interpretation of the UV analyses (as one
often sees in MANOVA), the MV analysis is
unwarranted
More to consider
y
Intro now, more details as we discuss each method
y
Assumptions– important for inferences beyond the sample
y
Normality: Basic assumption of General Linear Model;
concerned with an elliptical pattern of residuals for the data
◦ Skewness: Distribution of scores is tilted (asymmetrical)
x Direction established by tail
x greater skewness = less normality
◦ Kurtosis: Degree of peakedness of data
x 3 Types: leptokurtic (thin); mesokurtic (normal); platykurtic (flattened)
More to consider
y Linearity
◦ Data forms a relatively straight oval line when plotted y Homoscedasticity
◦ variance of 1 variable is equal at all levels of other variables
x understood through standard deviations across variables and scatter plots
◦ Referred to as homogeneity of variance in ANOVA methods
y Homogeneity of regression
◦ Regression slopes between covariate and DV are equal across groups of IV
◦ Do not want this statistic (F) to be significantly different—if so, violation of assumption for (M)ANCOVA
More to consider
y Multicollinearity
◦ Correlation coefficient (r) between predictors is noticeably large
◦ Causes instability in the statistical procedure
◦ Can’t differentiate which variables are contributing to outcome
◦ Singularity
x Redundant variables—brings discriminant in equation to zero y Orthogonality
◦ Allows no association among variables
◦ Not realistic in real world data
◦ May allow greater interpretability versus data that are too related
More to consider
y
Outliers
◦ Effect mean (inflate/deflate) disguising true relationship
◦ Distort data—create noise (error) lose power
◦ Transformations (log or square root) may be helpful with outliers
x Reshapes distribution creating a more normal distribution x However you now have a scale with which you are unfamiliar
and which you cannot generalize back to the original
Some distinctions
y Types of data
◦ Nominal/Categorical
◦ Ordinal
◦ Continuous
x Interval or Ratio
y The types of variables involved will say
much about what analyses are going to be appropriate and/or how one might
proceed with a particular analysis
Types of data
y
One thing to keep in mind is that these distinctions are largely arbitrary
y
One can dichotomize a continuous measure into categories
◦ A bad idea most of the time
y
An ordinal measure (e.g. likert question) has a mean/construct that actually falls along a
continuum
y
How the data is to be considered is largely left to
the researcher
Sample vs. Population
y
In typical research we are rarely dealing with a population
y
The goal in research is not to simply describe our data but to generalize to the real world from
which the sample is taken
◦ This is the purpose of conducting inferential analyses which require certain assumptions to be met in order to be utilized
y
Many analyses and data collection are for a
variety of reasons (not good) sample-specific, and not much use to the scientific community
y
Take care in the initial phase of research planning
to help guard against such a situation
The linear combination of variables
y Whether of IVs or DVs, a linear combination of variables is often necessary to interpret the data
◦ This idea is essential to thinking multivariately
y MultReg
◦ Finding the linear combination of IVs that best predicts the DV
y Manova
◦ What linear combination of DVs maximizes
the distinction between groups
How many variables
y
Considerations
◦ Cost
◦ Availability
◦ Meaningfulness
◦ Theory
y
For ease of understanding and efficiency we
typically want the fewest number of variables that will explain the most
◦ Ockham’s razor
Statistical power and effect size
y
A problem that has plagued the social sciences is the lack of power to find subtle effects
y
Some multivariate procedures will require relatively large amounts of data (e.g. SEM)
y
Power and sample size are a required
consideration before any attempt at research,
multivariate or otherwise, though typically sample size will be determined by the practicalities and limitations of the research
y
After the fact, emphasis should be placed on effect size and model fit, rather than p-values
y
More later…
The matrices of interest
y Data matrix
◦ What you see in SPSS or whatever program you’re using
◦ Includes the cases and their corresponding values for the variables of interest
y Correlation matrix- R
◦ Contains information about the linear relationship between variables
x Standardized covariance
◦ Symmetrical
◦ Square
◦ Typically only the bottom portion is shown as the top portion is its mirror image and the diagonal contains all ones (each variable is perfectly correlated with itself)
covxy
x y
r= s s
The matrices of interest
y Variance/Covariance matrix - Σ
◦ Square and symmetrical
◦ Variance of each variable is on the diagonal, covariances with other variables on the off- diagonals
y In some cases you will have the option to use correlations or covariances as the
unit of analysis, with some debate about
which is better under what circumstances
The matrices of interest
y
Sum of Squares and cross-products matrix - S
y
Precursor to the Variance/Covariance matrix (the values before division by N-1)
y
On the diagonal is a variable’s sum of the squared deviations from its mean
y
Off-diagonal elements are the sum of the
products of the deviation scores for the two
variables
Methods of analysis
y A host of methods are available to the researcher
y The kind of question asked will help guide one in choosing the appropriate analysis, however the data may be available to
multiple methods, and almost always is
Degree of relationship
y Bivariate r
◦ The degree of linear relationship between two variables
◦ Partial and semi-partial
y Multiple R
◦ The relationship of a set of variables to another (dependent) variable
y Canonical R
◦ The grandaddy
◦ Relationship between sets of variables
y Methods are also available to assess the relationship among non-continuous variables
◦ E.g. Chi-square, Multiway Frequency Analysis
Group Differences
y Very popular research question in social sciences (too popular really)
y Is group A different from B?
◦ The answer is always yes, and with a large enough sample, statistically significantly so
y Anova and related
y Manova the multivariate counterpart
y Repeated measures
Predicting group membership
y Turning the group difference question the other way around
y Discriminant function analysis
y Logistic regression
Structure
y
Data reduction and classification
y
Cluster analysis
◦ Seeks to identify homogeneous subgroups of cases or variables based on some measure of ‘distance’
◦ Identify a set of groups in which within-group variation is minimized and between-group variation is maximized
y
Principal components and Factor analysis
◦ Reduce a large number of variables to smaller
◦ Often used in psych for the development of inventories y
Structural equation modeling
◦ Where factor analysis and regression meet
Time course of events
y
How long is it before some event occurs?
y
How does a DV change over the course of time?
y
The former question can be answered with survival/failure analysis
◦ Survival rates for disease
◦ Time before failure for a particular electronic part y
The latter is often examined with time-series
analysis
◦ Many time periods are available for analysis
x E.g. monthly stock prices over the past five years
◦ Popular in the economics realm
Decision tree
Decision tree
Decision tree
y
Although such guides may be useful, as mentioned before, multiple analyses may be appropriate for the data under
consideration
y
The best plan of attack is to have a well-defined research question, and
collect data appropriate to
the analysis that will best
answer that question
Multivariate Methods: Quick Glance
Organizational Chart based on: Type of Research Focus
(Group differences or Correlational).
Research Question IVs: Number and Scale # & Scale Method
Research Focus IVs DVs Multivariate
Number & Scale Number & Scale Method Group Differences
1+ categorical & continuous 1 continuous ANCOVA
1+ categorical 2+ continuous MANOVA
2+ continuous 1+ categorical DFA
1+categ or cont 1 categorical LR Correlational
2+ continuous 1 continuous MR
2+ continuous 2+ continuous CC
- 2+ continuous PCA & FA
Note: Scale and number of Independent (IV) and Dependent (DV) categorical or continuous variables.
+ indicates 1 or more; ANCOVA = Analysis of Covariance; MANOVA = Multivariate Analysis of Variance; DFA = Discriminant Function Analysis; LR=Logistic Regression; MR = Multiple
Regression; CC = Canonical Correlation; PCA/FA = Principal Components/Factor Analysis
Summary of Methods
y The multivariate methods we will look at are a set of tools for analyzing multiple variables in an integrated and powerful way.
y They allow the examination of richer and perhaps more realistic designs than can be assessed with traditional
univariate methods that only analyze one outcome variable and usually just one or two independent variables (IVs)
y Compared to univariate methods, multivariate methods allow us to analyze a complex array of variables, providing greater assurance that we can come to some synthesizing conclusions with less error and more validity than if we were to analyze variables in isolation.
