multivariate analysis overview. introduction multivariate thinking ◦ body of thought processes...
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Multivariate AnalysisMultivariate AnalysisOverview
IntroductionIntroductionMultivariate thinking
◦Body of thought processes that illuminate the interrelatedness between and within sets of variables.
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 Why the multivariate approach?approach?Big idea- multiple response outcomesWith univariate analyses we have just one
dependent variable of interestAlthough any analysis of data involving more
than one variable could be seen as ‘multivariate’, we typically reserve the term for multiple dependent variables
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 Why MV over the univariate approach?approach?Complexity
◦The subject/data studied may be more complex than what univariate methods can offer in terms of analysis
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 Why MV over the univariate approach?approach?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 Different doses of caffeine test performance Causality is more easily deduced
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 Correlational
Why not MV?Why not MV?In the past the computations
were overwhelming even with smaller datasets, and so MV analyses were typically avoided
Now this is not a problem but there are still reasons to not do a MV analysis
Why not MV?Why not MV?Ambiguity
◦ MV analysis may result in a less clear understanding of the data
E.g. group differences on a linear combination of DVs (Manova)
Differences are easily interpreted in a univariate sense
◦ Ambiguity because of ignorance of the technique is not a valid reason however
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
No free lunch◦ MV analyses come with their own rules and assumptions
that may make analysis difficult or not as strong
MultivariateMultivariate Pros and Cons Pros and Cons SummarySummary
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
Disadvantages◦ Larger Ns are often required◦ More difficult to interpret◦ Less known about the robustness of
assumptions
Primary purposes of MV Primary purposes of MV analysisanalysisPrediction and explanationDetermining structure
PredictionPredictionThe 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?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 A caveat regarding ‘explanation’‘explanation’Determining variable importance can
be a suspect endeavorSomething that might be deemed a
statistically significant variable may not make the cut had the study been conducted again
Depending on a number of factors, results may be sample specific◦ i.e. you may not see the same ordering next
time
StructureStructure 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? Methods involve classification and/or data
reduction Latent variables (constructs)
◦ Example: Observed variables Giddiness, Silliness, Irrationality,
Possessiveness and Misunderstanding reduced to the underlying construct of ‘Love’
Interest may be in reducing variables (Factor analysis), emphasis on group membership (Cluster analysis), stimulus structure (MDS) etc.
Prediction and StructurePrediction and StructureBoth prediction and structure
may be the goal of analysis◦SEM and path analysis
How well does the model fit the data?
Multivariate ThemesMultivariate Themes
Multivariate ThemesMultivariate Themes
Things to considerThings to considerInitial variable choiceComes down to:
◦ Familiarity with previous research◦ Instrument used◦ Expertise with field of study◦ Common sense
Much of the ‘hard work’ consists of developing a plan of attack and deciding on how to study the problem
Initial Examination of DataInitial Examination of DataPreliminary 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
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 considerMore to consider Intro now, more details as we discuss each
methodAssumptions– important for inferences
beyond the sampleNormality: Basic assumption of General
Linear Model; concerned with an elliptical pattern of residuals for the data◦ Skewness: Distribution of scores is tilted
(asymmetrical) Direction established by tail greater skewness = less normality
◦ Kurtosis: Degree of peakedness of data 3 Types: leptokurtic (thin); mesokurtic (normal);
platykurtic (flattened)
More to considerMore to consider Linearity
◦ Data forms a relatively straight oval line when plotted Homoscedasticity
◦ variance of 1 variable is equal at all levels of other variables understood through standard deviations across
variables and scatter plots ◦ Referred to as homogeneity of variance in ANOVA
methods 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 considerMore to consider 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
Redundant variables—brings discriminant in equation to zero
Orthogonality ◦ Allows no association among variables◦ Not realistic in real world data◦ May allow greater interpretability versus data that
are too related
More to considerMore to considerOutliers
◦ Effect mean (inflate/deflate) disguising true relationship
◦ Distort data—create noise (error) lose power
◦ Transformations (log or square root) may be helpful with outliers Reshapes distribution creating a more normal
distribution However you now have a scale with which you
are unfamiliar and which you cannot generalize back to the original
Some distinctionsSome distinctionsTypes of data
◦Nominal/Categorical◦Ordinal◦Continuous
Interval or RatioThe 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 dataTypes of dataOne thing to keep in mind is that these
distinctions are largely arbitraryOne can dichotomize a continuous
measure into categories◦ A bad idea most of the time
An ordinal measure (e.g. likert question) has a mean/construct that actually falls along a continuum
How the data is to be considered is largely left to the researcher
Sample vs. PopulationSample vs. PopulationIn typical research we are rarely dealing
with a populationThe goal in research is not to simply
describe our data but to generalize to the real world
Many analyses and data collection are for a variety of reasons (not good) sample-specific, and not much use to the scientific community
Take care in the initial phase of research planning to help guard against such a situation
The linear combination of The linear combination of variablesvariablesWhether of IVs or DVs, a linear
combination of variables is often necessary to interpret the data◦This idea is essential to thinking
multivariatelyMultReg
◦Finding the linear combination of IVs that best predicts the DV
Manova◦What linear combination of DVs
maximizes the distinction between groups
How many variablesHow many variablesConsiderations
◦ Cost◦ Availability◦ Meaningfulness◦ Theory
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 Statistical power and effect sizeeffect sizeA problem that has plagued the social
sciences is the lack of power to find subtle effects
Some multivariate procedures will require relatively large amounts of data (e.g. SEM)
Power and sample size are a required consideration before any attempt at research, multivariate or otherwise
After the fact, emphasis should be placed on effect size and model fit, rather than p-values
More later…
The matrices of interestThe matrices of interest 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
Correlation matrix- R◦ Contains information about the linear relationship
between variables 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
rs s
The matrices of interestThe matrices of interestVariance/Covariance matrix - Σ
◦Square and symmetrical◦Variance of each variable is on the
diagonal, covariances with other variables on the off-diagonals
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 interestThe matrices of interestSum of Squares and cross-products
matrix - S Precursor to the Variance/Covariance
matrix (the values before division by N-1)
On the diagonal is a variable’s sum of the squared deviations from its mean
Off-diagonal elements are the sum of the products of the deviation scores for the two variables
Methods of analysisMethods of analysisA host of methods are available
to the researcherThe 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 relationshipDegree of relationship Bivariate r
◦ The degree of linear relationship between two variables
◦ Partial and semi-partial Multiple R
◦ The relationship of a set of variables to another (dependent) variable
Canonical R◦ The grandaddy◦ Relationship between sets of variables
Methods are also available to assess the relationship among non-continuous variables◦ E.g. Chi-square, Multiway Frequency Analysis
Group DifferencesGroup DifferencesVery popular research question in
social sciences (too popular really)Is group A different from B?
◦The answer is always yes, and with a large enough sample, statistically significantly so
Anova and relatedManova the multivariate
counterpartRepeated measures
Predicting group Predicting group membershipmembershipTurning the group difference
question the other way aroundDiscriminant function analysisLogistic regression
StructureStructureData reduction and classificationCluster 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
Principal components and Factor analysis◦ Reduce a large number of variables to smaller◦ Often used in psych for the development of
inventoriesStructural equation modeling
◦ Where factor analysis and regression meet
Time course of eventsTime course of eventsHow long is it before some event
occurs?How does a DV change over the course
of time?The former question can be answered
with survival/failure analysis◦ Survival rates for disease◦ Time before failure for a particular electronic
partThe latter is often examined with time-
series analysis◦ Many time periods are available for analysis
E.g. monthly stock prices over the past five years◦ Popular in the economics realm
Decision treeDecision tree
Decision tree
Decision treeDecision tree
Although such guides may be useful, as mentioned before, multiple analyses may be appropriate for the data under consideration
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 GlanceMultivariate 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 MethodsSummary of Methods The multivariate methods we will look at are a
set of tools for analyzing multiple variables in an integrated and powerful way.
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)
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.