Multivariate Statistics
Principal Components/Factor Analysis
Structural Equation Modeling
Principal Components &
Factor Analysis
• We have only one set of variables.
• Each is well correlated with some of the
others.
• We want to capture the variance, or
covariance, of the p variables
• Repackaging it into m components or
factors. [usually with m < p]
• Each of which is a weighted linear
combination of the variables.
Creation or Discovery?
• I tend to think of the components/factors
as being things we have created out of the
data.
• Psychologists are more likely to think of
them as estimates of underlying
dimensions (latent variables).
• I am skeptical about their being a concrete
reality that we can know, I think our reality
is created.
Data Reduction
• PCA/FA may be used to reduce the p
variables to a smaller set of m
components/factors for use in subsequent
analysis.
• Chia, Wuensch, Childers, Chuang, Cheng,
Cesar-Romero, & Nava (1994)
• Students in Mexico, Taiwan, and the US
• 45 item “family values” scale.
• My research associate wanted me to conduct
45 3-way ANOVAs (one on each item)
• I balked, insisted on reducing the 45 variables
to a smaller number of components.
• Then did a Culture x Sex x Age (under 20 vs.
over 20) ANOVA on each component.
• The loadings were used to name the
components.
The Seven Components
1. Family Solidarity (respect for the family)
2. Executive Male (men make decisions,
women are homemakers)
3. Conscience (important for family to
conform to social and moral standards)
4. Equality of the Sexes (minimizing sexual
stereotyping)
5. Temporal Farsightedness (interest in the
future and the past)
6. Independence (desire for material
possessions with freedom from parental
constraint)
7. Spousal Employment (each spouse
should make decisions about his/her own
job)
Results of the ANOVAs
• US students (especially the women) stood
out as being sexually egalitarian, wanting
independence.
• Younger US students put little importance
on family solidarity.
• Taiwanese students were distinguished by
scoring very high on the temporal
farsightedness component
• Taiwanese students were low on the
conscience component.
• Among Taiwanese students the men were
more sexually egalitarian than the women.
• and the women more concerned with
independence than were the men.
• The Mexican students were like the
Taiwanese in being concerned with family
solidarity,
• but not with sexual egalitarianism and
independence.
• The Mexican students were like the US
students in attaching more importance to
conscience and less to temporal
farsightedness than did the Taiwanese.
• Among the Mexican students the men
attached more importance to
independence than did the women.
Factor Analysis and Test
Development
• 21 items in Patel’s SBS
• Were designed to measure a single
dimension.
• FA indicated there were three dimensions.
– avoidance behaviors (such as moving away
from a gay) -- 13 items
– aggression from a distance (such as making
harassing phone calls) – 6 items
– up-close aggression (physical fighting) – 2
items
Item Analysis
• For a scale or subscale, see how well
each item correlates with the sum of
scores on other items.
• Find Cronbach alpha, the average splithalf reliability, corrected for attenuation.
• This is a conservative estimate of
reliability.
• Would deletion of any item increase
Cronbach alpha?
Homework Assignment
• Please read the document Cronbach's
Alpha and Maximized Lambda4.
• Follow the instructions there to conduct an
item analysis with SAS and with SPSS.
• Bring your output to class for discussion.
Path Analysis
• This is a simplification of Structural
Equation Modeling.
• Only measured variables are included, no
latent variables.
• We test to see how well our causal model
fits with the (nonexperimentally obtained)
data.
Sunita Patel’s Thesis
The data are the
same as those
discussed earlier
under the topic of
canonical
correlation.
Significant paths are
in red. The path
coefficients are
standardized, like
beta weights.
Structural Equation Modeling
• Essentially a combination of sequential
multiple regression and factor analysis.
• There are two parts of the model.
• The measurement model is how the
observed variables are related to latent
variables (factors).
• The structural model relates the latent
variables to one another.
Student Opinion of Instruction
Surveys
• Greenwald and Gillmore (1997)
• Measured variables are items on the
survey.
• Latent variables are
– Expected grade in course
– Work put into the course by the student
– The evaluation the student gives the
professor
Conclusions
• Professors who have lenient grading
policies (making Expected Grades high)
– Get good evaluations
– Do not motivate the students to put much
work into the course
• Professors with stringent grading policies
– Get poor evaluations
– Do motivate the students to work hard on the
course.
Confirmatory Factor Analysis
• A type of SEM.
• How well does an a priori factor model fit
the data?
• Example from Tabachnick & Fidell (2007)
– Measured variables are subscales of the
Wechsler Intelligence Scale for Children
(WISC)
– Latent variables are Verbal IQ and
Performance IQ.
Goodness of Fit
• There are several statistics that measure
how well the model fits the data, and
standards for how well is good enough.
• These are used in path analysis, CFA, and
other applications of SEM.
• One may also modify the model and see
how that affects the fit.
Adding the path in red
produced noticeable
improvements in the
goodness of fit
statistics.