Part 1 Other Multivariate Tests. Measuring Intelligence 1.Sometimes in multi-variate testing, we...

Part 1

Other Multivariate Tests

Measuring Intelligence1. Sometimes in multi-variate testing, we don’t know

exactly what the latent variable is.

2. I’d like to hear some of your definitions of intelligence?

3. There is very little consensus about the definition of intelligence. In your opinion, given the absence of consensus, is there any point in studying/measuring intelligence?

Measuring Intelligence

1. “It matters very little what the tests are, so long as they are numerous”. (Binet, 1911).

2. Critical Thinking Question: What was the reasoning behind the above quote from Binet?

3. To his credit, Binet’s tests generated scores that correlated well with the child’s school performance, and with the teacher’s evaluation of the child. That is, Binet’s test had predictive validity.

Alfred Binet (1857-1911)

Measuring Intelligence1. The test most commonly used to assess intelligence in adults is the

Wechsler Adult Intelligence Scale (Revised) WAIS-R.

2. One of the WAIS-R sub-tests is called the “verbal test”, although it measures many things, including general knowledge, vocabulary, comprehension, and arithmetic skills.

3. Another of the WAIS-R sub-tests, called the “performance test”, requires assembling parts into wholes, completing pictures, and rearranging pictures into a coherent sequence or story line.

4. Weirdly enough, there is a slight negative correlation between the verbal and performance sub-tests!


WAIS-R Verbal Test Questions


WAIS-R Performance Test Questions


SAT Questions


Progressive Matrix TestWhat advantage does this have some over other tests?


• Factor Analysis - a statistical method for studying the interrelations among various tests, the object of which is to discover what the tests have in common and whether these communalities can be ascribed to one or several factors that run through all or some of these tests.

Factor Analysis

• Technique for determining which variables tend to “clump together”– Which variables tend to be correlated with each

other and not with other variables• Clump of variables is called a factor• Degree to which variable is correlated with

a factor is called its factor loading

Factor Analysis

These variables“loadheavily”on Factor 1, but not on factor 2

These variables“loadheavily”on Factor 2, but not on factor 1Yes, “loading heavily” is a subjective call,

there are some stats that offer cut-off points.


How does this diagram relate to intelligence testingand factor analysis?

Measuring Intelligence1. When attempting to explain any phenomenon, scientists are

often attracted to the principle of parsimony: All other things being equal, the simplest explanation for a phenomenon is the best (i.e., the most elegant).

2. So, if performance on various “mental” tests rise and fall together, it may be parsimonious to attribute this pattern to a single underlying cause …


1. General Intelligence or g - according to Charles Spearman, a mental attribute that is called upon in any intellectual task a person has to perform.

2. Consistent with the notion of “g”, performance is positively correlated on tests about general knowledge, comprehension, arithmetic, and vocabulary.

Measuring Intelligence1. Some have speculated that ‘g’ might have two components…

2. Fluid Intelligence - The ability to respond to new and unusual problems, quickly and flexibly.

3. Crystallized Intelligence - The repertoire of previously acquired skills and information.

4. The two are strongly correlated (r=.6), so it might be reasonable to have one descriptor, ‘g’, for both.

5. However, there appears to be some ‘dissociation’… with age, crystallized intelligence increases while fluid intelligence decreases. (Anecdote: Rat’s with stem-cell brain transplants in hippocampus.)

Comparing Two Multivariate Approaches

1. Let’s compare and contrast two multivariate approaches…MANOVA and Factor Analysis

2. Both allow for the simultaneous evaluation of MULTIPLE dependent variables.

3. However, in MANOVA, the researcher specifies what the DVs (latent variables) are in advance! (Sort of like a hierarchical MR.)

4. The MANOVA (like all stats in the ANOVA family) addresses differences between means on a latent variable Between Subject Case – Differences between of among groups

Within Subject Case – Differences between among IV levels


1. By contrast, in (exploratory) Factor Analysis, the computer looks for correlations, and the researcher assigns names to the emergent ‘factors’ (latent variables) post hoc! (Sort of like a stepwise MR.)

2. The question addressed by (exploratory) Factor Analysis is NOT related to group differences, but rather…. what is the factor (latent variable) structure?

Examples: How many personality factors are there, and what are they?

How many musical-genre factors are there, and what are they?

How many psychological disorders are there, and what are they?Factor Analysis can show that whales or more similar to cows than to sharks.


1. One factor-analytic-like procedure is called “Principle Component Analysis”. This is an exploratory technique to reveal the primary axes (principle components) of a latent variable.

2. All exploratory factor-analytic-like procedures help us with the first of our four goals of science: DESCRIPTION!

3. Another goal of science is EXPLANATION, and that can entail empirical tests of a hypothesis ….

4. Confirmatory Factor Analysis – A multivariate technique for empirically testing hypotheses about the structure of, or relationships between, latent variables. (Like Hierarchical MR).

How to Read Results Involving Unfamiliar Statistical Techniques• Don’t panic!• Look for a p level• Look for indication of degree of association

or size of a difference• Reference an intermediate or advanced

statistics text• Take more statistics courses!

Procedures that Compare Groups

• Analysis of variance (ANOVA)

• Analysis of covariance (ANCOVA)

• Multivariate analysis of variance (MANOVA)

• Multivariate analysis of covariance (MANCOVA)

Part 2

Factor Analysis

Factor Analysis In SPSS

• Factor Analysis: A multivariate procedure– More than one dependent variable is involved

• One goal of factor analysis is to reduce the number of measured variables into a more manageable set of “factors”.– A factor is an abstract statistical construct – Factors can provide parsimonious descriptions


• Setting It Up– Be sure to have your measured variables

created in “variable view”.

Factor Analysis In SPSS• What to “click”

– Analyze Dimension Reduction –> Factor– “Slide” the variables of interest to the “variables” box.

Ignore the Selection variable box– Descriptives Button

• Click Coefficients (within the correlation matrix box)– Extraction Button

• Set the Method to Principal Components – Rotation Button

• Set the Method to Varimax– Ignore the Scores and Options Buttons


• Interpreting the SPSS output– Three of the output boxes are helpful

• Correlation Matrix Box– Shows the simple, “first order” correlations of all variables

• Total Variance Explained Box– Gives % of variance that each significant factor “accounts for”– The cumulative % is also shown for the entire model– Remember “loading” is the term for correlation

» A variable that “strongly loads” on a factor is strongly correlated with that factor.

• Rotated Component Box– Shows each variable’s “loading on” (correlation with) each factor.

Factor Analysis In SPSSCorrelation Matrix Box

Shows the first-order correlations among all variables in the model.

Factor Analysis In SPSSTotal Variance Explained Box

Components 1 and 2 are significant.Each “explains” (accounts for) about 50% of the variance.

The other remaining 8 components (#’s 3 through 10) are “computable”, but n.s.

Factor Analysis In SPSSRotated Component Box

Variables 1 through 5

“load strongly”an

Factor #1and

poorly on Factor #2

A “Component”

is aFactor

{}


“load strongly”an

Factor #2and

poorly on Factor #1

Factor Analysis In SPSSFactor 1

Factor 2


“load strongly”on

Factor #1and

poorly on Factor #2



Factor #2and

poorly on Factor #1

V1

V2

V3

V4

V5

V6

V8V7

V9V10

Factor Analysis In SPSSFactor 1

Factor 2



Factor #1and

poorly on Factor #2



Factor #2and

poorly on Factor #1

V1

V2

V3

V4

V5

V6

V8V7

V9V10

The “varimax” rotation method identifies orthogonal (90 deg) factors.


Factor 1

Factor 2



Factor #1and

“a little on” Factor #2



Factor #2and

“a little on”Factor #1

V1

V2

V3V4

V5

V6

V8V7

V9V10

“Oblique” rotation methods smaller angles (some non-zero correlation) between factors.


• Conclusion– In this example, we started with 10 measured

variables.– Factor Analysis parsimoniously reduced these 10

variables to two factors, i.e., two “latent variables”.• The two factors accounted for almost all the variance in

the data set of interest.– Now, it’s up to the researcher to provide a relevant

name for the two factors.• SPSS can’t do such interpretation for us!

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