Test 3: Polynomial and Factor Analysis Flashcards
In psychology the most common analysis is….
are linear models such as Linear Regression (straight lines).
What is the simplest way to identify if a curvilinear relationship is present?
The simplest way to identify if a curvilinear relationship
is present is by using polynomial terms in a linear
regression.
What are Polynomial Terms?
how many curvilinear relationships are present in the literature?
Basic level polynomial terms are multiples of IV:
X = Linear (same as Pearson Correlation 1x IV-DV) X2 = Quadratic (IV squared) X3 = Cubic (IV x IV x IV) X4 = Quartic (IV x IV x IV x IV) X5 = Quintic (IV x IV x IV x IV x IV)
o X is linear, anything else is curved i.e. curvilinear.
o Only 3-5% of all psychological literature reports
curvilinear relationships.
(4) Reasons why most Researchers don’t report
curvilinear relationships:
- When making prediction or proposing theories many
researchers fail to think of psychological phenomenon
in quadric terms. - Most researchers don’t even know how to compute
(calculate) a polynomial relationship using a linear
regression. - In addition, many researchers do not know how to
create a figure of a polynomial relationship. - Finally, many researchers do not know how to
interpret a polynomial relationship.
Two most famous examples of polynomial relationships in the literature?
what shapes are they?
(A) The Yerkes-Dodson Law:
o This is an unusual but possible data arrangement in
which it forms an invented-u shape.
o Famous study which identified that the relationship
between arousal and performance is NOT linear.
There is an optimal level of physiological arousal
that enhances performance but too much or too
little of it can hinder performance.
(B) Ebbinghaus Forgetting Curve:
o Identified that the rate of forgetting overtime is not
linear, people forget information rapidly at first but then
forgetting curve gets shallower and remains relatively
stable over time from a 1 hour since learning.
**In summary, there is somewhere between few and
some examples of polynomial relationships reported in
psychological data- mainly because researchers do not
look for them.
The “old school” method for computing a curvilinear relationship instead of putting it in a macro?
what variables are needed for graphing?
Set up variables in Jamovi and compute a regression:
a. Multiply the IV by itself to create the polynomial terms;
- X, X2, X3 etc.
b. Enter polynomial terms in ascending order (X, X2, X3
etc.) one step at a time i.e. a hierarchal regression.
c. There is a single DV (outcome variable) with no
transformations done.
d. Ask Jamovi to compute the R2, change in R2, M, SDs
so we can graph our curvilinear relationship and
interpret it (if one is present).
Forming a curvilinear hypothesis is ___ but important.
For example….
o Forming a hypothesis that is clear and ambiguous is
particularly hard to do for curvilinear relationships.
o Based on Pauls’ experience with both variables his
hypothesis is:
a. Mindfulness would exhibit an initial negative linear
relationship with depressive symptoms (Linear
Relationships).
b. Depressive symptoms will be lowest for medium-level
mindfulness scores (Curvilinear Relationship).
Note: curvilinear hypothesis has to components.
Considering whether a curvilinear relationship might be present? Steps we take…
why do we look at change in R2?
what variables are needed to graph it?
why do we need to reverse the order?
(A) Look at the scatterplot to determine if an additional
polynomial relationship may be present. One way is to
see if there a re lots of scores which fall a significant
distance from the regression line.
(B) Prepare the data into polynomial terms.
(C) Conduct Heirarchal Polynomial Regression Analysis,
adding each polynomial term in sequentially, one
step at a time.
(D) Look at the R2 (explained variance) to determine if
adding the polynomial terms produces a significant
change in explained variance in the DV (need to
double check the change is significant using the p-
value).
(E) Graph it: there are subtle variations in the basic forms
of polynomial relationships which make it crucial that
you always graph your relationship-to ensure your
interpretation and description of the relationship
between 2 variables is accurate.
(F) Interpret the graph:
(G) Do the Reverse Order:
Regressions are directional (unlike correaltions) thus,
the order of X-Y matter. To ensure you are getting an
accurate grasp of the exact relationship present you
will need to do the reverse order as well.
What are some key components in a polynomial graph to consider?
of curves
middle of x-axis
range of x-axis
in a curvilinear relationship the slope is not __
• Number of Curves:
- Linear = No Curve (X)
- Quadratic = 1x Curve (X2)
- Cubic = 2x Curve (X3)
- And so forth…
• Mid-point on the x-axis is the M (3.29) for Mindfulness
(IV)
• The x-axis ranges from -2SD to +2SD.
• The slope at the left side of the graph is much steeper
than the curve at the right side of the graph.
The slope of the line is not constantly (-.59) across the
range of mindfulness scores i.e. straight line: instead
we found evidence of a curvilinear relationship
(quadratic) which is steeper at low levels and flatter at
high levels of mindfulness.
Mindfulness is protective against depressive symptoms
up to a point (low to moderate levels).
*In other words, we have evidence that a quadratic
relationship better describes the association between
mindfulness and depressive symptoms. Thus, to report a
linear relationship would be in accurate and misleading
to the true association.
i.e. slope of -.59 only applies to a few individuals in the middle of the distribution!
Key Points about Polynomials?
> A polynomial relationship ___ the linear relationship.
Each polynomial term is….
important-if you don’t find a significant linear
relationship
o That linear relationships are not always the only
relationship present between two variables.
o A polynomial relationship qualifies the linear
relationship. In other words, demonstrates that the
slope of the line is not constant, as indicated by the
Pearson R, but provides a more exact description of the
curvilinear relationship between two variables.
o Each polynomial term is mathematically independent
from all others.
• E.g. X is independent from X2, X3 and X4 etc.
• Finding one significant relationship does not indicate
you will find a significant relationship in the next
sequential polynomial term.
• E.g. you can have a significant X, X2 and a non-
significant X3
• E.g. you can have a nonsignificant X, X2 and a
significant X3
*important-if you don’t find a significant linear relationship
one still should test for a quadratic relationship-because
polynomial terms are mathematically independent.
Possible Shapes of the Quadratic Slopes:
> two basic forms
> variations in…
o A quadratic can be either:
U- shaped
Inversed U-shape
o can be:
Tipping upward
Tipping downward
Level
*Although there are two basic shapes of quadratic
curves, they can have subtle variations. Thus, I cannot
stress the point enough, how important it is to ALWAYS
graph your relationship.
Polynomial’s and Concerns About Replicability:
A significant concern for many researchers about finding statistically significant curvilinear relationships is not being able to replicate these findings in a new setting, or with different participants.
Example: Looking at mindfulness and depressive symptoms in a community sample of adults (N = 483) that would replicate and extend the original finding.
Our output confirms that the original findings has been replicated with a community sample.
In fact, the R2 change value tells us that there was a 9.2% increase in explained variance.
Polynomial Graph (replication):
> protective
diminishing returns
floor effect
Graph is similar but distinctive and provides a more interesting finding because:
(A) Suggests mindfulness is protective for low and
medium levels of mindfulness BUT high levels of
mindfulness were not.
(B) This finding (and the previous one) is a pattern terms
“diminishing returns” i.e. a positive trait is generally
beneficial, but higher and higher levels of this
positive trait lead it becoming less and less effective
and to the point of being useless.
(C) Is there a “floor effect” present? No, the flattest part
of the curve is not in line with the minimum number
possible for depressive symptoms (i.e. 0).
Can you conduct a polynomial longitudinal study?
Yes.
e.g. Residualise the DV and include linear IV and
quadratic IV as predictor variables.
Remeber: longitudinal regressions calculate the CHANGE in DV overtime which is predicted by our IV[s].
Does the order of X-Y in a longitudinal polynomial regression matter?
(A) It does matter the direction of the analysis; X – Y or Y
– X.
(B) Why? Because the two directions of a regression are
not mathematically equivalent! The multiplicative term
(x2 or y2) is different.
(C) This may be bad news for some; because it doubles
the number of analyses you need to make if you are
trying to identify a curvilinear relationship between 2x
variables.
(D) Highlights the importance of collecting longitudinal
data so one can identify the direction of causality.
Quadratic examples:
> Caffiene and Gratitude
> Weed and Happiness
(A) Haivng up to 3 cups of coffee increase one’s
gratitude but begins to decrease after that.
(B) Having moderate to small amounts of weed makes
you happier but having high levels makes you less
happier.
Final Points on Polynomial Relationships:
> Assumption of ___
Can actually be found in ___, ___, ___ or ___.
o Most analyses in psychological research are based on
the assumption of linearity, i.e., that straight lines can
adequately describe the relationship
o However, curvilinear relationships can be found in all
kinds of analyses: concurrent or longitudinal;
experimental or subject variable. The possibilities are
endless.
o The assumption of linearity is often too simplistic and
can be misleading.
What Does Factor Analysis Do?
> how many clusters do we want?
> what external factors can fluctuate this number?
It’s an inferential analysis method which is prominent within the social sciences: education, criminology or business etc.
- It’s an analytic technique which attempts to
find groupings of items that constitute sub-
factors within a single measure. - Normally, we find 2-5 clusters of items.
- The number of clusters of items found
within a single measure depends on a) the
natural association between the items b)
the sophistication of the software and its
ability to identify clusters of items in the
data.
Factor Analysis Terminology is confusing:
PCA, EFA and Factor Anlaysis
(3) main terms that are commonly used interchangeably but there are subtle variations in their meaning:
• Factor Analysis:
- Is the original term coined for this type of
analysis back when the computation was
done by hand in the 1920’s. It took hours to
days to calculate a factor analysis by hand
and the term has stuck over the years.
- However, like all statistical methods they
have been developed over the years
resulting in there being many versions of
analysis (6-8) which fall under the factor
analysis umbrella.
• Principle Components Analysis (PCA):
- Is the preferred Factor Analysis method
and most commonly if studies say they
conducted a factor analysis they mean a
PCA.
• Exploratory Factor Analysis (EFA):
- In today’s lecture, we will be conducting
and referring to an EFA.
Factor Analysis: In Jamovi there are two basic types of Factor Analysis you can choose from-
(A) Scale Analysis:
- Reliability Analysisn i.e. Cronbach's
alpha
(B) Data Reduction:
- Principal Component Analysis (PCA)
- Exploratory Factor Analysis (EFA)
- Confirmatory Factor Analysis (CFA)
Note: PCA and EFA should give us the same results.
*when we want to reduce the data into a
small set of factors before conducting
further computations.
GERM: General Emotion Regulation Measure distinctive in terms of…
and asks what 3 things?
• GERM scale was developed to extend this
existing scale to capture individuals’ goals
or motivations for engaging in these
emotion regulation behaviors i.e. WHY
does someone engage in cognitive
reappraisal or expression suppression?
• To do this their scale asked individuals
“how much they:”
Try to experience emotion
Try to avoid experiencing an emotion
Actually, experience an emotion
Factor Anlaysis: Maximum Likelihood
> Is the most commonly used extraction
method that set parameters on multivariate data to
ensure the maximum likelihood of finding
associations within the data.
How we want to estimate any associations
between the 24 items, item to item,
therefore there is a lot of comparisons
being made.
Factor Analysis: Number of Factors-
> 2 methods…
what is a parallel analysis? what does it calculate?
Parallel Analysis:
• The parallel analysis is a stimulation, an
estimate of the size of eigenvalues based
on the number of items (24) in conjunction
with the number of participants (218).
• Thereby, it estimates the eigenvalues
occurring by chance.
• The critical points are where the estimated
eigenvalues (yellow dotted line) intercepts
with the blue line (observed eigenvalues).
(3) Goals of Factor Analysis:
The goal of “data reduction” techniques, like EFA, is to:
Identify the proper number of factors; not
too few (1) or too many 5+
The factors found should be internally
consistent; High Cronbach’s alpha i.e.
above .70.
The correlation[s] between factors should
be low to moderate (not high!);
Why do we not want a high correlation between our factors?
If we have an excessively high correlation between two factors (.90) then we are actually only measuring one factor not two and we should combine them to make a single factor.
Both assumption checks should be….
Significant (p-value < .05) or above .60.
What is the assmuption made when conducting a Factor Analysis?
The assumption of an EFA is that the list of items does NOT constitute a single factor. Rather we conduct an EFA when we predict that our scale is measuring 2+ factors (clusters of items) within the same scale.
How do we determine how many factors there are?
o Our decision on how many factors are
being measured by the same scale is
largely determined by our analysis of the
scree plot.
o A “Scree” is a mass of loose stones that
form or cover a slope on the mountain.
o The test was developed by Raymond
Cattell, and it helped determine the total
number of factors in a PCA.
a. Identify the kink/elbow (the point where
the mountain and the scree intercept).
b. Note: the number of factors is the
numerical value immediately to the left of
the kink/elbow.
What is an eigenvalue? How many are there in a factorial analysis?
Definition: “numerical values that signify how strongly items intercorrelate for a given factor”.
o Eigenvalues are computed as a way to
identify the clustering of items.
o We want only a few of the eigenvalues to
be higher in numerical value i.e. indicative
of a high intercorrelation.
o The scree plot reports the sizes of the
eigenvalues and we use the plot to identify
how many sub-factors are present in our
GERM scale.
Note: since there are 24 items in the GERM scale there will be 24 eigenvalues!!!
Why can Jamovi sometimes be wrong in telling us how many factors there are?
When the obtained eignenvalue and the theroretically predicted eigenvalues sit close together on the scatterplot Jamovi can accidently mistake it for being a significant eigenvalue.
What are the yellow dots on a scree-plot?
The parallel analysis
What are double loading items?
When items in the factor loadings overlap or are loaded into more than one factor.
what is uniqueness?
Calculates the unique variance of this item that is not shared with the other items.
If this number is too high than it indicates that the item does not group nicely into the factors identified.
e.g. contempt was .767 (with 3-factor structure) and .913 (2-factor structure).
Factor Analysis: Interesting Examination of Internal Reliability
- All 24 items: .83
- 12 try-positive items: .87
- 12 try-negative items: .89
*many researchers would have stopped once they saw their scale had reliable internal validity without examining a 2-factor structure. Based on Paul’s theoretical understanding on emotional regulation he continued with a factor analysis.
However, two important bits of information point us in the direction of two factors here:
1) the EFA clearly resulted in two
distinguishable factors, and
2) the correlation between PosTry and
PosNeg = -.061, p = .38.
What does this information tell us? It seems that all 24 of the ‘try to experience’ items loaded on a single factor of ‘trying to experience an emotion’, but trying to experience positive emotions was unrelated to trying to experience negative emotions.
(4) common errors people make in factor analysis?
- Using the ‘eigenvalues greater than one’
rule (will often mislead you, although it was
right here) - Misinterpreting the kink in the scree plot.
The proper number of factors is usually
the number to the left of the kink. - Looking at all indicators—no one output
determines the correct number of factors.
One must consider all indicators. - ‘Forcing’ a particular solution based on
your theoretical views (or biases). Let the
data tell you what is going on there.
what do you conduct a CFA for?
Confirmatory Factor Analysis (CFA):
- Used when you have an existing scale and
you would like to confirm the already
identified factor structure on a new set of
data.
- After gaining significant EFA findings I
would collect more data and conduct a
CFA to test the 2 factor solution on a
different set of data.
- You’re trying to ‘confirm’ whether this pre-
existing factor structure applies to your
data or not.
What unqiue information does a CFA give you that is not in a EFA?
> Model Fit: tells us that the predicted
model (two factors) fits the raw data well if
the p-value is < .05.
What do we want for CFA: Factor loadings and model fit?
(A) Factor Loadings
- We want all of these loadings to be p < .05
and of at least medium strength. They are
all good (except for contempt, which is
nevertheless significant).
(B) Correlation
- What is the correlation between PosTry and
NegTry? It is very similar to what I obtained
with a raw correlation before (-.06 vs. -.08,
both non-sig).
- Model fit tells me that the predicted model
(two factors) fits the raw data well.
what is an orthoginal or oblimin rotation?
> example of each
Rotation refers to a mathematical transformation of the data to minimise or maximise correlation between factors.
Orthogonal is the case where you force the factors to
be maximally uncorrelated, whereas oblique allows
more correlation among the factors. Varimax is
orthogonal, and is often used. Oblimin, for example, is an oblique rotation.
We use different types of rotation for different goals with measures. Rotation, if done correctly, clarifies relationships between factors.
