Factor analysis is a statistical technique used to reduce the dimensionality of a set of correlated variables by identifying underlying factors. It seeks to explain the variance between observed variables in terms of a smaller number of latent factors. The document describes how factor analysis works, including that it begins with a correlation matrix and aims to group highly correlated variables together into factors while variables with low correlations are separated into different factors. Factor analysis can help provide a clearer understanding of the relationships in a dataset and enable subsequent analyses using the identified factors.
Introduces and explains the use of multiple linear regression, a multivariate correlational statistical technique. For more info, see the lecture page at http://goo.gl/CeBsv. See also the slides for the MLR II lecture http://www.slideshare.net/jtneill/multiple-linear-regression-ii
Introduces and explains the use of multiple linear regression, a multivariate correlational statistical technique. For more info, see the lecture page at http://goo.gl/CeBsv. See also the slides for the MLR II lecture http://www.slideshare.net/jtneill/multiple-linear-regression-ii
Factor analysis in marketing research intentions to designate a large number of variables or questions by using a reduced set of underlying variables, called factors. Factor analysis is unsurpassed when cast-off to simplify complex data sets with many variables
Factor Analysis is a statistical tool that measures the impact of a few un-observed variables called factors on a large number of observed variables. It is often used to determine a linear relationship between variables before subjecting them to further analysis.
Multiple regression analysis is a powerful technique used for predicting the unknown value of a variable from the known value of two or more variables.
Aleksey Narko
II year Management
Econometrics Final Project
I took the data set about the wealth of nations and in particular the dependence between the population and total wealth of the country (nation).
Source: http://data.worldbank.org/data-catalog/wealth-of-nations
2011 WSB-NLU
Professor: Jacek Leskow
Descriptive statistics are methods of describing the characteristics of a data set. It includes calculating things such as the average of the data, its spread and the shape it produces.
Factor Analysis as a Tool for Survey Analysissajjalp
Factor analysis is particularly suitable to extract few factors from the large number of related variables
to a more manageable number, prior to using them in other analysis such as multiple regression or multivariate
analysis of variance. It can be beneficial in developing of a questionnaire. Sometimes adding more statements in the
questionnaire fail to give clear understanding of the variables. With the help of factor analysis, irrelevant questions
can be removed from the final questionnaire. This study proposed a factor analysis to identify the factors underlying
the variables of a questionnaire to measure tourist satisfaction. In this study, Kaiser-Meyer-Olkin measure of
sampling adequacy and Bartlett’s test of Sphericity are used to assess the factorability of the data. Determinant score
is calculated to examine the multicollinearity among the variables. To determine the number of factors to be
extracted, Kaiser’s Criterion and Scree test are examined. Varimax orthogonal factor rotation method is applied to
minimize the number of variables that have high loadings on each factor. The internal consistency is confirmed by
calculating Cronbach’s alpha and composite reliability to test the instrument accuracy. The convergent validity is
established when average variance extracted is greater than or equal to 0.5. The results have revealed that the factor
analysis not only allows detecting irrelevant items but will also allow extracting the valuable factors from the data
set of a questionnaire survey. The application of factor analysis for questionnaire evaluation provides very valuable
inputs to the decision makers to focus on few important factors rather than a large number of parameters.
Keywords: factor analysis, Kaiser-Meyer-Olkin, Bartlett’s test of Sphericity, determinant score, Kaiser’s criterion,
Scree test, Varimax
Factor analysis in marketing research intentions to designate a large number of variables or questions by using a reduced set of underlying variables, called factors. Factor analysis is unsurpassed when cast-off to simplify complex data sets with many variables
Factor Analysis is a statistical tool that measures the impact of a few un-observed variables called factors on a large number of observed variables. It is often used to determine a linear relationship between variables before subjecting them to further analysis.
Multiple regression analysis is a powerful technique used for predicting the unknown value of a variable from the known value of two or more variables.
Aleksey Narko
II year Management
Econometrics Final Project
I took the data set about the wealth of nations and in particular the dependence between the population and total wealth of the country (nation).
Source: http://data.worldbank.org/data-catalog/wealth-of-nations
2011 WSB-NLU
Professor: Jacek Leskow
Descriptive statistics are methods of describing the characteristics of a data set. It includes calculating things such as the average of the data, its spread and the shape it produces.
Factor Analysis as a Tool for Survey Analysissajjalp
Factor analysis is particularly suitable to extract few factors from the large number of related variables
to a more manageable number, prior to using them in other analysis such as multiple regression or multivariate
analysis of variance. It can be beneficial in developing of a questionnaire. Sometimes adding more statements in the
questionnaire fail to give clear understanding of the variables. With the help of factor analysis, irrelevant questions
can be removed from the final questionnaire. This study proposed a factor analysis to identify the factors underlying
the variables of a questionnaire to measure tourist satisfaction. In this study, Kaiser-Meyer-Olkin measure of
sampling adequacy and Bartlett’s test of Sphericity are used to assess the factorability of the data. Determinant score
is calculated to examine the multicollinearity among the variables. To determine the number of factors to be
extracted, Kaiser’s Criterion and Scree test are examined. Varimax orthogonal factor rotation method is applied to
minimize the number of variables that have high loadings on each factor. The internal consistency is confirmed by
calculating Cronbach’s alpha and composite reliability to test the instrument accuracy. The convergent validity is
established when average variance extracted is greater than or equal to 0.5. The results have revealed that the factor
analysis not only allows detecting irrelevant items but will also allow extracting the valuable factors from the data
set of a questionnaire survey. The application of factor analysis for questionnaire evaluation provides very valuable
inputs to the decision makers to focus on few important factors rather than a large number of parameters.
Keywords: factor analysis, Kaiser-Meyer-Olkin, Bartlett’s test of Sphericity, determinant score, Kaiser’s criterion,
Scree test, Varimax
Data Analysis & Interpretation and Report WritingSOMASUNDARAM T
Statistical Methods for Data Analysis (Only Theory), Meaning of Interpretation, Technique of Interpretation, Significance of Report Writing, Steps, Layout of Research Report, Types of Research Reports, Precautions while writing research reports
Forecasting Academic Performance using Multiple Linear Regressionijtsrd
Regression is one of the most powerful statistical methods used in educational researches. This paper shows the important instance of regression methodology called Multiple Linear Regression MLR and proposes a framework of the forecasting of the students' test scores, based on Intelligence Quotient IQ and the number of hours that the students studied. This paper was applied the aid of the Statistical Package for Social Sciences SPSS version 23 and PYTHON version 3.7. Yee Mon Khaing | Aung Cho "Forecasting Academic Performance using Multiple Linear Regression" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26517.pdfPaper URL: https://www.ijtsrd.com/computer-science/data-miining/26517/forecasting-academic-performance-using-multiple-linear-regression/yee-mon-khaing
The use of data and its modelling in science provides meaningful interpretation of real world problems. This presentation provides an easy to understand overview of data visualization and analytics , and snippets of data science applications using R - programming.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Cancer cell metabolism: special Reference to Lactate Pathway
Factor analysis in Spss
1. 1
Factor Analysis
Factor analysis attempts to bring inter-correlated
variables together under more general, underlying
variables.
More specifically, the goal of factor analysis is to
reduce “the dimensionality of the original space and
to give an interpretation to the new space, spanned
by a reduced number of new dimensions which are
supposed to underlie the old ones” (Rietveld & Van
Hout 1993:254).
Rietveld, T. & Van Hout, R. (1993). Statistical Techniques for the Study of Language and
Language Behaviour. Berlin – New York: Mouton de Gruyter.
Friday, November 11, 2016 05:15 PM
2. 2
Factor Analysis
Or to explain the variance in the observed variables
in terms of underlying latent factors” (Habing
2003).
Thus, factor analysis offers not only the possibility
of gaining a clear view of the data, but also the
possibility of using the output in subsequent
analyses (Field 2000; Rietveld & Van Hout 1993).
Field, A. (2000). Discovering Statistics using SPSS for Windows. London – Thousand Oaks
– New Delhi: Sage publications.
Rietveld, T. & Van Hout, R. (1993). Statistical Techniques for the Study of Language and
Language Behaviour. Berlin – New York: Mouton de Gruyter.
Friday, November 11, 2016 05:15 PM
3. 3
Factor Analysis
The starting point of factor analysis is a correlation
matrix, in which the inter-correlations between the
studied variables are presented. The dimensionality
of this matrix can be reduced by “looking for
variables that correlate highly with a group of other
variables, but correlate very badly with variables
outside of that group” (Field 2000: 424). These
variables with high inter-correlations could well
measure one underlying variable, which is called a
‘factor’.
Field, A. (2000). Discovering Statistics using SPSS for Windows. London – Thousand Oaks
– New Delhi: Sage publications.
Friday, November 11, 2016 05:15 PM
4. 4
Factor Analysis
Factor analysis is a method of dimension reduction.
It does this by seeking underlying unobservable
(latent) variables that are reflected in the observed
variables (manifest variables).
Friday, November 11, 2016 05:15 PM
5. 5
Factor Analysis
There are many different methods that can be used
to conduct a factor analysis
There are many different types of rotations that
can be done after the initial extraction of factors.
You also need to determine the number of factors
that you want to extract.
6. 6
Factor Analysis
Given the number of factor analytic techniques and
options, it is not surprising that different analysts
could reach very different results analysing the
same data set.
7. 7
Factor Analysis
However, all analysts are looking for a simple
structure.
Simple structure is a pattern of results such that
each variable loads highly onto one and only one
factor.
8. 8
Factor Analysis
Factor analysis is a technique that requires a large
sample size.
Factor analysis is based on the correlation matrix of
the variables involved, and correlations usually need
a large sample size before they stabilize.
9. 9
Factor Analysis
As a rule of thumb, a bare minimum of 10 observations
per variable is necessary to avoid computational
difficulties.
Number of Cases Prospects
50 very poor
100 poor
200 fair
300 good
500 very good
1000 excellent
Comrey and Lee (1992) A First Course In Factor Analysis
10. 10
Factor Analysis
In this example I have included many options, while
you may not wish to use all of these options, I have
included them here to aid in the explanation of the
analysis.
11. 11
Factor Analysis
In this example we examine students assessment of
academic courses. We restrict attention to 12 variables.
Item 13 INSTRUCTOR WELL PREPARED
Item 14 INSTRUCTOR SCHOLARLY GRASP
Item 15 INSTRUCTOR CONFIDENCE
Item 16 INSTRUCTOR FOCUS LECTURES
Item 17 INSTRUCTOR USES CLEAR RELEVANT EXAMPLES
Item 18 INSTRUCTOR SENSITIVE TO STUDENTS
Item 19 INSTRUCTOR ALLOWS ME TO ASK QUESTIONS
Item 20 INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS
Item 21 INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING
Item 22 I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION
Item 23 COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS
Item 24 COMPARED TO OTHER COURSES THIS COURSE WAS
Scored on a five point Likert scale, seven is better.
12. 12
Factor Analysis
In this example we examine students assessment of
academic courses. We restrict attention to 12 variables.
Scored on a five point Likert scale.
14. 14
Factor Analysis
Select variables 13-24 that is “instructor well
prepared” to “compared to other courses this course
was”. By using the arrow button.
Use the buttons at the side of the screen to set additional options.
15. 15
Factor Analysis
Use the buttons at the side of the previous screen to set
the Descriptives. Employ the Continue button to return to
the main Factor Analysis screen.
Note the request for a determinant.
16. 16
Factor Analysis
Use the buttons at the side of the main screen to set the
Extraction. Employ the Continue button to return to the
main Factor Analysis screen.
Note the request for Principal axis factoring, 3 factors
and a scree plot.
17. 17
Factor Analysis
Use the buttons at the side of the main screen to set the
Rotation (Varimax). Employ the Continue button to return
to the main Factor Analysis screen.
18. 18
Factor Analysis
Varimax rotation tries to maximize the variance of each of
the factors, so the total amount of variance accounted for
is redistributed over the three extracted factors.
19. 19
Factor Analysis
Select the OK button to proceed with the analysis, or
Paste to preserve the syntax.
Syntax for varimax and 3 factors, alternatives promax and 2
factor
/variables item13 item14 item15 item16 item17 item18 item19 item20
item21 item22 item23 item24
/print initial det kmo repr extraction rotation fscore univaratiate
/format blank(.30)
/plot eigen rotation
/criteria factors(3)
/extraction paf
/rotation varimax
/method = correlation.
20. 20
Factor Analysis
The descriptive statistics
table is output because we
used the univariate option.
Mean - These are the means
of the variables used in the
factor analysis.
Are they meaningful for a
Likert scale!
Norman, G. (2010). Likert scales, levels of
measurement and the “laws” of statistics.
Advances in health sciences education, 15(5),
625-632.
Descriptive Statistics
4.46 .729 1365
4.53 .700 1365
4.45 .732 1365
4.28 .829 1365
4.17 .895 1365
3.93 1.035 1365
4.08 .964 1365
3.78 .909 1365
3.77 .984 1365
3.61 1.116 1365
3.81 .957 1365
3.67 .926 1365
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Mean Std. Deviation Analysis N
21. 21
Factor Analysis
The descriptive statistics
table is output because we
used the univariate option.
Std. Deviation - These
are the standard
deviations of the variables
used in the factor
analysis.
Are they meaningful for a
Likert scale!
Descriptive Statistics
4.46 .729 1365
4.53 .700 1365
4.45 .732 1365
4.28 .829 1365
4.17 .895 1365
3.93 1.035 1365
4.08 .964 1365
3.78 .909 1365
3.77 .984 1365
3.61 1.116 1365
3.81 .957 1365
3.67 .926 1365
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Mean Std. Deviation Analysis N
22. 22
Factor Analysis
The descriptive statistics
table is output because we
used the univariate option.
Analysis N - This is the
number of cases used in
the factor analysis.
Note N is 1365.
Descriptive Statistics
4.46 .729 1365
4.53 .700 1365
4.45 .732 1365
4.28 .829 1365
4.17 .895 1365
3.93 1.035 1365
4.08 .964 1365
3.78 .909 1365
3.77 .984 1365
3.61 1.116 1365
3.81 .957 1365
3.67 .926 1365
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Mean Std. Deviation Analysis N
23. 23
Factor Analysis
The correlation matrix is included in the output
because we used the determinant option.
All we want to see in this table is that the
determinant is not 0.
If the determinant is 0, then there will be
computational problems with the factor analysis, and
SPSS may issue a warning message or be unable to
complete the factor analysis.
Correlation Matrixa
Determinant = .002a.
24. 24
Factor Analysis
Kaiser-Meyer-Olkin Measure of Sampling Adequacy
This measure varies between 0 and 1, and values closer
to 1 are better. A value of 0.6 is a suggested minimum.
KMO and Bartlett's Test
.934
8676.712
66
.000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity
25. 25
Factor Analysis
Bartlett's Test of Sphericity (see the ANOVA
slides) - This tests the null hypothesis that the
correlation matrix is an identity matrix. An identity
matrix is matrix in which all of the diagonal elements
are 1 and all off diagonal elements are 0 (indicates a
lack of correlation). You want to reject this null
hypothesis.
KMO and Bartlett's Test
.934
8676.712
66
.000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity
26. 26
Factor Analysis
Taken together, these tests provide a minimum
standard, which should be passed before a factor
analysis (or a principal components analysis) should be
conducted.
KMO and Bartlett's Test
.934
8676.712
66
.000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity
27. 27
Factor Analysis
Communalities - This
is the proportion of
each variable's
variance that can be
explained by the
factors (e.g., the
underlying latent
continua).
Communalities
.564 .676
.551 .619
.538 .592
.447 .468
.585 .623
.572 .679
.456 .576
.326 .369
.516 .549
.397 .444
.662 .791
.526 .632
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Initial Extraction
Extraction Method: Principal Axis Factoring.
28. 28
Factor Analysis
Initial - With principal
factor axis factoring, the
initial values on the diagonal
of the correlation matrix
are determined by the
squared multiple correlation
of the variable with the
other variables. For
example, if you regressed
items 14 through 24 on item
13, the squared multiple
correlation coefficient
would be 0.564.
Communalities
.564 .676
.551 .619
.538 .592
.447 .468
.585 .623
.572 .679
.456 .576
.326 .369
.516 .549
.397 .444
.662 .791
.526 .632
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Initial Extraction
Extraction Method: Principal Axis Factoring.
29. 29
Factor Analysis
Extraction - The values in
this column indicate the
proportion of each
variable's variance that can
be explained by the retained
factors. Variables with high
values are well represented
in the common factor space,
while variables with low
values are not well
represented. (In this
example, we don't have any
particularly low values.)
Communalities
.564 .676
.551 .619
.538 .592
.447 .468
.585 .623
.572 .679
.456 .576
.326 .369
.516 .549
.397 .444
.662 .791
.526 .632
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Initial Extraction
Extraction Method: Principal Axis Factoring.
30. 30
Factor Analysis
Factor - The initial number of factors is the same as
the number of variables used in the factor analysis.
However, not all 12 factors will be retained. In this
example, only the first three factors will be retained
(as we requested).
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.
31. 31
Factor Analysis
Initial Eigenvalues - Eigenvalues are the variances of the
factors. Because we conducted our factor analysis on the
correlation matrix, the variables are standardized, which
means that the each variable has a variance of 1, and the
total variance is equal to the number of variables used in
the analysis, in this case, 12.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.
32. 32
Factor Analysis
Initial Eigenvalues - Total - This column contains the
eigenvalues. The first factor will always account for the
most variance (and hence have the highest eigenvalue), and
the next factor will account for as much of the left over
variance as it can, and so on. Hence, each successive factor
will account for less and less variance.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.
33. 33
Factor Analysis
Initial Eigenvalues - % of Variance - This column contains
the percent of total variance accounted for by each factor
(6.249/12 = .52 or 52%).
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.
34. 34
Factor Analysis
Initial Eigenvalues - Cumulative % - This column contains
the cumulative percentage of variance accounted for by
the current and all preceding factors. For example, the
third row shows a value of 68.313. This means that the
first three factors together account for 68.313% of the
total variance.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.
35. 35
Factor Analysis
Extraction Sums of Squared Loadings - The number of
rows in this panel of the table correspond to the number
of factors retained. The values are based on the common
variance (of the retained factors). The values in this panel
of the table will always be lower than the values in the left
panel of the table, because they are based on the common
variance, which is always smaller than the total variance.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.
36. 36
Factor Analysis
Rotation Sums of Squared Loadings - The values in this
panel of the table represent the distribution of the
variance after the varimax rotation. Varimax rotation
tries to maximize the variance of each of the factors, so
the total amount of variance accounted for is
redistributed over the three extracted factors.
Note the more even split.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.
37. 37
Factor Analysis
The scree plot graphs the eigenvalue (variance) against the
factor number. You can see these values in the first two
columns of the variance explained table.
38. 38
Factor Analysis
From the third factor on, you can see that the line is almost
flat, meaning the each successive factor is accounting for
smaller and smaller amounts of the total variance.
You need to locate this,
so called, elbow!
In other words, when
the drop ceases and the
curve makes an elbow
toward a less steep
decline.
39. 39
Factor Analysis
Factor Matrix - This table
contains the unrotated
factor loadings, which are
the correlations between the
variable and the factor.
Because these are
correlations, possible values
range from -1 to +1. It is
usual to not report any
correlations that are less
than |.3|. As shown.
Factor Matrixa
.713 -.398
.703 -.339
.721
.648
.783
.740 .345
.616 .415
.550
.732
.613
.819 -.345
.695 -.386
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
1 2 3
Factor
Extraction Method: Principal Axis Factoring.
3 factors extracted. 7 iterations required.a.
40. 40
Factor Analysis
Factor - The columns under
this heading are the
unrotated factors that have
been extracted. As you can
see by the footnote provided
by SPSS, three factors were
extracted (the three
factors that we requested).
Factor Matrixa
.713 -.398
.703 -.339
.721
.648
.783
.740 .345
.616 .415
.550
.732
.613
.819 -.345
.695 -.386
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
1 2 3
Factor
Extraction Method: Principal Axis Factoring.
3 factors extracted. 7 iterations required.a.
41. 41
Factor Analysis
The plot shows the
items (variables) in
the rotated factor
space.
While this picture
may not be
particularly
helpful, when you
get this graph in
the SPSS output,
you can
interactively
rotate it.
43. 43
Factor Analysis
Another run of the factor analysis program is conducted
with a promax rotation. It is included to show how
different the rotated solutions can be, and to better
illustrate what is meant by simple structure.
As you will see with an oblique rotation, such as a promax
rotation, the factors are permitted to be correlated with
one another. With an orthogonal rotation, such as the
varimax shown above, the factors are not permitted to
be correlated (they are orthogonal to one another).
Oblique rotations, such as promax, produce both factor
pattern and factor structure matrices. For orthogonal
rotations, such as varimax and equimax, the factor
structure and the factor pattern matrices are the same.
44. 44
Factor Analysis
Use the buttons at the bottom of the screen to set the
alternate Rotation, employ the Continue button to return to
the main Factor Analysis screen.
46. 46
Factor Analysis
For a recent review see Factor Analysis at 100. Historical
Developments and Future Directions. By Robert Cudeck,
and Robert C. MacCallum (Eds.). Lawrence Earlbaum
Associates, Mahwah, NJ, 2007, xiii+381 pp., ISBN:978-0-
8058-5347-6 (hardcover), and, ISBN 978-0-8058-6212-6
(paperback).
47. 47
Factor Analysis
Summary
Factor Analysis like principal components is used to
summarise the data covariance structure in a smaller
number of dimensions. The emphasis is the
identification of underlying “factors” that might explain
the dimensions associated with large data variability.
A Beginner’s Guide to Factor Analysis: Focusing on
Exploratory Factor Analysis
An Gie Yong and Sean Pearce
Tutorials in Quantitative Methods for Psychology 2013
9(2) 79-94
48. 48
Factor Analysis
Principal Components Analysis and Factor Analysis share the search for
a common structure characterized by few common components, usually
known as “scores” that determine the observed variables contained in
matrix X.
However, the two methods differ on the characterization of the
scores as well as on the technique adopted for selecting their true
number.
In Principal Components Analysis the scores are the orthogonalised
principal components obtained through rotation, while in Factor
Analysis the scores are latent variables determined by unobserved
factors and loadings which involve idiosyncratic error terms.
The dimension reduction of matrix X implemented by each method
produces a set of fewer homogenous variables – the true scores –
which contain most of the model’s information.
49. 49
Factor Analysis
Summary
Principal Components is used to help understand the
covariance structure in the original variables and/or to
create a smaller number of variables using this
structure.
For Principal Components, see next weeks lecture.
50. 50
Factor Analysis
Overview of the steps in a
factor analysis. From: Rietveld
& Van Hout (1993: 291).
Rietveld, T. & Van Hout, R. (1993).
Statistical Techniques for the Study of
Language and
Language Behaviour. Berlin – New York:
Mouton de Gruyter.
51. 51
Factor Analysis
After having obtained the correlation matrix, it is time to decide which
type of analysis to use: factor analysis or principal component analysis. The
main difference between these types of analysis lies in the way the
communalities are used. In principal component analysis it is assumed that
the communalities are initially 1. In other words, principal component
analysis assumes that the total variance of the variables can be accounted
for by means of its components (or factors), and hence that there is no
error variance. On the other hand, factor analysis does assume error
variance. This is reflected in the fact that in factor analysis the
communalities have to estimated, which makes factor analysis more
complicated than principal component analysis, but also more conservative.
For further details see "Factor Analysis" Kootstra 2004
52. 52
SPSS Tips
Now you should go and try for yourself.
Each week our cluster (5.05) is booked for 2 hours
after this session. This will enable you to come and go
as you please.
Obviously other timetabled sessions for this module
take precedence.
Editor's Notes
Mike Cox, Newcastle University, me fecit 01/10/2015