Measurement and Dimensionality

Data Analysis for Psychology in R 3

Dr Josiah King

Psychology, PPLS

University of Edinburgh

Course Overview

multilevel modelling working with group structured data	regression refresher
	the multilevel model
	more complex groupings
	centering, assumptions, and diagnostics
	recap
factor analysis working with multi-item measures	measurement and dimensionality
	exploring underlying constructs (EFA)
	testing theoretical models (CFA)
	reliability and validity
	recap & exam prep

A confession

This week

• Measurement!
• Multi-item measures
• Scoring (Scale scores and “PCA”)

Measurement

Thorndike’s Credo

“Whatever exists at all exists in some amount. To know it thoroughly involves knowing its quantity as well as its quality.”
(Edward L. Thorndike. 1918)

What is measurement?

“The process of assigning numbers to represent properties”
(Campbell, 1920)

“The assignment of numbers to objects or events according to rules”
(Stevens, 1947)

• At the foundations of all scientific inquiry

• Too often taken for granted

• Key question: What do our numbers represent?

Quantifying the unquantifiable?

• Many things are experienced qualitatively, as matters of degree, and are not easily amenable to direct quantification

• We should always question whether our numbers truly represent a continuous quantity with homogeneous units

Measurement in Psychology

• Many psychological phenomena cannot be observed directly:

  • thoughts, feelings, behaviours etc.

• Grounded in natural language that we discuss every day

  • “aggression”, “intelligence”, “anxiety”

• Definitions are often fuzzy and lack consensus

• Result: Confusion and complexity in measurement, with scientific definitions often diverging from non-scientific definitions

What is a “construct”?

What

• The thing we are intending to quantify

• In psychology, education etc:

  • a (hopefully useful) abstraction about the world, derived from natural observations
  • often termed a “latent trait” or “ability” (typically denoted \(\theta\)).

Why

• Simplifies the world and provide a shared language for scientific study

• Study the same phenomena across diverse contexts.

  • e.g., what does “leadership” look like in hunter-gatherer societies, in the military and in the music industry?

Constructs, measures and data

Example

Are older people more satisfied with life? 112 people from 12 different dwellings (cities/towns) in Scotland. Information on their ages and some measure of life satisfaction.

d3 <- read_csv("https://uoepsy.github.io/data/lmm_lifesatscot.csv")
head(d3)

# A tibble: 6 × 4
    age lifesat dwelling size 
  <dbl>   <dbl> <chr>    <chr>
1    40      31 Aberdeen >100k
2    45      56 Glasgow  >100k
3    40      51 Glasgow  >100k
4    40      55 Dundee   >100k
5    40      41 Dundee   >100k
6    55      69 Perth    <100k

• Did anyone stop to think - What is lifesat (i.e., life satisfaction)?

• is someone scoring 69 more than twice as “satisfied with life” as someone scoring 31?

Impact of differences in perspectives

• Different operationalisations make it difficult to consolidate findings:

  • Jingle fallacy - Using same name to denote different things
  • Jangle fallacy - Using different names to denote same thing

“Nobody wants to use somebody else’s toothbrush”
(Elson et al., 2023)

Impact of differences in perspectives

• Different operationalisations make it difficult to consolidate findings:

  • Jingle fallacy - Using same name to denote different things
  • Jangle fallacy - Using different names to denote same thing

“Nobody wants to use somebody else’s toothbrush”
(Elson et al., 2023)

Psychometrics

• Scientific discipline concerned with the construction of psychological measurements

• Connects theoretical constructs to their domains of observable behaviours

Applications of psychometrics

• Education

  • Aptitude / ability tests (i.e., standard school tests)
  • Vocational tests

• Business

  • Selection (e.g., personality, skills)
  • Development (e.g., interests, leadership)
  • Performance (e.g., well-being, engagement)

• Health

  • Mental health symptoms e.g., anxiety
  • Clinical diagnoses e.g., personality disorders

• Key takeaway: People make life-changing decisions using psychometric evidence every day

Representational not actual measurement

• We cannot take our ruler and measure “life satisfaction”

• We create tests and hope responses tell us something about the construct we are interested in

• Important: Data is only ever item responses, not the construct itself

• Psychometrics is pseudo-representational: useful (hopefully) representation of target construct rather than ‘ground truth’ of universe

Testing and measurement are two distinct activities

“When tests are automatically granted the status of measurement, they are that much more easily appropriated as vehicles for social injustice, even when this may well have been the opposite of the intent of the test designer.”
(Derek Briggs 2021)

Alfred Binet (1857-1911) - ‘the father of the intelligence test’

• Binet’s purpose: diagnostic classification to be able to target educational interventions

• adopted and implicitly taken as ‘measurement’ and used for group comparisons by advocates of eugenics in 20th century USA.

Testing and measurement are two distinct activities

Concept	Definition	Scope	example
Measurement	Assigning numbers to numbers or events according to rules	Broad (theoretical/methodological)	Quantifying intelligence
Testing	Using a specific instrument/procedure to collect numbers	Narrow (practical/applied)	Administering WAIS to assess “IQ”

When we confuse testing with measurement:

• Risks reifying constructs that are partly socially defined (e.g., “intelligence”).

• Enables justifying inequalities (“low scores mean low ability”) without examining bias, cultural context, or systemic barriers.

Thorndike in the 21st Century…

Whatever exists at all exists in some amount. To know it thoroughly involves knowing the quality of how it has been quantified.

Criteria for good measurement

• To let us consider testing as getting close to measurement, a test must:

  • Assess what it is supposed to assess
  • Be consistent and reliable
  • Produce interpretable scores
  • Be relevant for specific populations
  • Differentiate between people in a fair way

• In this course we will cover first three and how psychologists evaluate them, last two are context-dependent

• We’ll talk about this more in later weeks, can you think of any examples of bad measurement?

So what…

• psychological constructs are complex, multifaceted, and abstract

• a single question (or “item”) will struggle to capture that complexity with any accuracy

• hence: multi-item measures

as always, more terminlogy

as always, more terminlogy

Some questions

Don’t do this at home!

“Dear [Insert LLM name here],

here is what we think ‘creativity’ is.

…
…

Can you give me 10 items for a questionnaire that capture the different aspects of creativity.

Give me statements that people can rate their agreement with.

Please and thank you!

Yours sincerely,
Josiah King”

• get responses from wooclap
• quick create items (via ELM)
  • BE CAREFUL. not ideal.
• we’ll come back to this this afternoon.
• by this afternoon - get them to fill in items and record L v R handed?

BREAK

Dimension Reduction

https://edin.ac/3WjlBWW

groups! dependencies! but not as we know it

longer, wider (faster, stronger?)

ppt	question	response
P001	Q1	1
P001	Q2	5
P001	Q3	1
P001	Q4	5
P001	Q5	4
P001	Q6	2
P002	Q1	5
P002	Q2	5
P002	Q3	4
P002	Q4	5
P002	Q5	4
P002	Q6	2
P003	Q1	4
P003	Q2	4
P003	Q3	4
P003	Q4	4
P003	Q5	4
P003	Q6	4
P004	Q1	2
P004	Q2	6
P004	Q3	4
P004	Q4	3
P004	Q5	2
P004	Q6	3
P005	Q1	3
P005	Q2	6
P005	Q3	4
P005	Q4	4
P005	Q5	3
P005	Q6	4
P006	Q1	4
P006	Q2	5
P006	Q3	4
P006	Q4	5
P006	Q5	5
P006	Q6	3
P007	Q1	4
P007	Q2	3
P007	Q3	3
P007	Q4	5
P007	Q5	6
P007	Q6	5
P008	Q1	4
P008	Q2	4
P008	Q3	3
P008	Q4	5
P008	Q5	4
P008	Q6	5
P009	Q1	5
P009	Q2	5
P009	Q3	3
P009	Q4	4
P009	Q5	3
P009	Q6	5
P010	Q1	4
P010	Q2	5
P010	Q3	3
P010	Q4	5
P010	Q5	4
P010	Q6	4
P011	Q1	2
P011	Q2	4
P011	Q3	3
P011	Q4	5
P011	Q5	4
P011	Q6	4
P012	Q1	4
P012	Q2	4
P012	Q3	2
P012	Q4	3
P012	Q5	5
P012	Q6	4
P013	Q1	3
P013	Q2	5
P013	Q3	1
P013	Q4	4
P013	Q5	2
P013	Q6	3
P014	Q1	6
P014	Q2	5
P014	Q3	4
P014	Q4	5
P014	Q5	5
P014	Q6	3
P015	Q1	5
P015	Q2	4
P015	Q3	4
P015	Q4	5
P015	Q5	4
P015	Q6	6
P016	Q1	3
P016	Q2	3
P016	Q3	2
P016	Q4	4
P016	Q5	3
P016	Q6	3
P017	Q1	4
P017	Q2	6
P017	Q3	4
P017	Q4	5
P017	Q5	6
P017	Q6	7
P018	Q1	5
P018	Q2	4
P018	Q3	4
P018	Q4	5
P018	Q5	4
P018	Q6	6
P019	Q1	5
P019	Q2	4
P019	Q3	4
P019	Q4	6
P019	Q5	1
P019	Q6	6
P020	Q1	4
P020	Q2	4
P020	Q3	4
P020	Q4	5
P020	Q5	4
P020	Q6	7
P021	Q1	6
P021	Q2	6
P021	Q3	4
P021	Q4	6
P021	Q5	3
P021	Q6	5
P022	Q1	3
P022	Q2	5
P022	Q3	3
P022	Q4	4
P022	Q5	4
P022	Q6	5
P023	Q1	4
P023	Q2	4
P023	Q3	5
P023	Q4	4
P023	Q5	3
P023	Q6	4
P024	Q1	5
P024	Q2	4
P024	Q3	3
P024	Q4	5
P024	Q5	4
P024	Q6	6
P025	Q1	4
P025	Q2	5
P025	Q3	5
P025	Q4	4
P025	Q5	4
P025	Q6	2
P026	Q1	3
P026	Q2	5
P026	Q3	3
P026	Q4	5
P026	Q5	1
P026	Q6	4
P027	Q1	4
P027	Q2	3
P027	Q3	2
P027	Q4	3
P027	Q5	2
P027	Q6	3
P028	Q1	5
P028	Q2	2
P028	Q3	3
P028	Q4	2
P028	Q5	4
P028	Q6	4
P029	Q1	3
P029	Q2	3
P029	Q3	2
P029	Q4	3
P029	Q5	1
P029	Q6	3
P030	Q1	4
P030	Q2	5
P030	Q3	3
P030	Q4	5
P030	Q5	3
P030	Q6	4
P031	Q1	3
P031	Q2	3
P031	Q3	1
P031	Q4	5
P031	Q5	2
P031	Q6	4
P032	Q1	4
P032	Q2	5
P032	Q3	2
P032	Q4	6
P032	Q5	3
P032	Q6	6
P033	Q1	4
P033	Q2	4
P033	Q3	3
P033	Q4	4
P033	Q5	2
P033	Q6	5
P034	Q1	3
P034	Q2	5
P034	Q3	2
P034	Q4	5
P034	Q5	2
P034	Q6	3
P035	Q1	3
P035	Q2	5
P035	Q3	5
P035	Q4	6
P035	Q5	5
P035	Q6	7
P036	Q1	3
P036	Q2	4
P036	Q3	3
P036	Q4	4
P036	Q5	5
P036	Q6	4
P037	Q1	4
P037	Q2	4
P037	Q3	3
P037	Q4	5
P037	Q5	3
P037	Q6	5
P038	Q1	4
P038	Q2	4
P038	Q3	4
P038	Q4	3
P038	Q5	4
P038	Q6	3
P039	Q1	3
P039	Q2	2
P039	Q3	2
P039	Q4	4
P039	Q5	2
P039	Q6	6
P040	Q1	4
P040	Q2	5
P040	Q3	2
P040	Q4	4
P040	Q5	3
P040	Q6	3
P041	Q1	5
P041	Q2	5
P041	Q3	4
P041	Q4	4
P041	Q5	5
P041	Q6	4
P042	Q1	7
P042	Q2	6
P042	Q3	5
P042	Q4	4
P042	Q5	3
P042	Q6	5
P043	Q1	7
P043	Q2	4
P043	Q3	3
P043	Q4	4
P043	Q5	5
P043	Q6	5
P044	Q1	4
P044	Q2	5
P044	Q3	5
P044	Q4	6
P044	Q5	6
P044	Q6	5
P045	Q1	3
P045	Q2	5
P045	Q3	2
P045	Q4	5
P045	Q5	4
P045	Q6	5
P046	Q1	1
P046	Q2	3
P046	Q3	5
P046	Q4	5
P046	Q5	2
P046	Q6	4
P047	Q1	5
P047	Q2	4
P047	Q3	4
P047	Q4	5
P047	Q5	4
P047	Q6	4
P048	Q1	6
P048	Q2	4
P048	Q3	2
P048	Q4	4
P048	Q5	2
P048	Q6	4
P049	Q1	4
P049	Q2	5
P049	Q3	2
P049	Q4	5
P049	Q5	3
P049	Q6	4
P050	Q1	2
P050	Q2	4
P050	Q3	3
P050	Q4	4
P050	Q5	4
P050	Q6	5
P051	Q1	3
P051	Q2	2
P051	Q3	3
P051	Q4	4
P051	Q5	2
P051	Q6	4
P052	Q1	4
P052	Q2	5
P052	Q3	4
P052	Q4	5
P052	Q5	6
P052	Q6	3
P053	Q1	5
P053	Q2	5
P053	Q3	4
P053	Q4	5
P053	Q5	3
P053	Q6	6
P054	Q1	4
P054	Q2	5
P054	Q3	3
P054	Q4	4
P054	Q5	3
P054	Q6	3
P055	Q1	5
P055	Q2	6
P055	Q3	3
P055	Q4	5
P055	Q5	4
P055	Q6	4
P056	Q1	4
P056	Q2	5
P056	Q3	1
P056	Q4	4
P056	Q5	3
P056	Q6	3
P057	Q1	4
P057	Q2	5
P057	Q3	4
P057	Q4	4
P057	Q5	5
P057	Q6	4
P058	Q1	4
P058	Q2	5
P058	Q3	4
P058	Q4	4
P058	Q5	3
P058	Q6	5
P059	Q1	5
P059	Q2	6
P059	Q3	2
P059	Q4	6
P059	Q5	4
P059	Q6	5
P060	Q1	3
P060	Q2	3
P060	Q3	2
P060	Q4	4
P060	Q5	4
P060	Q6	4
P061	Q1	5
P061	Q2	5
P061	Q3	5
P061	Q4	5
P061	Q5	5
P061	Q6	4
P062	Q1	5
P062	Q2	4
P062	Q3	4
P062	Q4	4
P062	Q5	5
P062	Q6	4
P063	Q1	4
P063	Q2	5
P063	Q3	3
P063	Q4	5
P063	Q5	4
P063	Q6	4
P064	Q1	5
P064	Q2	5
P064	Q3	3
P064	Q4	5
P064	Q5	4
P064	Q6	5
P065	Q1	2
P065	Q2	2
P065	Q3	2
P065	Q4	2
P065	Q5	2
P065	Q6	4
P066	Q1	6
P066	Q2	3
P066	Q3	3
P066	Q4	5
P066	Q5	2
P066	Q6	3
P067	Q1	2
P067	Q2	3
P067	Q3	3
P067	Q4	3
P067	Q5	4
P067	Q6	4
P068	Q1	3
P068	Q2	5
P068	Q3	3
P068	Q4	5
P068	Q5	4
P068	Q6	5
P069	Q1	6
P069	Q2	5
P069	Q3	6
P069	Q4	5
P069	Q5	6
P069	Q6	6
P070	Q1	3
P070	Q2	6
P070	Q3	4
P070	Q4	4
P070	Q5	5
P070	Q6	5
P071	Q1	3
P071	Q2	4
P071	Q3	3
P071	Q4	4
P071	Q5	3
P071	Q6	3
P072	Q1	4
P072	Q2	6
P072	Q3	3
P072	Q4	5
P072	Q5	6
P072	Q6	7
P073	Q1	3
P073	Q2	3
P073	Q3	4
P073	Q4	3
P073	Q5	5
P073	Q6	4
P074	Q1	4
P074	Q2	4
P074	Q3	2
P074	Q4	4
P074	Q5	3
P074	Q6	5
P075	Q1	4
P075	Q2	6
P075	Q3	2
P075	Q4	5
P075	Q5	5
P075	Q6	5
P076	Q1	4
P076	Q2	5
P076	Q3	3
P076	Q4	5
P076	Q5	4
P076	Q6	3
P077	Q1	3
P077	Q2	3
P077	Q3	2
P077	Q4	4
P077	Q5	1
P077	Q6	4
P078	Q1	6
P078	Q2	5
P078	Q3	6
P078	Q4	5
P078	Q5	5
P078	Q6	6
P079	Q1	4
P079	Q2	5
P079	Q3	3
P079	Q4	3
P079	Q5	6
P079	Q6	4
P080	Q1	1
P080	Q2	3
P080	Q3	2
P080	Q4	2
P080	Q5	1
P080	Q6	3
P081	Q1	4
P081	Q2	4
P081	Q3	3
P081	Q4	5
P081	Q5	3
P081	Q6	5
P082	Q1	2
P082	Q2	4
P082	Q3	3
P082	Q4	4
P082	Q5	2
P082	Q6	3
P083	Q1	5
P083	Q2	4
P083	Q3	4
P083	Q4	4
P083	Q5	3
P083	Q6	3
P084	Q1	2
P084	Q2	3
P084	Q3	2
P084	Q4	3
P084	Q5	3
P084	Q6	3
P085	Q1	4
P085	Q2	4
P085	Q3	4
P085	Q4	5
P085	Q5	4
P085	Q6	3
P086	Q1	4
P086	Q2	5
P086	Q3	4
P086	Q4	3
P086	Q5	3
P086	Q6	7
P087	Q1	6
P087	Q2	5
P087	Q3	6
P087	Q4	5
P087	Q5	7
P087	Q6	5
P088	Q1	6
P088	Q2	3
P088	Q3	5
P088	Q4	5
P088	Q5	3
P088	Q6	5
P089	Q1	4
P089	Q2	5
P089	Q3	3
P089	Q4	4
P089	Q5	5
P089	Q6	4
P090	Q1	5
P090	Q2	4
P090	Q3	4
P090	Q4	5
P090	Q5	3
P090	Q6	7
P091	Q1	7
P091	Q2	5
P091	Q3	5
P091	Q4	7
P091	Q5	6
P091	Q6	6
P092	Q1	4
P092	Q2	4
P092	Q3	3
P092	Q4	6
P092	Q5	3
P092	Q6	5
P093	Q1	2
P093	Q2	3
P093	Q3	2
P093	Q4	4
P093	Q5	2
P093	Q6	4
P094	Q1	3
P094	Q2	3
P094	Q3	1
P094	Q4	2
P094	Q5	3
P094	Q6	5
P095	Q1	5
P095	Q2	5
P095	Q3	4
P095	Q4	4
P095	Q5	4
P095	Q6	6
P096	Q1	6
P096	Q2	5
P096	Q3	3
P096	Q4	5
P096	Q5	3
P096	Q6	6
P097	Q1	4
P097	Q2	4
P097	Q3	3
P097	Q4	5
P097	Q5	5
P097	Q6	4
P098	Q1	5
P098	Q2	3
P098	Q3	3
P098	Q4	4
P098	Q5	4
P098	Q6	5
P099	Q1	5
P099	Q2	5
P099	Q3	5
P099	Q4	6
P099	Q5	6
P099	Q6	4
P100	Q1	3
P100	Q2	4
P100	Q3	3
P100	Q4	5
P100	Q5	4
P100	Q6	4
P101	Q1	5
P101	Q2	5
P101	Q3	4
P101	Q4	4
P101	Q5	3
P101	Q6	4
P102	Q1	4
P102	Q2	4
P102	Q3	1
P102	Q4	5
P102	Q5	4
P102	Q6	3
P103	Q1	6
P103	Q2	4
P103	Q3	5
P103	Q4	4
P103	Q5	3
P103	Q6	5
P104	Q1	5
P104	Q2	7
P104	Q3	4
P104	Q4	5
P104	Q5	4
P104	Q6	5
P105	Q1	5
P105	Q2	2
P105	Q3	2
P105	Q4	5
P105	Q5	2
P105	Q6	5
P106	Q1	5
P106	Q2	6
P106	Q3	4
P106	Q4	6
P106	Q5	4
P106	Q6	5
P107	Q1	6
P107	Q2	5
P107	Q3	6
P107	Q4	6
P107	Q5	5
P107	Q6	6
P108	Q1	5
P108	Q2	5
P108	Q3	2
P108	Q4	3
P108	Q5	1
P108	Q6	4
P109	Q1	4
P109	Q2	4
P109	Q3	3
P109	Q4	6
P109	Q5	3
P109	Q6	5
P110	Q1	5
P110	Q2	4
P110	Q3	2
P110	Q4	4
P110	Q5	4
P110	Q6	5
P111	Q1	5
P111	Q2	3
P111	Q3	3
P111	Q4	3
P111	Q5	3
P111	Q6	3
P112	Q1	5
P112	Q2	4
P112	Q3	3
P112	Q4	5
P112	Q5	3
P112	Q6	3
P113	Q1	5
P113	Q2	5
P113	Q3	3
P113	Q4	6
P113	Q5	4
P113	Q6	5
P114	Q1	4
P114	Q2	3
P114	Q3	4
P114	Q4	4
P114	Q5	3
P114	Q6	3
P115	Q1	5
P115	Q2	5
P115	Q3	4
P115	Q4	6
P115	Q5	5
P115	Q6	4
P116	Q1	5
P116	Q2	4
P116	Q3	5
P116	Q4	4
P116	Q5	3
P116	Q6	4
P117	Q1	5
P117	Q2	5
P117	Q3	4
P117	Q4	5
P117	Q5	5
P117	Q6	5
P118	Q1	7
P118	Q2	4
P118	Q3	4
P118	Q4	5
P118	Q5	5
P118	Q6	5
P119	Q1	5
P119	Q2	4
P119	Q3	3
P119	Q4	5
P119	Q5	3
P119	Q6	5
P120	Q1	4
P120	Q2	5
P120	Q3	4
P120	Q4	5
P120	Q5	5
P120	Q6	3
P121	Q1	4
P121	Q2	4
P121	Q3	3
P121	Q4	5
P121	Q5	5
P121	Q6	4
P122	Q1	4
P122	Q2	4
P122	Q3	4
P122	Q4	5
P122	Q5	5
P122	Q6	4
P123	Q1	6
P123	Q2	5
P123	Q3	4
P123	Q4	6
P123	Q5	6
P123	Q6	4
P124	Q1	4
P124	Q2	5
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P124	Q4	5
P124	Q5	5
P124	Q6	4
P125	Q1	3
P125	Q2	2
P125	Q3	2
P125	Q4	4
P125	Q5	2
P125	Q6	5
P126	Q1	3
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P126	Q3	2
P126	Q4	4
P126	Q5	2
P126	Q6	4
P127	Q1	5
P127	Q2	6
P127	Q3	4
P127	Q4	5
P127	Q5	6
P127	Q6	3
P128	Q1	4
P128	Q2	4
P128	Q3	3
P128	Q4	4
P128	Q5	3
P128	Q6	5
P129	Q1	5
P129	Q2	5
P129	Q3	3
P129	Q4	4
P129	Q5	4
P129	Q6	4
P130	Q1	7
P130	Q2	5
P130	Q3	4
P130	Q4	5
P130	Q5	6
P130	Q6	7
P131	Q1	4
P131	Q2	4
P131	Q3	2
P131	Q4	4
P131	Q5	3
P131	Q6	2
P132	Q1	4
P132	Q2	4
P132	Q3	3
P132	Q4	5
P132	Q5	4
P132	Q6	2
P133	Q1	6
P133	Q2	5
P133	Q3	2
P133	Q4	5
P133	Q5	3
P133	Q6	6
P134	Q1	2
P134	Q2	3
P134	Q3	1
P134	Q4	3
P134	Q5	5
P134	Q6	6
P135	Q1	4
P135	Q2	5
P135	Q3	3
P135	Q4	5
P135	Q5	5
P135	Q6	5
P136	Q1	4
P136	Q2	4
P136	Q3	4
P136	Q4	3
P136	Q5	5
P136	Q6	5
P137	Q1	4
P137	Q2	2
P137	Q3	3
P137	Q4	4
P137	Q5	3
P137	Q6	3
P138	Q1	3
P138	Q2	6
P138	Q3	5
P138	Q4	5
P138	Q5	5
P138	Q6	6
P139	Q1	5
P139	Q2	3
P139	Q3	3
P139	Q4	5
P139	Q5	4
P139	Q6	5
P140	Q1	2
P140	Q2	2
P140	Q3	3
P140	Q4	3
P140	Q5	4
P140	Q6	4
P141	Q1	4
P141	Q2	4
P141	Q3	4
P141	Q4	6
P141	Q5	5
P141	Q6	4
P142	Q1	5
P142	Q2	5
P142	Q3	4
P142	Q4	3
P142	Q5	3
P142	Q6	5
P143	Q1	6
P143	Q2	5
P143	Q3	4
P143	Q4	6
P143	Q5	6
P143	Q6	5
P144	Q1	4
P144	Q2	4
P144	Q3	3
P144	Q4	5
P144	Q5	2
P144	Q6	4
P145	Q1	3
P145	Q2	5
P145	Q3	4
P145	Q4	5
P145	Q5	4
P145	Q6	4
P146	Q1	5
P146	Q2	4
P146	Q3	4
P146	Q4	5
P146	Q5	5
P146	Q6	6
P147	Q1	5
P147	Q2	6
P147	Q3	3
P147	Q4	5
P147	Q5	5
P147	Q6	3
P148	Q1	5
P148	Q2	4
P148	Q3	4
P148	Q4	6
P148	Q5	4
P148	Q6	5
P149	Q1	5
P149	Q2	3
P149	Q3	3
P149	Q4	4
P149	Q5	2
P149	Q6	3
P150	Q1	5
P150	Q2	5
P150	Q3	3
P150	Q4	5
P150	Q5	4
P150	Q6	4
P151	Q1	3
P151	Q2	3
P151	Q3	4
P151	Q4	5
P151	Q5	3
P151	Q6	2
P152	Q1	5
P152	Q2	4
P152	Q3	2
P152	Q4	4
P152	Q5	5
P152	Q6	3
P153	Q1	3
P153	Q2	4
P153	Q3	3
P153	Q4	3
P153	Q5	2
P153	Q6	4
P154	Q1	4
P154	Q2	5
P154	Q3	3
P154	Q4	3
P154	Q5	5
P154	Q6	4
P155	Q1	4
P155	Q2	5
P155	Q3	3
P155	Q4	5
P155	Q5	5
P155	Q6	6
P156	Q1	2
P156	Q2	4
P156	Q3	4
P156	Q4	3
P156	Q5	1
P156	Q6	3
P157	Q1	4
P157	Q2	3
P157	Q3	5
P157	Q4	4
P157	Q5	2
P157	Q6	3
P158	Q1	6
P158	Q2	4
P158	Q3	4
P158	Q4	5
P158	Q5	5
P158	Q6	3
P159	Q1	3
P159	Q2	4
P159	Q3	2
P159	Q4	3
P159	Q5	4
P159	Q6	5
P160	Q1	4
P160	Q2	3
P160	Q3	3
P160	Q4	5
P160	Q5	3
P160	Q6	7
P161	Q1	5
P161	Q2	4
P161	Q3	2
P161	Q4	4
P161	Q5	4
P161	Q6	3
P162	Q1	5
P162	Q2	6
P162	Q3	5
P162	Q4	4
P162	Q5	4
P162	Q6	4
P163	Q1	4
P163	Q2	5
P163	Q3	2
P163	Q4	4
P163	Q5	4
P163	Q6	6
P164	Q1	4
P164	Q2	4
P164	Q3	3
P164	Q4	4
P164	Q5	3
P164	Q6	3
P165	Q1	3
P165	Q2	2
P165	Q3	2
P165	Q4	4
P165	Q5	3
P165	Q6	3
P166	Q1	3
P166	Q2	3
P166	Q3	2
P166	Q4	3
P166	Q5	2
P166	Q6	6
P167	Q1	6
P167	Q2	5
P167	Q3	4
P167	Q4	6
P167	Q5	5
P167	Q6	4
P168	Q1	4
P168	Q2	4
P168	Q3	5
P168	Q4	4
P168	Q5	6
P168	Q6	5
P169	Q1	5
P169	Q2	5
P169	Q3	4
P169	Q4	5
P169	Q5	4
P169	Q6	7
P170	Q1	4
P170	Q2	3
P170	Q3	1
P170	Q4	4
P170	Q5	3
P170	Q6	4
P171	Q1	5
P171	Q2	6
P171	Q3	5
P171	Q4	4
P171	Q5	4
P171	Q6	5
P172	Q1	6
P172	Q2	3
P172	Q3	2
P172	Q4	4
P172	Q5	2
P172	Q6	6
P173	Q1	4
P173	Q2	4
P173	Q3	4
P173	Q4	5
P173	Q5	4
P173	Q6	6
P174	Q1	2
P174	Q2	5
P174	Q3	3
P174	Q4	5
P174	Q5	6
P174	Q6	3
P175	Q1	7
P175	Q2	7
P175	Q3	6
P175	Q4	6
P175	Q5	5
P175	Q6	6
P176	Q1	3
P176	Q2	4
P176	Q3	4
P176	Q4	4
P176	Q5	4
P176	Q6	6
P177	Q1	5
P177	Q2	6
P177	Q3	5
P177	Q4	4
P177	Q5	5
P177	Q6	6
P178	Q1	7
P178	Q2	5
P178	Q3	6
P178	Q4	5
P178	Q5	5
P178	Q6	7
P179	Q1	4
P179	Q2	6
P179	Q3	4
P179	Q4	5
P179	Q5	5
P179	Q6	5
P180	Q1	4
P180	Q2	4
P180	Q3	3
P180	Q4	5
P180	Q5	4
P180	Q6	5
P181	Q1	5
P181	Q2	4
P181	Q3	4
P181	Q4	4
P181	Q5	4
P181	Q6	5
P182	Q1	4
P182	Q2	5
P182	Q3	2
P182	Q4	5
P182	Q5	4
P182	Q6	3
P183	Q1	3
P183	Q2	4
P183	Q3	3
P183	Q4	3
P183	Q5	4
P183	Q6	4
P184	Q1	3
P184	Q2	5
P184	Q3	3
P184	Q4	5
P184	Q5	3
P184	Q6	6
P185	Q1	5
P185	Q2	5
P185	Q3	4
P185	Q4	5
P185	Q5	3
P185	Q6	3
P186	Q1	4
P186	Q2	3
P186	Q3	4
P186	Q4	5
P186	Q5	4
P186	Q6	5
P187	Q1	3
P187	Q2	4
P187	Q3	4
P187	Q4	3
P187	Q5	4
P187	Q6	5
P188	Q1	4
P188	Q2	5
P188	Q3	4
P188	Q4	5
P188	Q5	4
P188	Q6	5
P189	Q1	5
P189	Q2	6
P189	Q3	3
P189	Q4	5
P189	Q5	4
P189	Q6	5
P190	Q1	4
P190	Q2	5
P190	Q3	3
P190	Q4	5
P190	Q5	3
P190	Q6	3
P191	Q1	4
P191	Q2	3
P191	Q3	3
P191	Q4	4
P191	Q5	5
P191	Q6	5
P192	Q1	7
P192	Q2	5
P192	Q3	6
P192	Q4	6
P192	Q5	6
P192	Q6	4
P193	Q1	2
P193	Q2	4
P193	Q3	4
P193	Q4	5
P193	Q5	3
P193	Q6	3
P194	Q1	4
P194	Q2	4
P194	Q3	3
P194	Q4	5
P194	Q5	4
P194	Q6	5
P195	Q1	5
P195	Q2	5
P195	Q3	3
P195	Q4	4
P195	Q5	4
P195	Q6	6
P196	Q1	3
P196	Q2	6
P196	Q3	4
P196	Q4	4
P196	Q5	3
P196	Q6	4
P197	Q1	3
P197	Q2	5
P197	Q3	4
P197	Q4	5
P197	Q5	3
P197	Q6	4
P198	Q1	3
P198	Q2	4
P198	Q3	2
P198	Q4	3
P198	Q5	3
P198	Q6	2
P199	Q1	4
P199	Q2	5
P199	Q3	3
P199	Q4	5
P199	Q5	4
P199	Q6	5
P200	Q1	5
P200	Q2	4
P200	Q3	2
P200	Q4	5
P200	Q5	4
P200	Q6	5
P201	Q1	5
P201	Q2	6
P201	Q3	5
P201	Q4	5
P201	Q5	4
P201	Q6	7
P202	Q1	3
P202	Q2	5
P202	Q3	2
P202	Q4	5
P202	Q5	3
P202	Q6	3
P203	Q1	5
P203	Q2	4
P203	Q3	3
P203	Q4	4
P203	Q5	4
P203	Q6	1
P204	Q1	5
P204	Q2	5
P204	Q3	4
P204	Q4	5
P204	Q5	4
P204	Q6	5
P205	Q1	6
P205	Q2	3
P205	Q3	4
P205	Q4	3
P205	Q5	3
P205	Q6	5
P206	Q1	6
P206	Q2	4
P206	Q3	3
P206	Q4	4
P206	Q5	5
P206	Q6	4
P207	Q1	5
P207	Q2	3
P207	Q3	3
P207	Q4	5
P207	Q5	2
P207	Q6	5
P208	Q1	6
P208	Q2	5
P208	Q3	3
P208	Q4	5
P208	Q5	4
P208	Q6	5
P209	Q1	4
P209	Q2	5
P209	Q3	3
P209	Q4	4
P209	Q5	3
P209	Q6	3
P210	Q1	4
P210	Q2	5
P210	Q3	3
P210	Q4	5
P210	Q5	3
P210	Q6	3
P211	Q1	4
P211	Q2	3
P211	Q3	5
P211	Q4	4
P211	Q5	2
P211	Q6	4
P212	Q1	3
P212	Q2	4
P212	Q3	3
P212	Q4	4
P212	Q5	2
P212	Q6	4
P213	Q1	4
P213	Q2	4
P213	Q3	2
P213	Q4	5
P213	Q5	3
P213	Q6	3
P214	Q1	5
P214	Q2	5
P214	Q3	2
P214	Q4	4
P214	Q5	3
P214	Q6	2
P215	Q1	5
P215	Q2	4
P215	Q3	4
P215	Q4	5
P215	Q5	4
P215	Q6	6
P216	Q1	2
P216	Q2	5
P216	Q3	2
P216	Q4	3
P216	Q5	2
P216	Q6	4
P217	Q1	5
P217	Q2	4
P217	Q3	3
P217	Q4	4
P217	Q5	3
P217	Q6	2
P218	Q1	3
P218	Q2	4
P218	Q3	2
P218	Q4	2
P218	Q5	3
P218	Q6	1
P219	Q1	2
P219	Q2	4
P219	Q3	2
P219	Q4	4
P219	Q5	3
P219	Q6	4
P220	Q1	5
P220	Q2	3
P220	Q3	2
P220	Q4	2
P220	Q5	4
P220	Q6	4
P221	Q1	5
P221	Q2	4
P221	Q3	4
P221	Q4	3
P221	Q5	4
P221	Q6	5
P222	Q1	5
P222	Q2	5
P222	Q3	5
P222	Q4	5
P222	Q5	4
P222	Q6	6
P223	Q1	4
P223	Q2	4
P223	Q3	4
P223	Q4	6
P223	Q5	5
P223	Q6	3
P224	Q1	5
P224	Q2	4
P224	Q3	3
P224	Q4	5
P224	Q5	3
P224	Q6	3
P225	Q1	4
P225	Q2	5
P225	Q3	3
P225	Q4	4
P225	Q5	3
P225	Q6	6
P226	Q1	3
P226	Q2	5
P226	Q3	3
P226	Q4	4
P226	Q5	4
P226	Q6	6
P227	Q1	4
P227	Q2	4
P227	Q3	5
P227	Q4	5
P227	Q5	3
P227	Q6	7
P228	Q1	4
P228	Q2	5
P228	Q3	4
P228	Q4	5
P228	Q5	4
P228	Q6	7
P229	Q1	4
P229	Q2	4
P229	Q3	2
P229	Q4	3
P229	Q5	3
P229	Q6	2
P230	Q1	3
P230	Q2	3
P230	Q3	2
P230	Q4	4
P230	Q5	3
P230	Q6	2
P231	Q1	4
P231	Q2	4
P231	Q3	2
P231	Q4	4
P231	Q5	4
P231	Q6	4
P232	Q1	6
P232	Q2	5
P232	Q3	4
P232	Q4	6
P232	Q5	5
P232	Q6	5
P233	Q1	5
P233	Q2	4
P233	Q3	4
P233	Q4	4
P233	Q5	3
P233	Q6	6
P234	Q1	5
P234	Q2	3
P234	Q3	4
P234	Q4	5
P234	Q5	4
P234	Q6	5
P235	Q1	5
P235	Q2	5
P235	Q3	3
P235	Q4	4
P235	Q5	4
P235	Q6	4
P236	Q1	7
P236	Q2	4
P236	Q3	5
P236	Q4	4
P236	Q5	5
P236	Q6	4
P237	Q1	3
P237	Q2	3
P237	Q3	5
P237	Q4	4
P237	Q5	5
P237	Q6	5
P238	Q1	4
P238	Q2	4
P238	Q3	3
P238	Q4	5
P238	Q5	3
P238	Q6	5
P239	Q1	3
P239	Q2	3
P239	Q3	2
P239	Q4	4
P239	Q5	3
P239	Q6	4
P240	Q1	5
P240	Q2	4
P240	Q3	3
P240	Q4	4
P240	Q5	6
P240	Q6	4
P241	Q1	2
P241	Q2	2
P241	Q3	2
P241	Q4	3
P241	Q5	5
P241	Q6	4
P242	Q1	6
P242	Q2	6
P242	Q3	5
P242	Q4	6
P242	Q5	5
P242	Q6	6
P243	Q1	2
P243	Q2	4
P243	Q3	3
P243	Q4	5
P243	Q5	3
P243	Q6	6
P244	Q1	4
P244	Q2	4
P244	Q3	3
P244	Q4	3
P244	Q5	3
P244	Q6	6
P245	Q1	5
P245	Q2	5
P245	Q3	4
P245	Q4	6
P245	Q5	5
P245	Q6	5
P246	Q1	4
P246	Q2	4
P246	Q3	3
P246	Q4	4
P246	Q5	4
P246	Q6	5
P247	Q1	7
P247	Q2	6
P247	Q3	4
P247	Q4	5
P247	Q5	5
P247	Q6	4
P248	Q1	5
P248	Q2	4
P248	Q3	3
P248	Q4	4
P248	Q5	3
P248	Q6	3
P249	Q1	2
P249	Q2	4
P249	Q3	2
P249	Q4	3
P249	Q5	4
P249	Q6	3
P250	Q1	5
P250	Q2	3
P250	Q3	1
P250	Q4	4
P250	Q5	4
P250	Q6	4
P251	Q1	4
P251	Q2	3
P251	Q3	3
P251	Q4	5
P251	Q5	2
P251	Q6	2
P252	Q1	6
P252	Q2	6
P252	Q3	4
P252	Q4	4
P252	Q5	3
P252	Q6	6
P253	Q1	3
P253	Q2	4
P253	Q3	3
P253	Q4	4
P253	Q5	3
P253	Q6	5
P254	Q1	4
P254	Q2	5
P254	Q3	5
P254	Q4	4
P254	Q5	3
P254	Q6	6
P255	Q1	3
P255	Q2	4
P255	Q3	4
P255	Q4	4
P255	Q5	2
P255	Q6	5
P256	Q1	6
P256	Q2	5
P256	Q3	2
P256	Q4	4
P256	Q5	4
P256	Q6	6
P257	Q1	2
P257	Q2	2
P257	Q3	2
P257	Q4	3
P257	Q5	4
P257	Q6	4
P258	Q1	3
P258	Q2	4
P258	Q3	3
P258	Q4	4
P258	Q5	6
P258	Q6	5
P259	Q1	6
P259	Q2	5
P259	Q3	4
P259	Q4	5
P259	Q5	5
P259	Q6	6
P260	Q1	4
P260	Q2	4
P260	Q3	4
P260	Q4	5
P260	Q5	4
P260	Q6	5
P261	Q1	5
P261	Q2	5
P261	Q3	3
P261	Q4	4
P261	Q5	5
P261	Q6	5
P262	Q1	1
P262	Q2	4
P262	Q3	5
P262	Q4	4
P262	Q5	1
P262	Q6	4
P263	Q1	3
P263	Q2	4
P263	Q3	2
P263	Q4	4
P263	Q5	4
P263	Q6	5
P264	Q1	4
P264	Q2	3
P264	Q3	2
P264	Q4	4
P264	Q5	3
P264	Q6	5
P265	Q1	4
P265	Q2	4
P265	Q3	3
P265	Q4	4
P265	Q5	4
P265	Q6	6
P266	Q1	4
P266	Q2	4
P266	Q3	2
P266	Q4	4
P266	Q5	3
P266	Q6	3
P267	Q1	6
P267	Q2	7
P267	Q3	6
P267	Q4	6
P267	Q5	5
P267	Q6	7
P268	Q1	3
P268	Q2	3
P268	Q3	2
P268	Q4	5
P268	Q5	2
P268	Q6	2
P269	Q1	3
P269	Q2	6
P269	Q3	3
P269	Q4	4
P269	Q5	4
P269	Q6	6
P270	Q1	4
P270	Q2	3
P270	Q3	2
P270	Q4	4
P270	Q5	1
P270	Q6	5
P271	Q1	5
P271	Q2	6
P271	Q3	4
P271	Q4	5
P271	Q5	3
P271	Q6	6
P272	Q1	5
P272	Q2	6
P272	Q3	4
P272	Q4	7
P272	Q5	3
P272	Q6	4
P273	Q1	3
P273	Q2	4
P273	Q3	4
P273	Q4	4
P273	Q5	4
P273	Q6	5
P274	Q1	4
P274	Q2	7
P274	Q3	4
P274	Q4	5
P274	Q5	6
P274	Q6	4
P275	Q1	3
P275	Q2	3
P275	Q3	2
P275	Q4	4
P275	Q5	2
P275	Q6	6
P276	Q1	3
P276	Q2	3
P276	Q3	2
P276	Q4	4
P276	Q5	2
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P277	Q1	4
P277	Q2	6
P277	Q3	3
P277	Q4	5
P277	Q5	4
P277	Q6	4
P278	Q1	4
P278	Q2	4
P278	Q3	3
P278	Q4	4
P278	Q5	5
P278	Q6	4
P279	Q1	5
P279	Q2	4
P279	Q3	3
P279	Q4	4
P279	Q5	3
P279	Q6	4
P280	Q1	4
P280	Q2	7
P280	Q3	4
P280	Q4	5
P280	Q5	4
P280	Q6	4
P281	Q1	5
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P281	Q4	5
P281	Q5	3
P281	Q6	6
P282	Q1	6
P282	Q2	4
P282	Q3	3
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P282	Q5	5
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P283	Q1	4
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P283	Q4	5
P283	Q5	3
P283	Q6	3
P284	Q1	3
P284	Q2	1
P284	Q3	3
P284	Q4	4
P284	Q5	3
P284	Q6	3
P285	Q1	4
P285	Q2	4
P285	Q3	3
P285	Q4	4
P285	Q5	4
P285	Q6	6
P286	Q1	5
P286	Q2	4
P286	Q3	3
P286	Q4	4
P286	Q5	2
P286	Q6	4
P287	Q1	4
P287	Q2	3
P287	Q3	4
P287	Q4	4
P287	Q5	4
P287	Q6	4
P288	Q1	5
P288	Q2	3
P288	Q3	3
P288	Q4	4
P288	Q5	3
P288	Q6	4
P289	Q1	5
P289	Q2	3
P289	Q3	4
P289	Q4	4
P289	Q5	3
P289	Q6	4
P290	Q1	4
P290	Q2	5
P290	Q3	5
P290	Q4	4
P290	Q5	4
P290	Q6	6
P291	Q1	4
P291	Q2	5
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P291	Q4	4
P291	Q5	5
P291	Q6	4
P292	Q1	4
P292	Q2	4
P292	Q3	3
P292	Q4	5
P292	Q5	3
P292	Q6	3
P293	Q1	3
P293	Q2	4
P293	Q3	4
P293	Q4	5
P293	Q5	3
P293	Q6	4
P294	Q1	4
P294	Q2	5
P294	Q3	3
P294	Q4	5
P294	Q5	3
P294	Q6	5
P295	Q1	5
P295	Q2	4
P295	Q3	4
P295	Q4	5
P295	Q5	6
P295	Q6	6
P296	Q1	5
P296	Q2	4
P296	Q3	3
P296	Q4	4
P296	Q5	4
P296	Q6	5
P297	Q1	4
P297	Q2	2
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P297	Q4	3
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P298	Q1	4
P298	Q2	2
P298	Q3	2
P298	Q4	1
P298	Q5	1
P298	Q6	5
P299	Q1	5
P299	Q2	5
P299	Q3	2
P299	Q4	6
P299	Q5	3
P299	Q6	3
P300	Q1	4
P300	Q2	4
P300	Q3	4
P300	Q4	5
P300	Q5	4
P300	Q6	3
P301	Q1	4
P301	Q2	7
P301	Q3	3
P301	Q4	5
P301	Q5	4
P301	Q6	5
P302	Q1	6
P302	Q2	5
P302	Q3	4
P302	Q4	6
P302	Q5	4
P302	Q6	3
P303	Q1	6
P303	Q2	5
P303	Q3	4
P303	Q4	6
P303	Q5	5
P303	Q6	5
P304	Q1	4
P304	Q2	3
P304	Q3	3
P304	Q4	3
P304	Q5	3
P304	Q6	4
P305	Q1	4
P305	Q2	5
P305	Q3	4
P305	Q4	5
P305	Q5	4
P305	Q6	5
P306	Q1	5
P306	Q2	7
P306	Q3	6
P306	Q4	6
P306	Q5	5
P306	Q6	5
P307	Q1	4
P307	Q2	3
P307	Q3	1
P307	Q4	3
P307	Q5	2
P307	Q6	3
P308	Q1	3
P308	Q2	4
P308	Q3	4
P308	Q4	4
P308	Q5	4
P308	Q6	6
P309	Q1	5
P309	Q2	3
P309	Q3	1
P309	Q4	4
P309	Q5	2
P309	Q6	4
P310	Q1	6
P310	Q2	5
P310	Q3	3
P310	Q4	6
P310	Q5	3
P310	Q6	7
P311	Q1	6
P311	Q2	6
P311	Q3	5
P311	Q4	4
P311	Q5	4
P311	Q6	5
P312	Q1	5
P312	Q2	5
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P312	Q4	5
P312	Q5	4
P312	Q6	5
P313	Q1	6
P313	Q2	5
P313	Q3	4
P313	Q4	6
P313	Q5	5
P313	Q6	6
P314	Q1	5
P314	Q2	5
P314	Q3	3
P314	Q4	3
P314	Q5	1
P314	Q6	2
P315	Q1	2
P315	Q2	5
P315	Q3	2
P315	Q4	5
P315	Q5	2
P315	Q6	4
P316	Q1	3
P316	Q2	2
P316	Q3	4
P316	Q4	3
P316	Q5	2
P316	Q6	4
P317	Q1	6
P317	Q2	5
P317	Q3	5
P317	Q4	6
P317	Q5	6
P317	Q6	5
P318	Q1	4
P318	Q2	5
P318	Q3	4
P318	Q4	5
P318	Q5	3
P318	Q6	3
P319	Q1	4
P319	Q2	3
P319	Q3	3
P319	Q4	4
P319	Q5	4
P319	Q6	5
P320	Q1	5
P320	Q2	4
P320	Q3	3
P320	Q4	5
P320	Q5	3
P320	Q6	6
P321	Q1	2
P321	Q2	3
P321	Q3	2
P321	Q4	4
P321	Q5	1
P321	Q6	4
P322	Q1	5
P322	Q2	7
P322	Q3	5
P322	Q4	5
P322	Q5	5
P322	Q6	5
P323	Q1	3
P323	Q2	5
P323	Q3	2
P323	Q4	4
P323	Q5	4
P323	Q6	5
P324	Q1	6
P324	Q2	6
P324	Q3	3
P324	Q4	5
P324	Q5	4
P324	Q6	6
P325	Q1	4
P325	Q2	4
P325	Q3	3
P325	Q4	4
P325	Q5	3
P325	Q6	3
P326	Q1	3
P326	Q2	3
P326	Q3	3
P326	Q4	5
P326	Q5	3
P326	Q6	5
P327	Q1	6
P327	Q2	4
P327	Q3	4
P327	Q4	2
P327	Q5	6
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P328	Q1	5
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P328	Q4	6
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P328	Q6	4
P329	Q1	4
P329	Q2	4
P329	Q3	4
P329	Q4	5
P329	Q5	4
P329	Q6	3
P330	Q1	6
P330	Q2	6
P330	Q3	3
P330	Q4	5
P330	Q5	4
P330	Q6	6
P331	Q1	5
P331	Q2	4
P331	Q3	4
P331	Q4	4
P331	Q5	4
P331	Q6	6
P332	Q1	3
P332	Q2	4
P332	Q3	3
P332	Q4	4
P332	Q5	6
P332	Q6	5
P333	Q1	4
P333	Q2	4
P333	Q3	5
P333	Q4	3
P333	Q5	3
P333	Q6	4
P334	Q1	3
P334	Q2	4
P334	Q3	1
P334	Q4	5
P334	Q5	1
P334	Q6	3
P335	Q1	4
P335	Q2	5
P335	Q3	4
P335	Q4	6
P335	Q5	4
P335	Q6	5
P336	Q1	3
P336	Q2	4
P336	Q3	3
P336	Q4	4
P336	Q5	3
P336	Q6	4
P337	Q1	2
P337	Q2	1
P337	Q3	4
P337	Q4	4
P337	Q5	1
P337	Q6	3
P338	Q1	2
P338	Q2	4
P338	Q3	2
P338	Q4	4
P338	Q5	4
P338	Q6	4
P339	Q1	4
P339	Q2	5
P339	Q3	3
P339	Q4	3
P339	Q5	3
P339	Q6	5
P340	Q1	5
P340	Q2	5
P340	Q3	4
P340	Q4	5
P340	Q5	4
P340	Q6	5
P341	Q1	3
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P341	Q3	3
P341	Q4	5
P341	Q5	5
P341	Q6	5
P342	Q1	4
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P342	Q3	2
P342	Q4	4
P342	Q5	2
P342	Q6	4
P343	Q1	5
P343	Q2	5
P343	Q3	5
P343	Q4	4
P343	Q5	4
P343	Q6	5
P344	Q1	5
P344	Q2	4
P344	Q3	4
P344	Q4	4
P344	Q5	3
P344	Q6	6
P345	Q1	4
P345	Q2	3
P345	Q3	3
P345	Q4	5
P345	Q5	4
P345	Q6	5
P346	Q1	4
P346	Q2	4
P346	Q3	2
P346	Q4	5
P346	Q5	5
P346	Q6	5
P347	Q1	6
P347	Q2	5
P347	Q3	3
P347	Q4	5
P347	Q5	5
P347	Q6	3
P348	Q1	4
P348	Q2	5
P348	Q3	5
P348	Q4	6
P348	Q5	5
P348	Q6	7
P349	Q1	3
P349	Q2	4
P349	Q3	4
P349	Q4	3
P349	Q5	3
P349	Q6	7
P350	Q1	2
P350	Q2	3
P350	Q3	3
P350	Q4	3
P350	Q5	2
P350	Q6	3
P351	Q1	5
P351	Q2	5
P351	Q3	4
P351	Q4	5
P351	Q5	3
P351	Q6	4
P352	Q1	5
P352	Q2	5
P352	Q3	4
P352	Q4	5
P352	Q5	4
P352	Q6	3
P353	Q1	3
P353	Q2	3
P353	Q3	4
P353	Q4	3
P353	Q5	3
P353	Q6	5
P354	Q1	6
P354	Q2	6
P354	Q3	4
P354	Q4	6
P354	Q5	5
P354	Q6	4
P355	Q1	2
P355	Q2	4
P355	Q3	3
P355	Q4	5
P355	Q5	3
P355	Q6	3
P356	Q1	4
P356	Q2	5
P356	Q3	4
P356	Q4	6
P356	Q5	5
P356	Q6	6
P357	Q1	5
P357	Q2	3
P357	Q3	2
P357	Q4	4
P357	Q5	2
P357	Q6	3
P358	Q1	4
P358	Q2	6
P358	Q3	4
P358	Q4	6
P358	Q5	3
P358	Q6	5
P359	Q1	3
P359	Q2	3
P359	Q3	2
P359	Q4	2
P359	Q5	1
P359	Q6	3
P360	Q1	4
P360	Q2	5
P360	Q3	4
P360	Q4	4
P360	Q5	3
P360	Q6	3
P361	Q1	3
P361	Q2	3
P361	Q3	2
P361	Q4	4
P361	Q5	1
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P362	Q1	5
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P362	Q3	4
P362	Q4	4
P362	Q5	1
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P363	Q1	4
P363	Q2	6
P363	Q3	5
P363	Q4	5
P363	Q5	4
P363	Q6	4
P364	Q1	3
P364	Q2	4
P364	Q3	5
P364	Q4	4
P364	Q5	4
P364	Q6	7
P365	Q1	4
P365	Q2	4
P365	Q3	6
P365	Q4	5
P365	Q5	5
P365	Q6	5
P366	Q1	3
P366	Q2	3
P366	Q3	3
P366	Q4	3
P366	Q5	3
P366	Q6	3
P367	Q1	5
P367	Q2	4
P367	Q3	4
P367	Q4	6
P367	Q5	4
P367	Q6	4
P368	Q1	3
P368	Q2	4
P368	Q3	3
P368	Q4	4
P368	Q5	1
P368	Q6	3
P369	Q1	4
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P369	Q3	3
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P369	Q5	5
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P370	Q1	4
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P370	Q3	2
P370	Q4	5
P370	Q5	4
P370	Q6	3
P371	Q1	5
P371	Q2	4
P371	Q3	3
P371	Q4	6
P371	Q5	4
P371	Q6	5
P372	Q1	4
P372	Q2	3
P372	Q3	3
P372	Q4	3
P372	Q5	3
P372	Q6	3
P373	Q1	3
P373	Q2	4
P373	Q3	2
P373	Q4	3
P373	Q5	5
P373	Q6	3
P374	Q1	6
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P374	Q3	6
P374	Q4	6
P374	Q5	6
P374	Q6	6
P375	Q1	4
P375	Q2	4
P375	Q3	4
P375	Q4	4
P375	Q5	3
P375	Q6	5
P376	Q1	4
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P376	Q4	6
P376	Q5	4
P376	Q6	4
P377	Q1	5
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P377	Q3	4
P377	Q4	3
P377	Q5	2
P377	Q6	4
P378	Q1	5
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P378	Q3	5
P378	Q4	4
P378	Q5	5
P378	Q6	4
P379	Q1	6
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P379	Q3	6
P379	Q4	6
P379	Q5	6
P379	Q6	5
P380	Q1	4
P380	Q2	4
P380	Q3	3
P380	Q4	4
P380	Q5	1
P380	Q6	5
P381	Q1	4
P381	Q2	4
P381	Q3	2
P381	Q4	3
P381	Q5	3
P381	Q6	4
P382	Q1	2
P382	Q2	3
P382	Q3	4
P382	Q4	5
P382	Q5	3
P382	Q6	3
P383	Q1	3
P383	Q2	6
P383	Q3	4
P383	Q4	6
P383	Q5	2
P383	Q6	5
P384	Q1	1
P384	Q2	4
P384	Q3	3
P384	Q4	4
P384	Q5	4
P384	Q6	3
P385	Q1	3
P385	Q2	3
P385	Q3	3
P385	Q4	3
P385	Q5	1
P385	Q6	6
P386	Q1	5
P386	Q2	4
P386	Q3	2
P386	Q4	4
P386	Q5	4
P386	Q6	7
P387	Q1	5
P387	Q2	5
P387	Q3	4
P387	Q4	4
P387	Q5	5
P387	Q6	4
P388	Q1	5
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P388	Q5	3
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P389	Q1	3
P389	Q2	3
P389	Q3	3
P389	Q4	5
P389	Q5	2
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P390	Q1	4
P390	Q2	3
P390	Q3	3
P390	Q4	4
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P391	Q1	6
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P391	Q5	5
P391	Q6	4
P392	Q1	4
P392	Q2	4
P392	Q3	3
P392	Q4	4
P392	Q5	1
P392	Q6	4
P393	Q1	2
P393	Q2	3
P393	Q3	3
P393	Q4	5
P393	Q5	4
P393	Q6	4
P394	Q1	3
P394	Q2	4
P394	Q3	1
P394	Q4	4
P394	Q5	3
P394	Q6	4
P395	Q1	5
P395	Q2	3
P395	Q3	4
P395	Q4	4
P395	Q5	4
P395	Q6	2
P396	Q1	4
P396	Q2	3
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P396	Q4	4
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P397	Q1	4
P397	Q2	3
P397	Q3	3
P397	Q4	4
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P398	Q1	5
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P399	Q1	3
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P400	Q1	5
P400	Q2	3
P400	Q3	3
P400	Q4	5
P400	Q5	5
P400	Q6	5
P401	Q1	3
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P401	Q3	2
P401	Q4	5
P401	Q5	2
P401	Q6	3
P402	Q1	6
P402	Q2	5
P402	Q3	2
P402	Q4	5
P402	Q5	1
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P403	Q1	4
P403	Q2	5
P403	Q3	4
P403	Q4	4
P403	Q5	4
P403	Q6	7
P404	Q1	2
P404	Q2	2
P404	Q3	3
P404	Q4	3
P404	Q5	4
P404	Q6	3
P405	Q1	3
P405	Q2	4
P405	Q3	3
P405	Q4	4
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P406	Q1	5
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P406	Q3	3
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P406	Q5	4
P406	Q6	3
P407	Q1	4
P407	Q2	4
P407	Q3	2
P407	Q4	5
P407	Q5	2
P407	Q6	4
P408	Q1	6
P408	Q2	6
P408	Q3	3
P408	Q4	6
P408	Q5	6
P408	Q6	5
P409	Q1	7
P409	Q2	7
P409	Q3	6
P409	Q4	6
P409	Q5	5
P409	Q6	7
P410	Q1	5
P410	Q2	4
P410	Q3	3
P410	Q4	4
P410	Q5	3
P410	Q6	5
P411	Q1	3
P411	Q2	7
P411	Q3	3
P411	Q4	5
P411	Q5	3
P411	Q6	5
P412	Q1	7
P412	Q2	7
P412	Q3	4
P412	Q4	5
P412	Q5	3
P412	Q6	7
P413	Q1	5
P413	Q2	4
P413	Q3	4
P413	Q4	6
P413	Q5	4
P413	Q6	4
P414	Q1	5
P414	Q2	5
P414	Q3	3
P414	Q4	4
P414	Q5	6
P414	Q6	5
P415	Q1	5
P415	Q2	5
P415	Q3	3
P415	Q4	5
P415	Q5	3
P415	Q6	4
P416	Q1	6
P416	Q2	4
P416	Q3	4
P416	Q4	5
P416	Q5	5
P416	Q6	4
P417	Q1	5
P417	Q2	4
P417	Q3	1
P417	Q4	3
P417	Q5	4
P417	Q6	2
P418	Q1	4
P418	Q2	5
P418	Q3	3
P418	Q4	5
P418	Q5	3
P418	Q6	4
P419	Q1	3
P419	Q2	3
P419	Q3	3
P419	Q4	3
P419	Q5	4
P419	Q6	5
P420	Q1	5
P420	Q2	5
P420	Q3	5
P420	Q4	4
P420	Q5	6
P420	Q6	7
P421	Q1	7
P421	Q2	6
P421	Q3	5
P421	Q4	6
P421	Q5	5
P421	Q6	6
P422	Q1	3
P422	Q2	4
P422	Q3	1
P422	Q4	4
P422	Q5	3
P422	Q6	4
P423	Q1	2
P423	Q2	5
P423	Q3	3
P423	Q4	4
P423	Q5	4
P423	Q6	4
P424	Q1	3
P424	Q2	4
P424	Q3	3
P424	Q4	3
P424	Q5	5
P424	Q6	3
P425	Q1	3
P425	Q2	4
P425	Q3	2
P425	Q4	3
P425	Q5	2
P425	Q6	2
P426	Q1	6
P426	Q2	7
P426	Q3	7
P426	Q4	7
P426	Q5	6
P426	Q6	7
P427	Q1	5
P427	Q2	5
P427	Q3	4
P427	Q4	5
P427	Q5	5
P427	Q6	5
P428	Q1	3
P428	Q2	4
P428	Q3	2
P428	Q4	4
P428	Q5	4
P428	Q6	4
P429	Q1	5
P429	Q2	4
P429	Q3	5
P429	Q4	3
P429	Q5	3
P429	Q6	4
P430	Q1	1
P430	Q2	4
P430	Q3	3
P430	Q4	5
P430	Q5	6
P430	Q6	4
P431	Q1	5
P431	Q2	3
P431	Q3	3
P431	Q4	5
P431	Q5	3
P431	Q6	4
P432	Q1	4
P432	Q2	5
P432	Q3	3
P432	Q4	3
P432	Q5	4
P432	Q6	6
P433	Q1	7
P433	Q2	5
P433	Q3	4
P433	Q4	5
P433	Q5	3
P433	Q6	5
P434	Q1	3
P434	Q2	3
P434	Q3	2
P434	Q4	4
P434	Q5	2
P434	Q6	4
P435	Q1	5
P435	Q2	4
P435	Q3	3
P435	Q4	5
P435	Q5	3
P435	Q6	5
P436	Q1	2
P436	Q2	4
P436	Q3	3
P436	Q4	4
P436	Q5	3
P436	Q6	4
P437	Q1	4
P437	Q2	3
P437	Q3	3
P437	Q4	4
P437	Q5	2
P437	Q6	3
P438	Q1	5
P438	Q2	6
P438	Q3	3
P438	Q4	5
P438	Q5	4
P438	Q6	5
P439	Q1	7
P439	Q2	3
P439	Q3	3
P439	Q4	3
P439	Q5	4
P439	Q6	5
P440	Q1	3
P440	Q2	4
P440	Q3	3
P440	Q4	5
P440	Q5	3
P440	Q6	6
P441	Q1	5
P441	Q2	4
P441	Q3	4
P441	Q4	5
P441	Q5	3
P441	Q6	3
P442	Q1	6
P442	Q2	3
P442	Q3	4
P442	Q4	5
P442	Q5	3
P442	Q6	5
P443	Q1	4
P443	Q2	3
P443	Q3	3
P443	Q4	5
P443	Q5	4
P443	Q6	6
P444	Q1	3
P444	Q2	4
P444	Q3	2
P444	Q4	4
P444	Q5	4
P444	Q6	5
P445	Q1	5
P445	Q2	4
P445	Q3	4
P445	Q4	4
P445	Q5	4
P445	Q6	3
P446	Q1	2
P446	Q2	4
P446	Q3	4
P446	Q4	5
P446	Q5	5
P446	Q6	6
P447	Q1	4
P447	Q2	5
P447	Q3	1
P447	Q4	4
P447	Q5	4
P447	Q6	2
P448	Q1	5
P448	Q2	7
P448	Q3	4
P448	Q4	5
P448	Q5	4
P448	Q6	5
P449	Q1	3
P449	Q2	5
P449	Q3	3
P449	Q4	5
P449	Q5	4
P449	Q6	5
P450	Q1	2
P450	Q2	3
P450	Q3	2
P450	Q4	3
P450	Q5	1
P450	Q6	4
P451	Q1	4
P451	Q2	4
P451	Q3	2
P451	Q4	2
P451	Q5	4
P451	Q6	4
P452	Q1	2
P452	Q2	3
P452	Q3	3
P452	Q4	4
P452	Q5	3
P452	Q6	4
P453	Q1	4
P453	Q2	5
P453	Q3	4
P453	Q4	4
P453	Q5	2
P453	Q6	3
P454	Q1	5
P454	Q2	5
P454	Q3	4
P454	Q4	5
P454	Q5	6
P454	Q6	4
P455	Q1	2
P455	Q2	4
P455	Q3	2
P455	Q4	5
P455	Q5	2
P455	Q6	4
P456	Q1	4
P456	Q2	3
P456	Q3	3
P456	Q4	4
P456	Q5	4
P456	Q6	4
P457	Q1	5
P457	Q2	6
P457	Q3	3
P457	Q4	6
P457	Q5	5
P457	Q6	7
P458	Q1	2
P458	Q2	5
P458	Q3	3
P458	Q4	3
P458	Q5	4
P458	Q6	4
P459	Q1	4
P459	Q2	4
P459	Q3	2
P459	Q4	4
P459	Q5	4
P459	Q6	3
P460	Q1	3
P460	Q2	6
P460	Q3	4
P460	Q4	6
P460	Q5	4
P460	Q6	5
P461	Q1	6
P461	Q2	6
P461	Q3	3
P461	Q4	6
P461	Q5	5
P461	Q6	5
P462	Q1	6
P462	Q2	6
P462	Q3	5
P462	Q4	5
P462	Q5	4
P462	Q6	3
P463	Q1	5
P463	Q2	3
P463	Q3	1
P463	Q4	5
P463	Q5	3
P463	Q6	4
P464	Q1	4
P464	Q2	4
P464	Q3	4
P464	Q4	4
P464	Q5	3
P464	Q6	4
P465	Q1	6
P465	Q2	5
P465	Q3	4
P465	Q4	5
P465	Q5	2
P465	Q6	5
P466	Q1	4
P466	Q2	4
P466	Q3	3
P466	Q4	3
P466	Q5	3
P466	Q6	4
P467	Q1	5
P467	Q2	6
P467	Q3	3
P467	Q4	7
P467	Q5	4
P467	Q6	4
P468	Q1	5
P468	Q2	6
P468	Q3	2
P468	Q4	3
P468	Q5	4
P468	Q6	4
P469	Q1	4
P469	Q2	5
P469	Q3	3
P469	Q4	4
P469	Q5	3
P469	Q6	5
P470	Q1	4
P470	Q2	6
P470	Q3	3
P470	Q4	5
P470	Q5	3
P470	Q6	5
P471	Q1	3
P471	Q2	4
P471	Q3	6
P471	Q4	3
P471	Q5	3
P471	Q6	4
P472	Q1	3
P472	Q2	2
P472	Q3	2
P472	Q4	4
P472	Q5	2
P472	Q6	2
P473	Q1	5
P473	Q2	5
P473	Q3	4
P473	Q4	5
P473	Q5	6
P473	Q6	5
P474	Q1	3
P474	Q2	4
P474	Q3	4
P474	Q4	4
P474	Q5	1
P474	Q6	4
P475	Q1	3
P475	Q2	3
P475	Q3	4
P475	Q4	4
P475	Q5	1
P475	Q6	4
P476	Q1	2
P476	Q2	4
P476	Q3	5
P476	Q4	4
P476	Q5	3
P476	Q6	3
P477	Q1	3
P477	Q2	4
P477	Q3	3
P477	Q4	4
P477	Q5	3
P477	Q6	6
P478	Q1	6
P478	Q2	4
P478	Q3	5
P478	Q4	6
P478	Q5	6
P478	Q6	5
P479	Q1	2
P479	Q2	5
P479	Q3	2
P479	Q4	4
P479	Q5	3
P479	Q6	5
P480	Q1	4
P480	Q2	4
P480	Q3	2
P480	Q4	5
P480	Q5	2
P480	Q6	3
P481	Q1	4
P481	Q2	3
P481	Q3	6
P481	Q4	4
P481	Q5	3
P481	Q6	4
P482	Q1	6
P482	Q2	6
P482	Q3	4
P482	Q4	6
P482	Q5	7
P482	Q6	6
P483	Q1	4
P483	Q2	3
P483	Q3	2
P483	Q4	3
P483	Q5	4
P483	Q6	4
P484	Q1	4
P484	Q2	5
P484	Q3	4
P484	Q4	5
P484	Q5	4
P484	Q6	5
P485	Q1	5
P485	Q2	5
P485	Q3	1
P485	Q4	3
P485	Q5	2
P485	Q6	5
P486	Q1	5
P486	Q2	6
P486	Q3	4
P486	Q4	7
P486	Q5	6
P486	Q6	5
P487	Q1	2
P487	Q2	3
P487	Q3	2
P487	Q4	3
P487	Q5	1
P487	Q6	3
P488	Q1	3
P488	Q2	3
P488	Q3	4
P488	Q4	6
P488	Q5	3
P488	Q6	6
P489	Q1	3
P489	Q2	5
P489	Q3	4
P489	Q4	3
P489	Q5	3
P489	Q6	3
P490	Q1	4
P490	Q2	4
P490	Q3	3
P490	Q4	4
P490	Q5	6
P490	Q6	6
P491	Q1	5
P491	Q2	5
P491	Q3	4
P491	Q4	5
P491	Q5	3
P491	Q6	6
P492	Q1	5
P492	Q2	5
P492	Q3	5
P492	Q4	7
P492	Q5	5
P492	Q6	5
P493	Q1	4
P493	Q2	4
P493	Q3	3
P493	Q4	5
P493	Q5	2
P493	Q6	4
P494	Q1	2
P494	Q2	4
P494	Q3	1
P494	Q4	4
P494	Q5	4
P494	Q6	5
P495	Q1	4
P495	Q2	5
P495	Q3	3
P495	Q4	5
P495	Q5	2
P495	Q6	6
P496	Q1	4
P496	Q2	4
P496	Q3	3
P496	Q4	4
P496	Q5	4
P496	Q6	6
P497	Q1	4
P497	Q2	4
P497	Q3	3
P497	Q4	3
P497	Q5	3
P497	Q6	4
P498	Q1	2
P498	Q2	3
P498	Q3	3
P498	Q4	4
P498	Q5	3
P498	Q6	3
P499	Q1	4
P499	Q2	5
P499	Q3	3
P499	Q4	4
P499	Q5	2
P499	Q6	4
P500	Q1	5
P500	Q2	4
P500	Q3	4
P500	Q4	3
P500	Q5	4
P500	Q6	4

ppt	life satisfaction
	Q1	Q2	Q3	Q4	Q5	Q6
P001	1	5	1	5	4	2
P002	5	5	4	5	4	2
P003	4	4	4	4	4	4
P004	2	6	4	3	2	3
P005	3	6	4	4	3	4
P006	4	5	4	5	5	3

Scale Scores?

Q	wording
Q1	Overall, I am satisfied with my life.
Q2	In most ways my life is close to my ideal.
Q3	The conditions of my life are excellent.
Q4	I have gotten the important things I want in life.
Q5	If I could live my life over, I would change almost nothing.
Q6	My life has turned out better than I expected it would.

ppt	Q1	Q2	Q3	Q4	Q5	Q6	LFSAT
P001	1	5	1	5	4	2	18
P002	5	5	4	5	4	2	25
P003	4	4	4	4	4	4	24
P004	2	6	4	3	2	3	20
P005	3	6	4	4	3	4	24
P006	4	5	4	5	5	3	26

• each question contains error (because of wording, respondents mood, the context, misunderstanding etc).

• combine multiple items (e.g., by averaging or summing them) and we hope that the random errors cancel out, and the signal of the true construct becomes clearer

Scale Scores?

Pros:

• a useful tool in applied settings - the number is always calculated the same way

Cons:

• ignores differences in item quality
  • all questions are equally representative of “life satisfaction”
• assumes ‘unidimensionality’
  • all questions capture 1 thing, and that 1 thing is “life satisfaction”

quick refresher

• Variance = Deviance around the mean of a single variable

• Covariance = Representation of how two variables change together

• Correlation = Standardised version of covariance

Covariance/Correlation Matrices

Assumption throughout this block: covariance between two items is because they measure the same thing(s) (in theory!)

Q	wording
Q1	Overall, I am satisfied with my life.
Q2	In most ways my life is close to my ideal.
Q3	The conditions of my life are excellent.
Q4	I have gotten the important things I want in life.
Q5	If I could live my life over, I would change almost nothing.
Q6	My life has turned out better than I expected it would.

lifesat_data6 |> 
  select(-ppt) |>
  cor()

     Q1   Q2   Q3   Q4   Q5   Q6
Q1 1.00 0.36 0.37 0.36 0.34 0.26
Q2 0.36 1.00 0.34 0.45 0.39 0.26
Q3 0.37 0.34 1.00 0.35 0.37 0.30
Q4 0.36 0.45 0.35 1.00 0.34 0.25
Q5 0.34 0.39 0.37 0.34 1.00 0.25
Q6 0.26 0.26 0.30 0.25 0.25 1.00

“variables” and “dimensions”

Q	wording
Q1	Overall, I am satisfied with my life.
Q2	In most ways my life is close to my ideal.

library(psych)
lifesat_data6 |>
  select(Q1:Q2) |>
  pairs.panels()

more variables and dimensions

Q	wording
Q1	Overall, I am satisfied with my life.
Q2	In most ways my life is close to my ideal.
Q3	The conditions of my life are excellent.
Q4	I have gotten the important things I want in life.
Q5	If I could live my life over, I would change almost nothing.
Q6	My life has turned out better than I expected it would.

lifesat_data6 |>
  select(-ppt) |>
  pairs.panels()

more variables and more dimensions

Q	wording
Q1	Overall, I am satisfied with my life.
Q2	In most ways my life is close to my ideal.
Q3	The conditions of my life are excellent.
Q4	I have gotten the important things I want in life.
Q5	If I could live my life over, I would change almost nothing.
Q6	My life has turned out better than I expected it would.
Q7	I feel happy and content most days.
Q8	I experience joy and pleasure regularly in my daily life.
Q9	I feel a sense of inner peace and satisfaction.
Q10	I wake up most mornings feeling positive about the day ahead.

lifesat_data10 |>
  select(-ppt) |>
  pairs.panels()

Eurgh… hard to see!

more variables and more dimensions (2)

Q	wording
Q1	Overall, I am satisfied with my life.
Q2	In most ways my life is close to my ideal.
Q3	The conditions of my life are excellent.
Q4	I have gotten the important things I want in life.
Q5	If I could live my life over, I would change almost nothing.
Q6	My life has turned out better than I expected it would.
Q7	I feel happy and content most days.
Q8	I experience joy and pleasure regularly in my daily life.
Q9	I feel a sense of inner peace and satisfaction.
Q10	I wake up most mornings feeling positive about the day ahead.

lifesat_data10 |>
  select(-ppt) |>
  cor() |> 
  heatmap()

Think: “buckets” of co-variation!

more variables and more dimensions (2)

Q	wording
Q1	Overall, I am satisfied with my life.
Q2	In most ways my life is close to my ideal.
Q3	I feel happy and content most days.
Q4	The conditions of my life are excellent.
Q5	I experience joy and pleasure regularly in my daily life.
Q6	I have gotten the important things I want in life.
Q7	I feel a sense of inner peace and satisfaction.
Q8	If I could live my life over, I would change almost nothing.
Q9	I wake up most mornings feeling positive about the day ahead.
Q10	My life has turned out better than I expected it would.

lifesat_data10 |>
  select(-ppt) |>
  cor() |> 
  heatmap()

Visibility of the “buckets” depends on ordering of the items!

how many dimensions can you think in?

Three variables measuring unrelated things:

Rate agreement on:

• Q1: I am the life and soul of the party
• Q2: I like penguins
• Q3: I enjoy studying statistics

Three variables perfectly measuring the exact same thing

Time spent looking at phone last week:

• In hours
• In days
• In weeks

Three variables measuring the same thing but differently

Rate agreement on:

• Q1: I think cake is the best food
• Q2: I feel great when I eat cake
• Q3: I often eat cake

how many dimensions can you think in?

Three variables measuring unrelated things:

Rate agreement on:

• Q1: I am the life and soul of the party
• Q2: I like penguins
• Q3: I enjoy studying statistics

Three variables perfectly measuring the exact same thing

Time spent looking at phone last week:

• In hours
• In days
• In weeks

Three variables measuring the same thing but differently

Rate agreement on:

• Q1: I think cake is the best food
• Q2: I feel great when I eat cake
• Q3: I often eat cake

how many dimensions can you think in? (2)

Three variables measuring unrelated things:

Rate agreement on:

• Q1: I am the life and soul of the party
• Q2: I like penguins
• Q3: I enjoy studying statistics

Three variables perfectly measuring the exact same thing

Time spent looking at phone last week:

• In hours
• In days
• In weeks

Three variables measuring the same thing but differently

Rate agreement on:

• Q1: I think cake is the best food
• Q2: I feel great when I eat cake
• Q3: I enjoy studying statistics

Principal Component Analysis (PCA)

In short:

• re-express covariances between \(k\) items as \(k\) dimensions

• dimensions are termed “components” and are orthogonal (perpendicular/uncorrelated)

• dimensions sequentially capture most variance

eigendecomposition

In short:

• The math behind PCA

It’s a lot
We don’t need to understand it.

source: https://it.memedroid.com/memes/detail/3218300/And-thats-why-my-life-is-so-hard

Some terms will come up…

• “eigenvector”
  • direction of each dimension (relative to our original variables)
  • which variables is the dimension related to?
• “eigenvalue”
  • magnitude of each dimension
  • how much variance is captured by the dimension?

PCA in R

lifesat_data6 |>
  select(-ppt) |>
  cor()

     Q1   Q2   Q3   Q4   Q5   Q6
Q1 1.00 0.36 0.37 0.36 0.34 0.26
Q2 0.36 1.00 0.34 0.45 0.39 0.26
Q3 0.37 0.34 1.00 0.35 0.37 0.30
Q4 0.36 0.45 0.35 1.00 0.34 0.25
Q5 0.34 0.39 0.37 0.34 1.00 0.25
Q6 0.26 0.26 0.30 0.25 0.25 1.00

library(psych)
principal(lifesat_data6[,-1], nfactors = 6, rotate = "none")  

• principal() can take a covariance matrix, a correlation matrix, or the raw data.
  
• rotate = "none" is saying that we want these dimensions to be orthogonal (in future weeks we’ll mix things up)

PCA in R (2)

lifesat_data6 |>
  select(-ppt) |>
  cor()

     Q1   Q2   Q3   Q4   Q5   Q6
Q1 1.00 0.36 0.37 0.36 0.34 0.26
Q2 0.36 1.00 0.34 0.45 0.39 0.26
Q3 0.37 0.34 1.00 0.35 0.37 0.30
Q4 0.36 0.45 0.35 1.00 0.34 0.25
Q5 0.34 0.39 0.37 0.34 1.00 0.25
Q6 0.26 0.26 0.30 0.25 0.25 1.00

library(psych)
principal(lifesat_data6[,-1], nfactors = 6, rotate = "none")  

the reduction: how many components do we keep?

how many/how much variation do you want to keep?

the reduction: how many components do we keep?

scree plots (these are the eigenvalues - the magnitudes)

the reduction: how many components do we keep?

parallel analysis

Parallel analysis suggests that the number of factors =  NA  and the number of components =  1 

minimum average partial

Very Simple Structure
Call: vss(x = x, n = n, rotate = rotate, diagonal = diagonal, fm = fm, 
    n.obs = n.obs, plot = plot, title = title, use = use, cor = cor)
VSS complexity 1 achieves a maximimum of 0.72  with  1  factors
VSS complexity 2 achieves a maximimum of 0.75  with  2  factors

The Velicer MAP achieves a minimum of 0.04  with  1  factors 
BIC achieves a minimum of  -44.6  with  1  factors
Sample Size adjusted BIC achieves a minimum of  -16.1  with  1  factors

Statistics by number of factors 
  vss1 vss2   map dof   chisq prob sqresid  fit RMSEA BIC SABIC complex  eChisq
1 0.72 0.00 0.042   9 1.1e+01 0.26     2.7 0.72 0.023 -45   -16     1.0 9.6e+00
2 0.46 0.75 0.124   4 9.0e-01 0.92     2.4 0.75 0.000 -24   -11     1.6 6.8e-01
3 0.46 0.70 0.227   0 7.7e-02   NA     2.0 0.78    NA  NA    NA     1.7 7.1e-02
4 0.37 0.63 0.448  -3 7.0e-12   NA     2.2 0.76    NA  NA    NA     2.2 5.8e-12
5 0.37 0.63 1.000  -5 3.6e-11   NA     2.2 0.77    NA  NA    NA     2.2 2.7e-11
6 0.37 0.63    NA  -6 3.6e-11   NA     2.2 0.77    NA  NA    NA     2.2 2.7e-11
     SRMR eCRMS eBIC
1 2.5e-02 0.033  -46
2 6.7e-03 0.013  -24
3 2.2e-03    NA   NA
4 2.0e-08    NA   NA
5 4.2e-08    NA   NA
6 4.2e-08    NA   NA

Scoring

Where each observation stands on each dimension.

# We're keeping only 1 component now:
mypca <- principal(lifesat_data6[,-1], nfactors = 1, rotate = "none")  

mypca$scores

         PC1
[1,]  1.0525
[2,] -0.2234
[3,]  1.0369
[4,] -0.8464
[5,] -0.2131
[6,] -0.0231
...   ...
...   ...

• so if the dimension is highly related to items 1, 2, and 3, and person \(i\) scores high on those items, then they will score high on the dimension.

Using Scores (1)

Let’s imagine we did a study where participants:

1. completed our life satisfaction questionnaire
2. completed the “balloon analogue risk task”
   • (essentially how far do you pump up a balloon - to measure risk taking behaviour)

RQ: Do people who are more satisfied with life engage with less risk taking?

Data:

head(mystudy)

# A tibble: 6 × 7
     Q1    Q2    Q3    Q4    Q5    Q6  BART
  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1     1     1     3     1     2     1    13
2     2     3     5     4     4     2     9
3     5     3     3     5     4     5    10
4     2     2     3     2     3     5    15
5     5     4     6     4     3     4     9
6     3     1     5     2     3     1     9

mod <- lm(BART ~ Q1 + Q2 + Q3 + Q4 + Q5 + Q6, data = mystudy)
summary(mod)

Call:
lm(formula = BART ~ Q1 + Q2 + Q3 + Q4 + Q5 + Q6, data = mystudy)

Residuals:
   Min     1Q Median     3Q    Max 
-5.899 -1.851 -0.037  1.557  7.988 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 12.69081    1.57212    8.07  1.6e-11 ***
Q1          -0.11956    0.26830   -0.45     0.66    
Q2          -0.03193    0.31441   -0.10     0.92    
Q3           0.00721    0.37090    0.02     0.98    
Q4          -0.22601    0.31598   -0.72     0.48    
Q5          -0.34299    0.38346   -0.89     0.37    
Q6           0.17688    0.21444    0.82     0.41    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.92 on 68 degrees of freedom
Multiple R-squared:  0.0797,    Adjusted R-squared:  -0.00155 
F-statistic: 0.981 on 6 and 68 DF,  p-value: 0.445

Using Scores (2)

Let’s imagine we did a study where participants:

1. completed our life satisfaction questionnaire
2. completed the “balloon analogue risk task”
   • (essentially how far do you pump up a balloon - to measure risk taking behaviour)

RQ: Do people who are more satisfied with life engage with less risk taking?

Data:

head(mystudy)

# A tibble: 6 × 7
     Q1    Q2    Q3    Q4    Q5    Q6  BART
  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1     1     1     3     1     2     1    13
2     2     3     5     4     4     2     9
3     5     3     3     5     4     5    10
4     2     2     3     2     3     5    15
5     5     4     6     4     3     4     9
6     3     1     5     2     3     1     9

library(car)
vif(mod)

  Q1   Q2   Q3   Q4   Q5   Q6 
1.81 1.71 1.53 1.93 1.87 1.08 

Using Scores (3)

Let’s imagine we did a study where participants:

1. completed our life satisfaction questionnaire
2. completed the “balloon analogue risk task”
   • (essentially how far do you pump up a balloon - to measure risk taking behaviour)

RQ: Do people who are more satisfied with life engage with less risk taking?

Data:

head(mystudy)

# A tibble: 6 × 8
     Q1    Q2    Q3    Q4    Q5    Q6  BART     PC1
  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>   <dbl>
1     1     1     3     1     2     1    13 -2.29  
2     2     3     5     4     4     2     9 -0.335 
3     5     3     3     5     4     5    10  0.0432
4     2     2     3     2     3     5    15 -1.35  
5     5     4     6     4     3     4     9  0.375 
6     3     1     5     2     3     1     9 -1.19  

mypca <- principal(mystudy[,1:6], nfactors = 1, rotate = "none")
mystudy$PC1 = mypca$scores[,1]

mod <- lm(BART ~ PC1, data = mystudy)
summary(mod)

Call:
lm(formula = BART ~ PC1, data = mystudy)

Residuals:
   Min     1Q Median     3Q    Max 
-6.135 -1.854 -0.058  1.597  7.297 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   10.547      0.329   32.10   <2e-16 ***
PC1           -0.714      0.331   -2.16    0.034 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.85 on 73 degrees of freedom
Multiple R-squared:  0.06,  Adjusted R-squared:  0.0471 
F-statistic: 4.66 on 1 and 73 DF,  p-value: 0.0342

Using Scale Scores

Let’s imagine we did a study where participants:

1. completed our life satisfaction questionnaire
2. completed the “balloon analogue risk task”
   • (essentially how far do you pump up a balloon - to measure risk taking behaviour)

RQ: Do people who are more satisfied with life engage with less risk taking?

Data:

head(mystudy)

# A tibble: 6 × 8
     Q1    Q2    Q3    Q4    Q5    Q6  BART SScore
  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl>
1     1     1     3     1     2     1    13      9
2     2     3     5     4     4     2     9     20
3     5     3     3     5     4     5    10     25
4     2     2     3     2     3     5    15     17
5     5     4     6     4     3     4     9     26
6     3     1     5     2     3     1     9     15

mystudy$SScore <- rowSums(mystudy[,1:6])

mod <- lm(BART ~ SScore, data = mystudy)
summary(mod)

Call:
lm(formula = BART ~ SScore, data = mystudy)

Residuals:
   Min     1Q Median     3Q    Max 
-6.164 -1.859 -0.164  1.532  7.053 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  13.0964     1.3748    9.53  1.9e-14 ***
SScore       -0.1086     0.0568   -1.91     0.06 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.86 on 73 degrees of freedom
Multiple R-squared:  0.0476,    Adjusted R-squared:  0.0346 
F-statistic: 3.65 on 1 and 73 DF,  p-value: 0.06

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