W11 Exercises

Q1 - EFA [14 marks]

Below are the results of a data reduction of a set of 12 items assessing environmental conscientiousness. Participants are asked to respond to each statement on a 5-point likert scale from “Strongly Disagree” to “Strongly Agree”.

Based on the results and the item descriptions below, provide an interpretation of the factor solution. Your description should include:

• Comment on the numerical solution [6 marks]
• Discussion of suitability of the selected number of factors [6 marks]
• An attempt to define the factors [2 marks]

item	wording
y1	I recycle regularly
y2	I use eco-friendly transportation
y3	I buy sustainable products
y4	I know how to reduce my carbon footprint
y5	Protecting resources matters to me
y6	I care about protecting the environment

item	wording
y7	I feel responsible for my environmental impact
y8	I am worried about climate change effects
y9	I know about the harm of single-use plastics
y10	I know how deforestation affects climate change
y11	I know about relevant environmental policies
y12	Wildlife destruction concerns me deeply

Solution 1.

• 12 items, 4 factors extracted. explains 29% variance
• Factors ML2 & ML1 both have \(\geq 3\) salient/primary loadings
  • salient = \(\geq |0.3|\)
• Factors ML3 & ML4: both have only 1 item with salient loading
• 3rd factor explaining only 6%, 4th factor only 2%
• SSloadings for ML3 & ML4 are <1
• 3 complex items (y4, y5, y6)
• some items (y5,y6) have no salient loadings
• Factors ML1,2,3 correlated weak-moderate
• Factor ML4 not strongly correlated with others
• probably overextracting (too many factors)
• main pointer = low variance expl of ML3 & ML4, plus not enough items
• complex items y5 & y6 are spread across ML3 & ML4 - a 3 factor solution may well make more sense
• item y4 = one to keep an eye on
• ML1 = “environmental knowledge”
• ML2 = “environmental behaviours”
• ML3/4 combined look like “environmental concern”

Q2 - SSloadings [6 marks]

Calculate the 6 values missing from the table below: SSloadings [2 marks] and proportion variance [2 marks] & cumulative variance [2 marks].

       PC1   PC2
item1 0.90  0.00
item2 0.90 -0.29
item3 0.90  0.30
item4 0.70  0.70
item5 0.81 -0.50

               PC1  PC2 
SS loadings     __   __ 
Proportion Var  __   __ 
Cumulative Var  __   __ 

Solution 2. Table filled in:

                 PC1   PC2
SS loadings    3.576 0.914
Proportion Var 0.715 0.183
Cumulative Var 0.715 0.898

How?
start by squaring all the numbers, and sum the columns to give us SSloadings:

        PC1    PC2
item1 0.810 0.0000
item2 0.810 0.0841
item3 0.810 0.0900
item4 0.490 0.4900
item5 0.656 0.2500
Sum   3.576 0.9141

divide SSloadings by 5 (because 5 observed variables) to get proportion variance

  PC1   PC2 
0.715 0.183 

those two numbers are then cumulatively summed for cumulative variance:

  PC1   PC2 
0.715 0.898 

Q3 - MLM [10 marks]

A company that makes “6-minute journals” is undertaking some research to showcase the effectiveness of their product in helping to alleviate unwanted feelings. They recruited 166 people signing up to one of 10 “anger management classes” in different cities, and asked them if they would like to have a free journal to help with reflection. 88 participants chose to take a journal, and 78 did not. Each participant filled out weekly assessments of anger levels for 10 weeks. Scores on the anger measure can range from 0 to 15, with changes of 3 being considered ‘clinically meaningful’.

To investigate if having a journal helps to reduce anger levels, the company fit a multilevel model to the data, with anger levels being modelled by week number (0 to 9, with 0 representing the first week participants filled in the anger assessment), whether the journal was used (“no”/“yes”, with “no” as the reference level).

Provide an interpretation of each of:

• the fixed effects [4 marks]
• the random effects [4 marks]
• the relevance of the findings, considering the context of the study design and researchers’ aims [2 marks]

Solution 3.

• anger for someone who doesn’t journal, at start (in “week 1”, or “week 0” is fine here, give benefit of doubt) is 10.22

• no significant change over the study period for those who don’t journal

• people to take the journal have lower anger at the start by -0.3

• journal group decreases in anger over the study by an additional -0.24 compared to no-journal group

• both participants and classes vary in starting anger levels and in change in anger over study period

• participants vary (both intercept and slopes of change) much more than classes

• high level of ppt variability relative to fixed slope

• ppts who start more angry decrease less (positive correlation intercepts and slopes)

• take journal = significant reduction in anger

• effect is small - over 10 weeks they only go down by -1.622

• difference in two groups at outset suggests two groups are not comparable

• self-selecting journal - maybe all we’re doing is splitting up people who do/don’t want to change

Q4 - hierarchical data structures [3 marks]

Provide example levels for each of the three types of study: Cross-Sectional, Repeated Measures, Longitudinal [3 marks]

level	cross-sectional	repeated measures	longitudinal
2	…	…	…
1	…	…	…

Solution 4. anything that makes sense here, obvious ones are:

level	cross-sec	rpt measures	longitudinal
2	department	people	people
1	people	experimental items	timepoints