These are the steps taken to perform factor analysis:
1. First a correlation matrix is produced: the researchers themselves do not need to look at the correlational matrix, although the program uses the information from this matrix to perform further calculations. In our example, however, the researchers say ‘before perform- ing factor analysis, we inspected the correlational matrix to check the strength of the correlations’.
2. Then a set of factors is extracted: in practice it is possible to extract as many factors as variables (if each variable was not highly correlated with any other), but this would defeat the object of factor analysis. We want to account for as much variance as possible, while keeping the number of factors extracted as small as possible. Although in our first example there were two clear factors, often things are not as simple as this, and the decision on how many factors to ‘keep’ is decided by the researchers on the basis of both statistical and theoretical criteria.
3. Number of factors to retain: when factor analysis is performed in SPSS, the program decides how many factors to keep on the basis of statistical criteria. Each factor that is extracted accounts for a certain amount of variance.
(a) Eigenvalues show the proportion of variance accounted for by each factor. The sum of the eigenvalues is the number of variables in the analysis. Any factor that has an eigenvalue of 1.00 is kept.
It is a useful rule, but what happens if one of the factors has an eigenvalue of 0.99?
Using this rule blindly would mean that the factor would not emerge, and yet it could be theoretically important. It is here that the researcher would consider whether to keep such a factor. If it is decided that the factor should be kept, then SPSS needs to be overruled.
(b) Scree plot. This is simply the number of factors plotted against variance accounted for.
Here is a scree plot:
8
6
4
2
01 2 3 4 5 6 7 8 9 10
Component number
Eigenvalue
The idea is that factors drop off to some level and then plateau. The rule is that you look at the plot and see where the plateau levels out, and then choose the number of factors just before this point. Here we would choose two components. This is fine if your scree plot looks like the one above, but not if it looks like this:
CHAPTER 14 Introduction to factor analysis 463
It is harder to say here how many components ‘should’ be kept.
(c) The third criterion is to look at how much variance the factors account for. It is good practice to try to account for approximately 75% of the variance. However, you need to try to explain the most variance with the least number of factors. Thus, all the criteria must be used together to decide how many factors to keep. A good researcher needs to take everything into consideration in coming to a decision on how many factors to retain.
Kristiansen and Kuczaj (2013) tell us that they used the criterion of eigenvalues over 1, which agreed with the scree plot which suggested that eight factors/components should be retained.
They tell us that these factors explained 59.7% of the total variance.
4. Unrotated factor loadings: the program gives information on the strength of the relationships between the items and the factors. However, researchers are usually interested only in the rotated factor matrix – this tends to be easier to interpret (see above). Kristiansen and Kuczaj tell us that they used varimax rotation (the default on SPSS).
5. Naming the factors: the researcher then looks at the rotated factor loadings, to see what the clusters of variables have in common. To do this, a decision must be taken on how strong a loading must be for it to be included in the naming process. As mentioned above, this tends to be fairly arbitrary and varies between 0.3 and 0.5. Factor 1 will normally account for the largest amount of variance, followed by Factor 2, and so on.
In the study on horses, the authors have told us which item relates to which dimension of the questionnaire. So for the first factor, the items are all to do with the horse feeling anxious or nervous. This is the strongest factor, and apart from one item ‘s/he is suspicious of others’, the items tally with the dimension from which they are drawn: it is clear that ‘neuroticism’ is a good label for Factor 1. This is, in fact, the label given by the authors.
They give the total eigenvalues for each factor/component and show us the amount of variance explained by each factor. So neuroticism accounts for 11.23% of the variation in scores.
Activity 14.4
Look at the rotated matrix above (Table 14.6). You already know that the authors have named Factor 1 as ‘neuroticism’. Have a go at naming the other seven factors. Of course there are no ‘right’ answers, but you might want to check whether your Factor names are similar to those of the authors. (See the Answers to exercises at the end of the book.)
5 4 3 2 1
01 2 3 4 5 6 7 8 9 10
Component number
Eigenvalue
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Personal reflection
Ellen Boddington and Professor Mark McDermott University of East London
ARTICLE: Predicting resistance to health education messages for cannabis use: the role of rebelliousness, autic mastery health value and ethnicity
Professor Mark McDermott says:
“I have a long-standing interest in the psychology of rebelliousness – the desire to oppose perceived require- ments. I was involved in a large-scale study published in 2009 of the relationship between rebelliousness and health behaviour. Afterward, it occurred to me that rebelliousness might be associated with young people’s resistance to health persuasion messages. So, Ellen and I embarked on a study to find out if this was the case.
We thought resistance to health persuasion messages about cannabis use, and in particular about smoking cannabis, was an appropriate variable to study in this context. It seemed clear to us that many people, irrespec- tive of age, are of the view that cannabis use is relatively harmless, when compared against the effects, for example, of alcohol consumption. However, we were aware of a growing scientific literature which shows that the recreational use of non-medicinal cannabis is associated with a variety of undesirable mental and physical health consequences, both in the short and longer term. With this background in mind, we were of the view that the range of psychological constructs in Reversal Theory (RT) would provide a rich theoretical framework in which to test our hypotheses, since rebelliousness is included as a key component of the theory.
Equipped with our independent RT, we needed a psychometrically sound dependent variable. As part of the study we designed and developed our own measure of resistance to health persuasion messages about cannabis use. As we suspected, it turned out there is more than one dimension to such resistance, with two being identified from our principal components analysis, one component being about the perceived risks to health from cannabis use, and the other being about the perceived need for society to take steps to reduce cannabis use within the general population.
Perhaps unsurprisingly, we found that self-reported frequency of cannabis use was most predictive of resist- ance to health persuasions messages about the risks associated with such activity: after all, people who are most frequently engaged in a behaviour are not likely to be receptive to reasons for giving it up. What did surprise us though, was that it is the proactive form of rebelliousness (or, ‘negativism’, as RT calls it), the sensation-seeking form of rebelliousness, that was predictive of resistance to messages, and not reactive rebelliousness, the latter form of negativism taking its impetus from disaffection. So, it is not the disaffected oppositional individual who rejects health messages about cannabis use. Rather, it is the user who is more disposed to excitement seeking forms of rebellion – who enjoys the thrill of engaging in something illicit – that predicts resistance to accepting the risks of cannabis use. Not wanting control or power over others (called ‘autic mastery’ in RT) was also found to be predictive, as was placing little value on health.
”
Example from the literature
Predicting resistance to health education messages for cannabis use: the role of rebelliousness, autic mastery health value and ethnicity
Boddington and McDermott (2013) carried out a factor analysis (principal components analysis, PCA) on 28 items as part of their study into resistance to health messages. They carried out a scree analysis
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to determine how many factors to extract; this showed a two-factor solution. They then carried out a PCA specifying two factors and varimax rotation. The authors say: ‘In order to produce two concise subscales, only items that had a factor analysis weighting of 7.6 were included in each. This cut-off meant that factor one consisted of 12 items and factor two six items’ (p. 4).
Their table of results is reproduced below:
Health messages resistance scale (for cannabis) items and associated loadings on the two factors
Scale item Factor loading
Factor 1.
Cannabis contains many cancer causing agents .76
Smoking cannabis can shorten your life .73
Some psychological problems are made worse by the use of cannabis .72 Long-term cannabis users have greater impairment in learning, memory and
attention than non-users .69
Cannabis smoke contains many harmful chemicals .69
Smoking cannabis can lead to fatal diseases .68
Breathing air which contains cannabis smoke can be bad for your health .67 The longer the individual has been using cannabis the greater the likelihood of illness .64 I believe the claims made by the cannabis literature which states the possible
adverse side-effects .64
People who smoke cannabis are more likely to develop coronary heart disease than
non-cannabis smokers .62
Heavy cannabis use can cause panic and paranoia through the disturbance of
perception caused by the drug .62
People who take cannabis are more likely to suffer from psychological problems than
non-users .62
Factor 2.
The Government should be more active in discouraging the smoking of cannabis .84 There needs to be more advertisements trying to persuade people not to smoke
cannabis .84
A lot more needs to be done to prevent people from starting to use cannabis .75 Penalties for first-time possession of cannabis should be more severe .74
Breathing other people’s smoke does not worry me .68
Advertisements trying to persuade me not to use cannabis would make me more
aware of the dangers of cannabis smoking .64
These two factors (components) were then used in further analyses, i.e. Pearson’s correlations and multiple regression.
Activity 14.5
Give sensible names for the two factors extracted by Boddington and McDermott.
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