Multidimensional measurement

25 Multidimensional measurement

and factor analysis

Introduction

styles and pupil achievement, it has been suggested that multidimensional typologies of teacher

However limited our knowledge of astronomy, behaviour should be developed. Such typologies, many of us have learned to pick out certain

clusterings of stars from the infinity of those that it is believed, would enable the researcher to group together similarities in teachers’ judgements about

crowd the Northern skies and to name them as the specific aspects of their classroom organization familiar Plough, Orion and the Great Bear. Few of

us would identify constellations in the Southern and management, and their ways of motivating, assessing and instructing pupils.

Hemisphere that are instantly recognizable by Techniques for grouping such judgements are those in Australia. many and various. What they all have in common Our predilection for reducing the complexity of is that they are methods for ‘determining the elements that constitute our lives to a more simple number and nature of the underlying variables order doesn’t stop at star gazing. In numerous among a large number of measures’, a definition ways, each and every one of us attempts to discern which Kerlinger (1970) uses to describe one of the patterns or shapes in seemingly unconnected best known grouping techniques, ‘factor analysis’. events in order to better grasp their significance We begin the chapter by illustrating elementary for us in the conduct of our daily lives. The linkage analysis which can be undertaken by hand, educational researcher is no exception.

As research into a particular aspect of and move to factor analysis, which is best left to the computer. Finally, we append a brief note on

human activity progresses, the variables being multilevel modelling and another about cluster explored frequently turn out to be more complex analysis, the latter as a way of organizing people or than was first realized. Investigation into the groups rather than variables. relationship between teaching styles and pupil

achievement is a case in point. Global distinctions between behaviour identified as progressive or traditional, informal or formal, are vague and

Elementary linkage analysis: an example

woolly and have led inevitably to research Elementary linkage analysis (McQuitty 1957) is findings that are at worse inconsistent, at best,

one way of exploring the relationship between the inconclusive. In reality, epithets such as informal

teacher’s personal constructs, that is, of assessing or formal in the context of teaching and learning

the dimensionality of the judgements that the relate to ‘multidimensional concepts’, that is,

teacher makes about his or her pupils. It seeks concepts made up of a number of variables.

to identify and define the clusterings of certain ‘Multidimensional scaling’, on the other hand, is a

variables within a set of variables. Like factor way of analysing judgements of similarity between

analysis, which we shortly illustrate, elementary such variables in order that the dimensionality

linkage analysis searches for interrelated groups of those judgements can be assessed (Bennett and

of correlation coefficients. The objective of the Bowers 1977). As regards research into teaching

search is to identify ‘types’. By type, McQuitty

560 MULTIDIMENSIONAL MEASUREMENT

(1957) refers to ‘a category of people or other are related to the original pair which initially objects (personal constructs in our example) such

constituted Cluster 1. that the members are internally self-contained in

4 Now identify any variables which are most being like one another’.

like the variables elicited in Step 3. Repeat Seven constructs were elicited from an infant

this procedure until no further variables are school teacher who was invited to discuss the

identified.

ways in which she saw the children in her class

5 Excluding all those variables which belong (see Chapter 20). She identified favourable and

within Cluster 1, repeat Steps 2 to 4 until all unfavourable constructs as follows: ‘intelligent’

the variables have been accounted for. (+), ‘sociable’ (+), ‘verbally good’ (+), ‘well

behaved’ (+), ‘aggressive’ (−), ‘noisy’ (−) and

Factor analysis

‘clumsy’ (−). Four boys and six girls were then selected at

Factor analysis is a method of grouping together random from the class register and the teacher

variables which have something in common. It was asked to place each child in rank order under

is a process which enables the researcher to take each of the seven constructs, using rank position

a set of variables and reduce them to a smaller

1 to indicate the child most like the particular number of underlying factors which account for construct, and rank position 10, the child least

as many variables as possible. It detects structures like the particular construct. The teacher’s rank

and commonalities in the relationships between ordering is set out in Box 25.1. Notice that on

variables. Thus it enables researchers to identify three constructs, the rankings have been reversed

where different variables in fact are addressing in order to maintain the consistency of Favourable

the same underlying concept. For example, one

1, Unfavourable = 10. variable could measure somebody’s height in Box 25.2 sets out the intercorrelations between

centimetres; another variable could measure the the seven personal construct ratings shown in

same person’s height in inches. The underlying Box 25.1 (Spearman’s rho is the method of

factor that unites both variables is height; it is a correlation used in this example).

latent factor that is indicated by the two variables. Elementary linkage analysis enables the

Factor analysis can take two main forms: researcher to cluster together similar groups of

exploratory factor analysis and confirmatory factor variables by hand.

analysis. The former refers to the use of factor analysis (principal components analysis in particular) to explore previously unknown

Steps in elementary linkage analysis

groupings of variables, to seek underlying patterns,

1 In Box 25.2, underline the strongest, that is clusterings and groups. By contrast confirmatory the highest, correlation coefficient in each

factor analysis is more stringent, testing a found column of the matrix. Ignore negative signs.

set of factors against a hypothesized model

2 Identify the highest correlation coefficient in of groupings and relationships. This section the entire matrix. The two variables having

introduces the most widely used form of factor this correlation constitute the first two of

analysis: principal components analysis. We refer Cluster 1.

the reader to further books on statistics for a fuller

3 Now identify all those variables which are discussion of factor analysis and its variants. most like the variables in Cluster 1. To do

The analysis here uses SPSS output, as it is the this, read along the rows of the variables

most commonly used way of undertaking principal which emerged in Step 2, selecting any of

components analysis by educational researchers. the coefficients which are underlined in the

As an example of factor analysis, one could have rows. Box 25.3 illustrates diagrammatically

the following variables in a piece of educational the ways in which these new cluster members

research:

FACTOR ANALYSIS 561

Chapter

Box 25.1

Rank ordering of ten children on seven constructs

SOCIABLE (favourable)

VERBALLY GOOD

CLUMSY

(favourable)

1 Richard

(unfavourable)

10 Alex

2 Caroline

9 Patrick

3 Heather

8 Karen

4 Janice

7 Tim

5 Patrick

6 Richard

6 Tim

5 Sharon

7 Alex

4 Jane

8 Sharon

3 Janice

9 Jane

2 Caroline

(unfavourable)

10 Karen

(favourable)

1 Heather

continued

562 MULTIDIMENSIONAL MEASUREMENT

Box 25.1

Continued

WELL BEHAVED

(favourable)

1 Janice 2 Jane 3 Sharon 4 Caroline 5 Heather 6 Richard 7 Tim 8 Karen 9 Patrick

(unfavourable)

10 Alex

Source: Cohen 1977

Box 25.2

Intercorrelations between seven personal constructs

73 −93 Verbally good

−81 Well behaved (decimal points omitted)

Source: Cohen 1977

student demotivation grouping together several variables under one or

poor student concentration

more common factor(s).

undue pressure on students To address factor analysis in more detail we

narrowing effect on curriculum provide a worked example. Consider the following

punishing the weaker students variables concerning school effectiveness:

overemphasis on memorization

testing only textbook knowledge.

the clarity of the direction that is set by the These seven variables can be grouped together

school leadership

under the single overarching factor of ‘negative

the ability of the leader to motivate and inspire effects of examinations’. Factor analysis, working

the educators

through multiple correlations, is a method for

the drive and confidence of the leader

FACTOR ANALYSIS 563

Chapter

Box 25.3

(latent variables) that can embrace several of The structuring of relationships among the seven

these variables, or of which the several variables personal constructs

are elements or indicators?’ Factor analysis will indicate whether there are. We offer a three-

badly behaved

aggressive

stage model for undertaking factor analysis. In

CLUSTER 1

what follows we distinguish factors from variables:

a factor is an underlying or latent feature under

CLUSTER 2 verbally good

intelligent

which groups of variables are included; a variable is one of the elements that can be a member of an

denotes a reciprocal relationship between two variables

underlying factor. In our example here we have 24 variables and, as we shall see, 5 factors.

Source: Cohen 1977

the consultation abilities or activities of the

Stage 1

leader

Let us imagine that we have gathered data from the example set by the leader

1,000 teachers in several different schools, and the commitment of the leader to the school

we wish to see how the 24 variables above can the versatility of the leader’s styles

be grouped, based on their voting (using ratio the ability of the leader to communicate clear, data by awarding marks out of ten for each of individualized expectations the variables). (This follows the rule that there

the respect in which the leader is held by staff

should be more subjects in the sample than there the staff’s confidence in the senior management are variables.) Bryman and Cramer (1990: 255) team suggest that there should be at least 5 subjects per

the effectiveness of the teamwork of the SMT

variable and a total of no fewer than 100 subjects the extent to which the vision for the school in the total sample. This analysis will be based on impacts on practice

SPSS processing and output, as Box 25.4. educators given opportunities to take on Although Box 25.4 seems to contain a lot leadership roles

of complicated data, in fact most of this need the creativity of the SMT

not trouble us at all. SPSS has automatically problem-posing, problem-identifying and found and reported 5 factors for us through problem-solving capacity of the SMT sophisticated correlational analysis, and it presents

the use of data to inform planning and school data on these 5 factors (the first 5 rows of the development

chart, marked ‘Component’). Box 25.4 takes the valuing of professional development in the

24 variables (listed in order on the left-hand school

column (Component)) and then it provides three staff consulted about key decisions

sets of readings: Eigenvalues, Extracted Sums of the encouragement and support for innova- Squared Loadings, and Rotated Sums of Squared tiveness and creativity

Loadings. Eigenvalues are measures of the variance everybody is free to make suggestions to inform

decision-making between factors. We are interested only in those

Eigenvalues that are greater than 1, since those the school works in partnership with parents

that are smaller than 1 generally are not of people take positive risks for the good of the interest to researchers as they account for less school and its development

than the variation explained by a single variable. staff voluntarily taking on coordination roles

Indeed SPSS automatically filters out for us the teamwork among school staff. Eigenvalues that are greater than 1, using the

Here we have 24 different variables. The question Kaiser criterion (in SPSS this is termed the Kaiser here is, ‘Are there any underlying groups of factors

Normalization).

564 MULTIDIMENSIONAL MEASUREMENT

Box 25.4

Initial SPSS output for principal components analysis

Total variance explained

Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings Component Total % of

Initial Eigenvalues

Cumulative Total % of

Cumulative

Total % of Cumulative

Extraction method: Principal components analysis

A scree plot can also be used at this stage, to Indeed, in using the scree plot one perhaps identify and comment on factors (this is available

has to look for the ‘bend in the elbow’ of at the click of a button in SPSS). A scree plot

the data (after factor one), and then regard shows each factor on a chart, in descending

those factors above the bend in the elbow as order of magnitude. For researchers the scree plot

being worthy of inclusion, and those below becomes interesting where it flattens out (like the

the bend in the elbow as being relatively rubble that collects at the foot of a scree), as

unimportant (Cattell 1966; Pallant 2001: 154). this indicates very clearly which factors account

However, this is draconian, as it risks placing for a lot of the variance, and which account

too much importance on those items above the for little. In the scree plot here (Box 25.5) one

bend in the elbow and too little importance on can see that the scree flattens out considerably

those below it. The scree plot adds little to the after the first factor, then it levels out a little

variance table presented in Box 25.4, though it for the next 4 factors, tailing downwards all the

does enable one to see at a glance which are the time. This suggests that the first factor is the

significant and less significant factors, or, indeed significant factor in explaining the greatest amount

which factors to focus on (the ones before the of variance.

scree levels off) and which to ignore.

FACTOR ANALYSIS 565

Chapter

Box 25.5

A scree plot

Eigenvalue 4

Component number

Next we turn to the columns labelled SPSS keeps a score of the cumulative amount of ‘Extraction Sums of Squared Loadings’. The

explanatory power of the 5 factors identified. In Extraction Sums of Squared Loadings contain

the column ‘Cumulative’ it tells us that in total two important pieces of information. First, in the

60.047 per cent of the total picture (of the 24 column marked ‘% of variance’ SPSS tells us how

variables) is accounted for – explained – by the much variance is explained by each of the factors

5 factors identified. This is a moderate amount identified, in order from the greatest amount of

of explanatory power, and researchers would be variance to the least amount of variance. So,

happy with this.

here the first factor accounts for 38.930 per cent However, the three columns under ‘Extraction of the variance in the total scenario – a very

Sums of Squared Loadings’ give us the initial, large amount – while the second factor identified

rather crude, unadjusted percentage of variance accounts for only 5.931 per cent of the total

of the total picture explained by the 5 factors variance, a much lower amount of explanatory

found. These are crude in the sense that the full power. Each factor is unrelated to the other, and so

potential of factor analysis has not been caught. the amount of variance in each factor is unrelated

What SPSS has done here is to plot the factors to, or explained by, the other factors; they are

on a two-dimensional chart (which it does not independent of each other. By giving us how

present in the data output) to identify groupings much variance in the total picture is explained by

of variables, the two dimensions being vertical each factor we can see which factors possess the

and horizontal axes as in a conventional graph most and least explanatory power – the power to

like a scattergraph. On such a two-dimensional explain the total scenario of 24 variables. Second,

chart some of the factors and variables could

566 MULTIDIMENSIONAL MEASUREMENT

be plotted quite close to each other, such that tapped, in that the rotation of the variables from discrimination between the factors would not

a two-dimensional to a three-dimensional chart

be very clear. However, if we were to plot the has been undertaken, thereby identifying more factors and variables on a three-dimensional chart

clearly the groupings of variables into factors, that includes not only horizontal and vertical

and separating each factor from the other much axes but also depth by rotating the plotted points

more clearly. We advise researchers to use the through 90 degrees, then the effect of this would

Rotation Sums of Squared Loadings rather than

be to bring closer together those variables that the Extraction Sums of Squared Loadings. With are similar to each other and to separate them

the Rotation Sums of Squared Loadings the more fully – in distance – from those variables that

percentage of variance explained by each factor is have no similarity to them, i.e. to render each

altered, even though the total cumulative per cent group of variables (factors) more homogeneous

(60.047 per cent) remains the same. For example, and to separate more clearly one group of variables

one can see that the first factor in the rotated (factor) from another group of variables (factor).

solution no longer accounts for 38.930 per cent The process of rotation keeps together those

as in the Extraction Sums of Squared Loadings, variables that are closely interrelated and keeps

but only 16.820 per cent of the variance, and them apart from those variables that are not closely

that factors 2, 3 and 4, which each accounted for related. This is represented in Box 25.6.

only just over 5 per cent of the variance in the This distinguishes more clearly one factor from

Extraction Sums of Squared Loadings now each another than that undertaken in the Extraction

account for over 11 per cent of the variance, and Sums of Squared Loadings. Rotation is undertaken

that factor 5, which accounted for 4.520 per cent by varimax rotation. This maximizes the variance

of the variance in the Extraction Sums of Squared between factors and hence helps to distinguish

Loadings now accounts for 8.556 per cent of the them from each other. In SPSS the rotation is

variance in the Rotated Sums of Squared Loadings. called orthogonal because the factors are unrelated

By this stage we hope that the reader has been to, and independent of, each other.

able to see that:

In the column ‘Rotation Sums of Squared

Factor analysis brings variables together into Loadings’ the fuller power of factor analysis is homogeneous and distinct groups, each of

which is a factor and each of which has an Eigenvalue of greater than 1.

Factor analysis in SPSS indicates the amount

Box 25.6

of variance in the total scenario explained Three-dimensional rotation

by each individual factor and all the factors together (the cumulative per cent).

The Rotation Sums of Squared Loadings is preferable to the Extraction Sums of Squared Loadings.

We are ready to proceed to the second stage.

Stage 2

Stage 2 consists of presenting a matrix of all of the relevant data for the researcher to be able to identify which variables belong to which factor (Box 25.7). SPSS presents what at first sight is a bewildering set of data, but the reader is advised

FACTOR ANALYSIS 567

Chapter

Box 25.7

The rotated components matrix in principal components analysis

Rotated component matrix a

Component

1 2 3 4 5 The clarity of the direction that is set by the

0.212 school leadership The ability of the leader to motivate and inspire

0.160 the educators The drive and confidence of the leader

0.222 The consultation abilities/activities of the leader

0.160 The example set by the leader

0.209 The commitment of the leader to the school

0.137 The versatility of the leader’s styles

5.668E-02 The ability of the leader to communicate clear,

0.205 individualized expectations The respect in which the leader is held by staff

0.240 The staff’s confidence in the SMT

0.279 The effectiveness of the teamwork of the SMT

8.104E-02 The extent to which the vision for the school

0.113 impacts on practice Educators given opportunities to take on

−2.66E-02 leadership roles The creativity of the SMT

0.189 Problem-posing, problem-identifying and

−3.21E-02 problem-solving capacity of SMT The use of data to inform planning and school

−3.79E-02 development

Valuing of professional development in the

7.013E-02 school Staff consulted about key decisions

0.167 The encouragement and support for

0.661 innovativeness and creativity Everybody is free to make suggestions to

0.642 inform decision-making The school works in partnership with parents

0.199 People take positive risks for the good of the

2.635E-02 school and its development Staff voluntarily taking on coordination roles

0.779 Teamwork among school staff

Extraction method: Principal components analysis Rotation method: Varimax with Kaiser Normalization

a. Rotation converged in six iterations.

to keep cool and to look at the data slowly, as, Across the top of the matrix in Box 25.7 we in fact, they are not complicated. SPSS often

have a column for each of the 5 factors (1–5) presents researchers with more data than they

that SPSS had found for us. The left-hand column need, overwhelming the researcher with data.

prints the names of each of the 24 variables with In fact the data in Box 25.7 are comparatively

which we are working. We can ignore those pieces straightforward.

of data which contain the letter ‘E’ (exponential),

568 MULTIDIMENSIONAL MEASUREMENT

as these contain figures that are so small as to be is still highly statistically significant, statistical able to be discarded. Look at the column labelled

significance being a combination of the coefficient ‘1’ (factor 1). Here we have a range of numbers

and the sample size.

that range from 0.114 for the variable ‘Teamwork We repeat this analysis for all 5 factors, deciding among school staff’ to 0.758 for the variable ‘The

the cut-off point, looking for homogeneous high drive and confidence of the leader’. The researcher

values and numerical distance from other variables now has to use his or her professional judgement

in the list.

to decide what the ‘cut-off’ points should be for inclusion in the factor. Not all 24 variables will appear in factor 1, only those with high values

Stage 3

(factor loadings – the amount that each variable By this time we have identified 5 factors. However, contributes to the factor in question). The decision

neither SPSS nor any other software package tells on which variables to include in factor 1 is not

us what to name each factor. The researcher has to

a statistical matter but a matter of professional devise a name that describes the factor in question. judgement. Factor analysis is an art as well as a

This can be tricky, as it has to catch the issue that science. The researcher has to find those variables

is addressed by all the variables that are included with the highest values (factor loadings) and

in the factor. We have undertaken this for all 5 include those in the factor. The variables chosen

factors, and we report this below, with the factor should not only have high values but also have

loadings for each variable reported in brackets. values that are close to each other (homogeneous)

and be some numerical distance away from the other variables. In the column labelled ‘1’ we can

Factor 1: Leadership skills in school management see that there are 7 such variables, and we set these

Cut-off point: 0.51

out in the example below. Other variables from the list are some numerical distance away from

Variables included:

the variables selected (see below) and also seem to

The drive and confidence of the leader (factor identified for inclusion in the factor. The variables

be conceptually unrelated to the seven variables

loading 0.758).

selected are high, close to each other and distant

The ability of the leader to motivate and inspire from the other variables. The lowest of these 7

the educators (factor loading 0.743). values is 0.513; hence the researcher would report

The use of data to inform planning and school that 7 variables had been selected for inclusion in

development (factor loading 0.690). factor 1, and that the cut-off point was 0.51 (i.e.

The example set by the leader (factor loading the lowest point, above which the variables have

The clarity of the direction set by the school gives considerable power to the factor. Hence we

been selected). Having such a high cut-off point

leadership (factor loading 0.559). have factor 1, which contains 7 variables.

The consultation abilities/activities of the Let us look at a second example, that of

leader (factor loading 0.548). factor 2 (the column labelled ‘2’). Here we

The commitment of the leader to the school can identify 4 variables that have high values

(factor loading 0.513). that are close to each other and yet some numerical distance away from the other variables (see example below). These 4 variables would

Factor 2: Parent and teacher partnerships in school constitute factor 2, with a reported cut-off point

development

of 0.445. At first glance it may seem that 0.445

Cut-off point: 0.44

is low; however, recalling that the data in the example were derived from 1,000 teachers, 0.445

Variables included:

FACTOR ANALYSIS 569

Chapter

The school works in partnership with parents

Everybody is free to make suggestions to inform (factor loading 0.804).

decision-making (factor loading 0.642).

People take positive risks for the good of the Each factor should usually contain a minimum of school and its development (factor loading

0.778). three variables, though this is a rule of thumb rather than a statistical necessity. Further, in

Teamwork among school staff (factor loading 0.642).

the example here, though some of the variables

The effectiveness of the teamwork of the SMT included have considerably lower factor loadings than others in that factor (e.g. in factor 2:

(factor loading 0.445). the effectiveness of the teamwork of the SMT

(0.445)), nevertheless the conceptual similarity Factor 3: Promoting staff development by creativity

to the other variables in that factor, coupled and consultation

with the fact that, with 1,000 teachers in the Cut-off point: 0.55

study, 0.445 is still highly statistically significant, combine to suggest that this still merits inclusion.

Variables included: As we mentioned earlier, factor analysis is an art as well as a science.

Staff consulted about key decisions (factor If one wished to suggest a more stringent level loading 0.854).

of exactitude then a higher cut-off point could

be taken. In the example above, factor 1 could 0.822).

The creativity of the SMT (factor loading

have a cut-off point of 0.74, thereby including

Valuing of professional development in the only 2 variables in the factor; factor 2 could have school (0.551).

a cut-off point of 0.77, thereby including only 2 variables in the factor; factor 3 could have a cut-off

Factor 4: Respect for, and confidence in, the senior point of 0.82, thereby including only 2 variables management team

in the factor; factor 4 could have a cut-off point of 0.80, thereby including only 2 variables in the

Cut-off point: 0.44 factor; and factor 5 could have a cut-off point of 0.77, thereby including only 1 variable in the

Variables included: factor. The decision on where to place the cut-off

The respect in which the leader is held by staff point is a matter of professional judgement when (factor loading 0.810).

reviewing the data.

The staff’s confidence in the SMT (factor In reporting factor analysis the above data would loading 0.809).

all be included, together with a short commentary,

The effectiveness of the teamwork of the SMT

for example:

(factor loading 0.443). In order to obtain conceptually similar and significant clusters of issues of the variables, principal

Factor 5: Encouraging staff development through components analysis with varimax rotation and participation in decision-making

Kaiser Normalization were conducted. Eigenvalues Cut-off point 0.64 equal to or greater than 1.00 were extracted. With regard to the 24 variables used, orthogonal rotation Variables included: of the variables yielded 5 factors, accounting for 16.820, 11.706, 11.578, 11.386 and 8.556 per cent

Staff voluntarily taking on coordination roles of the total variance respectively, a total of 60.047 (factor loading 0.779).

per cent of the total variance explained. The factor

The encouragement and support for innova- loadings are presented in table such-and-such. To tiveness and creativity (factor loading 0.661).

enhance the interpretability of the factors, only

570 MULTIDIMENSIONAL MEASUREMENT

variables with factor loadings as follows were selected upon a number of interrelated measures. We for inclusion in their respective factors: > 0.51

illustrate the use of factor analysis in a study of (factor 1), > 0.44 (factor 2), > 0.55 (factor 3),

occupational stress among teachers (McCormick > 0.44 (factor 4), and > 0.64 (factor 5). The

and Solman 1992).

factors are named, respectively: Leadership skills in Despite a decade or so of sustained research, the school management; Parent and teacher partnerships

concept of occupational stress still causes difficul- in school development; Promoting staff development by

ties for researchers intent upon obtaining objective creativity and consultation; Respect for, and confidence

measures in such fields as the physiological and in, the senior management team; and Encouraging staff

the behavioural, because of the wide range of indi- development through participation in decision-making.

vidual differences. Moreover, subjective measures (See http://www.routledge.com/textbooks/

such as self-reports, by their very nature, raise ques- 9780415368780 – Chapter 25, file SPSS Manual

tions about the external validation of respondents’ 25.1.)

revelations. This latter difficulty notwithstand- ing, McCormick and Solman (1992) chose the

Having presented the data for the factor analysis methodology of self-report as the way into the the researcher would then comment on what problem, dichotomizing it into, first, the teacher’s it showed, fitting the research that was being

conducted. view of self, and second, the external world as it is Factor analysis is based on certain assumptions

seen to impinge upon the occupation of teaching. which should be maintained in order to serve

Stress, according to the researchers, is considered fidelity to this technique, for example:

as ‘an unpleasant and unwelcome emotion’ whose negative effect for many is ‘associated with ill-

The data must be interval and ratio. ness of varying degree’ (McCormick and Solman

The sample size should be no fewer than around 1992). They began their study on the basis of the 150 persons. 1 following premises:

There should be at least 5 cases for each variable (Pallant (2001: 153) suggests 10 cases O Occupational stress is an undesirable and

for each variable). negative response to occupational experiences.

The relationships between the variables should O To be responsible for one’s own occupational stress can indicate a personal failing.

be linear.

Outliers should be removed. Drawing on attribution theory, McCormick and

The data must be capable of being factored. Solman (1992) consider that the idea of blame is To achieve this, several of the correlations

a key element in a framework for the exploration should be of 0.3 or greater, the Bartlett test

of occupational stress. The notion of blame for of sphericity (SPSS calculates this at the

occupational stress, they assert, fits in well with press of a button) should be significant at

tenets of attribution theory, particularly in terms the 0.05 level or better, and the Kaiser-

of attribution of responsibility having a self- Meyer-Olkin measure of sampling adequacy

serving bias. 2 Taken in concert with organizational (calculated automatically by SPSS) should be

facets of schools, the researchers hypothesized at 0.6 or above.

that teachers would ‘externalize responsibility for their stress increasingly to increasingly distant and identifiable domains’ (McCormick and

Factor analysis: an example

Solman 1992). Their selection of dependent and Factor analysis, we said earlier, is a way of

independent variables in the research followed determining the nature of underlying patterns

directly from this major hypothesis. among a large number of variables. It is particularly

McCormick and Solman (1992) developed a appropriate in research where investigators aim to

questionnaire instrument that included 32 items impose an ‘orderly simplification’ (Child 1970)

to do with occupational satisfaction. These were

FACTOR ANALYSIS: AN EXAMPLE 571

Chapter

scored on a continuum ranging from ‘strongly disagree’ to ‘strongly agree’. A further 38 items

Box 25.8

Factor analysis of responsibility for stress items had to do with possible sources of occupational

stress. Here, respondents rated the intensity of

Factor groupings of responsibility items with factor

the stress they experienced when exposed to

loadings and (rounded) percentages of teachers

each source. Stress items were judged on a scale

responding in the two most extreme categories of

ranging from ‘no stress’ to ‘extreme stress’. In yet much stress and extreme stress. another section of the questionnaire, respondents

Loading Percentage rated how responsible they felt certain nominated

Factor 1: School structure Superiors

0.85 persons or institutions were for the occupational 29

0.78 31 stress that they, the respondents, experienced.

School organization

0.77 13 These entities included self, pupils, superiors,

Peers

the Department of Education, the government

Factor 2: Bureaucratic

and society itself. Finally, the teacher-participants

authority Department of Education

0.89 70 were asked to complete a 14-item locus of control

0.88 66 scale, giving a measure of internality/externality.

Government

‘Internals’ are people who see outcomes as a

Factor 3: Teacher–student

function of what they themselves do; ‘externals’ see

relationships

outcomes as a result of forces beyond their control.

Students

0.60 The items included in this lengthy questionnaire 60

Society

0.50 20 arose partly from statements about teacher stress

Yourself

used in earlier investigations, but mainly as a result of hunches about blame for occupational

Source: McCormick and Solman 1992

stress that the researchers derived from attribution theory. As Child (1970) observes:

In most instances, the factor analysis is preceded by a In the factor analysis of the 8-item responsibility hunch as to the factors that might emerge. In fact, it

for stress measure, the researchers identified three would be difficult to conceive of a manageable anal-

factors. Box 25.8 shows those three factors with ysis which started in an empty-headed fashion. . . .

what are called their ‘factor loadings’. As we have Even the ‘let’s see what happens’ approach is pretty

seen, these are like correlation coefficients, ranging sure to have a hunch at the back of it somewhere. It is

from −1.0 to +1.0 and are interpreted similarly. this testing and the generation of hypotheses which

That is to say they indicate the correlation forms the principal concern of most factor analysts.

between the person/institution responsibility items

(Child 1970: 8)

shown in Box 25.8, and the factors. Looking at factor 1, ‘School structure’, for example, it can

The 90-plus-item inventory was completed by 387

be seen that in the 3 items loading there are, teachers. Separate correlation matrices composed in descending order of weight, superiors (0.85), of the inter-correlations of the 32 items on school organization (0.78) and peers (0.77). the satisfaction scale, the 8 items in the ‘School structure’ as a factor, the authors suggest, persons/institutions responsibility measure and the is easily identified and readily explained. But

38 items on the stress scale were factor analysed. The procedures followed by McCormick and

what of factor 3, ‘Teacher–student relationships’, which includes the variables students, society and

Solman (1992), Principal Components, which yourself? McCormick and Solman (1992) proffer were subsequently rotated, parallel those we have the following tentative interpretation: outlined earlier. (Readable accounts of factor

analysis may be found in Child 1970; Kerlinger An explanation for the inclusion of the variable 1970.)

‘yourself’ in this factor is not readily at hand.

572 MULTIDIMENSIONAL MEASUREMENT

Clearly, the difference between the variable ‘yourself’ factors were extracted and named as ‘Supervision’, and the ‘students’ and ‘society’ variables is that

‘Income’, ‘External demands’, ‘Advancement’ and only 20 per cent of these teachers rated themselves

‘School culture’. Again, space precludes a full as very or extremely responsible for their own

outline of the results set out in Box 25.10. Notice, stress, compared to 45 per cent and 60 per cent

however, an apparent anomaly in the first factor, respectively for the latter two. Possibly the degree of

‘Supervision’. Responses to items to do with responsibility which teachers attribute to themselves

teachers’ supervisors and recognition seem to for their occupational stress is associated with

indicate that in general, teachers are satisfied with their perceptions of their part in controlling

their supervisors, but feel that they receive too student behaviour. This would seem a reasonable

little recognition.

explanation, but requiring further investigation. Box 25.10 shows that 21 per cent of teacher-

(McCormick and Solman 1992)

respondents agree or strongly agree that they receive too little recognition, yet 52 per cent

Box 25.9 shows the factors derived from the agree or strongly agree that they do receive analysis of the 38 occupational stress items. The

recognition from their immediate supervisors.

5 factors extracted were named: ‘Student domain’, McCormick and Solman (1992) offer the ‘External (to school) domain’, ‘Time demands’,

following explanation:

‘School domain’ and ‘Personal domain’. While

a detailed discussion of the factors and their The difference can be explained, in the first instance, loadings is inappropriate here, we draw readers’

by the degree or amount of recognition given. That attention to some interesting findings. Notice, for

is, immediate supervisors give recognition, but not example, how the second factor, ‘External (to

enough. Another interpretation is that superiors school) domain’, is consistent with the factoring

other than the immediate supervisor do not give of the responsibility for stress items reported in

sufficient recognition for their work. Box 25.8. That is to say, the variables to do

(McCormick and Solman 1992) with the government and the Department of

Here is a clear case for some form of respondent Education have loaded on the same factor. The

validation (see Chapter 6 and 11). researchers venture this further elaboration of the

Having identified the underlying structures of point.

occupational stress and occupational satisfaction, when a teacher attributes occupational stress to the

the researchers then went on to explore the Department of Education, it is not as a member

relationships between stress and satisfaction by of the Department of Education, although such, in

using a technique called ‘canonical correlation fact, is the case. In this context, the Department of

analysis’. The technical details of this procedure are beyond the scope of this book. Interested

Education is outside ‘the system to which the teacher readers are referred to Levine (1984), who belongs’, namely the school. A similar argument can suggests that ‘the most acceptable approach be posed for the nebulous concept of Society. The to interpretation of canonical variates is the Government is clearly a discrete political structure. examination of the correlations of the original

variables with the canonical variate’ (Levine ‘School domain’, factor 4 in Box 25.9, consists

(McCormick and Solman 1992)

1984). This is the procedure adopted by of items concerned with support from the school

McCormick and Solman (1992). principal and colleagues as well as the general

From Box 25.11 we see that factors having nurturing atmosphere of the school. Of particular

high correlations with Canonical Variate 1 are interest here is that teachers report relatively low

Stress: Student domain (−0.82) and Satisfaction: levels of stress for these items.

External demands (0.72). Box 25.10 reports the factor analysis of the 32

The researchers offer the following interpreta- items to do with occupational satisfaction. Five

tion of this finding:

FACTOR ANALYSIS: AN EXAMPLE 573

Chapter

Box 25.9

Factor analysis of the occupational stress items Factor groupings of stress items with factor loadings and (rounded) percentages of teachers responding to the two

extremes of much stress and extreme stress

Loading

Percentage

Factor 1: Student domain Poor work attitudes of students

Difficulty in motivating students

Having to deal with students who constantly misbehave

Inadequate discipline in the school

Maintaining discipline with difficult classes

Difficulty in setting and maintaining standards

Verbal abuse by students

Students coming to school without necessary equipment

Deterioration of society’s control over children

Factor 2: External (to school) domain The Government’s education policies

The relationship which the Department of Education has with its schools

Unrealistic demands from the Department of Education

The conviction that the education system is getting worse

Media criticism of teachers

Lack of respect in society for teachers

Having to implement departmental policies

Feeling of powerlessness

Factor 3: Time demands Insufficient time for personal matters

Just not enough time in the school day

Difficulty of doing a good job in the classroom because of other delegated responsibilities

Insufficient time for lesson preparation and marking

Excessive curriculum demands

Difficulty in covering the syllabus in the time available

Demanding nature of the job

Factor 4: School domain Lack of support from the principal

Not being appreciated by the principal

Principal’s reluctance to make tough decisions

Lack of opportunity to participate in school decision-making

Lack of support from other colleagues

continued

574 MULTIDIMENSIONAL MEASUREMENT

Box 25.9

Continued

Percentage Lack of a supportive and friendly atmosphere

Loading

0.55 17 Things happen at school over which you have no control

0.41 36 Factor 5: Personal domain

Personal failings 0.76 13 Feeling of not being suited to teaching

0.72 10 Having to teach a subject for which you are not trained

Source: McCormick and Solman 1992

Box 25.10

Factor analysis of the occupational satisfaction items Factor groupings of satisfaction items with factor loadings and (rounded) percentages of teacher responses in the two

positive extremes of ‘strongly agree’ and ‘agree’ for positive statements, or ‘strongly disagree’ and ‘disagree’ for statements of a negative nature; the latter items were reversed for analysis and are indicated by ∗

Percentage Factor 1: Supervision My immediate supervisor does not back me up ∗

Loading

0.83 70 I receive recognition from my immediate supervisor

0.80 52 My immediate supervisor is not willing to listen ∗

0.78 68 My immediate supervisor makes me feel uncomfortable ∗

0.78 66 My immediate supervisor treats everyone equitably

0.68 62 My superiors do not appreciate what a good job I do ∗

0.66 39 I receive too little recognition ∗

0.51 21 Factor 2: Income

My income is less than I deserve ∗ 0.80 10 I am well paid in proportion to my ability

0.78 8 My income from teaching is adequate

0.78 19 My pay compares well with other non-teaching jobs

0.66 6 Teachers’ income is barely enough to live on ∗

0.56 24 Factor 3: External demands

Teachers have an excessive workload ∗ 0.72 5 Teachers are expected to do too many non-teaching tasks ∗

0.66 4 People expect too much of teachers ∗

0.56 4 There are too many changes in education ∗

0.53 10 I am satisfied with the Department of Education as an employer

0.44 12 People who aren’t teachers do not understand the realities in schools ∗

FACTOR ANALYSIS: AN EXAMPLE 575

Chapter

Box 25.10

Continued

Percentage Factor 4: Advancement

Loading

Teaching provides me with an opportunity to advance professionally

I am not getting ahead in my present position ∗

The Government is striving for a better education system

The Department of Education is concerned for teachers’ welfare

Factor 5: School culture I am happy to be working at this particular school

Working conditions in my school are good

Teaching is very interesting work

Source: McCormick and Solman 1992

Box 25.11

and vice versa. It may well be that, for some teachers, Correlations between (dependent) stress and

high demand in one of these is perceived as affecting (independent) satisfaction factors and canonical

their capacity to cope or deal with the demands of the variates

other. Certainly, the teacher who is experiencing the urgency of a struggle with student behaviour in the

Canonical variates

classroom, is unlikely to think of the requirements of

Stress factors persons and agencies outside the school as important.

Student domain

0.05 (McCormick and Solman 1992) External (to school) domain −0.15 −0.04

The outcomes of their factor analyses frequently

School domain

0.86 0.16 puzzle researchers. Take, for example, one of the

Personal domain

loadings on the third canonical variate. There,

Satisfaction factors

we see that the stress factor ‘Time demands’

Supervision

0.32 correlates negatively (−0.52). One might have

Income

0.45 0.13 0.12 supposed, McCormick and Solman (1992) say,

External demands

0.72 0.33 0.28 that stress attributable to the external domain

Advancement

would have correlated with the variate in the

School culture

same direction. But this is not so. It correlates positively at 0.80. One possible explanation, they

Source: adapted from McCormick and Solman 1992

suggest, is that an increase in stress experienced because of time demands coincides with a lowering of stress attributable to the external domain,

[This] indicates that teachers perceive that ‘non- as time is expended in meeting demands from teachers’ or outsiders expect too much of them

the external domain. The researchers concede, (External demands) and that stress results from poor

however, that this explanation would need more student attitudes and behaviour (Student domain).

close examination before it could be accepted. One interpretation might be that for these teachers,

McCormick and Solman’s (1992) questionnaire high levels of stress attributable to the Student

also elicited biographical data from the teacher- domain are associated with low levels of satisfaction

respondents in respect of sex, number of years in the context of demands from outside the school,

teaching, type and location of school and position

576 MULTIDIMENSIONAL MEASUREMENT

held in school. By rescoring the stress items on by secondary school teachers than by their

a scale ranging from ‘No stress’ (1) to ‘Extreme colleagues teaching younger pupils, not really a stress’ (5) and using the means of the factor scores,

very surprising result, the researchers observe, the researchers were able to explore associations

given that infant/primary schools are generally between the degree of perceived occupational

much smaller than their secondary counterparts stress and the biographical data supplied by

and that teachers are more likely to be part of a participants. Space precludes a full account of

smaller, supportive group. In the domain of Time McCormick and Solman’s (1992) findings. We

demands, females experienced more stress than illustrate some significant results in Box 25.12.

males, a finding consistent with that of other In the School domain more stress was reported

research. In the Personal domain, a significant difference was found in respect of the school’s location, the level of occupational stress increasing

Box 25.12

from the rural setting, through the country/city to Biographical data and stress factors

the metropolitan area.

To conclude, factor analysis techniques

5 are ideally suited to studies such as that

4 of McCormick and Solman (1992) in which

3 lengthy questionnaire-type data are elicited from a Stress 2 large number of participants and where researchers

1 are concerned to explore underlying structures and

Infants/

Secondary

relationships between dependent and independent

primary

variables. 3

School type

Inevitably, such tentative explorations raise as

Means for significant differences of biographical

many questions as they answer.

characteristics: External to school domain.