# 6.3 PRINCIPAL COMPONENTS ANALYSIS

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As in the previous section, my purpose is to provide you with an intuitive

rationale for the way in which principal components analysis works, just

sufficient for you to use appropriate software packages safely and without

over- or misinterpretation. I then suggest a procedure to follow when you do

this interpretation.

6.3.1 Rationale for Principal Components Analysis

Glance at Table 6.13. Scan the columns from top to bottom, as if they were

element ratings in a grid. Then run your eye along the rows from left to right,

as if they were construct ratings. What do you notice about them?

Looking Downwards. Not a lot is going on here. The ratings don’t change

much going from top to bottom, except in the two rightmost columns: 4 and 4

followed by 2s and a 1; 5 and 5 followed by 1s and 2s. There’s more variability

in the numbers in these last two columns.

Table 6.13 Just some numbers

1 2 3 3 4 5

2 3 3 3 4 5

1 2 3 3 2 1

1 2 2 2 2 1

2 2 3 3 2 2

1 1 2 2 1 1

Looking Across. There’s lots going on here.

. Firstly, can you see how the values increase from left to right in the first row

and the second? Next, notice how the values increase, then decrease again,

as you go from left to right, in the third, fourth, fifth, and sixth rows. The first

two rows follow one kind of pattern, while the last four rows follow a

different pattern.

. Something else to notice, too, is that the ratings are more varied in the first

two rows (they take values ranging across the full width of our conventional

5-point scale in the first row, and vary from 2 to 5 in the second). In contrast,

there’s less variability in the last four rows, with ratings taking values of 1, 2,

or 3 only.

There seem to be two distinct patterns of variability in this grid: two different

ways in which the values of the ratings vary.

Principal components analysis works by looking at the variability (technically,

the ‘variance’) in figures arranged in a table like Table 6.13. It identifies distinct

patterns of variability, following procedures which

. work out the extent to which the ratings in each row of the table are similar to

each other (using the correlation between each row and each other row), to

identify each distinct pattern

. work in a way which attributes as much as possible of the total variability to

each distinct pattern, using as few different patterns as possible.

The process is iterative. Firstly, the pattern which accounts for the largest

amount of variability is identified, reported, and removed – think of it as being

subtracted from the original table – set aside as it were. The next pattern is

identified likewise, and so on, until all of the variability has been accounted

for. These patterns of variability are called ‘components’.

And there seem to be two of them in our example in Table 6.13. Let’s call the

first one, which accounts for the kind of variability we see in rows 1 and 2,

‘component 1’; and the other, which accounts for most of the remaining

variability, ‘component 2’. (There are, in fact, two more components, but the

amount of variability they account for is tiny, as you can see in Table 6.14. This

was obtained by running Table 6.13 through a grid-analysis package.)

Principal components analysis packages always provide a variance table like

this one. This is useful, but the output which chiefly concerns us consists of

information about these patterns of variability in the form of a series of graphs

which plot the way in which the columns and rows (or, in our case, the

elements and the constructs) are arranged with respect to the principal

components. Figure 6.3 shows a graph of this kind.

Two dotted lines stand for the first two components; by convention, the

horizontal line represents the first component and the vertical line the second.

They’re vertical and horizontal, set at right angles to each other, because they

represent maximally distinct patterns in the data. The constructs are plotted as

straight lines whose angle with respect to each component reflects the extent to

which the construct is represented by the component, and whose length reflects

the amount of variance in the ratings on that construct.

In other words, each component is a statistical invention, whose purpose is to

represent, or stand for, as straightforwardly as possible, one of the different

patterns in the grid. The whole thing represents, in graphical terms, the

intuitive feeling we had when we looked at Table 6.13 and thought we

recognised two distinct patterns of variability – two sets of ratings with rather

differing variances. And as you can see in Figure 6.3, there are indeed two

groupings of lines representing constructs. Constructs A and B lie close to the

horizontal component line, which represents the pattern of variance which we

recognised in the first two rows of Table 6.13; constructs C, D, E, and F, lie very

close to the vertical line reflecting the pattern of variance in the bottom four

rows of Table 6.13.

Table 6.14 Percentage of variance accounted for by each component of Table 6.13

Component 1 Component 2 Component 3 Component 4

64.84 30.67 4.22 0.27

Figure 6.3 Principal components analysis graph for the data in Table 6.13

Because each component ‘stands for’ several constructs, the elements can be

positioned along each component, in place of their original position along each

construct; and there they are, plotted in Figure 6.3.

Now, because the resulting plot is a graphical representation, in which the

lengths of lines and the positions of points reflect the original ratings – because

the whole graph is a picture of patterns of similarity – distances matter and can

be meaningfully interpreted.

Constructs and Components

(a) The angle between any two construct lines reflects the extent to which the

ratings of elements on those constructs are correlated: the smaller the

angle, the more similar the ratings.

(b) The angle between a group of construct lines and the lines representing

the components reflects the extent to which the component can be taken to

represent the grouping of constructs in question: the smaller the angle, the

greater the extent.

In Figure 6.3, for example, the first (horizontal) component represents

construct A and construct B very well; it represents the remaining constructs

very poorly. The vertical component, on the other hand, stands for constructs

C, D, E, and F very well (they form a tight ‘fan’ angled very closely to the

vertical component).

Elements and Components

(a) The position of each element with respect to each component is exactly

like the position of a point on a graph: so far along the x-axis (first

component) and so far up or down the y-axis (second component). Of

itself, this doesn’t mean a lot; but what is enormously useful is that it gives

us a way of talking about:

(b) Relationships between elements – that is, the distance between any two

elements reflects the ratings each element received on all the constructs.

Any two elements which are close together in the graph received similar

ratings, any which are printed far apart would tend to show rather

different ratings in the original grid.

Look at the third and fourth columns in Table 6.13. They received identical

ratings. Now look at elements 3 and 4 in Figure 6.3: they are plotted in the same

position on the graph (in fact, I had to move the ‘element 3’ label in preparing

Figure 6.3, since the computer printout from which this figure was prepared

printed it directly on top of the ‘element 4’ label).

This property of a principal components analysis is particularly useful when

the differences between elements carry special meaning for us; for example,

when we want to compare how closely we construe ‘myself as I am’ and

‘myself as I wish to be’; any element and an ‘ideal’ supplied element; and so

on. We ask how far away from each other the two elements are, in the plot; we

draw inferences from what might need to change if ‘myself as I am’ were to

become more like ‘myself as I wish to be’. Along which constructs would

movement have to occur? That is, on which constructs would the ratings have

to be different for ‘myself as I am’ to move towards ‘myself as I wish to be’?

Finally, you need to know that a principal components analysis provides you

with plots showing the relationship between all the components, not just the

first two. You would examine these using the same approach as you used in

addressing Figure 6.3. As a rule of thumb, you ought to examine the plots for

all of the components that, between them, account for 80% of the variance.

Remember that distances matter, so that the horizontal and vertical

components within any graph are ‘to scale’, so far as variance goes.

Get a feeling for principal components by doing

Exercise 6.6.

6.3.2 Procedure for Interpretation of Principal Components

Analysis

The preceding rationale has been rather abstract. In Exercise 6.6, you were

looking at patterns in numbers, without knowing just what those numbers

stand for. No element names, just labels 1 to 6; no idea of what meanings the

constructs carry, just the blank and austere construct labels, each end of

constructs A to F. (Though, hopefully, things warmed up a little for you when

you addressed the last question of the exercise.)

Brr! You, the reader, havemy sympathy! And you also havemy congratulations.What

you’ve done is a necessary first step in understanding a principal components

analysis plot, but it’s the only step which is statistically faithful to the ratings provided

by your interviewee.

As soon as you start interpreting the principal components analysis any further

. by looking at what the actual elements are and where they lie with respect to the

principal components

. by looking at how the constructs are grouped and which components seem to

underlie them,

(all of which are essential if you’re to get the benefits of the analysis), youmove away

from the direct meaning offered to you by the interviewee, and into a realm in which

your own interpretations condition, influence, and possibly distort the information in

the original grid.

This becomes particularly important when you try to interpret the principal components

in terms of the constructs.What sort of component ‘underlies’ the constructs?

This sort of question is often resolved by trying to find a name for the component: a

labelwhich reflects themeaningin common betweenthe constructswhich lie closest

to that component ^ as at step 3 in the procedure below. And that act of naming

reflects yourown judgement.

Like any complex analysis, such as the cluster analysis we examined in Section 6.2,

principal components analysis requires you to make assumptionswhen you interpret

the original grid. Unlike cluster analysis, though, these assumptions are less visible.

They are less easily described to your interviewee in terms of comfortable analogies

like‘cutting up the grid with a pair of scissors’, and, unless the interviewee has some

grounding in statistics, your interpretations take on the flavour of ‘because I, the

expert, say so’.When this happens, you have less scope for negotiating a meaning

with your interviewee (and especially, for checking your understanding of the interviewee),

and, accordingly, it’s worth being rather cautious in the interpretations you

make using this analysis procedure. Be careful how you name the components. Do

so collaboratively if the interviewee understands the procedure.Don’t play the guru.

Point taken. And the following procedural guide seeks to take these comments

into account.

Let’s do it by moving right away from abstraction and looking at a familiar

grid. This is the one shown in Table 6.4, which we cluster-analysed in Section

6.2 with the results which appeared as Figure 6.2. We know a lot about it

already, which will help us to make sense of it using the present analysis. The

original grid is on the left, the table of variance accounted for on the right, and

the plot of the first two principal components at the lower right.

As always, the procedure starts with a familiarisation with the original grid: a

process analysis and the first three steps of an eyeball analysis. (1. What is the

interviewee thinking about? 2. How has the interviewee represented the topic?

and 3. How does s/he think?, as outlined in Section 5.3.2.)

Quite so. To avoid excessive overinterpretation, stick to the original grid as your

bedrock; that means reminding yourself of what the interviewee actually said, and

the circumstances inwhich s/he said it!

Figure 6.4 Store manager’s grid, variance accounted for, and plot of first two components

(1) Determine how many components you’ll need to work with. How many

components in the percentage variance table of Figure 6.4 do you need to look

at, to cover 80% of the variance? Here, the first two components account for

70.99 + 23.62 = 94% of the variance, so you can safely rely on just the one plot:

that of the first component against the second. (If you’d decided you needed to

look at the first three principal components, you’d need to work with the other

plots which the software has provided: the plot of component 1 against

component 2 as in this case, but also of component 1 against component 3, and

component 2 against component 3, teasing out the relationships for each plot

as follows.)

(2) Examine the shape of the lines representing the constructs: how tightly

are they spread? Are they well differentiated, or do they spread evenly all

round the plot like the spokes of a bicycle wheel? Here, the constructs

differentiate into two sheaves or fans of lines. The first consists of

. construct 3, ‘could be more interested in after sales – after sales well

handled’

. construct 1, ‘takes a while to learn the features of new lines – learns the new

models quickly’

. construct 4, ‘availability and choice knowledge poor – awareness of sizes,

colours, and availability’.

The second consists of

. construct 5, ‘takes it all very seriously – pleasant and easy-going’

. construct 2, ‘too forward in pushing a sale – good balance between active

selling and being helpful’.

(Notice how the supplied ‘overall effectiveness’ construct lies between these

two sheaves.)

(3) Identify any similarities in the meaning of these constructs:

. by inspection: does there appear to be any shared meaning? You’re very

tentatively ‘naming the components’. In Figure 6.4, the first grouping seems

to relate to product knowledge and interest in using that knowledge to make

a complete sale, while the second grouping appears to describe the

salesperson’s personal style while dealing with a customer.

. by examining their relationship to any supplied construct which might be

helpful in this respect. Here, construct 6, the ‘overall effectiveness’ construct,

aligns neither with the first grouping nor the second. No single construct or

group of constructs is particularly associated with effectiveness in a welldefined

way. In fact, you’d be tempted to argue, from the position of the

‘overall effectiveness’ construct, lying between the two groupings, that being

an effective salesperson depends neither on product knowledge nor on

personal style alone, but more or less equally on both.

(4) Note the position of any meaningful groupings with respect to the two

principal components: the vertical axis and the horizontal axis. So far, we’ve

almost ignored the components themselves, but they tell us something very

important. You’ll recall that components are derived sequentially, and in a

way which seeks to maximise their independence. So any sheaves or

groupings of constructs which lie near to one of the principal component

axes can be interpreted as, in some sense, independent of sheaves or groupings

which lie near to the other principal component axis. Can you find a meaning

for the groups of constructs based on this characteristic of the analysis?

In our own case, the position of the ‘product knowledge’ grouping near to the

first principal component, and the ‘style with customers’ grouping near to the

second, suggests that these are indeed two independent sets of constructs.

The psychologist might label this a ‘cognitive versus affective’ distinction,

while the trainer might think of this as indicative of two distinct sets of skills to

be learnt. A job analyst might view it as a statement about the structure of the

competency framework that pertains to this job, while someone interested in

knowledge management would view it as a statement reflecting the

interviewee’s experience and expertise in running a fashion section of the

department store.

It follows that you can’t decide, on the information present in the grid, which

of these is in any way definitive! That depends on your purpose, and the

context in which the grid interview was conducted. If you’re doing this grid

analysis in a research context as part of a dissertation, your supervisor will be

asking you about the analytic framework which underlies your use of the

repertory grid as a research technique! (Remember constructive alternativism?!)

Finally, be careful about overinterpretation. The two groups don’t sit squarely

on top of their components; they’re rotated through some 20 degrees

clockwise. This suggests that whatever these components are, while they’re

statistically independent of each other (they’re plotted at right angles), their

meaning isn’t completely separate. (There are further procedures in multivariate

statistics which might possibly clarify this, but these lie beyond the

scope of this guide.)

(5) Check your interpretations with the interviewee. Resist the temptation to

pronounce about them. For, example, if you’d decided in Figure 6.4 that the first

principal component is a ‘product knowledge and interest’ component and the

second is a ‘customer relations’ component, do not tell the interviewee that

that’s what they are. All you’ve done is to construe the interviewee’s

construing, and until you check it out, it has the status of a useful fiction,

regardless of the impressive statistical manipulations on which it’s based.

Instead, explore the links which you feel you’ve identified between the

constructs by asking questions to check whether those links make sense to the

interviewee.

Exactly. You’ll notice that we’ve both ganged up on you to reinforce the point.

A repertory grid isn’t a horoscope or a psychometric test, both of which depend

on something resembling a priest-and-peasant relationship between expert

interviewer and client. If your interview technique has been a good one, your interviewee

will be hanging on your every word, you’ll have noticed, and the temptation

might be there! But remember that the whole point of the interview is to work

collaboratively, and that’s as important in the analysis as in the original construct

elicitation.

Useful questions at this point might look like the following (with reference to

Figure 6.4):

. ‘D’you see those three lines labelled ‘‘availability and choice knowledge

poor’’; ‘‘takes a while to learn the features of new lines’’; ‘‘could be more

interested in after sales’’? They hang together so they look like they have

something in common. How do they differ from the other two (‘‘takes it all

very seriously’’ and ‘‘too forward in pushing a sale’’)? What is each set

saying about sales staff?’

. ‘Ann and Billie are overall the most effective; that’s how you rated them

originally. And d’you see where they lie along the line marked ‘‘overall a

less effective salesperson’’ and ‘‘overall an effective salesperson’’? Jane, on

the other hand, is a long way away from them on the graph. What would

need to change to move her along closer to Billie and Ann?’

This would be to encourage the interviewee to recognise that the position

along each of the lines representing constructs is related to the original ratings,

and that ‘movement along’ each of the lines is a way of interpreting the need

for change. One sort of answer to this last question, for example, might be,

‘Well, she’d need to improve her after sales, and show a lot more interest in

learning the various product lines. That’s more of an issue than getting her to

be relaxed and pleasant with the customers.’

This kind of exploration is particularly useful if you are trying to make the

interviewee’s tacit knowledge about the job explicit, asking her to think about

the way she views job expertise in a reflective way.

Convey your wisdoms and insights gently, check them, and build on

them! This can sometimes be difficult to do, since principal components

analysis does require some complicated statistical knowledge which it’s

difficult to share neatly and elegantly with the other person in the course

of your interview. That’s why my own choice of analysis technique is

cluster analysis as described in Section 6.2 above: the inferences it affords

are more easily explained to people who don’t have a grounding in

statistics.