Janforum

Publishing the results of factor analysis:interpretation and presentation

The use of factor analysis in the Journal of Advanced

Nursing is increasing. There were two papers published

using this technique in 1994 and 1995, three in 1996 and

1997 and by July 1998 there were ®ve papers. Factor

analysis is a statistical technique for the reduction to

underlying dimensions of multivariate data (Dillon &

Goldstein 1984), where many variables are simultaneously

measured in a group of individuals. As someone who

employs factor analysis regularly and always pays partic-

ular attention to studies in which it is used I would like to

propose some standards for the presentation of factor

analytical results which will both enhance their utility to

readers and demonstrate that those who use the technique

really understand its complexities.

Speci®cally I would like to address the following

aspects:

� reasons for using factor analysis;

� sample size;

� presentation of ®rst unrotated principal component;

� communalities;

� rotation, the presentation of Eigenvalues and explained

variance;

� selecting the number of factors from a principal com-

ponents analysis;

� presenting the rotated solution; and

� limitations of exploratory factor analysis.

For the purpose of this article all papers using factor

analysis in the Journal of Advanced Nursing between

January 1994 and July 1998 were reviewed.

Reasons for using factor analysis

Factor analysis encompasses a range of statistical tech-

niques for the reduction of multivariate data sets to fewer

underlying dimensions or factors (Hair et al. 1987). There

is considerable laxity in the literature where factor anal-

ysis is used over the use of the terms `dimensions',

`constructs' and `factors' (Kline 1994) and writers should

try to con®ne themselves to one term within a publication

and to explain what they mean by it. Whichever term is

used, a factor is essentially a undimensional construct or

dimension within a data set which is characterized by the

variables of which it is comprised.

There are two branches of factor analysis, explorato-

ry and con®rmatory. All of the studies except one

(Dunn & Burnett 1995) published in the Journal of

Advanced Nursing are purely exploratory and the term

exploratory factor analysis (e.g. principal components

analysis, common factor analysis, principal axis factor-

ing and maximum likelihood factor analysis) should be

more commonly used in order to distinguish these

techniques from con®rmatory factor analysis. The lim-

itation of exploratory factor analysis will be discussed

below.

Sample size

Popular textbooks on factor analysis give speci®c advice

on sample size (Child 1990, Kline 1994). However, spe-

ci®c comment on sample size is rarely made in papers

using factor analysis in the Journal of Advanced Nursing.

It is quite frequently the case that adequate sample sizes

are not reported (Zhan & Shen 1994, Lauri & SalanteraÈ

1995, Crow et al. 1996, Drummond & Rickwood 1997,

Persson et al. 1997, Sitzia et al. 1997, Gilleard & Reed

1998, Severinsson 1998). The required variable to subject

ratio lies between 1:5 and 1:10 (Kline 1994), with the

former being the absolute minimum and the latter being

suf®cient. In order to detect underlying dimensions

resulting from individual differences between subjects,

there must be a greater number of variables being mea-

sured than subjects in the study.

Presentation of ®rst unrotated principal component

The most commonly applied technique of exploratory

factor analysis is principal components analysis. Strictly

speaking this is not really a method of factor analysis but a

way of making a preliminary investigation of the correla-

tion between all of the variables in a set of data (Kline

1994). However, this is an easily applied method of

identifying factors and the results, especially where sam-

ple sizes are adequate, differ very little from other meth-

ods such as principal axis factoring. However, it is very

exceptional, where principal components analysis has

been used, to see the loadings for the ®rst unrotated

principal component being reported in the Journal of

Advanced Nursing (Watson & Deary 1994). Nevertheless, I

would advocate this practice, in addition to reporting the

variance extracted by this component, because useful

Journal of Advanced Nursing, 1998, 28(6), 1361±1363

Ó 1998 Blackwell Science Ltd 1361

information can be gleaned, such as whether or not there

is a general factor underlying the data.

Communalities

Similarly to the ®rst unrotated principal component, it is

very exceptional to see communalities being reported in

the Journal of Advanced Nursing and these are also useful

(Watson & Deary 1994). Communality represents the

amount of variance in one variable which is shared by

all the other variables and, generally speaking, these

should be high. Where communalities are low it can

indicate that variables are unreliable and it is helpful if the

reader can judge the contribution of each variable to the

overall analysis (Child 1990).

Rotation, the presentation of Eigenvaluesand explained variance

While it is common practice to present the Eigenvalues

(Bryman & Cramer 1997) and the amount of variance

extracted by the unrotated principal components in a table

along with each of the factors (Zee et al. 1994, Hicks 1996,

Persson et al. 1997, Andrew 1998, Rosenkoetter & Garris

1998, Servinsson 1998) for the rotated solution this is,

strictly speaking, incorrect. The Eigenvalues are derived

from the variance extracted by the unrotated principal

components Ð which merely serves as a guide to the

number of factors to be rotated Ð the Eigenvalues and the

variance extracted by the unrotated principal components

are only relevant to the unrotated solution.

Once a rotational procedure has been imposed the

variance is distributed differently across the rotated fac-

tors compared with the unrotated principal components.

In fact, that redistribution is the point behind the rotation

which is aimed at achieving a simpler and more easily

interpreted solution than is possible merely from the

principal unrotated components (Child 1990). The Eigen-

values and extracted variances should be presented sep-

arately from the rotated solution or merely reported in the

text (Watson & Deary 1994, Zhan & Shen 1994, Lauri &

SalanteraÈ 1995).

Selecting the number of factors from principalcomponents analysis

Admittedly there is no approved method of selecting the

number of factors to be rotated from a principal compo-

nents analysis. The choice for principal components anal-

ysis is between using Eigenvalues greater than unity or the

scree slope method. The scree slope method, reported only

once in the Journal of Advanced Nursing (Dunn & Bynett

1995) involves plotting the Eigenvalues against the number

of factors and selecting the number of factors by visual

inspection of the resulting curve. Using Eigenvalues great-

er than one is commonly employed because it requires no

judgement on behalf of the researcher.

However, it frequently leads to solutions which are not

parsimonious leading to a large number of factors which

cannot be interpreted. Using Eigenvalues greater than

unity is often suf®cient but I lodge a plea for a routine

combination of methods in order that the scree slope may

be used either to verify the number of factors selected

using Eigenvalues greater than unity or to rotate a more

economical number of factors.

Presenting the rotated solution

The practice, in presenting the results of the rotated

solution, of only showing the variables which load on the

rotated factors should be discouraged (Crow et al. 1996,

Hicks 1996, Andrew 1998, Manninen 1998, Rosenkoetter

& Garris 1998). This practice can be very misleading to the

reader because it may make the derived solution look

simpler than it really is. What is masked by the practice of

only showing those variables loading on the rotated

factors are those variables which either cross-load on

other factors or variables with loadings on factors other

than those to which they are ascribed which are relatively

high. Quite simply, the loadings of all variables on all

factors should be shown (van der Zee et al. 1994, Watson

& Deary 1994, Lauri & SalanteraÈ 1995, Clifford 1996,

Drummond & Rickwood 1997, Persson et al. 1997).

Limitations of exploratory factor analysis

Exploratory factor analysis explores multivariate data for

putative factorial solutions whereby the data may be

reduced. In contrast, con®rmatory factor analysis tests

hypothesized factorial solutions and provides estimates of

how closely hypothesized solutions explain the variance

in the data. In other words, the investigator can set up

models based on previous research, on intuition or on the

intercorrelations in the data and test how closely these ®t

the data (Dunn & Burnett 1995).

Researchers presenting the results of exploratory factor

analysis should acknowledge the limitations of this ap-

proach and refer to the superior, but vastly more complex,

technique of con®rmatory factor analysis. Nevertheless,

the value of exploratory factor analysis in exploring data

and providing putative models for testing should be

maintained, in my view.

Conclusion

Hopefully these views will stimulate others to contribute

to the discussion about the presentation of results from

multivariate statistical analysis and to a more informative

way of using such results in papers published in the

Journal of Advanced Nursing. The excitement, individu-

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1362 Ó 1998 Blackwell Science Ltd, Journal of Advanced Nursing, 28(6), 1361±1363

ality and interest in factor analysis should be in the

interpretation and not in the presentation.

Roger Watson

BSc PhD RGN CBiol MIBiol

Professor of Nursing,

Dublin City University,

Dublin, Republic of Ireland

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Ó 1998 Blackwell Science Ltd, Journal of Advanced Nursing, 28(6), 1361±1363 1363