Causal models and logical inference in epidemiological psychiatry P Bebbington The British Journal of Psychiatry 1980 136: 317-325 Access the most recent version at doi:10.1192/bjp.136.4.317
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BYiLJ. Psychiat.(1980), 136,317—325
Causal Models and Logical Inference in Epidemiological Psychiatry By PAUL BEBBINGTON SUMMARY The paper examines the nature of causality and how the concept can be interpreted and used in epidemiological psychiatry. The relationship of causality to logical relations is reiterated, and it is emphasized that these relationships in their pure form cannot apply to any one cause of an effect which is the outcome of several causes. In such a case the relation of the cause to the effect is one of weak implication. The fact of weak implication opens the door to the con struction of certain causal models, and the possibilities afforded by this situation are illustrated by reference to the work of Brow@ et al (1975). Weak implication may be an intransitive relationship, and this is shown to invalidate certain ways of testing such models and the conclusions which arise from them. Causal models and logical inferencein epidemiological psychiatry In this paper I will examine the concept of causality as it applies to psychiatric epidemio logy. The nature of epidemiological data makes it impossible to draw. inferences according to classical logic. This limitation of inference makes for particular possibilities in the construction of causal models and for particular difficulties in testing them out. I will elaborate this by reference to recent models proposed by Brown and his colleagues (1975, 1977), using data supplied in their papers.
Jones ci al, 1971). Causality would seem to be one particular way of organising and making sense of the multiplicity of information available, of coping with redundancy. In the development of modern logic by Dc Morgan, Boole, Peirce, Russell and Wittgenstein, the notion of impli cation was probably given form by this human tendency towards causal interpretation. How ever, the definition of the logical relation of implication does not now imply a causal con nection. Implication is a form of sentential logic, that is the logic of operations upon con stituent phrases which are themselves capable of standing alone as sentences. Implication can be represented in English by the conditional “¿If ‘¿a', then ‘¿b',― or symbolically by ‘¿a-+b'.The sentence represented by ‘¿a' is termed the ante cedent, ‘¿b' is the consequent, whilst —¿@ is an ‘¿operator'.Logic is a formal system whose properties depend upon the precise definition of the various operators; what would in science be called operational definition. ‘¿-÷‘ is defined by the truth table below, in which the truth of the overall proposition is dependent upon the truth of the component sentences operated upon.
Classical logic, causality and the real world It is evident that human beings readily inter pret events in terms of causality (Michotte, 1962), though whether this is innate or learned is conjectural (Olum, 1956). A whole sub specialty of social psychology has developed in the last 15 years devoted to the sti@dyof human attribution of causes to events (Kelly, 1973; * A
reworked
the Bronze 1977.
Medal
version
of an
of the Royal
essay
which
College
was
awarded
of Psychiatrists, 317
This
@
On•
I@ @II@ I@IUI@I III@I till IIllU@il ItII ZGS8-NDG-AK9J
318
CAUSAL MODELS AND LOGICAL INFERENCE
T-÷T T-÷F F—.T F-*F
=T =F =T =T
IN EPIDEMIOLOGICAL
PSYCHIATRY
represent a logical operation. This usage is totally distinct from the way in which an arrow is sometimes employed to indicate the direction of a causal connection. It is possible to con ceive a cause as being logically both necessary In this table the truth values of the ante and sufficient, that is both p—@q and q@-÷p,or as cedent are listed on the left of the arrow, those it is sometimes written p+ —¿*.q. This is also of the consequent on the right of it. The table termed reciprocal (as opposed to simple) accounts for all possible combinations of these implication. Such a situation may occasionally values and the corresponding truth value for the arise in epidemiology in the case of a very proposition as a whole is shown in the third column. It can be seen that an implication is poisonous toxin. The relationship is implicit in Koch's postulates for bacterial infections, though only false if a false consequent follows a true it is unlikely that these postulates are ever truly antecedent. fulfilled: indeed they have proved to be too Common linguistic usage may differ from rigorous for some thinkers in the field of chronic logical usage; a variety of common usages can be translated into the operator ‘¿if. . . then. . .‘ diseases (Yerushalmy and Palmer, 1959). Bac terial infections are good illustrations of neces and the common usage ‘¿if. . . then . . .‘ does not sary causes (although sometimes their necessity exactly coincide with the logical usage. This can best be illustrated with an example :— is a result of a semantic manipulation). The formal Jogic briefly described above is a If Stalin was a communist, then sugar is two-value system. We have used the traditional sweet. This, according to the truth table (for T—@.T) value-pair of truth and falsity, but the properties is true. This illustrates that the lay usage of of the system remain the same provided the two possible values are mutually exclusive. When ‘¿if. . . then . . .‘retains the connotation of a causal connection, whilst the logical usage is we look at the field of psychiatric epidemiology, the application of such a system seems inherently one of contingency only. We bridle at this useful. Many of our data are presented in a because the connection is not meaningful when dichotomous form: the 2 x 2 contingency table causality is not apparent. This emphasises that is the basis of much of our knowledge. Popu the concept of causality is partly one of imposed meaning. It is also in keeping with the nature of lations are divided on the basis of, say, ‘¿having causality as a metaphysical proposition: ‘¿all life events'/'not having ljfe events' versus ill/well. The final column of the truth table above represents events have causes' is a claim neither provable the truth values of the relationship of impli nor refutable (Popper, 1959). Causality has cation between two sentences. In the same way, two further requirements over and above those a contingency table may be regarded as repre of logical contingency, namely that cause pre senting the values of a relationship between two cedes effect and that it should be unidirec variables. The table can thus be used to test tional. However, causality may be elaborated the appropriateness of ‘¿â€”a― or of any .other by subsuming it as a special case of the logical relation as a model for the contingency. In the usage. hypothetical examples below, the value ‘¿zero' is Let p be a cause and q be an effect. If the the formal equivalent of ‘¿false',and other causal connection can be represented in the values are the formal equivalent of'true'. form p-+q, this implies that q inevitably follows from p (‘pimplies q), although q may follow Example 1 from other (‘not—¿ p') conditions. This rubric indicates that ‘¿p' is a sufficient cause of q. Some causal relationships can be represented q-+p. eventNo event200well100100 life In this case q only occi@irsif p occurs, although it need not do so. In this situation, p is said to be psychiatrically illLife a necessary cause of q. It should be emphasised that the symbol ‘¿â€”.‘ is here always used to.
319
PAUL BEBBINGTON
In example 1, there are no members of the category ‘¿psychiatricallyill/no life event'. This can be stated ‘¿it is false that those who have not had a life event are psychiatrically ill. The statement ‘¿the subject has not had a life event' renders false the statement ‘¿the subject has had a life event'. Hence the overall statement is false only when the ‘¿subject has had a life event' is false. For the relation ‘¿â€”p.' to hold between the statements ‘¿the subject is psychiatrically ill' and ‘¿the subject has had a life event', the latter has to be the consequent of the relation; the falsity of this latter statement is the sole determinant of the falsity of the relation as a whole. Hence ‘¿life events —¿. illness' is the appropriate model for this particular relationship. Example 2 Life event
No life event
30
10 100
ill
well
0
Example 3 Life event
No life event
50 0
0 100
In this illustration, persons with a life event are ill, persons without are not: life events are both a necessary and a sufficient cause of illness (life events@*—+illness). However, only the most sanguine (and in experienced) researcher would expect often to see relationships like this. The logicians them selves have always been aware of the dif ficulties of applying formal logic in the real world, and few have had the methodological optimism
of Mill
(1856)
in
setting
out
his
canons. Let us take some figures from Brown et al (1975, table Ia) as an example. In the ill group
I include
his patients
event8239well32146
and cases. For ‘¿life
of ‘¿severe events'.
eventNo
life
illLife
From this it will be concluded that while there is an association between life events and disorder, life events are neither sufficient nor necessary for psychiatric disorder. If we accept a methodology of causality, this situation can be represented as p + z—.qwhere z is a further causal factor unknown on the basis of the above data. In epidemiological psychiatry we are merely sampling from the pooi of possible causal factors. Because of this we usually find that our logical relations are those of ‘¿weak implication' (Boudon,
In this example all those with a life event are ill but so are some without. Life events are here a sufficient but not a necessary cause of psy chiatric illness (illness—s@life events).
ill well
events' I take his category Hence :—
1974).
However,
this
leads
to some
rather surprising situations. If we consider the situation of three variables linked thus: p—@q and q—@r,we may logically infer p—kr. The validity of this inference is characterised as the transitivity of the relationship. This does not apply when the implications p—@qand q—*.r are ‘¿weak', i.e. ‘¿if p, then q (generally)' and ‘¿if q then r (generally)'. To illustrate this, let us examine a hypothetical example relating social class, life events and psychiatric disorder. In the first case we take figures which are in accord with classical logic for the relationships ‘¿life events—*illness' (life events are a sufficient condition for illness) and ‘¿illness—+working class (working class status is a necessary condition of illness). The prediction is ‘¿life events.-*working class' (working class status is a necessary con dition of life events.) Case A classworkingNot classIllWellIllWell30
Life event No life eventWorking 100401000100 100 1000
00
320
CAUSAL MODELS AND LOGICAL
INFERENCE
If we collapse this table into tables of only two of the three variables,
IN EPIDEMIOLOGICAL
PSYCHIATRY
Case B
we obtain:
Working classill class classill
Not working
Working class 40
well 100i.e.
wellLife event No life event
0
100
20 1530 10
30Collapsing
Not working
well
ill
10 20
10 5
30
15
15
working class is necessary illness.Ill for this table as we did above, we
obtain:—Not Welllife event no 200i.e. life event
30 10
0
classill
i.e. working
events
No life
30
110
0
100
class status
is necessary
In
this
it should
to hold,
between illness and distribution of zeros)
for life
be
the
noted
logical
life events must differ
that,
purposes of illustration. in Case B, we will take a case of
weak implication. order
interaction
occurs
when
the
degree
of
direction of the association between two variables differs according to the level of a third variable. For example, the association between putting up an umbrella and remaining dry is much stronger if it is raining;
25
15
(generally)'.However ‘¿life events imply illness
classLife
(that is, the according to
makes the collapsing of the 2 x 2 x 2 table into separate 2 x 2 tables invalid except for
second
well35i.e.
No life
for
relationship
class. This is the strong logic equivalent of a second-order interaction* (see Everitt, 1977) and
* A
events
:—Not
case
However,
eventsill
30
events. Q.E.D.
transitivity
15
‘¿working class status implies illness generally'.Life
in addition:—Life
not working class
working
30 30
well 30i.e.
illness.But life events are sufficient for
eventsworkingclass
Working class
and the association
of comfort
and waterproof
clothing is likely to be reversed when the rain stops and the sun comes out.
Working class
events No life events
i.e. working events (at all).
working
30 30
25 20
class status does not imply life
The relation of weak implication may be either transitive or intransitive. Errors of argu ment arise when transitivity is assumed, as we shall see when we examine some of the con tentions of Brown et al (1975). Such an error is also the basis of the ‘¿ecologicalfallacy' that
relationships at one level of analysis must hold at another level (Susser, 1973). Transitivity in a relation of weak implication also implies
second
order interactions
order
interaction.
in weak relations
Second
open the
321
PAUL BEBBINGTON
door to more complicated causal models: the condition for the separation of the vulnerability model from, say, a synergy model is that relations should be weak. Let us examine the route followed by Brown et al (1975, 1977, 1978a) in their development of causal models of depression. They found that although life events and illness clustered in the working class portion of their population sur vey, one group of working class women who did not have an especially high rate of life events, namely those at home with young chil dren, did in fact have the highest rate of psy chiatric disorder. This led them to postulate the existence of vulnerability factors, and it is at this model, its validity, and its possible alter natives that we shall now look. Brown and his colleagues are actively con cerned with causal explanation in their work (e.g. Brown et al, 1973; Brown 1974). However, their elaboration of causal models is not equally explicit at all times: this is because most of their work is essentially exploratory rather than de voted to the testing of specific models. Hence some of their results are presented merely at face value, and the nature of the interaction of factors is not developed. In their first report of data from the Camberwell survey (Brown ci al 1975) the authors declare their intentions to (1) establish the causal link between life events and depressive disorders and (2) investigate social class and other group differences to see (a) whether differences in life event rates could account for differences in illness rates and (b) what other factors affect vulnerability to life events. This illustrates the emphasis on explor ation. The conceptualisation of causal links be tween variables is dealt with in detail by Susser (1973), and his categorisation of additional variables in an interaction is worth recapitu
lating.* I use his terminology. Additional variables can be conceptualised as either con founding or moderating. A confounding variable is one which is causally related both to the independent and to the dependent variable. It may in fact explain all the association be tween a hypothesised cause and a dependent variable. In this case the causal inference would be “¿spurious― (see also Brown, 1974), and control of the true causal (confounding) variable eliminates the originally observed association. Other confounding variables can reverse, minimise or strengthen the association between independent and dependent variable. Moderator variables, on the other hand, have a causal relation with the dependent but not with the independent variable. The moderative statusof a variableaffectingthe dependent variablemust be established by the demon stration of lack of association with the inde pendent variable. It is the possible modes of interaction of moderator variable and causal variable in their effect on the dependent vari able that allow a development of distinct causal models and isof particular interest to us here. These variables are further characterised by the phenomenon of joint effect, that is, their effect in association upon the dependent variable differs from the effect of each in isolation. In the ab sence of joint effect, two variables affecting the dependent variable are merely alternative causal factors. To illustrate this point let us take an example from Brown et al (1975), namely the case of life events and chronic difficulties men tioned above. The figures are taken from Table Ia in that paper. I have used them to calculate the corrected value (X) for the pro portion involved in the causal effect (see Brown et al 1973). It is of course, highly questionable to draw inferences from such small numbers. Neither do I comment here on the value of this method of analysis. However, the figures are of * It should be noted that Susser and Brown approach interest as an illustration. theanalysis ofpotential interactions bywayofthepossible Table I nature of the variable and of its causal effect. In this they differ from the approach of statisticians (see e.g. Everitt, 1977) who are concerned to explain the observed fre quency distribution. Because of this, the classification adopted by statisticians cuts across that proposed by Susser. Hence both a confounding variable and a modera
tor variable can show the phenomen of second-order interaction
with another
causal variable.
(‘x')Severe
Cases (‘x')Patients
events only Major difficulties only 7%Both26%29%
25% 11%13%
322
CAUSAL MODELS AND LOGICAL INFERENCE
IN EPIDEMIOLOGICAL
PSYCHIATRY
Hence, the interaction of the three variables witha youngestchildlessthan sixyearsolddid in the cases falls within the paradigm of 2 alter the class difference remain significant. However, native causal variables. This is not so of the ofwomen at this‘¿life stage'who experienceda paiienLs, in whom the independent effect of the life event or severe difficulty, the working class two causal variables is small indeed, and they group were much more likelyto develop illustrate the phenomena of joint effect quite psychiatric disturbance. neatly. These conclusions are dependent on the Itisapparent,therefore, thatlifeeventsfail fact, easily demonstrated, that in none of the to supplythe intervening variablewhich isto threegroups (patients, casesand normals)is explainthe association of classand disorder. there an association between life events and The layman might well ‘¿explain' the above major difficulties. This,incidentally, does not findings by adducingan increased susceptibility accordwithBrown's(1974)worry thata failure in working classwomen with young children. to show jointeffectbetween eventsand diffi This circular lineof reasoningis avoided by culties could arise because the latter may be Brown and his co-workers.The variables taken into account in the contextual rating of they use in theirattemptsto explainvulner the former. ability are defined without reference to illness However, the simple interactive moderator or totheriskfactors (class, life stage)discovered variable is not the only possibility. A variant of sofar. Thesefactors are:— it, which Susser (1973) terms a concomitant (1) the absenceof a confidingrelationship variable, arises out of Brown's work; this (withhusband) variable is: (2) unemployedstatus (1)independentoftheindependentvariable (3)lossofmotherbeforetheageof11 (2)exhibits jointeffect with theindependent (4) (forthe group with young children) the variable presenceof threeormore children under (3) hasno effect on itsown. theageof15inthehome. Sussergives, as a possible example,theinter Itisclearthatthesefactors are to servetwo action between smoking, exposure to asbestos purposes.They are to form the intervening and lung cancer: asbestos exposure increases the variableswhich explaindifferences in social risk of cancer, but apparently only in the classratesof disorder, and theyare to salvage smokers. Now this type of variable is interesting the somewhat shattered image of lifeeventsas because it is exactly what Brown and his col provokingfactors. The lastfunctionisrelated leagues mean by a vulnerability factor. A to theirincorporation into the vulnerability vulnerability factor, if it is to be distinguished model. from other causal variables in the field, must To what extentcan theybe saidto mediate increasethe effect of the independentvariable the association between classand disorder? but must not have an effect on itsown in pro The lackofa confiding relationship appearsto voking illness. From this certain predictions do so quiteneatly(Brown etal 1975,figure1, follow. However, before we investigate these page 237). However, the argument is complex predictions, we will examine the origins of the and runsasfollows :— (1)vulnerability in women with a young conceptof vulnerability in the work ofBrown childisgreater intheworkingclass. and hiscolleagues. Their concernwas in the (2)vulnerability isgreaterin women with refining of variables, of specifying the conditions impairedmaritalrelationships. under which interactions occur. Their hope was (3)impairedmaritalrelationships in women thatdifferential lifeeventrateswould explain with a young childaremore common in differential illness rates. Their discovery was that they did not. There is indeed a significant theworkingclass. Conclusion. For women with a young child, difference in event rates between working class the class difference in vulnerability is ex and middle class women. Brown et al (1975, plained by the class difference in poor Table III)furtherbroke down theirsampleby marital relationships. As this group of women ‘¿life stage': solely amongst thosewomen athome
PAUL BEBBINGTON
323
out with regard to presence or absence of dis (largely) account for overall social class dif order, or of the various vulnerability factors. ferences in rates of disorder, these too are explainedby the problem of poor marital Moreover, each of these manoeuvres increases the predictions which potentially arise from the relationships. This argument would only be conclusive if model. In summary, the predictions generated by the vulnerability model are as follows:— the relationships were those of strong impli (a) The association between life events and cation. They are not: the appropriate analysis disorder will be greater in the presence should be based upon a four dimensional of the vulnerability factor. contingencytable(vulnerability x classx poor (b) There will be an association between marital relationship x life stage). vulnerability factor and disorder only in The same proviso applies to the presence of the presence of life events. three or more children under age 15 in the (c) There will be an association between life home. The numbers involved in ‘¿loss of mother events and vulnerability factors only in before the age of 11' are tiny, and employment the cases. status apparently showed no class variation. It There are two likely alternative models which would be reasonable to conclude, therefore, may also be examined in terms of predictions of that as explanations of class differences these this type. factors remain at best subjudice. (i) Separate provoking agents with no joint We are now in a position to examine the effect. vulnerability model itself. Brown et al (1975, (a) There will be no change in the asso 1977) choose to investigate the appropriateness ciation of the first factor with disorder in of theirmodels by taking three-dimensional the presence of the second factor. contingency tables and partitioning them into (b) There will be no change in the asso two-dimensional ones. They hold that a vul ciation of the second factorwith disorder nerability model fits if it can be shown that there in the presence of the first factor. is an association between the vulnerability (c) There will be no association of the two factor and illness only in the presence of life factors with each other in either the events;or, to say the same thingin another well or the cases. way, thattheassociation betweenlife eventsand (ii) Separate provoking agents with joint effect illness is increased in the presence of the vul (moderator variables). nerability factor. However, there are a number (a) The association of the first factor with of difficulties in practice with this procedure of disorder will be increased in the pre the authors. Firstly, these predictions are sence of the second factor. couched essentially in terms of the degree of (b) The association of the second factor with association between the variables. To use the disorder will be increased in the pre y@ testas an index not of the significance of sence of the first factor. association but as a measure of the association (c) There will be a positive association of itself is an error, for the value of x2 is dependent the two factors in the cases and a nega upon both the degree of association and the tive association in the well. size of the tested sample. It could only be used As we have stated, Brown and his co-workers as an index of association comparing two samples test only the association (b), though it is some ifthesizeof thesampleswere equal. The second difficulty relatesto the arbi times a little difficult to see this from the manner in which they present their data. trariness of looking merely at predictions arising At first sight it would seem that these pre from the partition of the table in terms of the dictions afford a neat basis on which to dis level of one single factor and the associations criminate between models. Let us look at Brown expected between the two remaining factors. et al (1977) table IIIc as an example. Brown et al Brown et al (1975, 1977) use the presence or (b). I absence of life events to partition their 2 x 2 x 2 (1977) have already tested prediction will reiterate this in the interests of clarity. tables. A similar manoeuvre could be carried
324
CAUSAL MODELS AND LOGICAL INFERENCE
Table 2 Test of Predictions (a) Loss of mother lossCase before age 11No NormalSevere
such
NormalCase
life event
7
124
8
or major difficulty
None
235= 0 1526 8.45;
P
