SUPPLEMENTARY METHODS OF INVESTIGATION.
I.--THE MAINTENANCE OF AVERAGES.--SUPPLEMENT TO THE METHOD OF
DIFFERENCE.
A certain amount of law obtains among events that are usually spoken
of as matters of chance or accident in the individual case. Every kind
of accident recurs with a certain uniformity. If we take a succession
of periods, and divide the total number of any kind of event by the
number of periods, we get what is called the average for that period:
and it is observed that such averages are maintained from period to
period. Over a series of years there is a fixed proportion between
good harvests and bad, between wet days and dry: every year nearly
the same number of suicides takes place, the same number of crimes, of
accidents to life and limb, even of suicides, crimes, or injuries
by particular means: every year in a town nearly the same number of
children stray from their parents and are restored by the police:
every year nearly the same number of persons post letters without
putting an address on them.
This maintenance of averages is simple matter of observation, a datum
of experience, an empirical law. Once an average for any kind of event
has been noted, we may count upon its continuance as we count upon
the continuance of any other kind of observed uniformity. Insurance
companies proceed upon such empirical laws of average in length of
life and immunity from injurious accidents by sea or land: their
prosperity is a practical proof of the correctness and completeness
of the observed facts and the soundness of their inference to the
continuance of the average.
The constancy of averages is thus a guide in practice. But in
reasoning upon them in investigations of cause, we make a further
assumption than continued uniformity. We assume that the maintenance
of the average is due to the permanence of the producing causes. We
regard the average as the result of the operation of a limited sum
of forces and conditions, incalculable as regards their particular
incidence, but always pressing into action, and thus likely to operate
a certain number of times within a limited period.
Assuming the correctness of this explanation, it would follow that
_any change in the average is due to some change in the producing
conditions_; and this derivative law is applied as a help in the
observation and explanation of social facts. Statistics are collected
and classified: averages are struck: and changes in the average are
referred to changes in the concomitant conditions.
With the help of this law, we may make a near approach to the
precision of the Method of Difference. A multitude of unknown or
unmeasured agents may be at work on a situation, but we may accept the
average as the result of their joint operation. If then a new agency
is introduced or one of the known agents is changed in degree, and
this is at once followed by a change in the average, we may with
fair probability refer the change in the result to the change in the
antecedents.
The difficulty is to find a situation where only one antecedent has
been changed before the appearance of the effect. This difficulty may
be diminished in practice by eliminating changes that we have reason
to know could not have affected the circumstances in question.
Suppose, for example, our question is whether the Education Act
of 1872 had an influence in the decrease of juvenile crime. Such a
decrease took place _post hoc_; was it _propter hoc_? We may at once
eliminate or put out of account the abolition of Purchase in the Army
or the extension of the Franchise as not having possibly exercised any
influence on juvenile crime. But with all such eliminations, there may
still remain other possible influences, such as an improvement in
the organisation of the Police, or an expansion or contraction in
employment. "Can you tell me in the face of chronology," a leading
statesman once asked, "that the Crimes Act of 1887 did not diminish
disorder in Ireland?" But chronological sequence alone is not a proof
of causation as long as there are other contemporaneous changes of
condition that may also have been influential.
The great source of fallacy is our proneness to eliminate or isolate
in accordance with our prejudices. This has led to the gibe that
anything can be proved by statistics. Undoubtedly statistics may be
made to prove anything if you have a sufficiently low standard of
proof and ignore the facts that make against your conclusion. But
averages and variations in them are instructive enough if handled with
due caution. The remedy for rash conclusions from statistics is not no
statistics, but more of them and a sound knowledge of the conditions
of reasonable proof.
II.--THE PRESUMPTION FROM EXTRA-CASUAL COINCIDENCE.
We have seen that repeated coincidence raises a presumption of causal
connexion between the coinciding events. If we find two events going
repeatedly together, either abreast or in sequence, we infer that the
two are somehow connected in the way of causation, that there is a
reason for the coincidence in the manner of their production. It may
not be that the one produces the other, or even that their causes are
in any way connected: but at least, if they are independent one of the
other, both are tied down to happen at the same place and time,--the
coincidence of both with time and place is somehow fixed.
But though this is true in the main, it is not true without
qualification. We expect a certain amount of repeated coincidence
without supposing causal connexion. If certain events are repeated
very often within our experience, if they have great positive
frequency, we may observe them happening together more than once
without concluding that the coincidence is more than fortuitous.
For example, if we live in a neighbourhood possessed of many black
cats, and sally forth to our daily business in the morning, a
misfortune in the course of the day might more than once follow upon
our meeting a black cat as we went out without raising in our minds
any presumption that the one event was the result of the other.
Certain planets are above the horizon at certain periods of the year
and below the horizon at certain other periods. All through the year
men and women are born who afterwards achieve distinction in various
walks of life, in love, in war, in business, at the bar, in the
pulpit. We perceive a certain number of coincidences between
the ascendancy of certain planets and the birth of distinguished
individuals without suspecting that planetary influence was concerned
in their superiority.
Marriages take place on all days of the year: the sun shines on a good
many days at the ordinary time for such ceremonies; some marriages are
happy, some unhappy; but though in the case of many happy marriages
the sun has shone upon the bride, we regard the coincidence as merely
accidental.
Men often dream of calamities and often suffer calamities in real
life: we should expect the coincidence of a dream of calamity followed
by a reality to occur more than once as a result of chance. There are
thousands of men of different nationalities in business in London,
and many fortunes are made: we should expect more than one man of any
nationality represented there to make a fortune without arguing any
connexion between his nationality and his success.
We allow, then, for a certain amount of repeated coincidence without
presuming causal connexion: can any rule be laid down for determining
the exact amount?
Prof. Bain has formulated the following rule: "Consider the positive
frequency of the phenomena themselves, how great frequency of
coincidence must follow from that, supposing there is neither
connexion nor repugnance. If there be greater frequency, there is
connexion; if less, repugnance."
I do not know that we can go further definite in precept. The number
of casual coincidences bears a certain proportion to the positive
frequency of the coinciding phenomena: that proportion is to be
determined by common-sense in each case. It may be possible, however,
to bring out more clearly the principle on which common-sense proceeds
in deciding what chance will and will not account for, although our
exposition amounts only to making more clear what it is that we mean
by chance as distinguished from assignable reason. I would suggest
that in deciding what chance will not account for, we make regressive
application of a principle which may be called the principle of Equal
and Unequal Alternatives, and which may be worded as follows:--
Of a given number of possible alternatives, all equally
possible, one of which is bound to occur at a given time, we
expect each to have its turn an equal number of times in the
long run. If several of the alternatives are of the same
kind, we expect an alternative of that kind to recur with a
frequency proportioned to their greater number. If any of the
alternatives has an advantage, it will recur with a frequency
proportioned to the strength of that advantage.
Situations in which alternatives are absolutely equal are rare in
nature, but they are artificially created for games "of chance," as in
tossing a coin, throwing dice, drawing lots, shuffling and dealing a
pack of cards. The essence of all games of chance is to construct a
number of equal alternatives, making them as nearly equal as possible,
and to make no prearrangement which of the number shall come off. We
then say that this is determined by chance. If we ask why we believe
that when we go on bringing off one alternative at a time, each will
have its turn, part of the answer undoubtedly is that given by De
Morgan, namely, that we know no reason why one should be chosen rather
than another. This, however, is probably not the whole reason for our
belief. The rational belief in the matter is that it is only in the
long run or on the average that each of the equal alternatives will
have its turn, and this is probably founded on the experience of
actual trial. The mere equality of the alternatives, supposing them
to be perfectly equal, would justify us as much in expecting that
each would have its turn in a single revolution of the series, in one
complete cycle of the alternatives. This, indeed, may be described
as the natural and primitive expectation which is corrected by
experience. Put six balls in a wicker bottle, shake them up, and roll
one out: return this one, and repeat the operation: at the end of six
draws we might expect each ball to have had its turn of being drawn
if we went merely on the abstract equality of the alternatives. But
experience shows us that in six successive draws the same ball may
come out twice or even three or four times, although when thousands of
drawings are made each comes out nearly an equal number of times.
So in tossing a coin, heads may turn up ten or twelve times in
succession, though in thousands of tosses heads and tails are nearly
equal. Runs of luck are thus within the rational doctrine of chances:
it is only in the long run that luck is equalised supposing that the
events are pure matter of chance, that is, supposing the fundamental
alternatives to be equal.
If three out of six balls are of the same colour, we expect a ball of
that colour to come out three times as often as any other colour on
the average of a long succession of tries. This illustrates the second
clause of our principle. The third is illustrated by a loaded coin or
die.
By making regressive application of the principle thus ascertained by
experience, we often obtain a clue to special causal connexion. We are
at least enabled to isolate a problem for investigation. If we find
one of a number of alternatives recurring more frequently than the
others, we are entitled to presume that they are not equally possible,
that there is some inequality in their conditions.
The inequality may simply lie in the greater possible frequency of
one of the coinciding events, as when there are three black balls in
a bottle of six. We must therefore discount the positive frequency
before looking for any other cause. Suppose, for example, we find that
the ascendancy of Jupiter coincides more frequently with the birth of
men afterwards distinguished in business than with the birth of men
otherwise distinguished, say in war, or at the bar, or in scholarship.
We are not at liberty to conclude planetary influence till we have
compared the positive frequency of the different modes of distinction.
The explanation of the more frequently repeated coincidence may simply
be that more men altogether are successful in business than in war
or law or scholarship. If so, we say that chance accounts for the
coincidence, that is to say, that the coincidence is casual as far as
planetary influence is concerned.
So in epidemics of fever, if we find on taking a long average that
more cases occur in some streets of a town than in others, we are not
warranted in concluding that the cause lies in the sanitary conditions
of those streets or in any special liability to infection without
first taking into account the number of families in the different
streets. If one street showed on the average ten times as many cases
as another, the coincidence might still be judged casual if there were
ten times as many families in it.
Apart from the fallacy of overlooking the positive frequency, certain
other fallacies or liabilities to error in applying this doctrine of
chances may be specified.
1. We are apt, under the influence of prepossession or prejudice, to
remember certain coincidences better than others, and so to imagine
extra-casual coincidence where none exists. This bias works in
confirming all kinds of established beliefs, superstitious and other,
beliefs in dreams, omens, retributions, telepathic communications, and
so forth. Many people believe that nobody who thwarts them ever comes
to good, and can produce numerous instances from experience in support
of this belief.
2. We are apt, after proving that there is a residuum beyond what
chance will account for on due allowance made for positive frequency,
to take for granted that we have proved some particular cause for
this residuum. Now we have not really explained the residuum by the
application of the principle of chances: we have only isolated a
problem for explanation. There may be more than chance will account
for: yet the cause may not be the cause that we assign off-hand. Take,
for example, the coincidence that has been remarked between race and
different forms of Christianity in Europe. If the distribution of
religious systems were entirely independent of race, it might be
said that you would expect one system to coincide equally often
with different races in proportion to the positive number of their
communities. But the Greek system is found almost solely among
Slavonic peoples, the Roman among Celtic, and the Protestant among
Teutonic. The coincidence is greater than chance will account for. Is
the explanation then to be found in some special adaptability of the
religious system to the character of the people? This may be the right
explanation, but we have not proved it by merely discounting chance.
To prove this we must show that there was no other cause at work, that
character was the only operative condition in the choice of system,
that political combinations, for example, had nothing to do with it.
The presumption from extra-casual coincidence is only that there is
a special cause: in determining what that is we must conform to the
ordinary conditions of explanation.
So coincidence between membership of the Government and a classical
education may be greater than chance would account for, and yet the
circumstance of having been taught Latin and Greek at school may have
had no special influence in qualifying the members for their duties.
The proportion of classically educated in the Government may be
greater than the proportion of them in the House of Commons, and
yet their eminence may be in no way due to their education. Men of
a certain social position have an advantage in the competition for
office, and all those men have been taught Latin and Greek as a matter
of course. Technically speaking, the coinciding phenomena may be
independent effects of the same cause.
3. Where the alternative possibilities are very numerous, we are apt
not to make due allowance for the number, sometimes overrating it,
sometimes underrating it.
The fallacy of underrating the number is often seen in games of
chance, where the object is to create a vast number of alternatives,
all equally possible, equally open to the player, without his being
able to affect the advent of one more than another. In whist, for
example, there are some six billions of possible hands. Yet it is a
common impression that, one night with another, in the course of a
year, a player will have dealt to him about an equal number of good
and bad hands. This is a fallacy. A very much longer time is required
to exhaust the possible combinations. Suppose a player to have
2000 hands in the course of a year: this is only one "set," one
combination, out of thousands of millions of such sets possible. Among
those millions of sets, if there is nothing but chance in the matter,
there ought to be all proportions of good and bad, some sets all good,
some all bad, as well as some equally divided between good and bad.[1]
Sometimes, however, the number of possible alternatives is overrated.
Thus, visitors to London often remark that they never go there without
meeting somebody from their own locality, and they are surprised at
this as if they had the same chance of meeting their fellow-visitors
and any other of the four millions of the metropolis. But really the
possible alternatives of rencounter are far less numerous. The places
frequented by visitors to London are filled by much more limited
numbers: the possible rencounters are to be counted by thousands
rather than by millions.
[Footnote 1: See De Morgan's _Essay on Probabilities_, c. vi.,
"On Common Notions of Probability".]