**3.4 The Minimum Chi-Square Method**

The method is an extension of the chi-square goodness-of-fit test
described in Section 4.2. It will be seen that
it is closely related
to least squares and weighted least squares methods; the minimum
chi-square statistic has asymptotic properties similar to
ML. Pearson's (1900)
paper in which it was introduced is a foundation
stone of modern statistical analysis
^{(4)}; a comprehensive and readable
review (plus bibliography) was given by
Cochran (1952).

Consider observational data which can be binned, and a
model/hypothesis which predicts the population of each bin. The
chi-square statistic describes the goodness-of-fit of the data to the
model. If the *observed* numbers in each of *k* bins are
*O _{i}*, and the

(The parallel with weighted least squares is evident: the statistic is
the summed squares of the residuals weighted by what is effectively
the variance if the procedure is governed by Poisson statistics.) The
null hypotheses *H*_{0} (see
Section 4.1) is that
the proportion of objects falling in each category is
*E _{i}*; the chi-square procedure tests whether
the

The premise of the chi-square test then is that the deviations from
*E _{i}* are due to statistical fluctuations from limited
numbers of
observations per bin, i.e. ``noise'' or Poisson statistics, and the
chi-square distribution simply gives the probability that the chance
deviations from

Number of parameters | |||

Significance | |||

1 | 2 | 3 | |

0.68 | 1.00 | 2.30 | 3.50 |

0.90 | 2.71 | 4.61 | 6.25 |

0.99 | 6.63 | 9.21 | 11.30 |

There is good news and bad news about the chi-square test. First the
good: it is a test of which most scientists have heard, with which
many are comfortable, and from which some are even prepared to accept
the results. Moreover, because ^{2} is additive, the results of
different data sets which may fall in different bins, bin sizes, or
which may apply to different aspects of the same model, may be tested
all at once. The contribution to ^{2} of each bin may be examined and
regions of exceptionally good or bad fit delineated. In addition, ^{2}
is easily computed, and its significance readily estimated as
follows. The *mean* of the chi-square distribution equals the number of
degrees of freedom, while the *variance* equals twice the number of
degrees of freedom; see plots of the function in
Fig. 4. So as another
rule of thumb, if ^{2} should come out (for more than four bins) as
~ (number of bins - 1) then accept *H*_{0}. But if ^{2} exceeds twice (number
of bins - 1), *H*_{0} will probably be rejected.

Now the bad news: the data must be binned to apply the test, and the
bin populations must reach a certain size because it is obvious that
instability results as *E _{i}* -> 0. As another rule of
thumb then: > 80 per
cent of the bins must have

The minimum chi-square method of model-fitting consists of
minimizing the ^{2} statistic by varying the parameters of the
model. The premise on which this technique is based is obvious from
the foregoing - the model is assumed to be qualitatively correct, and
is adjusted to minimize (via ^{2}) the differences between the *E _{i}*
and

The essential question, having found appropriate parameters, is to estimate confidence limits for them. The answer is as given by Avni (1976): the region of confidence (significance level ) is defined by

where is from Table 1.

[It is interesting to note that (a) the depends only on the number
of parameters involved, and not on the goodness-of-fit (^{2}_{min}) actually
achieved, and (b) there is an alternative answer given by
Cline & Lesser (1970)
which must be in error: the result obtained by Avni has
been tested with Monte Carlo experiments by Avni himself and by
M. Birkinshaw (personal communication).]

By way of example, see Fig. 5. The model to
describe the distribution (Fig. 5a) requires
two parameters,
and *k*. Contours of
^{2} resulting from the
parameter search are shown in Fig. 5(b). When
the Avni prescription is applied, it gives ^{2}_{0.68} = ^{2}_{min} + 2.30, for
the value corresponding to 1
(significance level = 0.68); the contour
^{2}_{0.68} = 6.2
defines a region of confidence in the (, *k*) plane
corresponding to the 1 level of
significance. (Because the range of
interest for was
limited from other considerations to 1.9 < < 2.4,
the parameter search was not extended to define this contour fully.) A
cut along a line of constant is shown in Fig. 5(c); the calculation
of ^{2} defines upper and
lower values of *k* corresponding to = 1 for
this particular .

A last comment on the method of minimum chi-square. The procedure
has its limitations - loss of information due to binning and
inapplicability to small samples. However, it has one great advantage
over other model-fitting methods - the test of the optimum model is
carried out for free, as part of the procedure. For instance, the
example of Fig. 5 - there are seven bins, two
parameters and the
appropriate number of degrees of freedom is therefore four. The value
of ^{2}_{min} is
about 4, just as one would have hoped, and the optimum
model is thus a satisfactory fit.