NAG Library Routine Document
G11AAF computes statistics for a two-way contingency table. For a table with a small number of observations exact probabilities are computed.
|SUBROUTINE G11AAF (
||NROW, NCOL, NOBS, LDNOBS, EXPT, CHIST, PROB, CHI, G, DF, IFAIL)
||NROW, NCOL, NOBS(LDNOBS,NCOL), LDNOBS, IFAIL
||EXPT(LDNOBS,NCOL), CHIST(LDNOBS,NCOL), PROB, CHI, G, DF
For a set of
observations classified by two variables, with
levels respectively, a two-way table of frequencies with
columns can be computed.
To measure the association between the two classification variables two statistics that can be used are, the Pearson
, and the likelihood ratio test statistic,
are the fitted values from the model that assumes the effects due to the classification variables are additive, i.e., there is no association. These values are the expected cell frequencies and are given by
Under the hypothesis of no association between the two classification variables, both these statistics have, approximately, a
degrees of freedom. This distribution is arrived at under the assumption that the expected cell frequencies,
, are not too small. For a discussion of this point see Everitt (1977)
. He concludes by saying, ‘... in the majority of cases the chi-square criterion may be used for tables with expectations in excess of
in the smallest cell’.
In the case of the
approximation can be improved by using Yates' continuity correction factor. This decreases the absolute value of
tables with a small value of
the exact probabilities from Fisher's test are computed. These are based on the hypergeometric distribution and are computed using G01BLF
. A two tail probability is computed as
are the upper and lower one-tail probabilities from the hypergeometric distribution.
Everitt B S (1977) The Analysis of Contingency Tables Chapman and Hall
Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin
- 1: NROW – INTEGERInput
On entry: , the number of rows in the contingency table.
- 2: NCOL – INTEGERInput
On entry: , the number of columns in the contingency table.
- 3: NOBS(LDNOBS,NCOL) – INTEGER arrayInput
On entry: the contingency table
must contain , for and .
, for and .
- 4: LDNOBS – INTEGERInput
: the first dimension of the arrays NOBS
as declared in the (sub)program from which G11AAF is called.
- 5: EXPT(LDNOBS,NCOL) – REAL (KIND=nag_wp) arrayOutput
On exit: the table of expected values.
contains , for and .
- 6: CHIST(LDNOBS,NCOL) – REAL (KIND=nag_wp) arrayOutput
On exit: the table of contributions.
contains , for and .
- 7: PROB – REAL (KIND=nag_wp)Output
contains the two tail significance level for Fisher's exact test, otherwise PROB
contains the significance level from the Pearson
- 8: CHI – REAL (KIND=nag_wp)Output
On exit: the Pearson statistic.
- 9: G – REAL (KIND=nag_wp)Output
On exit: the likelihood ratio test statistic.
- 10: DF – REAL (KIND=nag_wp)Output
On exit: the degrees of freedom for the statistics.
- 11: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
on exit, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Note: G11AAF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
|On entry,||a value in , or all values in NOBS are zero.|
|On entry,||a table has a row or column with both values .|
At least one cell has expected frequency, , . The approximation may be poor.
For the accuracy of the probabilities for Fisher's exact test see G01BLF
The routine G01AFF
allows for the automatic amalgamation of rows and columns. In most circumstances this is not recommended; see Everitt (1977)
Multidimensional contingency tables can be analysed using log-linear models fitted by G02GBF
The data below, taken from Everitt (1977)
, is from
patients with brain tumours. The row classification variable is the site of the tumour: frontal lobes, temporal lobes and other cerebral areas. The column classification variable is the type of tumour: benign, malignant and other cerebral tumours.
The data is read in and the statistics computed and printed.
9.1 Program Text
Program Text (g11aafe.f90)
9.2 Program Data
Program Data (g11aafe.d)
9.3 Program Results
Program Results (g11aafe.r)