09-MY451-3waytables.Rmd

# Analysis of 3-way contingency tables {#c-3waytables}

In Section \@ref(s-descr1-2cat) and Chapter \@ref(c-tables) we discussed
the analysis of two-way contingency tables (crosstabulations) for
examining the associations between two categorical variables. In this
section we extend this by introducing the basic ideas of **multiway
contingency tables** which include more than two categorical variables.
We focus solely on the simplest instance of them, a **three-way table**
of three variables.

This topic is thematically related also to some of Chapter
\@ref(c-regression), in that a multiway contingency table can be seen as
a way of implementing for categorical variables the ideas of statistical
control that were also a feature of the multiple linear regression model
of Section \@ref(s-regression-multiple). Here, however, we will not
consider formal regression models for categorical variables (these are
mentioned only briefly at the end of the chapter). Instead, we give
examples of analyses which simply apply familiar methods for two-way
tables repeatedly for tables of two variables at fixed values of a third
variable.

The discussion is organised arond three examples. In each case we start
with a two-way table, and then introduce a third variable which we want
to control for. This reveals various features in the examples, to
illustrate the types of findings that may be uncovered by statistical
control.

**Example 9.1: Berkeley admissions**

Table \@ref(tab:t-berkeley1) summarises data on applications for admission to
graduate study at the University of California, Berkeley, for the fall
quarter 1973.^[These data, which were produced by the Graduate Division of UC
Berkeley, were first discussed in Bickel, P. J., Hammel, E. A., and
O’Connell, J. W. (1975), “Sex bias in graduate admissions: Data from
Berkeley”, *Science* **187**, 398–404. They have since become a
much-used teaching example. The version of the data considered here
are from Freedman, D., Pisani, R., and Purves, R., *Statistics*
(W. W. Norton, 1978).] The data are for five of the six departments with the
largest number of applications, labelled below Departments 2–5
(Department 1 will be discussed at the end of this section). Table
\@ref(tab:t-berkeley1) shows the two-way contingency table of the sex of the
applicant and whether he or she was admitted to the university.

  -----------------------------------------------------
  \                 Admitted   Admitted       \       \    
  Sex                     No        Yes   % Yes   Total
  ------------- ------------ ---------- ------- -------
  Male                  1180        686    36.8    1866

  Female                1259        468    27.1    1727

  Total                 2439       1154    32.1    3593
  -----------------------------------------------------

  : (\#tab:t-berkeley1)Table of sex of applicant vs. admission in the Berkeley admissions
  data. The column labelled ‘% Yes’ is the percentage of applicants
  admitted within each row. $\chi^{2}=38.4$, $df=1$, $P<0.001$.


The percentages in Table \@ref(tab:t-berkeley1) show that men were more
likely to be admitted, with a 36.8% success rate compared to 27.1% for
women. The difference is strongly significant, with $P<0.001$ for the
$\chi^{2}$ test of independence. If this association was interpreted
causally, it might be regarded as evidence of sex bias in the admissions
process. However, other important variables may also need to be
considered in the analysis. One of them is the academic department to
which an applicant had applied. Information on the department as well as
sex and admission is shown in Table \@ref(tab:t-berkeley2).

  -----------------------------------------------------------------------------
                               \            Admitted   Admitted       \       \    
  Department                   Sex                No        Yes   % Yes   Total
  ---------------------------- ---------- ---------- ---------- ------- -------
  2                            Male              207        353    63.0     560

                               Female              8         17    68.0      25

                               Total             215        370    63.2     585

  $\chi^{2}=0.25$, $P=0.61$

  3                            Male              205        120    36.9     325

                               Female            391        202    34.1     593

                               Total             596        322    35.1     918

  $\chi^{2}=0.75$, $P=0.39$

  4                            Male              279        138    33.1     417

                               Female            244        131    34.9     375

                               Total             523        269    34.0     792

  $\chi^{2}=0.30$, $P=0.59$

  5                            Male              138         53    27.7     191

                               Female            299         94    23.9     393

                               Total             437        147    25.2     584

  $\chi^{2}=1.00$, $P=0.32$

  6                            Male              351         22     5.9     373

                               Female            317         24     7.0     341

                               Total             668         46     6.4     714

  $\chi^{2}=0.38$, $P=0.54$

  Total                                         2439       1154    32.1    3593
  -----------------------------------------------------------------------------

  : (\#tab:t-berkeley2)Sex of applicant vs. admission by academic department in the
  Berkeley admissions data.

Table \@ref(tab:t-berkeley2) is a *three-way* contingency table, because each
of its internal cells shows the number of applicants with a particular
combination of three variables: department, sex and admission status.
For example, the frequency 207 in the top left corner indicates that
there were 207 male applicants to department 2 who were not admitted.
Table \@ref(tab:t-berkeley2) is presented in the form of a series of tables
of sex vs. admission, just like in the original two-way table
\@ref(tab:t-berkeley1), but now with one table for each department. These are
known as **partial tables** of sex vs. admission, **controlling for**
department. The word “control” is used here in the same sense as before:
each partial table summarises the data for the applicants to a single
department, so the variable “department” is literally held constant
within the partial tables.

Table \@ref(tab:t-berkeley2) also contains the marginal distributions of sex
and admission status within each department. They can be used to
construct the other two possible two-way tables for these variables, for
department vs. sex of applicant and department vs. admission status.
This information, summarised in Table \@ref(tab:t-berkeley3), is discussed
further below.

The association between sex and admission within each partial table can
be examined using methods for two-way tables. For every one of them, the
$\chi^{2}$ test shows that the hypothesis of independence cannot be
rejected, so there is no evidence of sex bias within any department. The
apparent association in Table \@ref(tab:t-berkeley1) is thus spurious, and
disappears when we control for department. Why this happens can be
understood by considering the distributions of sex and admissions across
departments, as shown in Table \@ref(tab:t-berkeley3). Department is clearly
associated with sex of the applicant: for example, almost all of the
applicants to department 2, but only a third of the applicants to
department 5 are men. Similarly, there is an association between
department and admission: for example, nearly two thirds of the
applicants to department 2 but only a quarter of the applicants to
department 5 were admitted. It is the combination of these associations
which induces the spurious association between sex and admission in
Table \@ref(tab:t-berkeley1). In essence, women had a lower admission rate
overall because relatively more of them applied to the more selective
departments and fewer to the easy ones.

  Of all applicants        Department 2     3     4     5     6
  ---------------------- -------------- ----- ----- ----- -----
  % Male                             96    35    53    33    52
  % Admitted                         63    35    34    25     6
  Number of applicants              585   918   792   584   714

  : (\#tab:t-berkeley3)Percentages of male applicants and applicants admitted by department
  in the Berkeley admissions data.


One possible set of causal connections leading to a spurious association
between $X$ and $Y$ was represented graphically by Figure \@ref(fig:f-xyzspurious). There are, however, other possibilities which may
be more appropriate in particular cases. In the admissions example,
department (corresponding to the control variable $Z$) cannot be
regarded as the cause of the sex of the applicant. Instead, we may
consider the causal chain Sex $\longrightarrow$ Department
$\longrightarrow$ Admission. Here department is an *intervening
variable* between sex and admission rather than a common cause of them.
We can still argue that sex has an effect on admission, but it is an
*indirect effect* operating through the effect of sex on choice of
department. The distinction is important for the original research
question behind these data, that of possible sex bias in admissions. A
direct effect of sex on likelihood on admission might be evidence of
such bias, because it might indicate that departments are treating male
and female candidates differently. An indirect effect of the kind found
here does not suggest bias, because it results from the applicants’ own
choices of which department to apply to.

In the admissions example a strong association in the initial two-way
table was “explained away” when we controlled for a third variable. The
next example is one where controlling leaves the initial association
unchanged.

**Example 9.2:** **Importance of short-term gains for investors
(continued)**

Table \@ref(tab:t-investors) showed a
relatively strong association between a person’s age group and his or
her attitude towards short-term gains as an investment goal. This
association is also strongly significant, with $P<0.001$ for the
$\chi^{2}$ test of independence. Table \@ref(tab:t-investors3) shows the
crosstabulations of these variables, now controlling also for the
respondent’s sex. The association is now still significant in both
partial tables. An investigation of the row proportions suggests that
the pattern of association is very similar in both tables, as is its
strength as measured by the $\gamma$ statistic ($\gamma=-0.376$ among
men and $\gamma=-0.395$ among women). The conclusions obtained from the
original two-way table are thus unchanged after controlling for sex.

  ----------------------------------------------------------------------------
  **MEN**                        \    Slightly           \        Very       \
  Age group             Irrelevant   important   Important   important   Total
  ---------------- --------------- ----------- ----------- ----------- -------
  Under 45                      29          35          30          22     116

                             0.250       0.302       0.259       0.190   1.000

  45–54                         83          60          52          29     224

                             0.371       0.268       0.232       0.129   1.000

  55–64                        116          40          28          16     200


                             0.580       0.200       0.140       0.080   1.000

  65 and over                  150          53          16          12     231

                             0.649       0.229       0.069       0.052   1.000

  Total                        378         188         126          79     771

                             0.490       0.244       0.163       0.102   1.000
  ----------------------------------------------------------------------------

  : (\#tab:t-investors3)Frequencies of respondents by age group and attitude towards
  short-term gains in Example 9.2, controlling for sex of respondent.
  The numbers below the frequencies are proportions within rows. $\chi^{2}=82.4$, $df=9$, $P<0.001$. $\gamma=-0.376$.

  ----------------------------------------------------------------------------
  **WOMEN**                      \    Slightly           \        Very       \  
  Age group             Irrelevant   important   Important   important   Total
  ---------------- --------------- ----------- ----------- ----------- -------
  Under 45                       8          10           8           4      30

                             0.267       0.333       0.267       0.133   1.000

  45–54                         28          17           5           8      58

                             0.483       0.293       0.086       0.138   1.000

  55–64                         37           9           3           4      53

                             0.698       0.170       0.057       0.075   1.000

  65 and over                   43          11           3           3      60

                             0.717       0.183       0.050       0.050   1.000

  Total                        116          47          19          19     201

                             0.577       0.234       0.095       0.095   1.000
  ----------------------------------------------------------------------------

  : (\#tab:t-investors3)Frequencies of respondents by age group and attitude towards
  short-term gains in Example 9.2, controlling for sex of respondent.
  The numbers below the frequencies are proportions within rows. $\chi^{2}=27.6$, $df=9$, $P=0.001$. $\gamma=-0.395$.

**Example 9.3**: ***The Titanic***

The passenger liner RMS *Titanic* hit an iceberg and sank in the North
Atlantic on 14 April 1912, with heavy loss of life. Table
\@ref(tab:t-titanic2) shows a crosstabulation of the people on board the
*Titanic*, classified according to their status (as male passenger,
female or child passenger, or member of the ship’s crew) and whether
they survived the sinking.^[The data are from the 1912 report of the official British Wreck
Commissioner’s inquiry into the sinking, available at
<http://www.titanicinquiry.org>.] The $\chi^{2}$ test of independence has
$P<0.001$ for this table, so there are statistically significant
differences in probabilities of survival between the groups. The table
suggests, in particular, that women and children among the passengers
were more likely to survive than male passengers or the ship’s crew.

  -----------------------------------------------------------
                                Survivor:         \         \    
  Group                               Yes        No     Total
  ---------------------------- ---------- --------- ---------
  Male passenger                      146       659       805

                                  (0.181)   (0.819)   (1.000)

  Female or child passenger           353       158       511

                                  (0.691)   (0.309)   (1.000)

  Crew member                         212       673       885

                                  (0.240)   (0.760)   (1.000)

  Total                               711      1490      2201

                                  (0.323)   (0.677)   (1.000)
  -----------------------------------------------------------

  : (\#tab:t-titanic2)Survival status of the people aboard the *Titanic*, divided into
  three groups. The numbers in brackets are proportions of survivors and
  non-survivors within each group. $\chi^{2}=418$, $\text{df}=2$, $P<0.001$.

We next control also for the class in which a person was travelling,
classified as first, second or third class. Since class does not apply
to the ship’s crew, this analysis is limited to the passengers,
classified as men vs. women and children. The two-way table of sex by
survival status for them is given by Table \@ref(tab:t-titanic2), ignoring
the row for crew members. This association is strongly significant, with
$\chi^{2}=344$ and $P<0.001$.

  ------------------------------------------------------
  \        \                   Survivor:       \       \    
  Class    Group                     Yes      No   Total
  -------- ---------------- ------------ ------- -------
  First    Man                        57     118     175

                                   0.326   0.674   1.000

           Woman or child            146       4     150

                                   0.973   0.027   1.000

           Total                     203     122     325

                                   0.625   0.375   1.000

  Second   Man                        14     154     168

                                   0.083   0.917   1.000

           Woman or child            104      13     117

                                   0.889   0.111   1.000

           Total                     118     167     285

                                   0.414   0.586   1.000

  Third    Man                        75     387     462

                                   0.162   0.838   1.000

           Woman or child            103     141     244

                                   0.422   0.578   1.000

           Total                     178     528     706

                                   0.252   0.748   1.000

  Total                              499     817    1316

                                   0.379   0.621   1.000
  ------------------------------------------------------

  :(\#tab:t-titanic3)Survival status of the passengers of the *Titanic*, classified by
  class and sex. The numbers below the frequencies are proportions
  within rows.

Two-way tables involving class (not shown here) suggest that it is
mildly associated with sex (with percentages of men 54%, 59% and 65% in
first, second and third class respectively) and strongly associated with
survival (with 63%, 41% and 25% of the passengers surviving). It is thus
possible that class might influence the association between sex and
survival. This is investigated in Table \@ref(tab:t-titanic3), which shows
the partial associations between sex and survival status, controlling
for class. This association is strongly significant (with $P<0.001$ for
the $\chi^{2}$ test) in every partial table, so it is clearly not
explained away by associations involving class. The direction of the
association is also the same in each table, with women and children more
likely to survive than men among passengers of every class.

The presence and direction of the association in the two-way Table
\@ref(tab:t-titanic2) are thus preserved and similar in every partial table
controlling for class. However, there appear to be differences in the
*strength* of the association between the partial tables. Considering,
for example, the ratios of the proportions in each class, women and
children were about 3.0 ($=0.973/0.326$) times more likely to survive
than men in first class, while the ratio was about 10.7 in second class
and 2.6 in the third. The contrast of men vs. women and children was
thus strongest among second-class passengers. This example differs in
this respect from the previous ones, where the associations were similar
in each partial table, either because they were all essentially zero
(Example 9.1) or non-zero but similar in both direction and strength
(Example 9.2).

We are now considering three variables, class, sex and survival.
Although it is not necessary for this analysis to divide them into
explanatory and response variables, introducing such a distinction is
useful for discussion of the results. Here it is most natural to treat
survival as the response variable, and both class and sex as explanatory
variables for survival. The associations in the partial tables in Table
\@ref(tab:t-titanic3) are then partial associations between the response
variable and one of the explanatory variables (sex), controlling for the
other explanatory variable (class). As discussed above, the strength of
this partial association is different for different values of class.
This is an example of a statistical **interaction**. In general, there
is an interaction between two explanatory variables if the strength and
nature of the partial association of (either) one of them on a response
variable depends on the value at which the other explanatory variable is
controlled. Here there is an interaction between class and sex, because
the association between sex and survival is different at different
levels of class.

Interactions are an important but challenging element of many
statistical analyses. Important, because they often correspond to
interesting and subtle features of associations in the data.
Challenging, because understanding and interpreting them involves
talking about (at least) three variables at once. This can seem rather
complicated, at least initially. It adds to the difficulty that
interactions can take many forms. In the *Titanic* example, for
instance, the nature of the class-sex interaction was that the
association between sex and survival was in the same direction but of
different strengths at different levels of class. In other cases
associations may disappear in some but not all of the partial tables, or
remain strong but in different directions in different ones. They may
even all or nearly all be in a different direction from the association
in the original two-way table, as in the next example.

**Example 9.4: Smoking and mortality**

A health survey was carried out in Whickham near Newcastle upon Tyne in
1972–74, and a follow-up survey of the same respondents twenty years
later.^[The two studies are reported in Tunbridge, W. M. G. et al. (1977).
“The spectrum of thyroid disease in a community: The Whickham
survey”. *Clinical Endocrinology* **7**, 481–493, and Vanderpump,
M. P. J. et al. (1995). “The incidence of thyroid disorders in the
community: A twenty-year follow-up of the Whickham survey”.
*Clinical Endocrinology* **43**, 55–69. The data are used to
illustrate Simpson’s paradox by Appleton, D. R. et al. (1996).
“Ignoring a covariate: An example of Simpson’s paradox”. *The
American Statistician* **50**, 340–341.] Here we consider only the $n=1314$ female respondents who
were classified by the first survey either as current smokers or as
never having smoked. Table \@ref(tab:t-whickham1) shows the
crossclassification of these women according to their smoking status in
1972–74 and whether they were still alive twenty years later. The
$\chi^{2}$ test indicates a strongly significant association (with
$P=0.003$), and the numbers suggest that a smaller proportion of smokers
than of nonsmokers had died between the surveys. Should we thus conclude
that smoking helps to keep you alive? Probably not, given that it is
known beyond reasonable doubt that the causal relationship between
smoking and mortality is in the opposite direction. Clearly the picture
has been distorted by failure to control for some relevant further
variables. One such variable is the age of the respondents.

  Smoker      Dead   Alive   Total
  -------- ------- ------- -------
  Yes          139     443     582
             0.239   0.761   1.000
  No           230     502     732
             0.314   0.686   1.000
  Total        369     945    1314
             0.281   0.719   1.000

  : (\#tab:t-whickham1)Table of smoking status in 1972–74 vs. twenty-year survival among
  the respondents in Example 9.4. The numbers below the frequencies are
  proportions within rows.

Table \@ref(tab:t-whickham2) shows the partial tables of smoking vs. survival
controlling for age at the time of the first survey, classified into
seven categories. Note first that this three-way table appears somewhat
different from those shown in Tables \@ref(tab:t-berkeley2),
\@ref(tab:t-investors3) and \@ref(tab:t-titanic3). This is because one variable,
survival status, is summarised only by the percentage of survivors
within each combination of age group and smoking status. This is a
common trick to save space in three-way tables involving dichotomous
variables like survival here. The full table can easily be constructed
from these numbers if needed. For example, 98.4% of the nonsmokers aged
18–24 were alive at the time of the second survey. Since there were a
total of 62 respondents in this group (as shown in the last column),
this means that 61 of them (i.e. 98.4%) were alive and 1 (or 1.6%) was
not.

The percentages in Table \@ref(tab:t-whickham2) show that in five of the
seven age groups the proportion of survivors is higher among nonsmokers
than smokers, i.e. these partial associations in the sample are in the
opposite direction from the association in Table \@ref(tab:t-whickham1). This
reversal is known as **Simpson’s paradox**. The term is somewhat
misleading, as the finding is not really paradoxical in any logical
sense. Instead, it is again a consequence of a particular pattern of
associations between the control variable and the other two variables.

  ----------------------------------------------------------------------------------------------
  \                   \% Alive after 20 years:            \   Number (in 1972--74):            \    
  Age group                            Smokers   Nonsmokers                 Smokers   Nonsmokers
  ---------------- --------------------------- ------------ ----------------------- ------------
  18–24                                   96.4         98.4                      55           62

  25–34                                   97.6         96.8                     124          157

  35–44                                   87.2         94.2                     109          121

  45–54                                   79.2         84.6                     130           78

  55–64                                   55.7         66.9                     115          121

  65–74                                   19.4         21.7                      36          129

  75–                                      0.0          0.0                      12           64

  All age groups                          76.1         68.6                     582          732
  ----------------------------------------------------------------------------------------------

  : (\#tab:t-whickham2)Percentage of respondents in Example 9.4 surviving at the time of
  the second survey, by smoking status and age group in 1972–74.


The two-way tables of age by survival and age by smoking are shown side
by side in Table \@ref(tab:t-whickham3). The table is somewhat elaborate and
unconventional, so it requires some explanation. The rows of the table
correspond to the age groups, identified by the second column, and the
frequencies of respondents in each age group are given in the last
column. The left-hand column shows the percentages of survivors within
each age group. The right-hand side of the table gives the two-way table
of age group and smoking status. It contains percentages calculated both
within the rows (without parentheses) and columns (in parentheses) of
the table. As an example, consider numbers for the age group 18–24.
There were 117 respondents in this age group at the time of the first
survey. Of them, 47% were then smokers and 53% were nonsmokers, and 97%
were still alive at the time of the second survey. Furthermore, 10% of
all the 582 smokers, 9% of all the 732 nonsmokers and 9% of the 1314
respondents overall were in this age group.

  -------------------------------------------------------------------------------------
             \ \                Row and column \%           \            \            \    
       % Alive Age group             Smokers            Nonsmokers    Total \%    Count
  ------------ ------------- ----------------------- --------------- ---------- -------
            97 18–24                   47                   53           100        117

                                      (10)                  (9)          (9)    

            97 25–34                   44                   56           100        281

                                      (21)                 (21)          (21)   

            91 35–44                   47                   53           100        230

                                      (19)                 (17)          (18)   

            81 45–54                   63                   38           100        208

                                      (22)                 (11)          (16)   

            61 55–64                   49                   51           100        236

                                      (20)                 (17)          (13)   

            21 65–74                   22                   78           100        165

                                       (6)                 (18)          (13)   

             0 75–                     17                   83           100         77

                                       (2)                  (9)          (6)    

            72 Total %                 44                   56           100    

                                      (100)                (100)        (100)   

           945 Total count             582                  732                    1314
  -------------------------------------------------------------------------------------

  : (\#tab:t-whickham3)Two-way contingency tables of age group vs. survival (on the left)
  and age group vs. smoking (on the right) in Example 6.4. The
  percentages in parentheses are column percentages (within smokers or
  nonsmokers) and the ones without parentheses are row percentages
  (within age groups).

Table \@ref(tab:t-whickham3) shows a clear association between age and
survival, for understandable reasons mostly unconnected with smoking.
The youngest respondents of the first survey were highly likely and the
oldest unlikely to be alive twenty years later. There is also an
association between age and smoking: in particular, the proportion of
smokers was lowest among the oldest respondents. The implications of
this are seen perhaps more clearly by considering the column
proportions, i.e. the age distributions of smokers and nonsmokers in the
original sample. These show that the group of nonsmokers was
substantially older than that of the smokers; for example, 27% of the
nonsmokers but only 8% of the smokers belonged to the two oldest age
groups. It is this imbalance which explains why nonsmokers, more of whom
are old, appear to have lower chances of survival until we control for
age.

The discussion above refers to the *sample* associations between smoking
and survival in the partial tables, which suggest that mortality is
higher among smokers than nonsmokers. In terms of statistical
significance, however, the evidence is fairly weak: the association is
even borderline significant only in the 35–44 and 55–64 age groups, with
$P$-values of 0.063 and 0.075 respectively for the $\chi^{2}$ test. This
is an indication not so much of lack of a real association but of the
fact that these data do not provide much power for detecting it. Overall
twenty-year mortality is a fairly rough measure of health status for
comparisons of smokers and nonsmokers, especially in the youngest and
oldest age groups where mortality is either very low or very high for
everyone. Differences are likely to be have been further diluted by many
of the original smokers having stopped smoking between the surveys. This
example should thus not be regarded as a serious examination of the
health effects of smoking, for which much more specific data and more
careful analyses are required.^[For one remarkable example of such studies, see Doll, R. et
al. (2004), “Mortality in relation to smoking: 50 years’
observations on male British doctors”, *British Medical Journal*
**328**, 1519–1528, and Doll, R. and Hill, A. B. (1954), “The
Mortality of doctors in relation to their smoking habits: A
preliminary report”, *British Medical Journal* **228**, 1451–1455.
The older paper is reprinted together with the more recent one in
the 2004 issue of *BMJ*.]

The Berkeley admissions data discussed earlier provide another example
of a partial Simpson’s paradox. Previously we considered only
departments 2–6, for none of which there was a significant partial
association between sex and admission. For department 1, the partial
table indicates a strongly significant difference in favour of women,
with 82% of the 108 female applicants admitted, compared to 62% of the
825 male applicants. However, the two-way association between sex and
admission for departments 1–6 combined remains strongly significant and
shows an even larger difference in favour of men than before. This
result can now be readily explained as a result of imbalances in sex
ratios and rates of admission between departments. Department 1 is both
easy to get into (with 64% admitted) and heavily favoured by men (88% of
the applicants). These features combine to contribute to higher
admissions percentages for men overall, even though within department 1
itself women are more likely to be admitted.

In summary, the examples discussed above demonstrate that many things
can happen to an association between two variables when we control for a
third one. The association may disappear, indicating that it was
spurious, or it may remain similar and unchanged in all of the partial
tables. It may also become different in different partial tables,
indicating an interaction. Which of these occurs depends on the patterns
of associations between the control variable and the other two
variables. The crucial point is that the two-way table alone cannot
reveal which of these cases we are dealing with, because the counts in
the two-way table could split into three-way tables in many different
ways. The only way to determine how controlling for other variables will
affect an association is to actually do so. This is the case not only
for multiway contingency tables, but for all methods of statistical
control, in particular multiple linear regression and other regression
models.

Two final remarks round off our discussion of multiway contingency
tables:

-   Extension of the ideas of three-way tables to four-way and larger
    contingency tables is obvious and straightforward. In such tables,
    every cell corresponds to the subjects with a particular combination
    of the values of four or more variables. This involves no new
    conceptual difficulties, and the only challenge is how to arrange
    the table for convenient presentation. When the main interest is in
    associations between a particular pair of two variables, the usual
    solution is to present a set of partial two-way tables for them, one
    for each combination of the other (control) variables. Suppose, for
    instance, that in the university admissions example we had obtained
    similar data at two different years, say 1973 and 2003. We would
    then have four variables: year, department, sex and
    admission status. These could be summarised as in Table
    \@ref(tab:t-berkeley2), except that each partial table for sex
    vs. admission would now be conditional on the values of both year
    and department. The full four-way table would then consist of ten
    $2\times 2$ partial tables, one for each of the ten combinations of
    two years and five departments, (i.e. applicants to Department 2 in
    1973, Department 2 in 2003, and so on to Department 6 in 2003).

-   The only inferential tool for multiway contingency tables discussed
    here was to arrange the table as a set of two-way partial tables and
    to apply the $\chi^{2}$ test of independence to each of them. This
    is a perfectly sensible approach and a great improvement over just
    analysing two-way tables. There are, however, questions which cannot
    easily be answered with this method. For example, when can we say
    that associations in different partial tables are different enough
    for us to declare that there is evidence of an interaction? Or what
    if we want to consider many different partial associations, either
    for a response variable with each of the other variables in turn, or
    because there is no single response variable? More powerful methods
    are required for such analyses. They are multiple regression models
    like the multiple linear regression of Section
    \@ref(s-regression-multiple), but modified so that they become
    approriate for categorical response variables. Some of these models
    are introduced on the course MY452.