6. Contingency tables – association of two (or more) categorical variables
Contingency tables – introduction
Contingency tables are tables that summarize frequencies (counts) of two (or more) categorical
variables. Their analysis allows to test (in)dependence between the two variables. Table 6.1 is a
contingency table summarizing frequencies of people of different eye and hair colors.
Table 6.1. Contingency table of two variables: eye and hair color with basic frequency statistics
(marginal sums and grand total).
Hair color
black brown blonde
marginal
sums
Eye color
blue 12 45 14 71
brown 51 256 84 391
marginal
sums 63 301 98
grand
total: 462
Basic analysis by goodness-of-fit test
Association between the variables (i.e. the null hypothesis that the variables are independent)
can be tested by a goodness-of-fit test. This is a universal approach suitable for tables of any
size and dimensions but its explanatory power is limited.
For goodness-of-fit test, we need expected frequencies which are calculated on the basis of
probability theory: P(event 1 and event 2) = P (event 1) x P (event 2), if the two events are
independent. In contingency tables, this can be used to calculate expected frequencies as the
product of ratios of corresponding marginal totals and the grand total.
For instance, expected probability of observing a blue-eyed and black-haired person in Table 6.1
can be calculated as P(blueE and blackH) = 63/462 x 71/462 = 0.02096. Multiplication of the
probability then gives the expected frequency Freq(e) = 0.02096 x 462 = 9.68.
The same approach can be used to calculate expected frequencies in all cells but is done
automatically by software nowadays. Goodness-of-fit test can consequently be computed (in
the same way as described in chapter 5). Note, however, that number of degrees of freedom is
determined as DF = (number of rows – 1) x (number of columns – 1)
In our example: We did not find a significant association between eye and hair color (χ2 =
0.785, DF = 2, p = 0.6755).
The goodness-of-fit test does not provide much more information on the result, though in case
of significant result, it may make sense to report also the difference between observed-
expected frequencies (i.e. the residuals), or their standardized values (residuals divided by
square root of corresponding expected frequencies) as supplementary information. In
particular, standardized residuals are useful as they indicate excess or deficiency of which
combinations cause association between the variables.
2x2 tables and their analysis
These tables represent a special and the simplest cases of contingency tables (Table 6.2).
Table 6.2. Structure of a 2x2 table.
Var2
level 1 level 2
Var 1 level 1 f11 f12 R1
level 2 f21 f22 R2
C1 C2 n
Their simplicity allows additional statistics to be computed to express how tight the association
between the two variables is. Most important of these is the phi-coefficient:
𝜑 =
𝑓11𝑓22 − 𝑓12𝑓21
√𝑅1𝑅2𝐶1𝐶2
= ±√
𝜒2
𝑛
where f, R C symbols correspond to cells in Table 6.2 and χ2 is the χ2 statistics of the table and n
is the grand total.
The phi-coefficient can thus be viewed as an average contribution of each observation to the
association between the variables. This implies its important advantage which lies in
comparability of the phi coefficients between datasets with unequal numbers of observations.
The 2x2 tables may seem trivial and not of much use. However, they and especially the phicoefficient
is frequently used in vegetation ecology to measure association between
occurrences of two species or as a fidelity measure of a species with a vegetation unit. In that
case Var1 describes frequency of given species and Var2 frequency of the vegetation unit in the
dataset.
Advanced analysis of contingency tables – odds and odds ratios
Odds and odds ratios are additional important statistics that can be used to analyze contingency
tables. They are defined for 2x2 tables only but can also be used in larger (in particular n x 2)
tables, which can be subdivided into a series of 2x2 tables. For table 6.1, we can calculate the
odds for the level 1 of Var1 as:
odds1 = p/(1-p) = (f11/R1)/(f12/R1)
where p is the probability of one outcome of the second variable and 1-p is probability of the
second outcome of the second variable. We can do the same for the second level of Var1 to get
odds2. Odds ratio then equals:
OR = odds1/odds2
Odds ratio directly indicates how probability of observing level 1 of Var1 changes with respect
to the levels of Var2.
OR values range between 0 and infinity, with OR < 1 indicating negative association, OR = 1
independence and OR > 1 positive association.
OR is a population parameter and the computation summarized above is actually its maximumlikelihood
estimation procedure. As a result, OR estimate has associated standard error and
confidence intervals (i.e. intervals within which the population OR lies with 95% probability). A
confidence interval directly indicates significance – if a confidence interval of OR contains 1, the
OR is not significantly different from 1 and thus independence between the two variables
cannot be rejected.
A worked example
Malaria is a dangerous disease widespread in tropical areas. It is caused by protozoans of the
genus Plasmodium and transmitted by mosquitos. To prevent infection, it is possible to take
prophylaxis, i.e. treatment which blocks the infection after mosquito bite. This is only possible
for short time journeys to areas with malaria since the prophylaxis drugs are not safe for longterm
use. Here we asked whether the prophylaxis is efficient and whether there is significant
difference between two types of prophylaxis. The data are summarized in Table 6.3.
Table 6.3. Table summarizing frequencies of travelers to the tropics infected by malaria (or not)
and anti-malaria prophylaxis they used.
Prophylaxis Infected by malaria Frequency
none (control) 0 40
none (control) 1 94
doxycycline 0 130
doxycycline 1 80
lariam 0 180
lariam 1 15
Note here, that contingency table can also have a form of table with individual factor
combinations and corresponding frequencies. This is actually a bit better for computation than
the cross-tabulated form.
Goodness of fit test demonstrates, that there is a significant association between the two
variables:
Chisq = 137.45, df = 2, p-value = 1.42e-30
Odds ratios summary then folow. Two odds ratios are produced comparing the second and third
level of to the first one (here control). lower and upper values indicate limits of confidence
intervals. We can see that both types of prophylaxis are associated with significantly decreased
infection rate.
infected
prophylax 0 p0 1 p1 oddsratio lower upper p.value
control 40 0.1142857 94 0.49735450 1.00000000 NA NA NA
doxy 130 0.3714286 80 0.42328042 0.26186579 0.16479825 0.41610692 6.790312e-09
lariam 180 0.5142857 15 0.07936508 0.03546099 0.01862937 0.06749997 8.847446e-34
To compare just the two prophylaxis types, we can select just the corresponding part of the
data for analysis (specifying this by square brackets in R). The result shows that taking Lariam is
associated with significantly lower infection rate than taking doxycycline.
infected
prophylax 0 p0 1 p1 oddsratio lower upper p.value
doxy 130 0.4193548 80 0.8421053 1.0000000 NA NA NA
lariam 180 0.5806452 15 0.1578947 0.1354167 0.07462922 0.2457171 1.531487e-13
In a paper/thesis, the result can by summarized as Table 6.4
Table 6.4. Summary of a contingency table analysis testing the association between malaria
prophylaxis and infection. Overall test of independence χ2 = 137.45, df = 2, p < 10-6
.
Odds ratio lower 95% conf. limit upper 95% conf. limit p
Lariam vs. none 0.035 0.019 0.067 < 10
-6
doxycycline vs. none 0.262 0.165 0.416 < 10
-6
Lariam vs. doxycycline 0.135 0.075 0.246 < 10
-6
Coincidence and causality
Note here, that significant results of contingency table analysis indicate significant association.
This can be caused either by coincidence or causality. Causality means that if we manipulate one
variable, the other also changes, i.e. one variable has a direct effect on the other. By contrast
coincidence may happen due to another variable affecting the two analyzed. Manipulation of
one variable has no effect on the other in case of coincidence.
Considering the malaria example, the travelers using prophylaxis are simultaneously more likely
to use mosquito repellents, which in reality can strongly decrease infection risk. Therefore, if
somebody from the no-prophylaxis travelers decided to take prophylaxis, it may have much
lower (or even no) effect that our analysis suggests.
People in general like causal explanations (and expect them). As a result, association is
frequently interpreted as causal relationship, which is however inappropriate. Association can
only suggest causality at maximum.
The only possible way to demonstrate causality is to use a manipulative experiment. In our
case, this would mean to select a group of people, assign them randomly into three groups
according to prophylaxis, send them to the tropics and see what happens. This type of research
would not be approved by any ethical committee however.
How to do in R
1. Chisq analysis of contingency tables
Option 1: apply chisq.test on matrix containing frequencies
Option 2: If the data are formatted in data frame as in Table
6.3, they can be converted to contingency table by function xtabs
data.table<-xtabs(freq~var1+var2, data=data.frame)
chisq.test can then be applied on the contingency table. If its
result is saved in an object:
test.res<-chisq.test(data.table)
running test.res$std.resid can then be used to display
standardized residuals.
2. Phi – coefficient
function phi (package psych) applied on a 2x2 matrix
3. Odds ratios
function epitab (package epitools) applied on contingency table
produced by xtabs. Square brackets can be used to select the
levels to compare.