OK, now we’re ready to do some analyses. This vignette focuses on relatively simple non-parametric tests and measures of association.

For tabular displays, the `CrossTable()`

function in the
`gmodels`

package produces cross-tabulations modeled after
`PROC FREQ`

in SAS or `CROSSTABS`

in SPSS. It has
a wealth of options for the quantities that can be shown in each
cell.

Recall the GSS data used earlier.

```
# Agresti (2002), table 3.11, p. 106
GSS <- data.frame(
expand.grid(sex = c("female", "male"),
party = c("dem", "indep", "rep")),
count = c(279,165,73,47,225,191))
(GSStab <- xtabs(count ~ sex + party, data=GSS))
## party
## sex dem indep rep
## female 279 73 225
## male 165 47 191
```

Generate a cross-table showing cell frequency and the cell contribution to \(\chi^2\).

```
# 2-Way Cross Tabulation
library(gmodels)
CrossTable(GSStab, prop.t=FALSE, prop.r=FALSE, prop.c=FALSE)
##
##
## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## |-------------------------|
##
##
## Total Observations in Table: 980
##
##
## | party
## sex | dem | indep | rep | Row Total |
## -------------|-----------|-----------|-----------|-----------|
## female | 279 | 73 | 225 | 577 |
## | 1.183 | 0.078 | 1.622 | |
## -------------|-----------|-----------|-----------|-----------|
## male | 165 | 47 | 191 | 403 |
## | 1.693 | 0.112 | 2.322 | |
## -------------|-----------|-----------|-----------|-----------|
## Column Total | 444 | 120 | 416 | 980 |
## -------------|-----------|-----------|-----------|-----------|
##
##
```

There are options to report percentages (row, column, cell), specify
decimal places, produce Chi-square, Fisher, and McNemar tests of
independence, report expected and residual values (pearson,
standardized, adjusted standardized), include missing values as valid,
annotate with row and column titles, and format as SAS or SPSS style
output! See `help(CrossTable)`

for details.

For 2-way tables you can use `chisq.test()`

to test
independence of the row and column variable. By default, the \(p\)-value is calculated from the asymptotic
chi-squared distribution of the test statistic. Optionally, the \(p\)-value can be derived via Monte Carlo
simulation.

```
(HairEye <- margin.table(HairEyeColor, c(1, 2)))
## Eye
## Hair Brown Blue Hazel Green
## Black 68 20 15 5
## Brown 119 84 54 29
## Red 26 17 14 14
## Blond 7 94 10 16
chisq.test(HairEye)
##
## Pearson's Chi-squared test
##
## data: HairEye
## X-squared = 138.29, df = 9, p-value < 2.2e-16
chisq.test(HairEye, simulate.p.value = TRUE)
##
## Pearson's Chi-squared test with simulated p-value (based on 2000
## replicates)
##
## data: HairEye
## X-squared = 138.29, df = NA, p-value = 0.0004998
```

`fisher.test(X)`

provides an **exact test**
of independence. `X`

must be a two-way contingency table in
table form. Another form, `fisher.test(X, Y)`

takes two
categorical vectors of the same length.

For tables larger than \(2 \times 2\)
the method can be computationally intensive (or can fail) if the
frequencies are not small.

```
fisher.test(GSStab)
##
## Fisher's Exact Test for Count Data
##
## data: GSStab
## p-value = 0.03115
## alternative hypothesis: two.sided
```

Fisher’s test is meant for tables with small total sample size. It
generates an error for the `HairEye`

data with \(n\)=592 total frequency.

Use the `mantelhaen.test(X)`

function to perform a
Cochran-Mantel-Haenszel \(\chi^2\) chi
test of the null hypothesis that two nominal variables are
*conditionally independent*, \(A \perp
B \; | \; C\), in each stratum, assuming that there is no
three-way interaction. `X`

is a 3 dimensional contingency
table, where the last dimension refers to the strata.

The `UCBAdmissions`

serves as an example of a \(2 \times 2 \times 6\) table, with
`Dept`

as the stratifying variable.

```
# UC Berkeley Student Admissions
mantelhaen.test(UCBAdmissions)
##
## Mantel-Haenszel chi-squared test with continuity correction
##
## data: UCBAdmissions
## Mantel-Haenszel X-squared = 1.4269, df = 1, p-value = 0.2323
## alternative hypothesis: true common odds ratio is not equal to 1
## 95 percent confidence interval:
## 0.7719074 1.0603298
## sample estimates:
## common odds ratio
## 0.9046968
```

The results show no evidence for association between admission and
gender when adjusted for department. However, we can easily see that the
assumption of equal association across the strata (no 3-way association)
is probably violated. For \(2 \times 2 \times
k\) tables, this can be examined from the odds ratios for each
\(2 \times 2\) table
(`oddsratio()`

), and tested by using
`woolf_test()`

in `vcd`

.

```
oddsratio(UCBAdmissions, log=FALSE)
## odds ratios for Admit and Gender by Dept
##
## A B C D E F
## 0.3492120 0.8025007 1.1330596 0.9212838 1.2216312 0.8278727
lor <- oddsratio(UCBAdmissions) # capture log odds ratios
summary(lor)
##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## A -1.052076 0.262708 -4.0047 6.209e-05 ***
## B -0.220023 0.437593 -0.5028 0.6151
## C 0.124922 0.143942 0.8679 0.3855
## D -0.081987 0.150208 -0.5458 0.5852
## E 0.200187 0.200243 0.9997 0.3174
## F -0.188896 0.305164 -0.6190 0.5359
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
woolf_test(UCBAdmissions)
##
## Woolf-test on Homogeneity of Odds Ratios (no 3-Way assoc.)
##
## data: UCBAdmissions
## X-squared = 17.902, df = 5, p-value = 0.003072
```

We can visualize the odds ratios of Admission for each department
with fourfold displays using `fourfold()`

. The cell
frequencies \(n_{ij}\) of each \(2 \times 2\) table are shown as a quarter
circle whose radius is proportional to \(\sqrt{n_{ij}}\), so that its area is
proportional to the cell frequency.

```
UCB <- aperm(UCBAdmissions, c(2, 1, 3))
dimnames(UCB)[[2]] <- c("Yes", "No")
names(dimnames(UCB)) <- c("Sex", "Admit?", "Department")
```

Confidence rings for the odds ratio allow a visual test of the null
of no association; the rings for adjacent quadrants overlap *iff*
the observed counts are consistent with the null hypothesis. In the
extended version (the default), brighter colors are used where the odds
ratio is significantly different from 1. The following lines produce
(**ref?**)(fig:fourfold1).

```
col <- c("#99CCFF", "#6699CC", "#F9AFAF", "#6666A0", "#FF0000", "#000080")
fourfold(UCB, mfrow=c(2,3), color=col)
```

Another `vcd`

function, `cotabplot()`

, provides
a more general approach to visualizing conditional associations in
contingency tables, similar to trellis-like plots produced by
`coplot()`

and lattice graphics. The `panel`

argument supplies a function used to render each conditional subtable.
The following gives a display (not shown) similar to (**ref?**)(fig:fourfold1).

When we want to view the conditional probabilities of a response
variable (e.g., `Admit`

) in relation to several factors, an
alternative visualization is a `doubledecker()`

plot. This
plot is a specialized version of a mosaic plot, which highlights the
levels of a response variable (plotted vertically) in relation to the
factors (shown horizontally). The following call produces (**ref?**)(fig:doubledecker), where
we use indexing on the first factor (`Admit`

) to make
`Admitted`

the highlighted level.

In this plot, the association between `Admit`

and
`Gender`

is shown where the heights of the highlighted
conditional probabilities do not align. The excess of females admitted
in Dept A stands out here.

The standard \(\chi^2\) tests for association in a two-way table treat both table factors as nominal (unordered) categories. When one or both factors of a two-way table are quantitative or ordinal, more powerful tests of association may be obtained by taking ordinality into account, using row and or column scores to test for linear trends or differences in row or column means.

More general versions of the CMH tests (Landis etal., 1978) (Landis, Heyman, and Koch 1978) are provided by
assigning numeric scores to the row and/or column variables. For
example, with two ordinal factors (assumed to be equally spaced),
assigning integer scores, `1:R`

and `1:C`

tests
the linear \(\times\) linear component
of association. This is statistically equivalent to the Pearson
correlation between the integer-scored table variables, with \(\chi^2 = (n-1) r^2\), with only 1 \(df\) rather than \((R-1)\times(C-1)\) for the test of general
association.

When only one table variable is ordinal, these general CMH tests are analogous to an ANOVA, testing whether the row mean scores or column mean scores are equal, again consuming fewer \(df\) than the test of general association.

The `CMHtest()`

function in `vcdExtra`

calculates these various CMH tests for two possibly ordered factors,
optionally stratified other factor(s).

** Example**:

Recall the \(4 \times 4\) table,
`JobSat`

introduced in @ref(sec:creating),

```
JobSat
## satisfaction
## income VeryD LittleD ModerateS VeryS
## < 15k 1 3 10 6
## 15-25k 2 3 10 7
## 25-40k 1 6 14 12
## > 40k 0 1 9 11
```

Treating the `satisfaction`

levels as equally spaced, but
using midpoints of the `income`

categories as row scores
gives the following results:

```
CMHtest(JobSat, rscores=c(7.5,20,32.5,60))
## Cochran-Mantel-Haenszel Statistics for income by satisfaction
##
## AltHypothesis Chisq Df Prob
## cor Nonzero correlation 3.8075 1 0.051025
## rmeans Row mean scores differ 4.4774 3 0.214318
## cmeans Col mean scores differ 3.8404 3 0.279218
## general General association 5.9034 9 0.749549
```

Note that with the relatively small cell frequencies, the test for
general give no evidence for association. However, the the
`cor`

test for linear x linear association on 1 df is nearly
significant. The `coin`

package contains the functions
`cmh_test()`

and `lbl_test()`

for CMH tests of
general association and linear x linear association respectively.

There are a variety of statistical measures of *strength* of
association for contingency tables— similar in spirit to \(r\) or \(r^2\) for continuous variables. With a
large sample size, even a small degree of association can show a
significant \(\chi^2\), as in the
example below for the `GSS`

data.

The `assocstats()`

function in `vcd`

calculates
the \(\phi\) contingency coefficient,
and Cramer’s V for an \(r \times c\)
table. The input must be in table form, a two-way \(r \times c\) table. It won’t work with
`GSS`

in frequency form, but by now you should know how to
convert.

```
assocstats(GSStab)
## X^2 df P(> X^2)
## Likelihood Ratio 7.0026 2 0.030158
## Pearson 7.0095 2 0.030054
##
## Phi-Coefficient : NA
## Contingency Coeff.: 0.084
## Cramer's V : 0.085
```

For tables with ordinal variables, like `JobSat`

, some
people prefer the Goodman-Kruskal \(\gamma\) statistic (Agresti 2002, 2.4.3) based on a comparison of
concordant and discordant pairs of observations in the case-form
equivalent of a two-way table.

A web article by Richard Darlington, [http://node101.psych.cornell.edu/Darlington/crosstab/TABLE0.HTM] gives further description of these and other measures of association.

The `Kappa()`

function in the `vcd`

package
calculates Cohen’s \(\kappa\) and
weighted \(\kappa\) for a square
two-way table with the same row and column categories (Cohen 1960). Normal-theory \(z\)-tests are obtained by dividing \(\kappa\) by its asymptotic standard error
(ASE). A `confint()`

method for `Kappa`

objects
provides confidence intervals.

```
data(SexualFun, package = "vcd")
(K <- Kappa(SexualFun))
## value ASE z Pr(>|z|)
## Unweighted 0.1293 0.06860 1.885 0.059387
## Weighted 0.2374 0.07832 3.031 0.002437
confint(K)
##
## Kappa lwr upr
## Unweighted -0.005120399 0.2637809
## Weighted 0.083883432 0.3908778
```

A visualization of agreement (Bangdiwala
1987), both unweighted and weighted for degree of departure from
exact agreement is provided by the `agreementplot()`

function. (**fig?**)(fig:agreesex) shows the
agreementplot for the `SexualFun`

data, produced as shown
below.

The Bangdiwala measures (returned by the function) represent the proportion of the shaded areas of the diagonal rectangles, using weights \(w_1\) for exact agreement, and \(w_2\) for partial agreement one step from the main diagonal.

```
unlist(agree)
## Bangdiwala Bangdiwala_Weighted weights1 weights2
## 0.1464624 0.4981723 1.0000000 0.8888889
```

In other examples, the agreement plot can help to show
*sources* of disagreement. For example, when the shaded boxes are
above or below the diagonal (red) line, a lack of exact agreement can be
attributed in part to different frequency of use of categories by the
two raters– lack of *marginal homogeneity*.

Correspondence analysis is a technique for visually exploring
relationships between rows and columns in contingency tables. The
`ca`

package gives one implementation. For an \(r \times c\) table, the method provides a
breakdown of the Pearson \(\chi^2\) for
association in up to \(M = \min(r-1,
c-1)\) dimensions, and finds scores for the row (\(x_{im}\)) and column (\(y_{jm}\)) categories such that the
observations have the maximum possible correlations.% ^{1}

Here, we carry out a simple correspondence analysis of the
`HairEye`

data. The printed results show that nearly 99% of
the association between hair color and eye color can be accounted for in
2 dimensions, of which the first dimension accounts for 90%.

```
library(ca)
ca(HairEye)
##
## Principal inertias (eigenvalues):
## 1 2 3
## Value 0.208773 0.022227 0.002598
## Percentage 89.37% 9.52% 1.11%
##
##
## Rows:
## Black Brown Red Blond
## Mass 0.182432 0.483108 0.119932 0.214527
## ChiDist 0.551192 0.159461 0.354770 0.838397
## Inertia 0.055425 0.012284 0.015095 0.150793
## Dim. 1 -1.104277 -0.324463 -0.283473 1.828229
## Dim. 2 1.440917 -0.219111 -2.144015 0.466706
##
##
## Columns:
## Brown Blue Hazel Green
## Mass 0.371622 0.363176 0.157095 0.108108
## ChiDist 0.500487 0.553684 0.288654 0.385727
## Inertia 0.093086 0.111337 0.013089 0.016085
## Dim. 1 -1.077128 1.198061 -0.465286 0.354011
## Dim. 2 0.592420 0.556419 -1.122783 -2.274122
```

The resulting `ca`

object can be plotted just by running
the `plot()`

method on the `ca`

object, giving the
result in @ref(fig:ca-haireye). `plot.ca()`

does not allow
labels for dimensions; these can be added with `title()`

. It
can be seen that most of the association is accounted for by the
ordering of both hair color and eye color along Dimension 1, a dark to
light dimension.

Agresti, Alan. 2002. *Categorical Data Analysis*. 2nd ed.
Hoboken, New Jersey: John Wiley & Sons.

Bangdiwala, Shrikant I. 1987. “Using SAS Software
Graphical Procedures for the Observer Agreement Chart.”
*Proceedings of the SAS User’s Group International
Conference* 12: 1083–88.

Cohen, J. 1960. “A Coefficient of Agreement for Nominal
Scales.” *Educational and Psychological Measurement* 20:
37–46.

Landis, R. J., E. R. Heyman, and G. G. Koch. 1978. “Average
Partial Association in Three-Way Contingency Tables: A Review and
Discussion of Alternative Tests,” *International Statistical
Review* 46: 237–54.

Related methods are the non-parametric CMH tests using assumed row/column scores (@ref(sec:CMH), the analogous

`glm()`

model-based methods (@ref(sec:CMH), and the more general RC models which can be fit using`gnm()`

. Correspondence analysis differs in that it is a primarily descriptive/exploratory method (no significance tests), but is directly tied to informative graphic displays of the row/column categories.↩︎