Chapter 33 Three-Way Tables: A 2x2xK Table and a Mantel-Haenszel Analysis

The material I discuss in this section is attributable to Jeff Simonoff and his book Analyzing Categorical Data. The example is taken from Section 8.1 of that book.

A three-dimensional or three-way table of counts often reflects a situation where the rows and columns refer to variables whose association is of primary interest to us, and the third factor (a layer, or strata) describes a control variable, whose effect on our primary association is something we are controlling for in the analysis.

33.1 Smoking and Mortality in the UK

In the early 1970s and then again 20 years later, in Whickham, United Kingdom, surveys yielded the following relationship between whether a person was a smoker at the time of the original survey and whether they were still alive 20 years later.

whickham1 <- matrix(c(443, 139, 502, 230), byrow=TRUE, nrow=2)
rownames(whickham1) <- c("Smoker", "Non-Smoker")
colnames(whickham1) <- c("Alive", "Dead")
pander(addmargins(whickham1))

	Alive	Dead	Sum
Smoker	443	139	582
Non-Smoker	502	230	732
Sum	945	369	1314

Here’s the two-by-two table analysis.

twoby2(whickham1)

2 by 2 table analysis: 
------------------------------------------------------ 
Outcome   : Alive 
Comparing : Smoker vs. Non-Smoker 

           Alive Dead    P(Alive) 95% conf. interval
Smoker       443  139       0.761     0.725    0.794
Non-Smoker   502  230       0.686     0.651    0.718

                                   95% conf. interval
             Relative Risk: 1.1099    1.0381    1.187
         Sample Odds Ratio: 1.4602    1.1414    1.868
Conditional MLE Odds Ratio: 1.4598    1.1335    1.884
    Probability difference: 0.0754    0.0266    0.123

             Exact P-value: 0.003 
        Asymptotic P-value: 0.0026 
------------------------------------------------------

chisq.test(whickham1)


    Pearson's Chi-squared test with Yates' continuity correction

data:  whickham1
X-squared = 9, df = 1, p-value = 0.003

There is a significant association between smoking and mortality (\(\chi^2\) = 8.75 on 1 df, p = 0.003), but it isn’t the one you might expect.

The odds ratio is 1.46, implying that the odds of having lived were 46% higher for smokers than for non-smokers.
Does that mean that smoking is good for you?

Not likely. There is a key “lurking” variable here - a variable that is related to both smoking and mortality that is obscuring the actual relationship - namely, age.

33.2 The `whickham` data including age, as well as smoking and mortality

The table below gives the mortality experience separated into subtables by initial age group.

age <- c(rep("18-24", 4), rep("25-34", 4), 
         rep("35-44", 4), rep("45-54", 4), 
         rep("55-64", 4), rep("65-74", 4), 
         rep("75+", 4))
smoking <- c(rep(c("Smoker", "Smoker", "Non-Smoker", "Non-Smoker"), 7))
status <- c(rep(c("Alive", "Dead"), 14))
counts <- c(53, 2, 61, 1, 121, 3, 152, 5,
            95, 14, 114, 7, 103, 27, 66, 12,
            64, 51, 81, 40, 7, 29, 28, 101,
            0, 13, 0, 64)
whickham2 <- data.frame(smoking, status, age, counts) %>% tbl_df()
whickham2$smoking <- factor(whickham2$smoking, levels = c("Smoker", "Non-Smoker"))
whickham2.tab1 <- xtabs(counts ~ smoking + status + age, data = whickham2)
whickham2.tab1

, , age = 18-24

            status
smoking      Alive Dead
  Smoker        53    2
  Non-Smoker    61    1

, , age = 25-34

            status
smoking      Alive Dead
  Smoker       121    3
  Non-Smoker   152    5

, , age = 35-44

            status
smoking      Alive Dead
  Smoker        95   14
  Non-Smoker   114    7

, , age = 45-54

            status
smoking      Alive Dead
  Smoker       103   27
  Non-Smoker    66   12

, , age = 55-64

            status
smoking      Alive Dead
  Smoker        64   51
  Non-Smoker    81   40

, , age = 65-74

            status
smoking      Alive Dead
  Smoker         7   29
  Non-Smoker    28  101

, , age = 75+

            status
smoking      Alive Dead
  Smoker         0   13
  Non-Smoker     0   64

The odds ratios for each of these subtables, except the last one, where it is undefined are as follows:

Age Group	Odds Ratio
18-24	0.43
25-34	1.33
35-44	0.42
45-54	0.69
55-64	0.62
65-74	0.87
75+	undefined

Thus, for all age groups except 25-34 year olds, smoking is associated with higher mortality.

Why? Not surprisingly, there is a strong association between age and mortality, with mortality rates being very low for young people (2.5% for 18-24 year olds) and increasing to 100% for 75+ year olds.

There is also an association between age and smoking, with smoking rates peaking in the 45-54 year old range and then falling off rapidly. In particular, respondents who were 65 and older at the time of the first survey had very low smoking rates (25.4%) but very high mortality rates (85.5%). Smoking was hardly the cause, however, since even among the 65-74 year olds mortality was higher among smokers (80.6%) than it was among non-smokers (78.3%). A flat version of the table (ftable in R) can help us with these calculations.

ftable(whickham2.tab1)

                  age 18-24 25-34 35-44 45-54 55-64 65-74 75+
smoking    status                                            
Smoker     Alive         53   121    95   103    64     7   0
           Dead           2     3    14    27    51    29  13
Non-Smoker Alive         61   152   114    66    81    28   0
           Dead           1     5     7    12    40   101  64

33.2.1 The Cochran-Mantel-Haenszel Test

So, the marginal table looking at smoking and mortality combining all age groups isn’t the most meaningful summary of the relationship between smoking and mortality. Instead, we need to look at the conditional association of smoking and mortality, given age, to address our interests.

The null hypothesis would be that, in the population, smoking and mortality are independent within strata formed by age group. In other words, H₀ requires that smoking be of no value in predicting mortality once age has been accounted for.

The alternative hypothesis would be that, in the population, smoking and mortality are associated within the strata formed by age group. In other words, H_A requires that smoking be of at least some value in predicting mortality even after age has been accounted for.

We can consider the evidence that helps us choose between these two hypotheses with a Cochran-Mantel-Haenszel test, which is obtained in R through the mantelhaen.test function. This test requires us to assume that, in the population and within each age group, the smoking-mortality odds ratio is the same. Essentially, this means that the association of smoking with mortality is the same for older and younger people.

mantelhaen.test(whickham2.tab1, conf.level  = 0.90)


    Mantel-Haenszel chi-squared test with continuity correction

data:  whickham2.tab1
Mantel-Haenszel X-squared = 5, df = 1, p-value = 0.02
alternative hypothesis: true common odds ratio is not equal to 1
90 percent confidence interval:
 0.490 0.875
sample estimates:
common odds ratio 
            0.655

The Cochran-Mantel-Haenszel test statistic is 5.44 (after a continuity correction) leading to a p value of 0.02, indicating strong rejection of the null hypothesis of conditional independence of smoking and survival given age.
The estimated common conditional odds ratio is 0.65. This implies that (given age) being a smoker is associated with a 35% lower odds of being alive 20 years later than being a non-smoker.
A 90% confidence interval for that common odds ratio is (0.49, 0.87), reinforcing rejection of the conditional independence (where the odds ratio would be 1).

33.2.2 Checking Assumptions: The Woolf test

We can also obtain a test (using the woolf_test function, in the vcd library) to see if the common odds ratio estimated in the Mantel-Haenszel procedure is reasonable for all age groups. In other words, the Woolf test is a test of the assumption of homogeneous odds ratios across the six age groups.

If the Woolf test is significant, it suggests that the Cochran-Mantel-Haenszel test is not appropriate, since the odds ratios for smoking and mortality vary too much in the sub-tables by age group. Here, we have the following log odds ratios (estimated using conditional maximum likelihood, rather than cross-product ratios) and the associated Woolf test.

## Next two results use the vcd library

vcd::oddsratio(whickham2.tab1, log = TRUE)

log odds ratios for smoking and status by age 

  18-24   25-34   35-44   45-54   55-64   65-74     75+ 
-0.6502  0.2247 -0.8407 -0.3461 -0.4742 -0.0993  1.5640

vcd::woolf_test(whickham2.tab1)


    Woolf-test on Homogeneity of Odds Ratios (no 3-Way assoc.)

data:  whickham2.tab1
X-squared = 3, df = 6, p-value = 0.8

As you can see, the Woolf test is not close to statistically significant, implying the common odds ratio is at least potentially reasonable for all age groups (or at least the ones under ages 75, where some data are available.)

33.2.3 Without the Continuity Correction

By default, R presents the Mantel-Haenszel test with a continuity correction, when used for a 2x2xK table. In virtually all cases, go ahead and do this, but as you can see below, the difference it makes in this case is modest.

mantelhaen.test(whickham2.tab1, correct=FALSE, conf.level = 0.90)


    Mantel-Haenszel chi-squared test without continuity correction

data:  whickham2.tab1
Mantel-Haenszel X-squared = 6, df = 1, p-value = 0.02
alternative hypothesis: true common odds ratio is not equal to 1
90 percent confidence interval:
 0.490 0.875
sample estimates:
common odds ratio 
            0.655