Statistics Minus the Maths

Now that we know there is a statistically significant relationship between diet and health outcome (since we rejected the null hypothesis of no relationship), we will naturally wonder what kind of relationship exists. Specifically, which diet is associated with better health? To answer this, we must look to the specific values within the cells and describe the patterns we observe. One straightforward way to do this is to make note of where the observed frequency differs notably from the expected frequency.

From Table 9.3.2, we can see that for those on the AHA diet, the frequencies for cancers, fatal heart disease, and non-fatal heart disease are all higher than expected. At the same time, the frequency for “healthy” is lower than expected on the AHA diet. Results for the Mediterranean diet are the exact opposite: cancers and both types of heart disease occur less frequently than expected, and the healthy outcome occurs more than expected. Thus, the Mediterranean diet is unambiguously associated with better outcomes than the AHA diet.

Though it is easy to make sense of results in this manner based on a comparison of observe to expected frequencies, the typical contingency table you encounter will probably not display expected frequencies. Instead, it is common to include percentages by row or by column, which can also be used to draw conclusions about the nature of a relationship between variables. For example, in Table 9.3.3, the expected frequencies from the prior table have been replaced by percentages. The percentages shown in this table are calculated by row, meaning that they are found by dividing the frequency in each cell by the total for that row, shown in the final column.

Drawing conclusions based on column or row percentages takes some care, and it is easy to make mistakes if you are not careful. For example, consider the AHA/Cancers cell, which shows 4.95%. Because this seems like a small percentage, it is tempting to conclude that being on the AHA diet is not associated with having cancer. However, that conclusion would be wrong. Since the percentages have been calculated by row, the key to finding meaningful patterns will be to compare across the percentages found within the same column. In this case, we see that the 4.95% in the AHA-Cancers cell is over twice as large as the 2.32% shown in the Mediterranean/Cancers cell right below it. To make sense of these percentages, remember that we have calculated percentages by row, meaning that the frequency in the cell is divided by the total at the far right of the table. So the 4.95% means that among all those on the AHA diet (303 individuals in total), 4.95% have cancer. By comparison, only around 2% (2.32% to be precise) of those on a Mediterranean diet have cancer. Both percentages are small, because cancer is (fortunately) a relatively rare outcome; at the bottom of the table we see that only 3.64% of participants overall have cancer. Still, since the percentage is higher among those on the AHA diet than on the Mediterranean diet, we can say that the AHA diet is positively associated with having cancer.

If we continue moving down the table column by column, we see that rates of heart disease are higher among those on the AHA diet (7.92% for fatal and 8.25% for non-fatal) than among those on the Mediterranean diet (4.64% and 2.65%). Finally, around 90% of those on the Mediterranean diet are healthy, while just under 79% of those on the AHA diet are.

We can reach an equivalent conclusion by examining percentages that are calculated by column, as shown in Table 9.3.4. Here, percentages are calculated by dividing the frequency in each cell by the total shown at the bottom of the column.

The AHA/Cancers cell shows 68.18%, indicating that around 68% of those who have cancer are on the AHA diet. In the case of this particular dataset, the split of participants between the two diets is approximately 50-50 (as shown by the percentages in the final column), so finding that the percentage of those with cancer who are on the AHA diet is well above 50% actually does indicate that individuals on the AHA diet are overrepresented among those with cancer. But remember, you cannot generally assume that 50% is a reasonable baseline for comparison; if, for example, 80% of study participants were on the AHA diet, seeing any percentage smaller than 80% for the AHA-Cancers would indicate that those on the AHA diet were underrepresented among those with cancer.

As a general rule, when percentages are calculated by columns, the key to finding meaningful patterns is making comparisons across the columns (within the same row), such as noticing that the 68.18% in the AHA-Cancers cell is greater than the 50.08% in the AHA Total cell. One can also see that while people on the AHA diet make up a clear majority of those with heart disease (both fatal and non-fatal), among those who are healthy only around 47% are on the AHA diet. As a clear contrast, those on the Mediterranean diet make up around 53% of the healthy subjects but just 32% of those with cancer, 37% of those with fatal heart disease, and 24% of those with non-fatal heart disease. Thus, we see consistent indication that outcomes are better for subjects on the Mediterranean diet, as opposed to the AHA one.

As you can now see, regardless of whether we examine (1) expected versus observed outcomes, (2) percentages by row, or (3) percentages by column, we reach the same conclusion: health outcomes are uniformly better for those on the Mediterranean diet. All three approaches are equally valid ways of evaluating associations from a contingency table, and you normally need use only one. We examined all three here for illustrative purposes since each approach is one you may encounter in your own analysis or in a research report.

A key assumption of this Chi Square test is that each subject contributes data to only one cell. Therefore, the sum of all cell frequencies in the table must be the same as the number of subjects in the experiment. Consider an experiment in which each of 16 subjects attempted two anagram problems. The data are shown in Table 9.3.5.

It would not be valid to use the Chi Square test on these data since each subject contributed data to two cells: one cell based on their performance on Anagram 1 and one cell based on their performance on Anagram 2. The total of the cell frequencies in the table is 32, but the total number of subjects is only 16.