The confounding variable, gender, should be controlled for by studying boys and girls separately instead of ignored when combining. This results in the apparent relationship in the combined table. In this hypothetical example, boys tended to consume more sugar than girls, and also tended to be more hyperactive than girls. This phenomenon is known as Simpson’s Paradox, which describes the apparent change in a relationship in a two-way table when groups are combined. This difference appears large enough to suggest that a relationship does exist between sugar intake and activity level. However, when we consider the data when the two groups are combined, the hyperactivity rates do differ: 43% for Low Sugar and 59% for High Sugar. What we observe by these percentages is exactly what we would expect if no relationship existed between sugar intake and activity level. Notice how the rates for Boys (67%) and Girls (25%) are the same regardless of sugar intake. GirlsĬombined results of hyperactivity studys for boys and girls. Results of hyperactivity study for girls. The following tables list these hypothetical results: Results of hyperactivity study for boys. Hypothetically, suppose sugar and hyperactivity observational studies have been conducted first separately for boys and girls, and then the data is combined. This would be interpreted then as for those who say they do not smoke 57.42% are Females – meaning that for those who do not smoke 42.58% are Male (found by 100% – 57.42%). take for example 120 divided by 209 to get 57.42%. Alternatively, we could compute the conditional probabilities of Gender given Smoking by calculating the Row Percents i.e. The data under Cell Contents tells you what is being displayed in each cell: the top value is Count and the bottom value is Percent of Column. We can see from this display that the 94.49% conditional probability of No Smoking given the Gender is Female is found by the number of No and Female (count of 120) divided by then number of Females (count of 127). This should result in the following two-way table with column percents:Īlthough you do not need the counts, having those visible aids in the understanding of how the conditional probabilities of smoking behavior within gender are calculated.
We can calculate these marginal probabilities using either Minitab or SPSS: For example, the conditional percentage of No given Female is found by 120/127 = 94.5%.
These conditional percentages are calculated by taking the number of observations for each level smoke cigarettes (No, Yes) within each level of gender (Female, Male). a variable that we use to explain what is happening with another variable). This tells the conditional distribution of smoke cigarettes given gender, suggesting we are considering gender as an explanatory variable (i.e. Since there were more females (127) than males (99) who participated in the survey, we should report the percentages instead of counts in order to compare cigarette smoking behavior of females and males. The marginal distribution on the right (the values under the column All) is for Smoke Cigarettes only (disregarding Gender). The marginal distribution along the bottom (the bottom row All) gives the distribution by gender only (disregarding Smoke Cigarettes). This should result in the following two-way table:
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