CHAPTER 6: Two-Way Tables* Basic Practice of Statistics - 3rd Edition CHAPTER 6: Two-Way Tables* *This material is important in statistics, but it is needed later in this book only for Chapter 24. You may omit it if you do not plan to read Chapter 24 or delay reading it until you reach Chapter 24. Basic Practice of Statistics 7th Edition Lecture PowerPoint Slides Chapter 5

In Chapter 6, We Cover … Marginal distributions Conditional distributions Simpson’s paradox

Categorical Variables Review: Categorical variables place individuals into one of several groups or categories. The values of a categorical variable are labels for the different categories. The distribution of a categorical variable lists the count or percent of individuals who fall into each category. When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. Two-way table – Describes two categorical variables, organizing counts according to a row variable and a column variable.

Two-Way Table What are the variables described by this two-way table? Job outside home Stay home No preference Total Women, no college 81 104 10 195 Women, with college 173 115 15 303 Men, no college 92 32 2 126 Men, with college 299 8 388 1012 What are the variables described by this two-way table? How many young adults were surveyed?

Marginal Distribution The marginal distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table. Note: Percents are often more informative than counts, especially when comparing groups of different sizes. To examine a marginal distribution: Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. Make a graph to display the marginal distribution.

Marginal Distribution Job outside home Stay home No preference Total Women, no college 81 104 10 195 Women, with college 173 115 15 303 Men, no college 92 32 2 126 Men, with college 299 8 388 1012 Examine the marginal distribution of gender/ education. Response Percent Women, no college 195/1012= 19.3% Women, with college 303/1012 = 29.9% Men, no college 126/1012 = 12.5% Men, with college 388/1012 = 38.3%

Conditional Distribution Marginal distributions tell us nothing about the relationship between two variables. A conditional distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. To examine or compare conditional distributions: Select the row(s) or column(s) of interest. Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). Make a graph to display the conditional distribution. Use a side-by-side bar graph or segmented bar graph to compare distributions.

Conditional Distribution Job outside home Stay home No preference Total Women, no college 81 104 10 195 Women, with college 173 115 15 303 Men, no college 92 32 2 126 Men, with college 299 8 388 1012 Calculate the conditional distribution of job preference for women and men with no college. Response Women no college Job outside home 81/195 = 41.5% Stay home 104/195 = 53.3% No preference 10/195 = 5.1% Men no college 92/126 = 73.0% 32/126 = 25.4% 2/126 = 1.6%

Simpson’s Paradox When studying the relationship between two variables, there may exist a lurking variable that creates a reversal in the direction of the relationship when the lurking variable is ignored, as opposed to the direction of the relationship when the lurking variable is considered. The lurking variable creates subgroups, and failure to take these subgroups into consideration can lead to misleading conclusions regarding the association between the two variables. An association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group. This reversal is called Simpson’s paradox.

Simpson’s Paradox Consider the survival rates for the following groups of victims who were taken to the hospital, either by helicopter, or by road: A higher percentage of those transported by helicopter died. Does this mean that this (more costly) mode of transportation isn’t helping? Counts Helicopter Road Victim died 64 260 Victim Survived 136 840 Total 200 1100 Percents Died Survived Helicopter 32% 68% Road 24% 76%

Simpson’s Paradox Consider the survival rates when broken down by type of accident. Serious accidents Counts Helicopter Road Died 48 60 Survived 52 40 Total 100 Percents Died Survived Helicopter 48% 52% Road 60% 40% Less-serious accidents Percents Died Survived Helicopter 16% 84% Road 20% 80% Counts Helicopter Road Died 16 200 Survived 84 800 Total 100 1000

Simpson’s Paradox Lurking variable: Accidents were of two sorts—serious (200) and less serious (1100). Helicopter evacuations had a higher survival rate within both types of accidents than did road evacuations. This is not evidence of the inefficacy of helicopter evacuation! This is an example of Simpson’s paradox. When the lurking variable (type of accident: serious or less serious) is ignored, the data seem to suggest road evacuations are safer than helicopter. However, when the type of accident is considered, the association is reversed and suggests helicopter evacuations are, in fact, safer.