Two-way tables organize data about two categorical variables (factors) obtained from a two-way design. (There are now two ways to group the data). - PowerPoint PPT Presentation

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Two-way tables organize data about two categorical variables (factors) obtained from a two-way design. (There are now two ways to group the data).

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Title: Two-way tables organize data about two categorical variables (factors) obtained from a two-way design. (There are now two ways to group the data).


1
Two-way tables
  • Two-way tables organize data about two
    categorical variables (factors) obtained from a
    two-way design. (There are now two ways to group
    the data).

2
Marginal distributions
  • We can look at each categorical variable
    separately in a two-way table by studying the row
    totals and the column totals. They represent the
    marginal distributions, expressed in counts or
    percentages (They are written as if in a margin.)

3
Relationships between categorical variables
  • The marginal distributions summarize each
    categorical variable independently. But the
    two-way table actually describes the relationship
    between both categorical variables.
  • The cells of a two-way table represent the
    intersection of a given level of one categorical
    factor with a given level of the other
    categorical factor.
  • Because counts can be misleading (for instance,
    one level of one factor might be much less
    represented than the other levels), we prefer to
    calculate percents or proportions for the
    corresponding cells. These make up the
    conditional distributions.

4
Conditional distributions
  • The counts or percents within the table represent
    the conditional distributions. Comparing the
    conditional distributions allows you to describe
    the relationship between both categorical
    variables.

29.30 11071 37785 cell total .
column total
5
  • The conditional distributions can be graphically
    compared using side by side bar graphs of one
    variable for each value of the other variable.

Here the percents are calculated by age range
(columns).
6
Music and wine purchase decision
What is the relationship between type of music
played in supermarkets and type of wine
purchased?
  • We want to compare the conditional distributions
    of the response variable (wine purchased) for
    each value of the explanatory variable (music
    played). Therefore, we calculate column percents.

We calculate the column conditional percents
similarly for each of the nine cells in the table
7
For every two-way table, there are two sets of
possible conditional distributions.
8
Simpsons paradox
  • An association or comparison that holds for all
    of several groups can reverse direction when the
    data are combined (aggregated) to form a single
    group. This reversal is called Simpsons paradox.

Example Hospital death rates
Here patient condition was the lurking variable.
9
  • TO REVIEW
  • Two-way tables consist of counts obtained by
    crosstabulating two categorical variables - the
    goal is to understand the relationship or
    association between these two variables.
  • The first method of looking for the relationship
    is to compute percentages - there are three
    types
  • those based on the grand total in the table (the
    joint distribution of the two variables)
  • those based on the column totals and those based
    on the row totals (the conditional distributions)
  • To look for association, consider all the
    percentages above but usually percent with
    respect to the explanatory variable's totals.

10
  • HOMEWORK READ SECTION 2.5 start 2.6
  • Go over examples 2.27-2.33, starting on p. 142.
  • Do the exercises 2.105-2.110
  • Use technology to compute the various
    distributions (joint and conditional) - in JMP,
    Analyze -gt Fit Y by X gives the 2-way tables
  • page 152ff 2.111-2.113, 2.119, 2.121
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