Transcript Summarizing the Relationship Between Two Variables with Tables

Summarizing the Relationship
Between Two Variables with
Tables
Chapter 6
Looking at Tables
• Tables are useful for examining the
relationship between:
– variables measured at the nominal or ordinal
level,
– or variables measured at the interval or ratio
level with a small number of discrete values.
Some Terminology and
Conventions
• Two-way tables, Crosstabulations
(Crosstabs)
• Column (explanatory or independent)
• Row (response or dependent)
• Although the book does not always do
this, there is a soft convention of stating a
table title as “Dependent by Independent”.
E.g. Table 2: Election Needed Now by Province.
• Cells: As in a spread sheet a table is divided
into cells aligned along rows and columns.
• Marginal Distributions: These are the numbers
that summarize the rows and the columns at the
side and bottom of a table
• Conditional Distributions: The book gives a very
complicated explanation for what this is. In
reality it is just the percentage of the cases in a
cell or cells. This can be the percentage of
cases along the horizontal row or the vertical
column.
Here is an example of a Two Way table made with the
“crosstab” procedure in SPSS. Question. Is there a
difference between the number of bathrooms that homes in
urban and rural Ontario have?
• Look at row 1.
Reading Across:
we see 94% of the homes
with 1 bathroom are
Urban 6% are Rural
• Look at Column 1.
Reading Down:
we see 52.2% of urban
homes have 1 bath-room,
36.8% have 2 bathrooms
11.0% have three or more
bathrooms.
The percentages give us a way to ‘eyeball’ the data and estimate if
there is a difference, but to ask if these difference are meaningful, we
must go further and calculate some statistics.
• The Chi Sq. Test is a common one to use in a table with nominal
variables such as this.
• It measures the difference between the number of cases we expect
to see in each cell and the number of cases we actually observe in
each cell of our table.
• The Chi Sq. value itself has little meaning for us. What matters is
whether or not the value is significant. In this case it is > .05.
• Therefore we reject the hypothesis that the results we see are
meaningfully different from what we could expect through simple
probability
• Therefore we also reject that there is any meaningful difference
between the number of bathrooms in urban and rural homes.
Simpson’s Paradox
That lurking variable thing again
• Example 6.4 in your book gives you a look
at a problem called Simpson’s Paradox.
• An association or comparison that holds
for all or several groups can reverse
direction when the data are combined to
form a single group (Moore pg. 169).
• As Moore further notes, this is usually the
sign of a “lurking” variable.
• In order to check for lurking variables we
can subdivide tables by a further
categorical variable (such as was done in
the book where the data was divided into
serious and less serious accidents).
Basic Cross Tab
CRIC/G&M New CanadaSurvey 2003
And now with control tables for
urban/rural