Crosstabs

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Crosstabs
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When to Use Crosstabs as a Bivariate Data
Analysis Technique

• For examining the relationship of two CATEGORIC
  variables
• For example, do men and women (variable is
  GENDER) have different distributions for the
  variable SUPER BOWL WATCHING often, sometimes,
  never?
• Ask class for more examples!
• Both variables must be categoric nominal,
  ordinal, or dichotomous.

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What is a Crosstab?

• A cross-tab is a table that displays a joint
  distribution a distribution of cases for two
  variables.
• It is also called a contingency table.
• Each cell displays the observed count of the
  cases that fall into a specific category of the
  IV AND a specific category of the DV.

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How to Set Up a Crosstab

• Decide which variable is the IV and which is the
  DV. (If you cant decide, make two tables, one
  for each option.)
• Place the IV in the columns. Each COLUMN
  represents a CATEGORY OF THE IV.
• The DV goes in the rows. Each ROW represents a
  CATEGORY OF THE DV.

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Cells and Marginals 1

• After the observed counts are placed in the
  cells, a new final column is created at the far
  right side of the table to display the row totals
  (or row marginals), adding the cell counts
  ACROSS.
• And a new row is created at the bottom of the
  table to display the column totals (or column
  marginals), adding the counts down the columns.

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Example Super Bowl Watching by Gender

• SUPER BOWL WATCHING BY GENDER

Watching Men Women Row Total
Often 20 30 50
Sometimes 15 20 35
Never 10 30 40
Column Total 45 80 125
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Crosstabs Features

• Here the IV (gender) is in the columns two
  columns are for the two categories of the
  variable (men and women) and the last column
  shows the total in each row.
• Three rows are for the three Super Bowl
  watching categories often, sometimes, and
  never. The bottom row shows the totals in each
  column.
• The number at the far lower right of the table
  shows the total number of cases.

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Cells and Marginals 2

• The marginals the row and column totals are
  fixed they must correspond to the frequency
  distributions of the variables that exist
  regardless of the relationship.
• The row totals are the frequency distribution of
  the DV.
• The column totals are the frequency distribution
  of the IV.
• More than one distribution is possible for the
  cell frequencies. Do they show a relationship?

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Next?

• The next step is to compute percentages for the
  table, computing within categories of the IV
  (i.e., in the direction of the IV GENDER, in
  this case).
• What of men watch often? What of men watch
  sometimes? What of men watch never? And then
  what are the corresponding percentages for women?
• Each columns percentages must add up to 100.
• Are these two distributions (one for men and
  one for women) similar to each other or not?

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ONLY COMPUTE PERCENTAGES DOWN THE COLUMNS!

• We could percentage the table across rows, down
  columns, or with each cell count as a proportion
  of the total or all of these displayed in the
  same table.
• RESIST THE TEMPTATION TO PERCENTAGE EVERYTHING!
  COMPUTE PERCENTAGES ONLY DOWN THE COLUMNS, IN THE
  DIRECTION OF THE IV.

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WHY?

• If percentages are computed down the columns, we
  can see at a glance if the IV categories have
  similar DV distributions.
• If we do it any other way, the percentages will
  be confusing and hard to read.

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What Do We Do Now?

• Look carefully at the DV percentage distributions
  do they look similar for the categories of the
  IV?
• Compute chi-square to see if the differences are
  significant.
• Compute measures of association to see if the
  relationship is strong.

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Significance

• The chi-square test of significance helps to
  confirm what we should be able to see by looking
  at the percentages Do the IV categories have
  different proportional () distributions into the
  DV categories?
• The null hypothesis is that their DV
  distributions are all the same.

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Significance of the Relationship

• The test of significance for a crosstab is called
  chi-square. (Thats a K sound in chi!)
• The chi-square test statistic measures the
  discrepancy between the observed distribution and
  a hypothetical null hypothesis distribution.
• The null hypothesis is that all categories of the
  IV have an identical distribution into the DV
  categories. (Men and women have identical
  distributions into the super bowl watching
  categories.)

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How to Calculate Chi-SquareStep 1 The Expected
Counts 1

• Compute expected counts for each cell, under the
  hypothesis of identical proportional DV
  distributions for all IV categories.
• In each cell, the expected count is calculated as
  the row total for the row the cell is in
  multiplied by the column total for the column the
  cell is in, and divided by the total sample size.
• E (row total)(column total) / N

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Expected Counts 2

• E CR/N
• The logic is that the row total divided by the
  grand total (sample size) represents the
  proportion of cases in the column that we would
  expect to find in this particular cell if there
  were no differences among the IV categories in
  their DV distributions (if men and women watched
  the Super Bowl in the same proportions).
• To get the expected count of cases in the cell,
  we apply that proportion row total / N to the
  column total by multiplying
• (column total)(row total) / N expected count in
  the cell

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Now Compute Chi-Square

• After we have computed all of the expected
  counts, we compute the overall discrepancy
  between these expected counts and the observed
  counts we found in our data. That is the value of
  chi-square
• ?2 ? (O E)2 / E
• In each cell, we subtract the expected count from
  the observed count and square the difference. We
  divide by the expected count for the cell. Then
  we repeat this calculation for all the cells of
  the table and sum the results.

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Chi-Square and Degrees of Freedom

• If there are big discrepancies between the
  expected (null hypothesis) counts and the
  observed counts, chi-square will be relatively
  large.
• We have to calculate the degrees of freedom for
  our chi-square df (r 1)(c 1) where r is
  the number of row (DV) categories and c is the
  number of column (IV) categories.

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OK, Look It Up!

• Once the df is calculated, we can look up our
  computed chi-square in the table of its
  distribution and see if it exceeds the critical
  value for the degrees of freedom.
• If it does, we can say that we have found a
  statistically significant difference among the IV
  categories for their DV distributions (men and
  women report significantly different levels of
  Super Bowl watching).

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Monkey Business with Chi-Square

• The chi-square test statistic is very sensitive
  to sample size.
• A large sample is quite likely to produce a
  significant chi-square result even when the
  differences in the distributions are pretty
  small.

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Measures of Association

• A measure of association is a measure of the
  strength of the relationship between the
  variables in the crosstab.
• Some of them are more appropriate for ordinal
  data, some for nominal data.
• They have different ranges and require care in
  interpretation.
• Check the How To section of the text for more
  details.

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Elaboration The Third Variable

• Researchers often introduce a third variable into
  their crosstab analysis.
• Like the variables in the original bivariate
  analysis, the third variable in a crosstab needs
  to be categoric. It should not have too many
  categories (two or three)!
• It is introduced as a layer variable, and
  software will produce a separate crosstab and
  test of significance of the initial bivariate
  analysis FOR EACH CATEGORY OF THE LAYER VARIABLE.

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Third Variable Example

• For example, in our GENDER/SUPER BOWL example, we
  could introduce a variable called AGE as the
  third variable in an elaboration. It could be
  coded as young or old for its two categories.
• We might find that the initial GENDER/SUPER BOWL
  relationship is different for older people and
  younger people. For example, among younger
  people, there might be very little difference
  between men and women in their Super Bowl
  watching, while there is a gender difference in
  viewing habits among the older respondents.