Cross Tabs and Chi-Squared

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Cross Tabs and Chi-Squared

• Testing for a Relationship Between
  Nominal/Ordinal Variables

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Cross Tabs and Chi-Squared

• The test you choose depends on level of
  measurement
• Independent Dependent Statistical Test
• Dichotomous Interval-ratio Independent Samples
  t-test
• Dichotomous
• Nominal Nominal Cross Tabs
• Dichotomous Dichotomous
• Nominal Interval-ratio ANOVA
• Dichotomous Dichotomous
• Interval-ratio Interval-ratio Correlation and
  OLS Regression
• Dichotomous

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Cross Tabs and Chi-Squared

• We are asking whether there is a relationship
  between two nominal (or ordinal) variablesthis
  includes dichotomous variables.
• (Even though one may use cross tabs for ordinal
  variables, it is generally better to treat them
  as interval variables and use more powerful
  statistical techniques whenever you can.)

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Cross Tabs and Chi-Squared

• Cross tabs and Chi-Squared will tell you whether
  classification on one nominal or ordinal variable
  is related to classification on a second nominal
  or ordinal variable.
• For Example
• Are rural Americans more likely to vote
  Republican in presidential races than urban
  Americans?
• Classification of Region Party Vote
• Are white people more likely to drive SUVs than
  blacks or Hispanics?
• Race Type of Vehicle

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Cross Tabs and Chi-Squared

• The statistical focus will be on the number of
  people in a sample who are classified in
  patterned ways on two variables.
• Why?
• Means and standard deviations are meaningless for
  nominal variables.

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Cross Tabs and Chi-Squared

• The procedure starts with a cross
  classification of the cases in categories of
  each variable.
• Example
• Data on male and female support for SJSU football
  from 650 students put into a matrix
• Yes No Maybe Total
• Female 185 200 65 450
• Male 80 65 55 200
• Total 265 265 120 650

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Cross Tabs and Chi-Squared

• In the example, I can see that the campus is
  divided on the issue.
• But are there associations between sex and
  attitudes?
• Example
• Data on male and female support for SJSU football
  from 650 students put into a matrix
• Yes No Maybe Total
• Female 185 200 65 450
• Male 80 65 55 200
• Total 265 265 120 650

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Cross Tabs and Chi-Squared

• But are there associations between sex and
  attitudes?
• An easy way to get more information is to convert
  the frequencies to percentages.
• Example
• Data on male and female support for SJSU football
  from 650 students put into a matrix
• Yes No Maybe Total
• Female 185 (41) 200 (44) 65 (14) 450 (99)
• Male 80 (40) 65 (33) 55 (28) 200 (101)
• Total 265 (41) 265 (41) 120 (18) 650 (100)
• percentages do not add to 100 due to rounding

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Cross Tabs and Chi-Squared

• We can see that in the sample men are less likely
  to oppose football, but no more likely to say
  yes than womenmen are more likely to say
  maybe
• Example
• Data on male and female support for SJSU football
  from 650 students put into a matrix
• Yes No Maybe Total
• Female 185 (41) 200 (44) 65 (14) 450 (99)
• Male 80 (40) 65 (33) 55 (28) 200 (101)
• Total 265 (41) 265 (41) 120 (18) 650 (100)
• percentages do not add to 100 due to rounding

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Cross Tabs and Chi-Squared

• Data on male and female support for SJSU football
  from 650 students put into a matrix
• Yes No Maybe Total
• Female 185 (41) 200 (44) 65 (14) 450 (99)
• Male 80 (40) 65 (33) 55 (28) 200 (101)
• Total 265 (41) 265 (41) 120 (18) 650 (100)
• percentages do not add to 100 due to rounding
• Using percentages to describe relationships is
  valid statistical analysis These are
  descriptive statistics! However, they are not
  inferential statistics.
• What can we say about the population?
• Could we have gotten sample statistics like these
  from a population where there is no association
  between sex and attitudes about starting
  football?
• This is where the Chi-Squared Test of
  Independence comes in handy.

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Cross Tabs and Chi-Squared

• The whole idea behind the Chi-Squared test of
  independence is to determine whether the patterns
  of frequencies in your cross classification table
  could have occurred by chance, or whether they
  represent systematic assignment to particular
  cells.
• For example, were women more likely to answer
  no than men or could the deviation in responses
  by sex have occurred because of random sampling
  or chance alone?

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Cross Tabs and Chi-Squared

• A number called Chi-Squared, ?2, tells us whether
  the numbers in our sample deviate from what would
  be expected by chance.
• Its formula
• fo observed frequency in each cell fe expected
  frequency in each cell
• A bigger ?2 will result as our sample data
  deviates more and more from what would be
  expected by chance.
• A big ?2 will imply that there is a relationship
  between our two nominal variables.

?2 ? ((fo - fe)2 / fe)
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Cross Tabs and Chi-Squared
?2 ? ((fo - fe)2 / fe)

• Calculating ?2 begins with the concept of a
  deviation of observed data from what is expected
  by chance alone.
• Deviation in ?2 Observed frequency Expected
  frequency
• Observed frequency is just the number of cases in
  each cell of the cross classification table. For
  example, 185 women said yes, they support
  football at SJSU. 185 is the observed frequency.
• Expected frequency is the number of cases that
  would be in a cell of the cross classification
  table if people in each group of one variable had
  a propensity to answer the same as each other on
  the second variable.

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Cross Tabs and Chi-Squared
?2 ? ((fo - fe)2 / fe)

• Data on male and female support for SJSU football
  from 650 students
• Yes No Maybe Total
• Female 185 200 65 450
• Male 80 65 55 200
• Total 265 265 120 650
• Expected frequency (if our variables were
  unrelated)
• Since females comprise 69.2 of the sample, wed
  expect 69.2 of the Yes answers to come from
  females, 69.2 of the No answers to come from
  females, and 69.2 of the Maybe answers to come
  from females. On the other hand, 30.8 of the
  Yes, No, and Maybe answers should come from
  Men.
• Therefore, to calculate expected frequency for
  each cell you do this
• fe cells row total / table total cells
  column total or
• fe cells column total / table total cells
  row total
• The idea is that you find the percent of persons
  in one category on the first variable, and
  expect to find that percent of those people in
  the other variables categories.

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Cross Tabs and Chi-Squared
?2 ? ((fo - fe)2 / fe)

• Data on male and female support for SJSU football
  from 650 students
• Yes No Maybe Total
• Female 185 200 65 450
• Male 80 65 55 200
• Total 265 265 120 650
• Now you know how to calculate the expected
  frequency (and the observed frequency is
  obvious).
• fe1 (450/650) 265 183.5 fe4 (200/650)
  265 81.5
• fe2 (450/650) 265 183.5 fe5 (200/650)
  265 81.5
• fe3 (450/650) 120 83.1 fe6 (200/650)
  120 36.9
• You already saw how to calculate the deviations
  too.
• Dc fo fe
• D1 185 183.5 1.5 D4 80 81.5
  -1.5
• D2 200 183.5 16.5 D5 65 81.5 -16.5
  
• D3 65 83.1 -18.1 D4 55 36.9
  18.1

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Cross Tabs and Chi-Squared
?2 ? ((fo - fe)2 / fe)

• Data on male and female support for SJSU football
  from 650 students
• Yes No Maybe Total
• Female 185 200 65 450
• Male 80 65 55 200
• Total 265 265 120 650
• Deviations
• Dc fo fe
• D1 185 183.5 1.5 D4 80 81.5
  -1.5
• D2 200 183.5 16.5 D5 65 81.5 -16.5
  
• D3 65 83.1 -18.1 D4 55 36.9
  18.1
• Now, we want to add up the deviations
• What would happen if we added these deviations
  together?
• To get rid of negative deviations, we square each
  one (like in computing standard deviations).

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Cross Tabs and Chi-Squared
?2 ? ((fo - fe)2 / fe)

• Data on male and female support for SJSU football
  from 650 students
• Yes No Maybe Total
• Female 185 200 65 450
• Male 80 65 55 200
• Total 265 265 120 650
• Deviations
• Dc fo fe
• D1 185 183.5 1.5 D4 80 81.5
  -1.5
• D2 200 183.5 16.5 D5 65 81.5 -16.5
  
• D3 65 83.1 -18.1 D4 55 36.9
  18.1
• To get rid of negative deviations, we square each
  one (like in standard deviations).
• (D1)2 (1.5)2 2.25 (D4)2 (-1.5)2
  2.25
• (D2)2 (16.5)2 272.25 (D5)2 (-16.5)2
  272.25
• (D3)2 (-18.1)2 327.61 (D6)2 (18.1)2
  327.61

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Cross Tabs and Chi-Squared
?2 ? ((fo - fe)2 / fe)

• Squared Deviations
• (D1)2 (1.5)2 2.25 (D4)2 (-1.5)2
  2.25
• (D2)2 (16.5)2 272.25 (D5)2 (-16.5)2
  272.25
• (D3)2 (-18.1)2 327.61 (D6)2 (18.1)2
  327.61
• Just how large is each of these squared
  deviations?
• The next step is to give the deviations a
  metric. The deviations are compared relative
  to the what was expected. In other words, we
  divide by what was expected.
• Youve already calculated what was expected in
  each cell
• fe1 (450/650) 265 183.5 fe4 (200/650)
  265 81.5
• fe2 (450/650) 265 183.5 fe5 (200/650)
  265 81.5
• fe3 (450/650) 120 83.1 fe6 (200/650)
  120 36.9
• Relative Deviations-squaredSmall values indicate
  little deviation from what was expected, while
  larger values indicate much deviation from what
  was expected
• (D1)2 / fe1 2.25 / 183.5 0.012 (D4)2 /
  fe4 2.25 / 81.5 0.028
• (D2)2 / fe2 272.25 / 183.5 1.484 (D5)2 / fe5
  272.25 / 81.5 3.340
• (D3)2 / fe3 327.61 / 83.1 3.942 (D6)2 /
  fe6 327.61 / 36.9 8.878

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Cross Tabs and Chi-Squared
?2 ? ((fo - fe)2 / fe)

• Relative Deviations-squaredSmall values indicate
  little deviation from what was expected, while
  larger values indicate much deviation from what
  was expected
• (D1)2 / fe1 2.25 / 183.5 0.012 (D4)2 /
  fe4 2.25 / 81.5 0.028
• (D2)2 / fe2 272.25 / 183.5 1.484 (D5)2 / fe5
  272.25 / 81.5 3.340
• (D3)2 / fe3 327.61 / 83.1 3.942 (D6)2 /
  fe6 327.61 / 36.9 8.878
• The next step will be to see what the total
  relative deviations-squared are
• Sum of
• Relative Deviations-squared 0.012 1.484
  3.942 0.028 3.340 8.878 17.684
• This number is also what we call Chi-Squared or
  ?2.
• So
• Of what good is knowing this number?

?2 ? ((fo - fe)2 / fe)
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Cross Tabs and Chi-Squared

• This value, ?2, would form an identifiable shape
  in repeated sampling if the two variables were
  unrelated to each other.
• That shape depends only on the number of rows and
  columns. We technically refer to this as the
  degrees of freedom.
• For ?2, df (rows 1)(columns 1)

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Cross Tabs and Chi-Squared

• For ?2, df (rows 1)(columns 1)
• ?2 distributions

df 5
FYI This should remind you of the normal
distribution, except that, it changes shape
depending on the nature of your variables.
df 10
df 20
df 1
1 5 10 20
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Cross Tabs and Chi-Squared
Think of the Power!!!!

• We can use the known properties of the ?2
  distribution to identify the probability that we
  would get our samples ?2 if our variables were
  unrelated!
• This is exciting!

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Cross Tabs and Chi-Squared

• If our ?2 in a particular analysis were under the
  shaded area or beyond, what could we say about
  the population given our sample?

5 of ?2 values
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Cross Tabs and Chi-Squared

• Answer Wed reject the null, saying that it is
  highly unlikely that we could get such a large
  chi-squared value from a population where the two
  variables are unrelated.

5 of ?2 values
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Cross Tabs and Chi-Squared

• So, what is the critical ?2 value?

5 of ?2 values
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Cross Tabs and Chi-Squared

• That depends on the particular problem because
  the distribution changes depending on the number
  of rows and columns.

df 5
df 10
df 20
df 1
1 5 10 20
Critical ?2 s
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Cross Tabs and Chi-Squared

• According to Table C, df 1, critical ?2
  3.84
• with ?-level .05, if df 5, critical ?2
  11.07
• df 10, critical ?2 18.31
• df 20, critical ?2 31.41

df 5
df 10
df 20
df 1
1 5 10 20
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Cross Tabs and Chi-Squared

• In our football problem above, we had a
  chi-squared of 17.68 in a cross classification
  table with 2 rows and 3 columns.
• Our chi-squared distribution for that table would
  have
• df (2 1) (3 1) 2.
• According to Table C, with ?-level .05,
  Critical Chi-Squared is 5.99.
• Since 17.68 gt 5.99, we reject the null.
• We reject that our sample could have come from a
  population where sex was not related to attitudes
  toward football.

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Cross Tabs and Chi-Squared

• Now lets get formal
• 7 steps to Chi-squared test of independence
• Set ?-level (e.g., .05)
• Find Critical ?2 (depends on df and ?-level)
• The null and alternative hypotheses
• Ho The two nominal variables are independent
• Ha The two variables are dependent on each
  other
• Collect Data
• Calculate ?2 ?2 ? ((fo - fe)2 / fe)
• Make decision about the null hypothesis
• Report the P-value

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Cross Tabs and Chi-Squared

• Afterwards, what have you found?
• If Chi-Squared is not significant, your variables
  are unrelated.
• If Chi-Squared is significant, your variables are
  related.
• Thats All!
• Chi-Squared cannot tell you anything like the
  strength or direction of association. For purely
  nominal variables, there is no direction of
  association.
• Chi-Squared is a large-sample test. If dealing
  with small samples, look up appropriate tests. (A
  condition of the test no expected frequency
  lower than 5 in each cell)
• The larger the sample size, the easier it is for
  Chi-Squared to be significant.
• 2 x 2 table Chi-Square gives same result as
  Independent Samples t-test for proportion and
  ANOVA.

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Cross Tabs and Chi-Squared

• If you want to know how you depart from
  independence, you may
• Check percentages (conditional distributions) in
  your cross classification table.
• Do a residual analysis
• The difference between observed and expected
  counts in a cell behaves like a significance test
  when divided by a standard error for the
  difference.
• That s.e. ?fe(1-cells row ?)(1 cells
  column ?)
• fo fe
• Z s.e.

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Cross Tabs and Chi-Squared

• Residual Analysis
• Lets do cell 5! s.e. ?fe(1-cells row ?)(1
  cells column ?)
• fo fe 5 row ? 200/650
  .308, column ? 265/650 .408
• Z s.e. s.e.
  ?81.5 (.692) (.592) 5.78
• Z 65 81.5 / 5.78 -2.85 2.85 gt 1.96, there
  is a significant difference in cell 5
• Data on male and female support for SJSU football
  from 650 students
• Yes No Maybe Total
• Female 185 200 65 450
• Male 80 65 55 200
• Total 265 265 120 650
• fe1 (450/650) 265 183.5 fe4 (200/650)
  265 81.5
• fe2 (450/650) 265 183.5 fe5 (200/650)
  265 81.5
• fe3 (450/650) 120 83.1 fe6 (200/650)
  120 36.9
• Deviations
• Dc fo fe
• D1 185 183.5 1.5 D4 80 81.5
  -1.5

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Cross Tabs and Chi-Squared

• Further topics you could explore
• Strength of Association
• Discussing outcomes in terms of difference of
  proportions
• Reporting Odds Ratios (likelihood of a group
  giving one answer versus other answers or the
  group giving an answer relative to other groups
  giving that answer)
• Strength and Direction of Association for Ordinal
  Variables
• Gamma (an inferential statistic, so check for
  significance)
• Ranges from -1 to 1
• Valence indicates direction of relationship
• Magnitude indicates strength of relationship
• Chi-squared and Gamma can disagree when there is
  a nonrandom pattern that has no direction.
  Chi-squared will catch it, gamma wont.
• Kendalls tau-b
• Somers d