Chi-Square Part II

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Chi-Square Part II

• Fenster

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Chi-Square Part II

• Let us see how this works in another example.

Attitudes towards Research Attitudes towards Research Attitudes towards Research
Attitudes Towards Statistics Favorable Neither favorable nor unfavorable Unfavorable Row Totals
Favorable 9 26 13 48
Neither favorable nor unfavorable 19 75 83 177
Unfavorable 16 56 110 182
Col. Totals 44 157 206 407
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Chi-Square Part II

• It has been argued that people with favorable
  attitudes towards research tend to have favorable
  attitudes towards statistics.
• Question If we knew the attitudes towards
  research of a respondent, can we predict the
  attitude toward statistics?

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Chi-Square Part II

• Step 2
• H1 Knowledge of attitudes toward research does
  help us predict attitudes towards statistics.
• Step 1
• HO Knowledge of attitudes toward research does
  not help us predict attitudes towards statistics.

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Chi-Square Part II

• Selecting a significance level Lets use ?.05.
  This gives us a ?2 critical of 9.488. Your book
  says the ?2 critical of 9.5.
• Step 4 Collect and summarize sample data.
• We will use the chi-square test with 4 degrees of
  freedom.
• Why four? df(r-1) X (c-1)
• We have 3 rows and 3 columns.
• so we get df (3-1) X (3-1) 2 X 24

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Chi-Square Part II

• If we find a ?2 greater than or equal to 9.5 we
  reject the null hypothesis and conclude that
  attitudes towards research can predict attitudes
  towards statistics.
• If we find a ?2 less than 9.5 we fail to reject
  the null hypothesis and conclude attitudes
  towards research cannot predict attitudes towards
  statistics.

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Calculation of Expected Frequencies

• Expected frequencies (Row total) X (Column
  Total)
• Grand Total

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Calculation of Expected Frequencies

• Cell a Favorable attitudes towards both
  research and statistics.
• (44) X (48) 5.18
• 407

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Calculation of Expected Frequencies

• Cell b Neither favorable or unfavorable
  attitudes towards research, favorable attitudes
  towards statistics.
• (157) X (48) 18.51
• 407

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Calculation of Expected Frequencies

• Cell c Unfavorable attitudes towards research,
  favorable attitudes towards statistics
• (206) X (48) 24.29
• 407

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Calculation of Expected Frequencies

• Cell d Favorable attitudes towards research,
  neither favorable or unfavorable attitudes
  towards statistics
• (44) X (177) 19.13
• 407

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Calculation of Expected Frequencies

• Cell e - Neither favorable or unfavorable
  attitudes towards both statistics and research
• (157) X (177) 68.27
• 407

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Calculation of Expected Frequencies

• Cell f Unfavorable attitudes towards research,
  neither favorable or unfavorable attitudes
  towards statistics
• (206) X (177) 89.58
• 407

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Calculation of Expected Frequencies

• Cell g Favorable attitudes towards research,
  unfavorable attitudes towards statistics
• (44) X (182) 19.67
• 407

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Calculation of Expected Frequencies

• Cell h - Neither favorable or unfavorable
  attitudes towards research, unfavorable attitudes
  towards statistics
• (157) X (182) 70.20
• 407

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Calculation of Expected Frequencies

• Cell i Unfavorable attitudes towards both
  research and statistics
• (206) X (182) 92.11
• 407

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So we set up our chi-square table
Cell f observed f expected f observed-f expected (i.e., RESIDUALS) (f observed-f expected)2 (f observed-f expected)2/f expected
a 9 5.18 3.82 14.59 2.81
b 26 18.51 7.49 56.1 3.03
c 13 24.29 -11.29 127.46 5.25
d 19 19.13 -0.13 0.17 0.008
e 75 68.27 6.73 45.29 0.6
f 83 89.58 -6.58 43.29 0.5
g 16 19.67 -3.67 13.46 0.67
h 56 70.20 -14.2 201.64 2.87
i 110 92.11 17.89 320.05 3.5
Total 407 407.00 0.00 20.2
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Hypothesis Testing with Chi-Square

• Step 5 Making a decision
• ?2 observed 20.2
• ?2 critical 9.488.
• Decision REJECT HO, and conclude that attitudes
  towards research allow us to predict attitudes
  towards statistics.

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Hypothesis Testing with Chi-Square

• Notes about chi-square
• (1) S (f observed - f expected)0.
• The RESIDUALS ALWAYS SUM TO ZERO.
• If S (f observed - f expected) does not equal
  zero (within rounding error), you have made a
  calculation error. Recheck your work.

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Hypothesis Testing with Chi-Square

• The chi-square test itself cannot tell us
  anything about directionality. One way to get
  directionality in the chi-square is to look at
  the (f observed- f expected) column. We see that
  certain cells occur much less frequently than we
  would expect.

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Hypothesis Testing with Chi-Square

• For example cell c (unfavorable attitudes towards
  research but favorable attitudes towards
  statistics) occurs much less frequently than we
  would expect on the basis of chance.

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Analysis of Residuals
Cell f observed f expected f observed-f expected (i.e., RESIDUALS) (f observed-f expected)2 (f observed-f expected)2/f expected
a 9 5.18 3.82 14.59 2.81
b 26 18.51 7.49 56.1 3.03
c 13 24.29 -11.29 127.46 5.25
d 19 19.13 -0.13 0.17 0.008
e 75 68.27 6.73 45.29 0.6
f 83 89.58 -6.58 43.29 0.5
g 16 19.67 -3.67 13.46 0.67
h 56 70.20 -14.2 201.64 2.87
i 110 92.11 17.89 320.05 3.5
Total 407 407.00 0.00 20.2
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Hypothesis Testing with Chi-Square

• We can also see that three cells that capture
  consistency of attitudes between research and
  statistics (cell a favorable attitudes for both,
  cell e neither favorable or unfavorable attitudes
  towards both, cell i unfavorable attitudes for
  both) all have a positive values for (f observed-
  f expected).
• Those three cells are consistent with the
  (unstated and untested) hypothesis that
  individuals tend to have similar attitudes for
  both research and statistics

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Hypothesis Testing with Chi-Square

• Only by examining the (f observed- f expected)
  can we give any statement on the directionality
  of the relationship. We could also analyze the
  column percentages as we move across categories
  of the independent variable to give us insight on
  directionality.

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Hypothesis Testing with Chi-Square

• 3) In this example, why do we get statistical
  significance? We can say that the cells d, e, f
  and g do not contribute to the statistical
  significance of the overall relationship. The
  individual chi-square values for these four cells
  are all very small. The overall relationship is
  significant because of the other cells.

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Analysis of Residuals
Cell f observed f expected f observed-f expected (i.e., RESIDUALS) (f observed-f expected)2 (f observed-f expected)2/f expected
a 9 5.18 3.82 14.59 2.81
b 26 18.51 7.49 56.1 3.03
c 13 24.29 -11.29 127.46 5.25
d 19 19.13 -0.13 0.17 0.008
e 75 68.27 6.73 45.29 0.6
f 83 89.58 -6.58 43.29 0.5
g 16 19.67 -3.67 13.46 0.67
h 56 70.20 -14.2 201.64 2.87
i 110 92.11 17.89 320.05 3.5
Total 407 407.00 0.00 20.2
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Hypothesis Testing with Chi-Square

• Chi-square allows us to decompose the overall
  relationship into its component parts. This
  decomposition allows us to assess whether all
  categories contribute to the significance of the
  overall relationship.

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Hypothesis Testing with Chi-Square

• Limitations for ?2
• So far we have stressed the virtues for ?2 such
  as weak assumptions, and a statistical
  significance test appropriate for nominal level
  data. This is why chi-square is so popular.
• There are two limitations for ?2, one minor and
  one major.

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Hypothesis Testing with Chi-Square

• Minor Limitation
• When the expected cell frequency is less than 5,
  ?2 rejects the null hypothesis too easily. (Note
  this means the EXPECTED frequency and NOT the
  OBSERVED frequency).
• Solution Use Yates' correction
• Yates correction
• Take the (f observed- f expected) -0.5

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Hypothesis Testing with Chi-Square

• Major Limitation
• We have set up a null hypothesis that there is no
  relationship between two variables and have tried
  to reject this hypothesis.
• We refer to a relationship as being statistically
  significant when we have established, subject to
  the risk of type I error, that there is a
  relationship between two variables.
• But does rejecting the null hypothesis mean the
  relationship is significant in the sense of being
  a strong or an important one?
• Not necessarily.

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Hypothesis Testing with Chi-Square

• Remember significance levels are dependent upon
  sample size.
• Let us say that you wanted to investigate the
  relationship between gender and level of
  tolerance. You had no money to investigate this
  relationship, so you handed out questionnaires
  around UML and found the following

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Hypothesis Testing with Chi-Square
Gender Gender
Attitudes towards racial tolerance Males Females Row Totals
High 24 26 50
Low 26 24 50
Column Totals 50 50 100
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Hypothesis Testing with Chi-Square

• Is there a significant relationship between
  gender and attitudes towards racial tolerance?
• Let us use a.05.
• We have one degree of freedom.
• ?2 critical3.8. ?2 observed0.16.
• Since ?2 observed (0.16) lt ?2 critical (3.8), we
  FAIL to reject the null hypothesis and conclude
  that gender does not help us predict to attitudes
  towards racial tolerance.

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Now let us say you had an extremely ambitious
study and you found the following relationship
Gender Gender
Attitudes towards racial tolerance Males Females Row Totals
High 2400 2600 5000
Low 2600 2400 5000
Column Totals 5000 5000 10000
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Hypothesis Testing with Chi-Square

• Is there a significant relationship between
  gender and attitudes towards racial tolerance?
• Let us use a.05.
• We have one degree of freedom.
• ?2 critical3.8, ?2 observed16.0.
• Since ?2 observed (16.0) gt ?2 critical (3.8), we
  easily reject the null hypothesis and conclude
  that gender does help us predict to attitudes
  towards racial tolerance.

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Hypothesis Testing with Chi-Square

• ?2 is sensitive to the number of cases in the
  sample. Even though the proportions in the cells
  remain unchanged, the new ?2 is 100 times the old
  chi-square because we have 100 times the number
  of cases.

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Hypothesis Testing with Chi-Square

• Corrections for the sample size problem
• Pearson's contingency coefficient (You can ask
  for the Contingency Coefficient with SPSS
  CROSSTABS output).

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Hypothesis Testing with Chi-Square

• C ?2
• ?2 N
• where Ntotal number of cases in sample
• Problem with C Cannot attain 1.0 in perfect
  relationship.
• As the syllabus says, there is no ideal solution
  to the sample size problem with chi-square.