Chi Square 2 II

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Chi Square (?2 ) II

• Two-way Tests

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CHI-SQUARE ?2

• Marginal Values Row Column Totals
• ?2 a Row by Column Association Test
• ?2 Goodness of Fit Model
• ?2 Test of Independence
• Expected Values (fe) RM CM / Total N

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Testing for a Gender Effect for Approval of
Clinton
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Frequency Data Approval by Gender
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Hypothesis Testing

• Step 1 State Hypothesis
• What is the Null?
• What is the Alternative (Research) Hypothesis?
• Last time, we just imputed equal probabilities
  from what we know about probability theory
• This time, we cant just say 25 in each cell
  because the sample sizes for men and women are
  different. We will show later how to get fe

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Frequency Table Approval for Clinton by Gender
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FIRST STEP COMPUTE PERCENTAGE TABLE

• Row margins
• 335/908 36.9 Disapprove
• 573/908 63.1 Approve
• NOTE Most Citizens Approve of Clinton
• BUT We Are Testing for a Gender EffectAre
  Women more Supportive Than Men?

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PERCENTAGIZE TABLE
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INTERPRETATION

• Row Marginal Most Citizens (63) Approved of
  Clinton.
• Column Marginal There Are More Men in the
  Sample (54).Step 1 Hypotheses
• Null HypothesisH0 Women Men 0 Chance
• Cell Percents Show Women More Supportive? Null
  Hypothesis Challenged.

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STEP 2 WHAT IS THE DISTRIBUTION?

• Frequencies
• Categorizing same individuals in two
  waysApproval and Gender
• Looking at the Effect of an Independent Variable
  (Gender) on Dependent Variable (Approval).

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Step 3 DETERMINE LEVEL OF SIGNIFICANCE

• Set alpha at risk level of 5 (?.05) for 95
  confidence.

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STEP 4 DETERMINE CRITICAL VALUE OF ?2

• Degrees of Freedom
• ( rows 1) ( Columns 1)
• Here (2 1) (2 1) 1 1 1
• Look up Critical Value of ?2 at 5 risk with 1
  df and find ?2 3.84

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CHI-SQUARE DISTRIBUTION
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STEP 5 MAKE DECISION

• Question Is the proportion of men and women
  approving Clinton different from what you would
  expect by chance?
• Assuming the null hypothesis is true what
  would be the expected values?
• Need to Compute Expected values.
• Ideas?

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CALCULATING EXPECTED VALUES

• Look at the Marginal ValuesNote 63 of all
  respondents approve
• Therefore, assuming the null is true that there
  is no gender difference what would you expect
  the cell percentages to look like?

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ASSUMING THE NULL HYPOTHESIS IS TRUE

• 37 of women should disapprove as well as 37 of
  men, sampling error
• 63 of women as well as 63 of men should
  approve, sampling error
• The proportions of men and women approving and
  disapproving should be the same sampling error

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CONSTRUCT A NULL TABLE of fe Values
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TWO METHODS FOR COMPUTING EXPECTED VALUES

• Text Method (Easiest)
• Row Margin Column Margin
• Total N
• Cell a 335 418 / 908 140,030 / 908 154
• Cell b 335 490 / 908 164,150 / 908 181
• Cell c 573 418 / 908 239,514 / 908 264
• Cell d 573 490 / 908 280,770 / 908 309

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2. Null Percentage MethodIf null is true, the
Percentage of Men and Women should be the same.
Then compute the frequency based on that
percentage. Purely for Mathematical Completeness

• Cell a .369 418 154
• Cell b .369 490 181
• Cell c .631 418 264
• Cell d .631 490 309

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CHI-SQUARE Frequency Observed (Expected)
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KEY QUESTIONS

• How closely do fo values match fe values?
• Do the squared fo fe differences fit the null
  hypothesis?
• Or, are the differences between observations and
  chance expectations so deviant as to justify
  rejection of the null?

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CHI-SQUARE
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CALCULATE CHI-SQUARE
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Compare Chi-Square Values

• Look in table for alpha at .05 with 1 df
• Critical value 3.84
• Chi-square computed from data 10.07

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STEP 6 STATE CONCLUSION

• Computed value of chi-square greater than
  critical value, therefore, reject the null
  hypothesis.
• Substantive interpretation?

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THE VALUE OF A VERB

• Elizabeth Loftus study of mock jury 100 Ss view
  video of car accident.
• ½ Ss asked How fast was car A going when it
  hit car B?
• ½ Ss asked How fast was car A going when it
  smashed into car B?
• Later asked Did you see any glass on the
  roadway?

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MOCK JURY DATA
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COMPUTE CHI-SQUARE
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MAKE DECISION

• Critical Value of ?2 at ? .05 with 1 df is ?2
  3.84
• Computed Value of ?2 4.58
• Therefore Reject the null hypothesis.