Lab 7

1
Lab 7

• Chi-Square

2
Overview

• Frequency Data
• Two Uses of ?2
• ?2 Test of Goodness of Fit
• ?2 Test of Association
• Assumptions of ?2
• SPSS Crosstabs
• Post-Hoc Interpretations
• Assignment 7

3
Frequency Data
4
Frequency Data

• What are frequency (or count) data?
• There are two types of data in the world
• Categorical
• Continuous

5
Frequency Data

• In all of the previous labs, we have dealt with
  continuous data (as the DV)
• Also called measurement data because the data
  are derived from a measurement process (i.e.,
  measuring each participants score on a specific
  variable)
• For example, height, IQ, self-esteem,
  neuroticism, etc.
• These variables are continuous because there are
  no distinct breaks between numbers in the scale
  in other words, fractional values are meaningful

6
Frequency Data

• But what if we want to compare different kinds of
  things, not just degrees of difference on the
  same dimension?
• In this case, we have to use categorical data
• For example, eye color, sex, University major,
  etc.
• These variables are categorical because they
  identify discrete categories that have no
  numerical relationship
• Thus, the only meaningful measure is the
  frequency of observations of each category in a
  given sample Hence the term frequency data

7
Two Uses of ?2
8
Two Uses of ?2

• The general logic of ?2 involves the comparison
  between expected and observed frequencies of a
  category

O Observed frequency E Expected frequency
9
Two Uses of ?2

• As with all significance tests, the computed ?2
  is then compared to a critical ?2 (e.g., at a
  .05)
• The critical value is found in the ?2
  distribution adjusted for the degrees of freedom

10
Two Uses of ?2

• There are two common uses of the ?2 significance
  test
• Tests of goodness of fit are used to determine if
  the distribution (shape) of a sample matches an
  expected/theoretical distribution
• For example, is the distribution of class marks
  as expected?
• Tests of association are used to determine if
  there is a relation between two categorical
  variables
• For example, is there an association between sex
  (M vs. F) and smoking (Y vs. N) in a given
  sample?

11
Two Uses of ?2

• Note that both tests are essentially the same, in
  that they assess the match between expected and
  observed frequencies of particular categoriesbut
• With goodness of fit tests, you usually want the
  ?2 to be non-significant, which indicates a good
  match to the expected distribution (i.e., the
  difference is not significant)
• With tests of association, you usually want the
  ?2 to be significant, which indicates a
  significant relation between two categorical
  variables

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?2 Test of Goodness of Fit
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?2 Test of Goodness of Fit

• In goodness of fit tests, your observed
  frequencies come from one variable
• For example, marks A, B, C, D, F
• Expected frequencies come from an
  expected/theoretical distribution
• For example 10 A, 30 B, 30 C, 25 D, 5 F

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?2 Test of Goodness of Fit

• The same formula is used to compute ?2
• For goodness of fit tests, df (number of
  categories 1)

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?2 Test of Goodness of Fit

• ?2(4) 8.61, ns
• Therefore, the observed frequencies of marks are
  not significantly different from the expected
  frequencies

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?2 Test of Association
17
?2 Test of Association

• Lets say you work at a clothing store in the
  mall
• One day, you start to wonder if people tend to
  buy clothes that match their eye color
• In other words, is there an association between
  preferred clothing color and eye color?
• Because both of these variables are categorical,
  we have to test this association using ?2

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?2 Test of Association

• As it happens, your store is holding a t-shirt
  sale, and the shirts come in four colors
• Brown, Blue, Green, and Black
• For one week, you record the eye color of
  everyone who purchases a t-shirt (N 200) and
  the color of the shirt

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?2 Test of Association
Observed Frequencies
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?2 Test of Association

• To begin, you need to figure out what the
  expected number of t-shirts purchased would be
  for each eye/shirt combination
• If there is no association, the proportion of any
  shirt color to any eye color would be equal to
  the proportion of that shirt color to the total
  number of shirts in the sample

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?2 Test of Association

• For example, to figure out the expected frequency
  of brown shirts purchased by brown-eyed people
• In this case, E11 15.90, which is quite a bit
  lower than the observed frequency of 27
• Now compute every expected frequency the same way

22
?2 Test of Association
Expected Frequencies
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?2 Test of Association

• To calculate the ?2, you plug the observed and
  expected frequencies for each combination into
  the formula and add up the all of the values
• For example, starting with the brown/brown
  combination

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?2 Test of Association

• For tests of association, the degrees of freedom
  (number of rows 1) x (number of columns 1)
• In our case, df 3 x 3 9
• Now, compare your calculated ?2 against the
  critical value at 9 df
• There is a significant association between eye
  color and color of shirt purchased, ?2(9)
  54.31, p lt .001
• NOTE This is not a causal relationship, just an
  association between two variables nothing was
  manipulated

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Assumptions of ?2
26
Assumptions of ?2

• There are two major assumptions underlying ?2
• 1) The sampling distributions of the deviations
  of the observed frequencies from the expected
  frequencies (i.e., O E) are normal
• This assumption is satisfied if expected values
  are not too small
• Potential for violation if df 1
• In other words, if the table was a 2 x 2, you
  might need to use the Yates correction (a more
  conservative estimate of ?2)
• See p. 170 for details on the Yates correction

27
Assumptions of ?2

• There are two major assumptions underlying ?2
• 2) Each individual can only contribute once to
  the frequency count (independent observations)

28
SPSS Crosstabs
29
SPSS Crosstabs

• The easiest way to enter frequency data is to use
  the weighting option in SPSS
• Create a variable for both of your measurements
  (e.g., eye_color and shirt_color) and define the
  value labels
• Create a variable for the observed frequency and
  enter your data

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SPSS Crosstabs
31
SPSS Crosstabs

• From the Data menu, select Weight Cases and
  tell SPSS to weight the cases by your frequency
  variable
• This is an easy way to tell SPSS how many
  observations belong in each category
• Once the data are entered and weighted, select
  Crosstabs from the Analyze Descriptives menu
• See pp. 172-173 for details on running the
  analysis

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SPSS Crosstabs

• Use this table to double-check your numbers and
  make sure everything looks right

33
SPSS Crosstabs

• This table summarizes your data and is equivalent
  to the one we calculated by hand

34
SPSS Crosstabs

• This table provides the calculated ?2 (the
  Pearson Chi-Square) with the associated df and
  p value
• The other tests are not important for us

35
Post-Hoc Interpretations
36
Post-Hoc Interpretations

• The significant ?2 value means that the observed
  frequencies differ from the expected frequencies
  more than would be expected on the basis of
  chance
• But that doesnt tell us which particular
  categories are driving the effect, so we need to
  interpret our results in more detail

37
Post-Hoc Interpretations

• There is no clear-cut method for post-hoc
  interpretation of ?2 data
• Three methods will be discussed here
• 1) Examining the cells
• 2) Specific contrasts
• 3) Simple effects

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Post-Hoc Interpretations

• 1) Examining the cells
• Visually inspect the table for large residuals

39
Post-Hoc Interpretations

• 1) Examining the cells
• Calculate a partial ?2 for just these six cells

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Post-Hoc Interpretations

• 1) Examining the cells
• That means that these six cells accounted for
  (44.66 / 54.31 ) 82.23 of the ?2 value
• If you look back at the table, you can see that
  four of these cells were consistent with our
  prediction (matching shirt/eye color)

41
Post-Hoc Interpretations

• 1) Examining the cells
• We can also treat the standardized adjusted
  residuals like z-scores to determine which
  observed frequencies are significantly different
  from the expected frequency
• But with so many contrasts, we have to control
  Type I error using the Bonferonni adjustment
• Divide a by the number of cells in the table (16
  in this case)
• a .05 / 16 .003

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Post-Hoc Interpretations

• 1) Examining the cells
• Find the critical z-value for this a-level
• http//www.fourmilab.ch/rpkp/experiments/analysis/
  zCalc.html
• Critical z 2.75
• Now compare all of the adjusted residuals in your
  table to this critical value any that exceed it
  are significant
• For example, the adjusted residual for the
  brown/brown combination is 3.90, so this observed
  frequency is significantly different from the
  expected frequency at the .05 level
• For each significant contrast, explain how it
  fits (or fails to fit) with your research question

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Post-Hoc Interpretations

• 2) Specific contrasts
• This method involves forming 2 x 2 tables from
  your overall table
• These tables can be formed by
• Selecting a specific 2 x 2 subset from your
  overall table, or
• Collapsing across categories to form a more
  general 2 x 2 table

44
Post-Hoc Interpretations

• 2) Specific contrasts
• Once you have your 2 x 2 table, you compute a ?2
  for it
• Correct a using the Bonferonni adjustment (see
  formula on p. 181)
• Compare the new ?2 to the critical ?2 at that
  a-level

45
Post-Hoc Interpretations

• 2) Specific contrasts
• For example, lets say you were interested in
  only blue vs. brown eyes and blue vs. brown shirts

46
Post-Hoc Interpretations
47
Post-Hoc Interpretations

• 2) Specific contrasts
• The obtained ?2 is 9.24
• Use the formula on p. 181 to calculate the total
  number of possible 2 x 2 tables within a 4 x 4
  table

48
Post-Hoc Interpretations

• 2) Specific contrasts
• Perform the Bonferroni adjustment using k
• a .05 / 36 .001
• Compare your obtained ?2 (9.24) to the critical
  ?2 at df 1 (because its from a 2 x 2 table) at
  this a-level

49
Post-Hoc Interpretations

• 2) Specific contrasts

50
Post-Hoc Interpretations

• 2) Specific contrasts
• It turns out that using the Bonferonni
  adjustment, the specific contrast between blue
  vs. brown eyes and blue vs. brown shirts is not
  significant, ?2(1) 9.24, ns

51
Post-Hoc Interpretations

• 2) Specific contrasts
• Another way of forming a 2 x 2 table is to
  collapse across categories in a meaningful way
• For example, you might lump blue and green
  together, and lump brown and black together, for
  both eye and shirt color
• Something like spring colors vs. fall colors
• If you use this method, you dont need to use the
  Bonferonni adjustment just set a at .01

52
Post-Hoc Interpretations

• 3) Simple effects
• With this method, you compare specific
  proportions by holding one factor constant and
  varying the other
• Similar to the simple main effects analysis from
  ANOVA
• For example, we could compare all pairs of
  shirt-colors for blue-eyed customers, then for
  green-eyed customers, etc.
• This method has low power and is quite
  conservative, so the first two methods are
  recommended instead

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Assignment 7
54
Assignment 7

• Question 1
• Introduction
• Discuss assumptions
• Report 2 x 4 ?2 analysis
• Interpret overall ?2 results
• Question 2
• Use post-hoc method 1 (with z-score comparisons)
• Not necessarily one right answer just use common
  sense and provide the information that is most
  important to evaluate the researchers hypothesis

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Assignment 7

• ?2 Table
• http//www.itl.nist.gov/div898/handbook/eda/sectio
  n3/eda3674.htm
• Z-Score Table
• http//www.fourmilab.ch/rpkp/experiments/analysis/
  zCalc.html