Are My Groups the Same or Different

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Are My Groups the Same or Different?

• John T. Drea
• Professor of Marketing
• Western Illinois University

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Analysis of Variance (ANOVA)

• Used when the means of more than two groups are
  to be compared.
• Most ANOVA problems have a nominal variable as
  the independent variable (ex a grouping
  variable) and an interval or ratio-level variable
  as the dependent variable.
• Example Comparing people in Milwaukee, 0-10
  miles outside city, 11-25 miles, and 25 (the
  groups - independent variable) on their attitude
  towards the attending Brewers games (the
  dependent variable)
• The null hypothesis for comparing these four
  groups would be written as

X1 X2 X3 X4
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ANOVA

• ANOVA compares the variances to make inferences
  about the means.
• The variance among the means of the groups will
  be large if the groups significantly differ from
  one another.
• F-ratio - the ratio of the between group mean
  square to the within group mean square (larger
  F-ratios indicate a greater difference)

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ANOVA

• ANOVA tell us if there are significant
  differences between three or more groups, but it
  doesnt tell us if there is a significant
  difference between Group 1 and Group 2, or
  between Group 2 and Group 3, or between Group 1
  and Group 3 -- only that groups 1, 2, and 3 are
  different.
• To determine this, the ideal way is to use
  orthogonal contrasts to compare the differences
  between particular groups.

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Orthogonal Contrasts

• This allows SPSS to compare the means of only
  designated variables.
• Requires the use of contrast codes.
• Codes should equal zero for a given contrast.
• Example
• You want to compare the means of three groups, A,
  B, and C, for a particular variable, X. A
  one-way ANOVA will tell you if significant
  differences exist for X between A, B, and C. But
  if you want to know whether A is significantly
  different than B, you would need to run an
  orthogonal contrast.

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Orthogonal Contrasts

• To compare Groups A and B, you would enter a
  contrast coefficient of 1 for Group A and a
  contrast code of -1 for Group B. Since you are
  not comparing Group C at this time, your code for
  this group is 0. Your contrast codes would look
  like this

1 -1 0
This compares Groups A and B with a t-test.
To compare Groups A and C, your contrast
codes would be like this
1 0 -1
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Orthogonal Contrasts

• To compare each of the three groups, you would
  enter three layers of contrast codes, like this

1 -1 0 Compares A and B 1 0 -1 Compares A and
C 0 1 -1 Compares B and C
Now, suppose you wanted to also compare and see
if the means of a combined Group A and Group B
were significantly different than the mean of
Group C. You could use the following contrast
codes
1 1 -2 Compares A and B with C
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Orthogonal Contrasts

• Problem Determine the contrast codes you would
  need for the following problem, where Milwaukee
  1, 0-10 miles 2, 11-25 miles 3, and 25
  miles 4
• You want to determine if there are significant
  difference in attitudes toward the attending
  Brewers games between these three groups, and you
  also want to make the following specific
  comparisons
• Milw. vs. 0-10, Milw. vs. 11-25, Milw. vs. 25
• Milw. vs. everyone outside of Milwaukee
• What are the contrast codes you need to make
  these comparisons?

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Chi-square

• Used to assess the statistical significance of
  the observed association in a crosstab.
• How different are the actual numbers in each cell
  from the expected numbers in each cell? The
  expected number are calculated through the
  following formula

fexpected (nrow ncolumn)/n
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Chi-square

• The chi-square statistic is calculated by
  examining the differences between the expected
  and the actual numbers in each cell of the table.

?2 S(fobserved fexpected)2/ fexpected
To check this manually, you would need to
calculate the degrees of freedom df(r-1)(c-1)
(i.e., a 3x3 table would have 4 degrees of
freedom), then check the chi-square distribution
table in Appendix B or you can let SPSS do it
for you.
A significance value of greater than .05
indicates the null hypothesis (the actual values
are not different from the expected values)
cannot be rejected.