Measuringtesting for association in tables: Cramers V and Odds Ratios

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Measuring/testing for association in tables
Cramers V and Odds Ratios

• Monday 28th November 2005

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Strength of Association

• Chi-square tells us whether there is a
  significant association between two variables
  (or whether there exists an association that
  would be unlikely to be found by chance).
• However it does not tell us in a clear way how
  strong this association is since the size of
  chi-square depends on the sample size.
• Today we will look at
• Cramérs V. This is a PRE statistic that tells us
  the strength of associations in contingency
  tables. (the meaning of PRE will be explained).
  It also enables us to compare different
  associations and decide which is stronger.
• Odds Ratios. These tell us the relative odds of
  an event occurring for different categories or
  groups of cases (or people). As such theyre
  relatively easily understood ways to discuss the
  strength of association in contingency tables.

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PRE

• Strength of association is generally measured by
  PRE or Proportional Reduction in Error.
• Another way of saying this is, how much better
  our prediction of the dependent variable will be
  if we know something about the independent
  variable.
• PRE statistics range from 0 to 1. Where
• values of 0 to 0.25 indicate a non-existent to
  weak association
• values between 0.26 and 0.50 indicate
  moderate association
• values between 0.51 and 0.75 indicate a
  moderate to strong association
• values between 0.76 and 1 indicate perfect
  association.

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Example Sex, Age and SportData from Young
Peoples Social Attitudes Study 2003 (available
on module website)
Boys (men)
Girls (women)
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What can we say about these tables?

• It looks like boys play sport with clubs more
  than girls.
• And it looks like both boys and girls become less
  likely to play sport with clubs as they get
  older.
• So,
• Does age significantly affect sports club
  membership for boys? For girls?
• Is the effect of age stronger/weaker for boys
  than it is for girls?

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1. Does age significantly affect sports club
membership for boys?
To answer this question we work out chi-square,
by calculating ?2 (Observedij
Expectedij)2 Expectedij (62
(95164)/303)2 (33 (95139)/303)2 (68
(125164)/303)2
(95164)/303 (95139)/303
(125164)/303) (57 (125139)/303)2
(34 (83164)/303)2 (49 (83139)/303)2
(125139)/303 (83164)/303
(83139)/303) (62-51.4)2/51.4
(33-43.6)2/43.6 (68-67.7)2/67.7
(57-57.3)2/57.3 (34-44.9)2/44.9
(49-38.1)2/38.1 2.2 2.6 0 0 2.6
3.1 10.5
?
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1. Does age significantly affect sports club
membership for boys?

• Chi-square 10.5.
• df (r-1)(c-1) 12 2
• If we look up the .05 value for 2 degrees of
  freedom its 5.99 and the .01 value is 9.21.
  Therefore this is significant at (plt 0.01).
• And the computer printout, below, confirms that
  in fact the p value is .005 (n.b. chi-square
  without rounding is 10.541).

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And girls?

• The similar chi-square test for girls shows that
  age has a significant affect on whether or not
  they play sport (plt.001)

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2. Is the effect of age stronger/weaker for boys
than it is for girls?

• The chi-square statistic is bigger for girls than
  for boys, however the sample of girls is also
  bigger (360 versus 303) so this may have affected
  it.
• To work out the strength of association we need
  to correct for both sample size and for the table
  shape (as this also affects the magnitude of
  chi-square statistics). A frequently used measure
  of association is Cramérs V
• ?2 .
• N (L-1)
• (where ?2 is chi-square, N is the sample size,
  and L is the lesser of the number of rows and
  number of columns).
• Note In any table where either the number of
  rows or columns is equal to 2, Cramérs V is
  equal to another measure of association, referred
  to as phi (or F).

?
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Comparing strength of association between age and
involvement in sport for boys and girls
Boys (men) ?2 10.541
Girls (women) ?2 23.394
Cramérs V for the two tables are therefore
10.5 0.186 and 23.9
0.258 303 (2-1) 360 (2-1)
?
?
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Cramérs V

• We can also see Cramérs V in SPSS printout (to
  the right boys are above and girls, below).

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Whats it mean?

• We can say that age has a significant but
  relatively weak affect on boys participation in
  sport.
• And that age has a significant and slightly
  stronger affect on girls participation in sport.
  
• Thus, both girls and boys are likely to decrease
  their participation in sports clubs as they get
  older but this effect is more pronounced among
  girls than among boys.
• And there is a (small) gender difference in the
  relationship between age and participation in
  sport.

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Odds Ratios
The odds Ratio is the odds of an outcome given
membership of group a divided by the odds of that
outcome given membership of group b. Or, looking
at the table below, the odds of playing sport if
youre male (or membership of group male),
divided by the odds of playing sport if youre
female (or membership of group female).
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Working out the odds.

• The odds of an event occurring can be worked out
  by the number of times that it occurs divided by
  the number of times that it does not occur.

ODDS OF A MALE PLAYING SPORT IN A CLUB 164/139
1.18 This means that a male is 1.18 times more
likely to be a member of a sports club than not
to be. ODDS OF A FEMALE PLAYING SPORT IN A CLUB
114/246 0.46 This means that a female is 0.46
times as likely to be a member of a sports club
as not to be (or less than 50 as likely). n.b.
its often easier to talk about it the other way
round (i.e. 246/114 2.15 therefore women are
about two times as likely not to take part in
sports clubs as they are to take part). The ODDS
RATIO is the odds of a male playing sport divided
by the odds of a woman playing sport 1.18 / 0.46
2.57 Therefore the odds that a male is part of
a sport club is over two and a half times as
great as the odds of a female being part of a
sports club.
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Odds Ratios

• When Odds Ratios are equal to 1 the two groups
  are identical (the odds of the given outcome are
  the same for both groups).
• When odds ratios get close to 0 or to infinity
  the groups are totally different (the odds of the
  given outcome are very high (or certain) for one
  group and very low (or zero) for the other).
• We will come back to odds ratios when we look at
  logistic regression and log-linear analysis.
• Note a mathematic description of the odds ratio
  is(a/c) / (b/d)where the independent
  variable is in the columns this is the same
  as(ad) / (bc)