Comparing Two Population Proportions

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Comparing Two Population Proportions

• Goal Compare two populations/treatments wrt a
  nominal (binary) outcome
• Sampling Design Independent vs Dependent Samples
• Methods based on large vs small samples
• Contingency tables used to summarize data
• Measures of Association Absolute Risk, Relative
  Risk, Odds Ratio

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Contingency Tables

• Tables representing all combinations of levels of
  explanatory and response variables
• Numbers in table represent Counts of the number
  of cases in each cell
• Row and column totals are called Marginal counts

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2x2 Tables - Notation
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Example - Firm Type/Product Quality

• Groups Not Integrated (Weave only) vs
  Vertically integrated (Spin and Weave) Cotton
  Textile Producers
• Outcomes High Quality (High Count) vs Low
  Quality (Count)

Source Temin (1988)
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Notation

• Proportion in Population 1 with the
  characteristic of interest p1
• Sample size from Population 1 n1
• Number of individuals in Sample 1 with the
  characteristic of interest X1
• Sample proportion from Sample 1 with the
  characteristic of interest
• Similar notation for Population/Sample 2

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Example - Cotton Textile Producers

• p1 - True proportion of all Non-integretated
  firms that would produce High quality
• p2 - True proportion of all vertically
  integretated firms that would produce High
  quality

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Notation (Continued)

• Parameter of Primary Interest p1-p2, the
  difference in the 2 population proportions with
  the characteristic (2 other measures given below)
• Estimator
• Standard Error (and its estimate)
• Pooled Estimated Standard Error when p1p2p

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Cotton Textile Producers (Continued)

• Parameter of Primary Interest p1-p2, the
  difference in the 2 population proportions that
  produce High quality output
• Estimator
• Standard Error (and its estimate)
• Pooled Estimated Standard Error when p1p2p

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Confidence Interval for p1-p2 (Wilsons Estimate)

• Method adds a success and a failure to each group
  to improve the coverage rate under certain
  conditions
• The confidence interval is of the form

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Example - Cotton Textile Production
95 Confidence Interval for p1-p2
Providing evidence that non-integrated producers
are more likely to provide high quality output
(p1-p2 gt 0)
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Significance Tests for p1-p2

• Deciding whether p1p2 can be done by
  interpreting plausible values of p1-p2 from the
  confidence interval
• If entire interval is positive, conclude p1 gt p2
  (p1-p2 gt 0)
• If entire interval is negative, conclude p1 lt p2
  (p1-p2 lt 0)
• If interval contains 0, do not conclude that p1 ?
  p2
• Alternatively, we can conduct a significance
  test
• H0 p1 p2 Ha p1 ? p2 (2-sided) Ha
  p1 gt p2 (1-sided)
• Test Statistic
• P-value 2P(Z?zobs) (2-sided) P(Z?
  zobs) (1-sided)

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Example - Cotton Textile Production
Again, there is strong evidence that
non-integrated performs are more likely to
produce high quality output than integrated firms
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Measures of Association

• Absolute Risk (AR) p1-p2
• Relative Risk (RR) p1 / p2
• Odds Ratio (OR) o1 / o2 (o p/(1-p))
• Note that if p1 p2 (No association between
  outcome and grouping variables)
• AR0
• RR1
• OR1

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Relative Risk

• Ratio of the probability that the outcome
  characteristic is present for one group, relative
  to the other
• Sample proportions with characteristic from
  groups 1 and 2

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Relative Risk

• Estimated Relative Risk

95 Confidence Interval for Population Relative
Risk
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Relative Risk

• Interpretation
• Conclude that the probability that the outcome is
  present is higher (in the population) for group 1
  if the entire interval is above 1
• Conclude that the probability that the outcome is
  present is lower (in the population) for group 1
  if the entire interval is below 1
• Do not conclude that the probability of the
  outcome differs for the two groups if the
  interval contains 1

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Example - Concussions in NCAA Athletes

• Units Game exposures among college socer players
  1997-1999
• Outcome Presence/Absence of a Concussion
• Group Variable Gender (Female vs Male)
• Contingency Table of case outcomes

Source Covassin, et al (2003)
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Example - Concussions in NCAA Athletes
There is strong evidence that females have a
higher risk of concussion
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Odds Ratio

• Odds of an event is the probability it occurs
  divided by the probability it does not occur
• Odds ratio is the odds of the event for group 1
  divided by the odds of the event for group 2
• Sample odds of the outcome for each group

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Odds Ratio

• Estimated Odds Ratio

95 Confidence Interval for Population Odds Ratio
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Odds Ratio

• Interpretation
• Conclude that the probability that the outcome is
  present is higher (in the population) for group 1
  if the entire interval is above 1
• Conclude that the probability that the outcome is
  present is lower (in the population) for group 1
  if the entire interval is below 1
• Do not conclude that the probability of the
  outcome differs for the two groups if the
  interval contains 1

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Osteoarthritis in Former Soccer Players

• Units 68 Former British professional football
  players and 136 age/sex matched controls
• Outcome Presence/Absence of Osteoathritis (OA)
• Data
• Of n1 68 former professionals, X1 9 had OA,
  n1-X159 did not
• Of n2 136 controls, X2 2 had OA, n2-X2134 did
  not

Interval gt 1
Source Shepard, et al (2003)
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Fishers Exact Test

• Method of testing for association for 2x2 tables
  when one or both of the group sample sizes is
  small
• Measures (conditional on the group sizes and
  number of cases with and without the
  characteristic) the chances we would see
  differences of this magnitude or larger in the
  sample proportions, if there were no differences
  in the populations

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Example Echinacea Purpurea for Colds

• Healthy adults randomized to receive EP (n1.24)
  or placebo (n2.22, two were dropped)
• Among EP subjects, 14 of 24 developed cold after
  exposure to RV-39 (58)
• Among Placebo subjects, 18 of 22 developed cold
  after exposure to RV-39 (82)
• Out of a total of 46 subjects, 32 developed cold
• Out of a total of 46 subjects, 24 received EP

Source Sperber, et al (2004)
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Example Echinacea Purpurea for Colds

• Conditional on 32 people developing colds and 24
  receiving EP, the following table gives the
  outcomes that would have been as strong or
  stronger evidence that EP reduced risk of
  developing cold (1-sided test). P-value from SPSS
  is .079.

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Example - SPSS Output
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McNemars Test for Paired Samples

• Common subjects being observed under 2 conditions
  (2 treatments, before/after, 2 diagnostic tests)
  in a crossover setting
• Two possible outcomes (Presence/Absence of
  Characteristic) on each measurement
• Four possibilities for each subjects wrt outcome
• Present in both conditions
• Absent in both conditions
• Present in Condition 1, Absent in Condition 2
• Absent in Condition 1, Present in Condition 2

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McNemars Test for Paired Samples
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McNemars Test for Paired Samples

• H0 Probability the outcome is Present is same
  for the 2 conditions
• HA Probabilities differ for the 2 conditions
  (Can also be conducted as 1-sided test)

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Example - Juveniles Tried as Adults

• Subjects - 2097 pairs of juveniles matched on
  prior criminal record and severity of current
  crime
• Condition Adult vs Juvenile Court (one of each
  in pair)
• Outcome Whether juvenile was re-arrested during
  follow-up

Source Bishop et al (1996)
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Example - Juveniles Tried as Adults

• H0 Tendency to for rearrest is not different
  between children tried as adults as those tried
  as juveniles
• HA Tendencies differ

Evidence that tendencies differ (higher risk of
rearrest among juveniles tried in adult court)
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Data Sources

• Temin, P. (1988). Product Quality and Vertical
  Integration in the Early Cotton Textile
  Industry, The Journal of Economic History,
  48(4), pp891-907
• Covassin, T., C.B. Swanik, and M.L. Sachs (2003).
  Sex Differences and the Incidence of Concussions
  Among Collegiate Athletes, Journal of Athletic
  Training, 38(3) pp238-244.
• Shepard, G.J., A.J. Banks, and W.G. Ryan (2003).
  Ex-Professional Association Footballers Have an
  Increased Prevalence of Osteoarthritis of the Hip
  Compared with Age Matched Controls Desite Not
  Having Sustained Notable Hip Injuries, British
  Journal of Sports Medicine, 37, pp80-81.
• Sperber, S.J., L.P. Shah, R.D. Gilbert, et al
  (2004). Echinacea purpurea for Prevention of
  Experimental Rhinovirus Colds, Clinical
  Infectious Diseases, 38, pp1367-1371.
• Bishop,D.M, C.E. Frazier, L. Lanza-Kaduce, L.
  Winner (1996). The Transfer of Juveniles to
  Criminal Court Does it Make a Difference? Crime
  Delinquency, 42, pp171-191.