Methods%20for%20Proportions%20Relations%20between%20Categorical%20Variables

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Methods for ProportionsRelations between
Categorical Variables

• Chapter 10

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Goals for Chapter 10

• 1. Standard deviations for proportion differences
• 2. Confidence intervals and hypothesis tests for
  proportion differences
• 3. Contingency Tables for several proportions
• 4. Statistical Significance
• the Chi-Square statistic for a contingency table
• 5. Relative Risk, Increased Risk, Odds Ratio

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Difference in Sample Proportions--standard
deviation, confidence interval

• (notation pi xi/Ni is observed proportion for
  sample i, i 1 or 2 for two samples note text
  uses ?hat, ? with carat over it)
• estimated s.d. for proportion difference
• s?1- ?2 ?p1(1-p1)/N1 p2(1-p2)/N2
• Confidence Interval for proportion difference
• p1-p2 -z1-?/2 s?1- ?2 ?? ?1- ?2 ? p1-p2 z1-?/2
  s?1- ?2
• Hypothesis test for proportion difference
• use z-statistic z (p1-p2)/ s?1- ?2
• formula above assumes null hypothesis,
  H0 population proportion difference
  is 0.

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Example for Proportion Differences

• A study classified pregnant women according to
  whether they smoked and whether they were able to
  get pregnant during the first cycle they tried.
• RESULTS

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Calculating Conditional Percentages

• What is proportion of women who smoke who also
  become pregnant during the first cycle? 29 /100
  29
• What is proportion of women who dont smoke who
  also become pregnant during the first cycle?
  198/48640.7

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Statistical Significance of 2x2 Tables--1

• Strength of Relation Compare percents or rates
  of those who do with those who dont
• Example smokers pregnant for 1st cycle, 29
• nonsmokers pregnant 1st cycle,
  41
• Therefore 41 / 29 1.4 times as likely to
  become pregnant during 1st cycle if nonsmoker.
• Size of Study Sample How does the number of
  subjects affect the significance of the result?
• Clearly the result becomes more significant (not
  result of chance) as sample size increases.
• If there had been only 59 women in the study,
  difference in proportions would be much less
  significant.

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Assessing Statistical Significance of Tables--2

• We use the Chi-Square Statistic to determine
  whether differences between proportions is
  real or due to chance.
• The Chi-square statistic shows how the
  distribution of observed proportions compared to
  those expected on the basis of pure chance
  varies for example, if we tossed snake-eyes in
  craps on every throw we might think the dice were
  loaded
• For previous example, if there were no difference
  between smokers and nonsmokers, we would expect
  the proportions for both to be the same as in the
  total
• First cycle227/586 0.387 or 38.7 on the
  basis of this expected proportion, we calculate
  the numbers
• .

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Calculating the Chisquare Statistic-Expected
Values

• Since the total number of smokers is 100, there
  would be 100x 0.387 38.7 smokers pregnant in
  the first cycle if there no difference between
  smokers and non.
• Since the total number of pregnant during the
  first cycle is 227, there would be 227-38.7
  188.3 nonsmokers pregnant during the first cycle,
• .

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Calculating the Chisquare Statistic--Differences

• Once the Expected values for each cell are
  calculated, we take the differences between the
  observed and expected values for each cell i,
  observedi - expectedi
• Note that we only have to calculate one
  difference the differences in rows or columns
  have to sum to zero.

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Calculating the Chisquare Statistic-2x2 Tables

• Once the differences, Di, and expected values,
  Ei, for each cell are calculated, then the
  chisquare statistic is evaluated from the
  formula.
• ?2 ? Di2 / Ei,
• ?2 (9.7)2 1/38.7 1/61.3 1/188.3
  1/297.7 4.78
• This value is greater than 3.84, the critical
  value for chisquare
• at a 95 significance level

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Two X Two Tables and Chi-Square Statistics

• Example Are males more likely to be
  underachievers? Students classified as
  underachievers if grades in high school below
  the prediction given by a reading test at Age 12.

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2x2 Table and Chisquare Statistic Example,cont.

• Calculation of Chisquare Statistic for previous
  example
• 1. Compute expected values boys under
  (39/69)x34 19.2
  girls under 34 - 19.2 14.8
  boys over 39 - 19.2 19.8
  girls over 30 - 14.8 15.2
• 2. Take the difference between observed and
  expected, square it, and divide by expected for
  each cell boys under (-6.8)2/ 19.2 2.41
  girls under (6.8)2 / 14.8 3.12 boys over
  (6.8)2 / 19.8 2.34 girls over (-6.8)2/
  15.2 3.04.
• 3. Sum the terms calculated in 2 to get the
  Chisquare statistic Chisquare
  2.41 3.12 2.34 3.04 10.91
• 4. Compare the calculated Chisquare statistic
  with 3.84 to determine significance (at the 95
  level). In this example, 10.91 is much greater
  than 3.84 so results (difference in proportion)
  is statistically significant.
  

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Risk and Odds

• Both the Risk and Odds give information about
  the likelihood of a positive response to a
  categorical variable, but their numerical values
  differ. Example 2x2 Table gives results for
  stopping smoking after eight weeks use of either
  a nicotine patch or placebo Note that the risk
  of continuing to smoke after using the nicotine
  patch is 0.47 or 47 compared to the greater
  risk for the placebo use, 0.80 or 80 . Thus
  the RISK is equivalent to the conditional
  probability for the outcome variable, given a
  response variable. The ODDS is the ratio of
  these conditional probabilities for the two
  outcome variables and can be less than or greater
  than one.

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Relative Risk and 2x2 Tables

• Every 2x2 table will have two explanatory
  variables (eg, for the previous slide, whether a
  nicotine patch or a placebo was used.
• The ratio of the risks for these two variables is
  called the RELATIVE RISK.
• Example
• RR, relative risk of continuing to smoke if
  placebo rather than nicotine patch used
  
  RR 0.80 / 0.47 80 / 47 1.70
  
  

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Odds Ratio and 2x2 Tables

• The ratio of the odds for the two explanatory
  variables is called, as might be expected, the
  Odds Ratio (OR). If the odds are very small then
  the Odds Ratio and Relative Risk are
  approximately equal.
• Examples
• stopping smoking with nicotine patch
• Odds(placebo) 96/24 4.0
• Odds (nicotine) 64/56 1.1
• Thus OR (96/24) / (64/ 56) 3.5,
• (compared to RR 1.7)

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Simpsons Paradox and Hidden Variables

• Example 1972 admissions rates for graduate
  programs, UC (Berkeley)--found overall that
  percent of women applicants admitted was less
  than the percent of men applicants, even though
  women percentages were higher for individual
  departments. (see Exercise 13). The paradox can
  be explained by different overall selectivity in
  each program and different proportions of men and
  women applying to each program.

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Goodness of Fit--Comparing Observed with
Theoretical Proportions

• Procedure
• tabulate observed frequencies,Ni,for each
  category
• tabulate expected (theoretical) frequencies, Ei
• take difference between corresponding observed
  and theoretical frequencies, Di Ni-Ei
• calculate Chi-square statistic by formula
• ?2 ? Di2 / Ei,
• with the degrees of freedom (df) k-1,
  where k is the number of categories (one
  proportion)
• Example Exercises 10-14,10.15 (p 402).

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Comparing Several Proportions, Categories

• Procedure
• Set up a k (no. of explanatory categories) by r
  (number of response proportions) contingency
  table
• tabulate number for each cell in the table,
  marginal totals for each category, Nk, response
  variable, Nr, and grand total, N.
• For the cell in the table corresponding to the
  kth category, rth response variable, the expected
  number (if the proportion would be the same for
  all response variables) is Ekr (Nk Nr) / N
• Calculate the difference Dkr Nkr - Ekr and
  calculate a Chi-square statistic from the formula
• ?2 ? Dkr2 / Ekr
• Degrees of freedom, df (k-1)(r-1)

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Comparing Several Proportions, Categories

• Example, Exercise 10.18, p.408
• A study of potential age discrimination considers
  promotions among middle managers in a large
  company. The data are
• promoted 9 under 30 29 in 30 to 39 category 32
  in 40 to 49 and 10 in 50 and over category
  (total 80)
• not promoted 41 under 30 41 in 30 to 39
  category,
• 48 in 40 to 49, and 49 in 50 and over (total
  170)