1 / 45

Chapter 10

Chapter 10. Categorical Data Analysis. Inference for a Single Proportion ( p ). Goal: Estimate proportion of individuals in a population with a certain characteristic ( p ). This is equivalent to estimating a binomial probability

sebastian-sykes
Télécharger la présentation

Chapter 10

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 10 Categorical Data Analysis

  2. Inference for a Single Proportion (p) • Goal: Estimate proportion of individuals in a population with a certain characteristic (p). This is equivalent to estimating a binomial probability • Sample: Take a SRS of n individuals from the population and observe y that have the characteristic. The sample proportion is y/n and has the following sampling properties:

  3. Large-Sample Confidence Interval for p • Take SRS of size n from population where p is true (unknown) proportion of successes. • Observe y successes • Set confidence level (1-a) and obtain za/2 from z-table

  4. Example - Ginkgo and Azet for AMS • Study Goal: Measure effect of Ginkgo and Acetazolamide on occurrence of Acute Mountain Sickness (AMS) in Himalayan Trackers • Parameter: p= True proportion of all trekkers receiving Ginkgo&Acetaz who would suffer from AMS. • Sample Data: n=126trekkers received G&A, y=18 suffered from AMS

  5. Wilson’s “Plus 4” Method • For moderate to small sample sizes, large-sample methods may not work well wrt coverage probabilities • Simple approach that works well in practice (n10): • Pretend you have 4 extra individuals, 2 successes, 2 failures • Compute the estimated sample proportion in light of new “data” as well as standard error:

  6. Example: Lister’s Tests with Antiseptic • Experiments with antiseptic in patients with upper limb amputations (John Lister, circa 1870) • n=12 patients received antiseptic y=1 died

  7. Significance Test for a Proportion • Goal test whether a proportion (p) equals some null value p0H0: p=p0 Large-sample test works well when np0 and n(1-p0)  5

  8. Ginkgo and Acetaz for AMS • Can we claim that the incidence rate of AMS is less than 25% for trekkers receiving G&A? • H0: p=0.25 Ha: p < 0.25 Strong evidence that incidence rate is below 25% (p < 0.25)

  9. 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

  10. 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

  11. Outcome Present Outcome Absent Group Total Group 1 y1 n1-y1 n1 Group 2 y2 n2-y2 n2 Outcome Total y1+y2 (n1+n2)-(y1+y2) n1+n2 2x2 Tables - Notation

  12. High Quality Low Quality Group Total Not Integrated 33 55 88 Vertically Integrated 5 79 84 Outcome Total 38 134 172 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)

  13. 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: y1 • Sample proportion from Sample 1 with the characteristic of interest: • Similar notation for Population/Sample 2

  14. 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

  15. 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 p1=p2=p:

  16. 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 p1=p2=p:

  17. Significance Tests for p1-p2 • Deciding whether p1=p2 canbe done by interpreting “plausible values” of p1-p2 from the confidence interval: • If entire interval is positive, conclude p1 > p2 (p1-p2 > 0) • If entire interval is negative, conclude p1 < p2 (p1-p2 < 0) • If interval contains 0, do not conclude that p1 p2 • Alternatively, we can conduct a significance test: • H0: p1 = p2Ha: p1 p2 (2-sided) Ha: p1 > p2 (1-sided) • Test Statistic: • RR: |zobs|  za/2 (2-sided) zobs  za (1-sided) • P-value: 2P(Z|zobs|) (2-sided) P(Z zobs) (1-sided)

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

  19. Associations Between Categorical Variables • Case where both explanatory (independent) variable and response (dependent) variable are qualitative • Association: The distributions of responses differ among the levels of the explanatory variable (e.g. Party affiliation by gender)

  20. Contingency Tables • Cross-tabulations of frequency counts where the rows (typically) represent the levels of the explanatory variable and the columns represent the levels of the response variable. • Numbers within the table represent the numbers of individuals falling in the corresponding combination of levels of the two variables • Row and column totals are called the marginal distributions for the two variables

  21. Example - Cyclones Near Antarctica • Period of Study: September,1973-May,1975 • Explanatory Variable: Region (40-49,50-59,60-79) (Degrees South Latitude) • Response: Season (Aut(4),Wtr(5),Spr(4),Sum(8)) (Number of months in parentheses) • Units: Cyclones in the study area • Treating the observed cyclones as a “random sample” of all cyclones that could have occurred Source: Howarth(1983), “An Analysis of the Variability of Cyclones around Antarctica and Their Relation to Sea-Ice Extent”, Annals of the Association of American Geographers, Vol.73,pp519-537

  22. Example - Cyclones Near Antarctica For each region (row) we can compute the percentage of storms occuring during each season, the conditional distribution. Of the 1517 cyclones in the 40-49 band, 370 occurred in Autumn, a proportion of 370/1517=.244, or 24.4% as a percentage.

  23. Example - Cyclones Near Antarctica Graphical Conditional Distributions for Regions

  24. Guidelines for Contingency Tables • Compute percentages for the response (column) variable within the categories of the explanatory (row) variable. Note that in journal articles, rows and columns may be interchanged. • Divide the cell totals by the row (explanatory category) total and multiply by 100 to obtain a percent, the row percents will add to 100 • Give title and clearly define variables and categories. • Include row (explanatory) total sample sizes

  25. Independence & Dependence • Statistically Independent: Population conditional distributions of one variable are the same across all levels of the other variable • Statistically Dependent: Conditional Distributions are not all equal • When testing, researchers typically wish to demonstrate dependence (alternative hypothesis), and wish to refute independence (null hypothesis)

  26. Pearson’s Chi-Square Test • Can be used for nominal or ordinal explanatory and response variables • Variables can have any number of distinct levels • Tests whether the distribution of the response variable is the same for each level of the explanatory variable (H0: No association between the variables • r = # of levels of explanatory variable • c = # of levels of response variable

  27. Pearson’s Chi-Square Test • Intuition behind test statistic • Obtain marginal distribution of outcomes for the response variable • Apply this common distribution to all levels of the explanatory variable, by multiplying each proportion by the corresponding sample size • Measure the difference between actual cell counts and the expected cell counts in the previous step

  28. 1 2 … c Total 1 n11 n12 … n1c n1. 2 n21 n22 … n2c n2. … … … … … … r nr1 nr2 … nrc nr. Total n.1 n.2 … n.c n.. Pearson’s Chi-Square Test • Notation to obtain test statistic • Rows represent explanatory variable (r levels) • Cols represent response variable (c levels)

  29. Pearson’s Chi-Square Test • Observed frequency (nij): The number of individuals falling in a particular cell • Expected frequency (Eij): The number we would expect in that cell, given the sample sizes observed in study and the assumption of independence. • Computed by multiplying the row total and the column total, and dividing by the overall sample size. • Applies the overall marginal probability of the response category to the sample size of explanatory category

  30. Pearson’s Chi-Square Test • Large-sample test (all Eij > 5) • H0: Variables are statistically independent (No association between variables) • Ha: Variables are statistically dependent (Association exists between variables) • Test Statistic: • P-value: Area above in the chi-squared distribution with (r-1)(c-1) degrees of freedom. (Critical values in Table 8)

  31. Example - Cyclones Near Antarctica Observed Cell Counts (nij): Note that overall: (1876/9165)100%=20.5% of all cyclones occurred in Autumn. If we apply that percentage to the 1517 that occurred in the 40-49S band, we would expect (0.205)(1517)=310.5 to have occurred in the first cell of the table. The full table of Eij:

  32. Example - Cyclones Near Antarctica Computation of

  33. Example - Cyclones Near Antarctica • H0: Seasonal distribution of cyclone occurences is independent of latitude band • Ha: Seasonal occurences of cyclone occurences differ among latitude bands • Test Statistic: • RR: cobs2 c.05,62 = 12.59 • P-value: Area in chi-squared distribution with (3-1)(4-1)=6 degrees of freedom above 71.2 • From Table 8, P(c222.46)=.001  P< .001

  34. SPSS Output - Cyclone Example P-value

  35. Misuses of chi-squared Test • Expected frequencies too small (all expected counts should be above 5, not necessary for the observed counts) • Dependent samples (the same individuals are in each row, see McNemar’s test) • Can be used for nominal or ordinal variables, but more powerful methods exist for when both variables are ordinal and a directional association is hypothesized

  36. 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): • AR=0 • RR=1 • OR=1

  37. 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:

  38. Relative Risk • Estimated Relative Risk: 95% Confidence Interval for Population Relative Risk:

  39. 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

  40. 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)

  41. Example - Concussions in NCAA Athletes There is strong evidence that females have a higher risk of concussion

  42. 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:

  43. Odds Ratio • Estimated Odds Ratio: 95% Confidence Interval for Population Odds Ratio

  44. 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

  45. 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, y1 =9 had OA, n1-y1=59 did not • Of n2= 136 controls, y2 =2 had OA, n2-y2=134 did not Interval > 1 Source: Shepard, et al (2003)

More Related