Data Day 2012_Barboza_Data Storytelling

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Data Day 2012_Barboza_Data Storytelling

  1. 1. Storytelling with Data Gia Elise Barboza, JD, PhD Assistant Professor Department of African American Studies and Health Science Northeastern University
  2. 2. The Research Approach• The scientific method o Theory (rational choice theory) o Hypothesis (a deficit approach) o Data Analysis (incorrect method) o Empirical Observation (cognitive bias) o Research Conclusions (illogical conclusion)• Reclaim our story using data o When? o Why? o How?
  3. 3. Take-Aways• Disaggregation leads to very different conclusions (this is Simpson’s Paradox)• Numbers are social constructs, the same data can be “used” to show the exact opposite results (this is NOT lying with stats) o Know which method leads to invariance• You must not rely on one data source to tell a story, stories are woven together via multiple sources of data o Quantitative (census, surveillance) and qualitative data (focus group, interview)
  4. 4. Why Reclaim Your Story?• Existing uses of, lack of access to and manipulation of data harms communities particularly on the basis of o Race; o Gender; and/or o Sexual orientation?• Research & data analyses often … o Are biased o Ignore community input • Research is done “on” the “community” not in it o Manipulated and (mis)used for political purposes• To reclaim our story its essential to be numerate and to understand limitations of research and data o Policy outcomes
  5. 5. When to Reclaim Your Story?• When to reclaim 1. The data shows no disparity but I want to use the same data to show that a disparity indeed exists 2. The data seem accurate but there is another side to the story (similar to (1) but different data are used) 3. My intuition does not match the data & the story being told is inaccurate (the data is being manipulated)
  6. 6. How to Reclaim?• Understand data “issues”• What is the story you are trying to reclaim? Three basic story lines to reclaim o Story 1 [the data shows no disparity but I want to use the same data to show that a disparity indeed exists]: State accounts of the impact of health care reform legislation do not address the needs of certain groups and disparities in access to insurance coverage remain o Story 2 [the data is reasonable but there is another side to the story]: Unravel the image of the welfare queen as an inaccurate portrayal of unwed motherhood and teenage pregnancy (among other things) o Story 3 [my intuition does not match the data]: Reclaiming the real story behind the Boston “miracle” by arguing that youth empowerment was responsible for the decrease in crime
  7. 7. Story 1• Story 1 [The data shows no disparity but I want to use the same data to show disparity exists]• In order to understand this story, we have to understand proportions, percents, odds and odds ratios and how they are easily manipulated o They are also the most misunderstood and misapplied o Helps get a sense of how data are social constructs o Involves understanding different interpretations of disparity using proportions, percents, odds and odds ratios
  8. 8. Data Issues: Is there a Disparity?• Many issues involve a comparison between two percents (or proportions/ probabilities). Consider the example of a test taken by men and women on which men are observed to have a higher pass rate.• How would you describe the disparity in pass (or fail) rates (and what are the pros and cons of each way)?1. The difference in pass (fail) rates (the “gap”)2. The ratio of the pass (fail) rates3. The odds ratio
  9. 9. The difference in pass(fail) rates• Assume that the passage rate for men is 97% and the passage rate for women is 90% o What is the difference in pass rates?; What is the difference in fail rates? o If 100 persons take the test, 70% of them female, how would you compute the number of women adversely affected by the exam? • Using a 7% difference I can show you that there will be about 5 women adversely affected (.07 * 70) • If there are 1000 people who take the exam, 50 women are adversely affected • If 10000 people take the exam, 500 women are affected o When impact is an issue, this is the best method to use o The problem is that a 7% difference between 97% and 90% should not be treated the same as a 7% point difference between 10% and 3% since the latter evinces a much larger disparity (the next slide shows why)
  10. 10. The ratio of the pass/fail rates • Assume that the passage rate for men is 97% and the passage rate for women is 90% o What is the ratio of the pass rates? • Men are 1.07 times as likely to pass? .97 = 1.07 .90 o What is the ratio of the fail rates? • Interpretation? .10 = 3.33 .03Interpretation? Which statistic would you use toshow disparate impact?
  11. 11. Odds Ratio of the Pass (Fail) Rates• The odds ratio of the pass rates is defined as the odds of passing for men divided by the odds of passing for women.• Calculate the odds ratio in pass rates o First calculate the odds of passing for men pM .97 .97 Odds = = = = 32.3: 1 1  pM 1  .97 .03 • Interpretation? o Second calculate the odds of passing for women pW .90 .90 Odds = = = = 9: 1 1  pW 1  .90 .10 Divide the two • Odds ratio = 32.3/9 = 3.6
  12. 12. Odds Ratio of the Pass (Fail) Rates• The odds ratio of the fail rates is defined as the odds of failing for women divided by the odds of failing for men.• Calculate the odds ratio in fail rates o First calculate the odds of failing for women pW .10 .10 1 Odds = = = =  1: 9 1  pW 1  .10 .90 9 • Interpretation? o Second calculate the odds of failing for men pM .03 .03 3 Odds = = = =  3 : 97 1  pM 1  .03 .97 97 o Divide the two 1 • Odds ratio = 9 = 97 = 3.6 3 27 97
  13. 13. Two Quick Examples• MCAS proficiency rates• Infant Mortality rates
  14. 14. MCAS proficiency by Race/Ethnicity
  15. 15. Infant Mortality Survival RatioYear White Black Ratio (B/W) White Black (B/W)1992 5.9 19 3.22 994.1 981 0.991993 5.9 15 2.54 994.1 985 0.991994 7.2 12.5 1.74 992.8 987.5 0.991995 4.7 11.9 2.53 995.3 988.1 0.991996 6.7 9.9 1.48 993.3 990.1 1.001997 9.5 12.8 1.35 990.5 987.2 1.001998 4 12 3.00 996 988 0.991999 5.6 13.5 2.41 994.4 986.5 0.992000 2.8 13.6 4.86 997.2 986.4 0.992001 5.1 13.5 2.65 994.9 986.5 0.99
  16. 16. Health Insurance Disparity• Massachusetts is often touted as having the lowest uninsurance rate in the nation o This is true! o BUT, I want to show disparity remains by race using the same data the state uses, in particular that • Coverage for some groups is as bad as it is in the state with the worst coverage rate, Texas • For some groups, things are actually worse after the legislative reforms in 2006 • The health care gap has at best remained the same for some groups
  17. 17. Empirical evidence for the success of health care reformin MA in 2006
  18. 18. “Among major subpopulations, the largest increases were observed among Hispanics (14.2%),persons with less than a high school diploma (12.0%), and persons making <$25,000 (11.9%).” Based on Number Insured Based on Number Uninsured Pre-law Post-law % change Pre-law Post-law % change White, non-Hispanic 93 97.3 4.6 7 2.7 61.4 Black, non-Hispanic 88.2 92.7 5.1 11.8 7.3 38.1 Hispanic 77.9 89 14.2 22.1 11 50.2 Asian 90.5 98.4 8.7 9.5 1.6 83.2 English 84.6 93.3 10.2 15.4 6.7 56.5 Spanish 69.1 81.8 18.4 30.9 18.2 41.1Less than high school diploma or GED 79.1 88.6 12.0 20.9 11.4 45.5 At least high school diploma or GED 92.2 96.8 5.0 7.8 3.2 59.0 < $25,000 79.5 89 11.9 20.5 11 46.3 $25,000 – $74,999 91 96.2 5.7 9 3.8 57.8 >= $75,000 97.4 99.4 2.1 2.6 0.6 76.9
  19. 19. Recall Hispanics had the largest gains! Table 1. Persons Aged 18 – 64 by Health Insurance Coverage andIn 2010, the Health Status by Latino Origin, Massachusetts, 2003 & 2010 (Annualpercent of Social and Economic Supplement (ASEC))foreign- No Health Health Status:born non-citizens Insurance (%) Poor (%)without Massachusetts 10.8 1.6insurance Non-Latino 10.1 1.5coverage is 2003 All Latino 17.9 2.6actually Foreign Born 27.9 3.6greater than US Born 8.1 1.7it was in Foreign Born Non-Citizen 41.0 3Texas,which is thestate with Massachusetts 5.2 1.8the lowest Non-Latino 4.4 1.4overall 2010 All Latino 12.5 5.4coverage Foreign Born 15.5 9.6rate. US Born 9.8 1.5 Foreign Born Non-Citizen 30.4 8.7
  20. 20. How many additional Latinos actually need to be covered to achieve parity with the general population?Table 4. Total Population, Total Population Uninsured, Percentage and Number of Individuals needed toachieve parity with State Overall Coverage Rate by Ethnicity, 2003 to 2010 Number of Percentage of Additional Additional Number Individuals Coverage Needed to Number Who Need Total Population (Drop in Achieve Uninsured Coverage in Percent State order to Uninsured Percent achieve state since 2003) drop in % 2003 2010 2003 2010Massachusetts 5,615,372 5,621,910 606,460 292,339 0.482 -- --Non-Latino 5,123,402 5,076,744 517,464 223,377Latino 491,970 545,166 88,063 68,146 0.774 42,449 25,695 US born 248,274 285,810 20,110 28,009 1.390 9,693 18,315 Foreign born 243,696 259,357 67,991 40,200 0.591 32,774 7,425 Foreign born non-citizen 131,574 95,170 53,945 28,932 0.536 26,003 2,927
  21. 21. If the red line is above the blue line the gap relative to whitesis worse in the post-law period! The health care gap is worse for blacks comparedto whites
  22. 22. Story 2: The demonization of black female sexuality• This is a story about how we use outliers to represent a group BUT the value we assign to the outlying behavior differs according to race (& gender) o When behavior is done by blacks is bad, we conclude all blacks do it, and engage the blame frame to impute moral failure o When behavior done by whites is bad, we excuse it as an aberration• Popular notions regarding black women, sexuality, and welfare are hypocritical at best o I want reclaim the story by exposing the hypocrisy around this issue. But how? • Juxtapose popular images; and • Disaggregate data around this issue • Flip the script on morality
  23. 23. “Tangle of Pathology?” Who Do We Value? 72% of black babies are born to unwed mothers
  24. 24. Who’s the Baby Daddy?• As of 2010… o “The black communitys 72 percent rate eclipses that of most other groups: 17 percent of Asians, 29 percent of whites, 53 percent of Hispanics and 66 percent of Native Americans were born to unwed mothers in 2008, the most recent year for which government figures are available. The rate for the overall U.S. population was 41 percent.”• In the 60s, when Moynihan characterized black family life as a “tangle of pathology” the black “illegitimacy” rate was 24%! (29 > 24!) o Logic dictates that the pathology still exists today… and includes whites.
  25. 25. Reinforcing the Blame Frame71.4%, 54.6% and 16.3% ofblack, Hispanic and whitewomen, respectively, who were pregnant were not married The percentage of whitepregnant women aged 18– 24 who have either never been married or who are unmarried but in a relationship isapproximately the same as that for black pregnant women overall, 70%! The right question…Blaming one group for their own failure seems to be a constant theme…
  26. 26. Same behavior, different result?• Next, consider the percentage of women who have either never been married or who are unmarried but in a relationship who report having EVER used contraception after having unprotected sex.• More than 50% of white women used contraception after unprotected sex compared to only 16% of black women.• Whites engage in the same behavior, they just have different observable results o But what is more immoral? Having a baby out of wedlock or using the morning after pill to induce abortion. Other arguments suggest abortion is worse…
  27. 27. • In another story the Boston Globe reported the teen birthrate fell to a record low, dropping 6% from 2008 to 39.1 births per 1000 in 2009.• “The decline in teen births is really quite amazing”!!! Let’s see just how amazing it is…“Rates fell significantly for all race and Hispanic origin groups between 2008 and2009, with declines ranging from 4 to 6 percent (for non-Hispanic white, non-Hispanic black, and ASIAN teenagers). The rate for Hispanic teenagers aged 15-19fell 10 percent in 2009 to 70.1 births per 1,000, the lowest rate ever reported for thisgroup in the two decades for which rates for Hispanic teenagers are available. Therate for API teenagers dropped 10 percent. Rates for all groups reached historiclows [2].” http://www.cdc.gov/nchs/data/nvsr/nvsr59/nvsr59_03.pdf
  28. 28. Dispositionism, Situationism or Some Combination of Both?: Whites’ Perceptions of Welfare Policy and Attitudes towards the Poor, Barboza (2011).Table 2. Attitudes Towards Welfare Recipients Among Individuals Who Believe That Most Welfare• There is a significant relationship between theRecipients Are Black Most Welfare Most Welfare Recipients Are perceived welfare recipients race and Recipients Are Black WhiteIn your opinion, do you think that most people who receive money from welfare today could get• A belief that people on welfare are morally differentalong without it if they tried, or do you think that most of them really need this help? than those not on welfare. Get along without it 50.2 45.5 Really need the help 49.8 54.5 o Individuals who perceive welfare recipients as black are significantly more likely to claim that welfare recipients have lower moral valuesIn general, do you think people on welfare have higher, lower, or about the same moral values as• A belief that welfare recipients do not really want toother Americans? Higher 2.3 2.9 work. Lower 35.5 24.3 o Individuals who perceive welfare recipients as black are more likely to About the same 62.2 72.8P = .001 claim that welfare recipients don’t want to work.Do you think that most welfare recipients today really want to work or not? Yes 42.6 59.4 No 57.4 40.6P < .000
  29. 29. It’s all really smoke and mirrors… Living with mother ‘This is a defining time for • 291 75.2 America,” says Mitt Romney. In labor force President Barack Obama, he78.7 229 Not in labor force argues, wants to turn 62 21.3 America into a “European-Source: 2005-2009 American Community Survey 5-Year Estimatessociety” in like entitlement which “government providesSurvey: American Community Survey Geographic Area: Census Tract 902, SuffolkCounty, Massachusetts every citizens with the same or similar rewards.” In contrast, Romney supports “an opportunity society, free people living under a limited government.” • Read more: http://www.politico.com/new“I come from a single family household. My mother was a single mother of 3 andshe had 3 jobs. She was able to go to RCC so we s/stories/0112/71791.html#ixzz grew up a lot by ourselves. I started 1ka0UKvU3 extremelyworking at the age of 13. I worked all through college…. I wasindependent.” (Focus group, January, 2011).
  30. 30. Story 3• Story 3: Based on your experience, the “data” being presented is wrong, I want the data to reflect my intuition• Story 3 illustrates: o How to use qualitative data to validate quantitative findings o That relying on existing data leads to the reification of misinformation o How overly simple analyses and a lack of depth leads to incorrect conclusions
  31. 31. The Problem of Youth Violence• We are retelling the story of the Boston miracle o Operation ceasefire & the 10-point coalition o In reclaiming the story, we show that these factors usually attributed to the decrease in youth violence in the 1990s are actually responsible for increases in youth violence today o but how?
  32. 32. Steps used in reclaiming the story• Define the crisis o Start at the beginning and the end -- with existing accounts of the decline in youth violence in the 1990s and the upsurge in youth violence in the 2000s o Use data in a way that is impactful• Reframe the issue as something to be solved by using a positive youth development framework o Focus on prevention• Show the impact of PYD on the problem of youth violence both o Show how PYD impacted crime in the 1990s o Show how it impacts crime today using community level inputs• Private funding is essential o Argue that the reason for the increase is the lack of federal funding and that we need private foundation money in order to solve the problem
  33. 33. The :: Miracle• Between 1996 and 2003, Boston experienced a drop of almost 90% in violent acts among youth, with shootings decreasing from a high of 550 in 1990 to a low of 133 in 1997.• Then, in 2003 a reversal in this pattern began. By 2004, there were almost three times as many shootings as there had been in 1997.• Where are we now?
  34. 34. Defining the Problem of Youth Violence• Homicide ranks as the 7th leading cause of death for both blacks and Hispanics but only 30th among whites. o The death rate from firearms is more than five times higher for black males than it is for white males (23.5 per 100,000 vs. 4.3 per 100,000).
  35. 35. Youth* Violence in Boston1 Young Black Male Dies Every Two Weeks in Young Blacks Boston* Ages 15 - 24
  36. 36. Youth Development in the Workplace Mentoring1. Mentoring Public2. Youth Workforce Development andand Supports Education Family Awareness & Policy3. Family Supports and Mental Health Initiatives Mental Health4. Conflict Resolution and Social Skill Development5. Community Capacity-Building Community Conflict Youth Resolution and6. Public Awareness & Policy Capacity- Workforce Initiatives Social Skill Building Development DevelopmentBut where’s the leverage point??
  37. 37. Table 4: Predicted Probability of Youth Employment By SelectedCharacteristics Holding Parental Work, School Enrollment and PovertyStatus Constant Black White Hispanic Male Female Male Female Male FemaleOverall (16-24) .35 .38 .51 .54 .45 .4816 .15 .16 .25 .27 .21 .2317 .19 .21 .31 .33 .26 .2818 .24 .26 .38 .41 .33 .35Below Poverty .23 .26 .37 .40 .32 .34LineAbove Poverty .38 .41 .54 .57 .48 .51LineSource: 2008-10 American Community Survey, Census Bureau
  38. 38. Newt is Right! “Inner city” youth should be working!Clean Toilets? Research Assistant at NEU?
  39. 39. Perceptions of Positive Role ModelsTable ES.6 Categories of Positive Role Models in Youth Lives • Youth who perceive Gang Professionals Family & Members & professionals and Friends Drug Dealers family & friends toPeers .488Mother .825 be positiveFather .762 influences areTeacher .633Pastor or Church Leaders .552 significantly lessThe Police .602 likely to engage inHealth Care Professionals .693Social Workers .817 bad behaviorStreet Workers .714Drug Dealers .880Gang MembersExtraction Method: Principal Component Analysis. .887 • Youth who perceiveRotation Method: Varimax with Kaiser Normalization. gang members to be positive influences are significantly more likely to engage in bad behavior.
  40. 40. Can’t just remove negative you must replace it with positive behaviors Kaplan-Meier Survival Estimate of Gang Membership Kaplan-Meier Survival Estimate Gang Exit 1 1 .75 .75 Survivorsip Probability .5 .5 .25 .25 0 0 0 5 10 15 20 0 5 10 15 20 Analysis Time (Age Became Gang Member) Analysis Time (Age Left Gang) 95% CI Survivor function 95% CI Survivor functionWhy get involved with a gang? “There are constant let down[s]. If I can’t resort to the people around me, why have them aroundme? Frustrated, don’t know who to turn to, so I turn to the street, someone owes me…”Why turn to the street? Because “being part of something is better than not being a part of anything.” This theme of wanting to belong wasechoed by many, gang involved or not. Survey results indicate that 16.3% of youth feel the need to rep their hood to others, 11% said they havebeen or currently are involved in a gang, and 22% said they are associated with a gang. 1 in 4 current or former gang members said theybecame involve between the ages of 8 and 10 years old.And this story by a former gang member corroborates the empirical evidence, “I joined a gang at 9 years old, I had my first tattoo at 9 years old, I always wanted to be a part of something, they took care ofpicked up my first gun at 9 years old.me, made sure I had sneakers, made sure I went to school and had money in mypocket.” When I commented that the gang had a positive role despite the fact it came with negative behaviors, his response to me was,“Nothing is free. That’s what I learned at an early age.”
  41. 41. How did employment change behavior? Post-survey FindingsIn general, do you think this In general, do you think thisprogram has helped you?... program has helped community 60% of youth say that either their age or having Open up new doors for by?... a CORI is the biggest 84% barrier the y face in your future? Providing People with acquiring employment 60% Learn about others’ Opportunities and only 13% say that experiences and how to there are a lot of high 88% Providing Role Models for quality jobs available for respect differences of 25% teens. Nevertheless, only opinion Children and Young Adults 11% of youth taking the Approach a problem by post-survey believed that the application process communicating without Giving People Money 19% was “extremely hard” 79% while 42% claimed it was anger to come up with a “not too hard” or “not good solution Providing the Labor to hard at all.” Moreover, 14% the majority of youth From hanging around in Clean up the Community received a job offer the street and being 76% Making People in the within 3 months. 13% unsafe Community Safer Providing new Providing people in the Connecting the experiences, empathy, community with Disconnected to problem solving, opportunities and a Job quickly providing a safe place access to role models and easily
  42. 42. The 6C’s of Youth DevelopmentConnection Competence Character Confidence Caring Contribution Organizational Opportunities to Sense of belonging Community Mentoring Empathy Decision-making learn and grow and responsibility participation Constant Meaning of Hard Positive identity Learning about Civic Skill building Responsibility supervision Work development others’ experiences Efficacy, self- Three-tiered Community Higher levels of Understanding worth, positive mentoring Engagement thriving “self” view of the future “I learned School Engagement how to get with “I need to “Makes me feel others” “Being a“Being part of it.” work on my good to help role model patience in people who need for a “I was told dealing with it.” younger Im good in work tasks.” girl.” math and I never was before.”
  43. 43. Table 1. Quotations Demonstrating the Impact of Employment on Positive YouthDevelopment Behaviors Youth Want to Program Impact on Youth Behaviors2 Change1PYD Concept “Id change the people I hang “Being part of it!”; “ out with.” “Being a role model for a younger girl.”Connection (a sense of “We want people to listen to “On this job, you get to communicate with safety, structure, and us and hear our struggles.” new people.” belonging; positive “I’m on my own, have to do “Feels real good to get stuff off our chestsbonds with people and it all myself.” and just vent” social institutions) “Taking time to know me” “This job kept me off the streets.”1Allparticipating youth were asked, “If you could change one thing about yourself, what would it be?” In addition, BHS youth were asked, “What doyou want to get out of this program?” 2 On the post-survey, all youth were asked to provide an open-ended response to the following, “What is the mostrewarding thing that happened to you this summer?” (q21). In addition, focus group respondents were asked to describe the mai n benefits ofemployment. Youth want to change, and the emphasis on PYD in an employment context allows them to begin to change their attitudes and behaviors…
  44. 44. Relationship Between Meaningful Employment and YVPCauses of Youth Crime Positive Youth Development Quotations Illustrating Concept the Relationship between PYD, YVP and MYERelationship issues “I think it’s the peopleNegative Socialization they put in front of us… Connectionw/out positive socialization my worksite supervisor,or role models she’s been there…” PYD concepts are linked directly to what the community attributes as being the causes of youth crime. All this to say that meaningful employment can change behaviors associated with criminogenic risk, but what does this have to do with the Boston miracle?
  45. 45. Rethinking the “Miracle”: Finding Unity and Peace in Boston – Replacing the Miracle with a Methodical Approach to Reduce Youth Violence, Barboza, et al. (2012)• It “ is not that the Boston model of the 1990s has failed, but rather that the City of Boston and the Boston Police failed to pursue the policies and practices that had been so successful during the late 1990s.” (Braga, et al. 2008). o The largest influx of federal funds in the history of the Commonwealth o An economy conducive to youth employment o An unprecedented amount of collaboration across all sectors including religious, family, schools, criminal justice, social service, health and non-profit organizations Massachusetts Arrest Rate for Violent Crime, 1986 – 2007 (Source: Fox) o By 2003 there were “dramatic cuts in community programs aimed towards at-risk children” (Fox, 2008). o More arrests of youth under 18 during a 20 year period o The number of black males incarcerated over this time period skyrocketed o This created a short-term moratorium on violence, the large number of well-funded youth development programs for at-risk youth -- is the real story behind the Boston miracle.
  46. 46. A call for Public Funding…• In a 2004 public opinion poll of African Americans, respondents were asked their opinion on a variety of social and political issues thought to be of particular salience to blacks living in the United States. o Not a single black respondent who was asked to describe the most pressing issue that a presidential candidate should discuss during the presidential campaign selected an answer that pertains to youth: youth crime/violence/gangs, drugs/youth, youth (other), parenting or lack of discipline, or youth values/respect. o Top responses included jobs/unemployment, the economy, health care, and war. This is best attributed to the fact that 41% said they were “very concerned” that in the next year they or someone in their household will be unemployed and looking for a job• These same adults were asked to state the one thing that would be most impactful for reducing youth crime and violence. o Most often cited was the importance of community programs, followed by holding parents legally responsible for their children’s actions. o Programmatic activities and family support are clearly viewed as important to reduce youth violence even though youth violence itself is not viewed as being an important issue for government to address. o Since politicians are responsive to their constituents, it seems likely that alternative modes of funding are essential to combat youth violence in the long run.
  47. 47. A Primer: Odds and Odds Ratios• The odds of something occurring (versus not occurring) is defined as the probability of it occurring divided by the probability it does not occur p Odds= 1− p• Example: Let the probability of rain be 1/4. What are the odds of observing rain (vs. not rain)?
  48. 48. Odds and Odds Ratios• Example: Let the probability of rain be 1/4. What are the odds of observing rain (vs. not rain)? 1 1 Odds = 4 1 1 4 = == 3 :   1 3 3  1    4 4• Interpretation: for every day it rains it will not rain three days
  49. 49. Relationship between probability and odds Probability: Probability = 1/4 Number of days it will rain over total number of days Odds: Number of days it will rain to the number of days it will NOT rain Rain Not rain
  50. 50. Odds and Odds Ratios• Odds Ratio: the ratio of two odds p1 Odds1 1− p1 Odds Ratio= = Odds 2 p2 1− p 2Interpretation: the odds of an event occurring in Group 1 versus the odds of it occurring in Group 2

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