Count Variables

# Count Variables

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## Count Variables

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1. Poisson & Negative Binomial Regression“Now I've got heartaches by the number,Troubles by the score,Every day you love me less,Each day I love you more” (Ray Price)

2. Count Variables • Number of times a particular event occurs to each case, usually within a given: • Time period (e.g., number of hospital visits per year) • Population size (e.g., number of registered sex offenders per 100,000 population), or • Geographical area (e.g., number of divorces per county or state) • Whole numbers that can range from 0 through +

3. Count DVs • Number of hospital visits, outpatient visits, services used, divorces, arrests, criminal offenses, symptoms, placements, children fostered, children adopted

4. Overview • Poisson regression • Basic model for count DVs • Negative binomial regression • Alternative to Poisson regression • Less restrictive assumptions, and so greater generality

5. Single (Dichotomous) IV Example • DV = number of foster children adopted • IV = marital status, 0 = unmarried, 1 = married • N = 285 foster mothers • Is there a difference in the number of foster children adopted by unmarried and married foster mothers?

6. Distribution of Count DVs • Typically skewed positively with large percentage of 0 values

7. Number of Foster Children Adopted

8. Descriptive Statistics • Table 5.1 • Why is a t-test for independent groups not appropriate here?

9. Strength & Direction of Relationships • Being married increased the mean number of children adopted by a factor of 1.47 (47%) • 1.112 / .754 = 1.47 • 100(1.47 – 1.00) = 47%

10. Question & Answer • Is there a difference in the number of foster children adopted by unmarried and married foster mothers? • Yes. The mean number of children adopted by unmarried mothers is .75 and by married mothers 1.11. So, being married increased the mean number of children adopted by a factor of 1.47 (47%). • But, analysis incorrect because…

11. Exposure • Opportunity for event to occur • Length of time, population size, geographical area, or other domain of interest • Number of years fostering varied across mothers, and so opportunity to adopt foster children varied • Unmarried mothers, M = 8.803 • Married mothers, M = 7.254

12. Rate • Count per unit of… • Time (e.g., number of children adopted per year) • Population (e.g., number of registered sex offenders per 100,000) • Geographical area (e.g., number of children below the poverty rate per state)

13. Rate (cont’d) •  =  / E •  (lambda), mean population rate • Sometimes referred to as the incidence rate •  (mu), mean population count • Sometimes referred to as incidence • E, exposure

14. Rate (cont’d) • Example • rateUnmarried = .754 / 8.803 = .086 • .086 children adopted yearly (rate) • rateMarried = 1.112 / 7.254 = .153 • .153 children adopted yearly (rate)

15. Incidence Rate Ratio (IRR) • IRR = Married / Unmarried • Quantifies the direction and strength of relationship between IVs and DV • Being married increased the yearly adoption rate by a factor of 1.78 (78%) • 153 / .086 = 1.78 • 100(1.78 – 1.00)

16. Incidence Rate Ratio (IRR) (cont’d) • IRR = 1 • Numerator group and denominator group have same incidence rate • IRR > 1 • Numerator group has a higher incidence rate than denominator group • IRR < 1 • Numerator group has a lower incidence rate than the denominator group • Potential range from 0 through +

17. Comparing IRR > 1 & IRR < 1 • Compute reciprocal of one of the IRRs • e.g., IRR of 2.00 and an IRR of .50 • Reciprocal of .50 is 2.00 (1 / .50 = 2.00) • IRRs are equal in size (but not in direction of the relationship)

18. Question & Answer • Is there a difference in the number of foster children adopted by unmarried and married foster mothers? • Yes. The yearly adoption rate for unmarried mothers is .09 and for married mothers .15. So, being married increased the yearly adoption rate by a factor of 1.78 (78%).

19. Poisson Regression

20. Single (Dichotomous) IV Example (ignoring exposure) • DV = number of foster children adopted • IV = marital status, 0 = unmarried, 1 = married • N = 285 foster mothers • Is there a difference in the number of foster children adopted by unmarried and married foster mothers?

21. Statistical Significance • Tables 5.2, 5.3 • Relationship between marital status and children adopted is statistically significant (Wald 2 = 5.846, p = .016) • H0:  = 0,  0,  ≤ 0, same as • H0: IRR = 1, IRR 0, IRR ≤ 0 • Likelihood ratio 2 better than Wald

22. Slope • B = slope • Positive slope, positive relationship • IRR > 1 • Negative slope, negative relationship • IRR < 1 • 0 slope, no linear relationship • IRR = 1

23. Slope (cont’d) • B = .388 • Positive relationship between marital status and children adopted • Married mothers adopt more children

24. IRR & Percentage Change • Exp(B) = IRR = 1.474 • % change = 100(1.474 - 1) = 47% • Married mothers adopt more children • Being married increased the yearly adoption rate by a factor of 1.47 (47%)

25. Poisson Model • ln() = α + 1X1 + 1X2 + … kXk, or • ln() =  • ln(), log of mean count (“log link”) • e.g., log of mean number of children adopted • , abbreviation for linear predictor (right hand side of this equation) • k = number of independent variables

26. Inverse (reverse) Link •  = e •  is the mean count • e.g., mean number of children adopted

27. ln() to  ln(mean) = -.282 + (.388)(XMarried) • Single mothers • ln(mean) = -.282 + (.338)(0) = -.282 • mean = e-.282 = .754 • mean = .75 children adopted • Married mothers • ln(mean) = -.282 + (.388)(1) = .106 • mean = e.106 = 1.112 • mean = 1.11 children adopted

28. Question & Answer • Is there a difference in the number of foster children adopted by unmarried and married foster mothers? • Yes. The mean number of children adopted by unmarried mothers is .75 and by married mothers 1.11. So, being married increased the mean number of children adopted by a factor of 1.47 (47%). • But, analysis incorrect because…

29. Single (Dichotomous) IV Example (with exposure) • Use SPSS to create an “offset” variable • Natural log of the exposure variable • Exposure variable must be > 0 • compute lnYearsFostered = ln(YearsFostered). • Enter offset variable into the regression analysis

30. Statistical Significance • Tables 5.4, 5.5 • Relationship between marital status and yearly adopton rate is statistically significant (Wald 2 = 13.131, p < .001)

31. IRR & Percentage Change • Exp(B) = IRR = 1.789 • % change = 100(1.789 - 1) = 79% • Married mothers adopt more children per year • Being married increased the yearly adoption rate by a factor of 1.79 (79%)

32. ln() to  ln(rate) = -2.457 + (.582)(XMarried) • Single mothers • ln(rate) = -2.457 + (.582)(0) = -2.457 • rate = e-2.457 = .086 • .09 children adopted yearly (rate) • Married mothers • ln(rate) = -2.457 + (.582)(1) = -1.875 • rate = e-1.875 = .153 • .15 children adopted yearly (rate)

34. Question & Answer • Is there a difference in the number of foster children adopted by unmarried and married foster mothers? • Yes. The yearly adoption rate for unmarried mothers is .09 and for married mothers .15. So, being married increased the yearly adoption rate by a factor of 1.79 (79%).

35. Single (Quantitative) IV Example • DV = number of foster children adopted • IV = Perceived responsibility for parenting (scale scores transformed to z-scores) • Offset variable = log of years fostered • N = 285 foster mothers • Do foster mothers who feel a greater responsibility to parent foster children adopt more foster children?

36. Statistical Significance • Tables 5.6, 5.7 • Relationship between parenting responsibility and yearly adoption rate is statistically significant (Wald 2 = 10.045, p = .002)

37. IRR & Percentage Change • Exp(B) = IRR = 1.202 • % change = 100(1.202 - 1) = 20% • Mothers with greater parenting responsibility adopt more children per year • For every one-standard deviaiton increase in parenting responsibility the yearly adoption rate increases by a factor of 1.20 (20%)

38. ln() to  ln(rate) = -2.008 + (.184)(XzParentRole) • e.g., mean parenting responsibility (z = 0): • ln(rate) = -2.008 + (.184)(0) = -2.008 • rate = e-2.008 = .13 • .13 children adopted yearly (rate)

39. Figure • zParentRole.xls

40. Question & Answer • Do foster mothers who feel a greater responsibility to parent foster children adopt more foster children? • Yes. For every one-standard deviation increase in parenting responsibility the yearly adoption rate increases by a factor of 1.20 (20%). The yearly adoption rate is .09 for mothers two standard deviations below the mean, .13 for mothers with the mean, and .19 for mothers two standard deviations above the mean.

41. Multiple IV Example • DV = number of foster children adopted • IV = Perceived responsibility for parenting (scale scores transformed to z-scores) • IV = marital status, 0 = unmarried, 1 = married • Offset variable = log of years fostered • N = 285 foster mothers • Do foster mothers who take more responsibility for parenting adopt more foster children per year, controlling for marital status?

42. Statistical Significance • Table 5.8 • Relationship between set of IVs and yearly adoption rate is statistically significant (2 = 27.792, p < .001) • H0: 1 = 2 = k = 0, same as • H0: IRR1 = IRR2 = IRRk = 1

43. Statistical Significance • Table 5.9 • Relationship between parenting responsibility and yearly adoption rate is statistically significant, controlling for marital status (2 = 11.853, p = .001) • Relationship between marital status and yearly adoption rate is statistically significant, controlling for parenting responsibility (2 = 16.520, p < .001)

44. Statistical Significance • Table 5.10 • Relationship between parenting responsibility and yearly adoption rate is statistically significant, controlling for marital status (Wald 2 = 11.576, p = .001) • Relationship between marital status and yearly adoption rate is statistically significant, controlling for parenting responsibility (Wald 2 = 14.433, p < .001)

45. IRR & Percentage Change: Parenting Responsibility • Exp(B) = IRR = 1.219 • % change = 100(1.219 - 1) = 22% • Mothers with greater parenting responsibility adopt more children per year, controlling for marital status • For every one-standard deviaiton increase in parenting responsibility the yearly adoption rate increases by a factor of 1.22 (22%), controlling for marital status

46. IRR & Percentage Change: Marital Status • Exp(B) = IRR = 1.842 • % change = 100(1.842 - 1) = 84% • Married mothers adopt more children per year, controlling for parenting responsibility • Being married increased the yearly adoption rate by a factor of 1.84 (84%), controlling for parenting responsibility

47. ln() to  ln(rate) = -2.498 + (.198)(XzParentRole) + (.611)(XMarried) • e.g., mean parenting responsibility (z = 0) and unmarried mothers: • ln(rate) = -2.498 + (.198)(0) + (.611)(0) = -2.498 • rate = e-2.498 = .08 • .08 children adopted yearly (rate)

48. Figure • Married & zParentRole.xls

49. Effect of Standardized Parenting Responsibility and Marital Status on Adoption Rate