Technical Appendix: Methodology for Calculating Odds Ratio from Probits

If we were to leave the results in probit form, some readers may not be satisfied by just 38

knowing the direction and significance of the differences in probability. Rather, these readers 39

may be satiated by a better understanding of the comparisons of two probabilities in the form of 40

Odds Ratios – particularly, epidemiologists and other public health practitioners and 41

researchers more familiar with Odds Ratios than probits. This author agrees that such 42

comparisons are useful and, perhaps, practical. 43

Selecting a standard subject was necessary in order to compute and compare the odds of 44

worse health. We were interested in assessing the odds of worse health through the relative 45

direct effect of bicyclist typology and, separately, the odds of worse health through the relative 46

indirect effect of physical activity. Though the standard subject could have been arbitrarily 47

selected, ours was selected based on 1) among categorical covariates, the category with the 48

greatest number of respondents and 2) the mean age. This resulted in choosing a standard 49

subject who was female, white, Non-Hispanic, student, who had not commute bicycled during 50

past semester and who was 30 years old. Though our calculations revolve around this single 51

subject, they could reasonably be replicated for other subjects who may be of interest to 52

different parties or stakeholders (i.e. using the mean age of students instead of the entire 53

sample). 54

Our outcome variables were all ordinal. For this reason, we had to decide on a consistent 55

threshold to use for each outcome. Regarding General Health Status, we decided to estimate the 56

probability of good, fair, or poor self-reported health versus very good or excellent self-reported 57

health. This decision was made for at least two reasons. First, only a small proportion of the 58

sample appeared to think they were of either Fair (6.5%) or Poor (0.5%) health. Second, we 59

sought to conceptually distinguish persons who perceived themselves of being of a better health 60

status to persons who perceived themselves as not being one of the better health statuses. For all 61

of the Unhealthy Days measures, we estimated the probability of more than three unhealthy 62

days during the past 30 days. Again, this decision was two-fold. First, the vast minority of 63

participants reported eight or more unhealthy days for two of the measures (Physically 64

Unhealthy Days and Activity Limited Days). Second, we sought to distinguish between persons 65

who appear did not seem to experience much of a burden due to health and may have reported 66

only a few unhealthy days for any of the items and those who seemed to be somewhat more 67

burdened by poor health and may have reported at least four unhealthy days. 68

For the estimation of the probits, there had to be a reference group selected. We chose to 69

show the results when using the Strong and Fearless as the reference group throughout the 70

manuscript. Results when the other bicyclist typologies served as the reference group are 71

shown in Supplemental Tables 3, 4a-4d, and 5. The Strong and Fearless was selected as the 72

reference group of interest mainly due to health expectations for this bicyclist typology. We 73

were interested in estimating the probability and odds of worse health. To aide in the 74

interpretability, we wanted the results to consistently be relative to the group we expected to 75

have the best health outcomes. Also, by choosing this group as the referent, we can primarily 76

produce Odds Ratios that are greater than one, which are generally more easily understood by 77

the anticipated audience. 78

Bryan, J. Michael 111

After establishing a standard subject and determining what we were estimating the 79

probability of, we had to establish the calculations of marginal effects. We used Muthen’s 80

methodology for estimating the total indirect effect and direct effect in the form of odds ratios 81

[48]. The general procedure was: 82

1. Calculate the standard normal distribution value by solving the estimated 83

equation(s) for each equation, 84

2. Calculate the probability of the outcome of interest from the standard 85

distribution value, 86

3. Calculate the odds of the outcome of interest, and 87

4. Calculate the odds ratios specific to the indirect effect and direct effect. 88

The first condition, Φ(1,1), estimated the z-score of the outcome of interest for standard 89

subjects who performed physical activity three or more times during the past week and 90

identified as the bicyclist typology of interest, keeping in mind that these calculation were 91

relative to the bicyclist typology serving as the reference group. The equation below depicts 92

how we solved for this first condition: 93 𝐶𝐷𝐹(𝐵𝑖𝑐𝑦𝑐𝑙𝑖𝑠𝑡 𝑇𝑦𝑝𝑜𝑙𝑜𝑔𝑦 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡: 1,1) = |𝛽₀₁+ 𝛽₁𝑋₁+ 𝛽₂𝑋₂+ 𝛽₃𝑋₃+ 𝛽₄𝑋₄+ 𝛽₅𝑋₅+ 94 𝛽₁₀𝑋₁₀+ 𝛽₁₁𝑋₁₁+ 𝛽₁₂(𝛾𝑝₀₂+ 𝛾𝑝₁𝑋₁+ 𝛾𝑝₂𝑋₂+ 𝛾𝑝₃𝑋₃+ 𝛾𝑝₄𝑋₄+ 𝛾𝑝₅𝑋₅+ 𝛾𝑝₁₀𝑋 + 𝛾𝑝₁₁𝑋) + 95 𝛽₁₃(𝛾𝑐₁𝑋₁+ 𝛾𝑐₂𝑋₂+ 𝛾𝑐₃𝑋₃+ 𝛾𝑐₄𝑋₄+ 𝛾𝑐₅𝑋₅+ 𝛾𝑐₁₀𝑋₁₀+ 𝛾𝑐₁₁𝑋₁₁)|/√𝛽₁2𝜎₂2_{+ 1, where} 96

CDF = Standard Normal Distribution value (z-score). 97

𝛽₀₁ = Threshold value for calculating probability of previously described outcome of 98

interest. 99

𝛽₁, 𝛽2, 𝛽3, 𝛽4, 𝛽5= parameter estimate for regressing outcome of interest on each bicyclist

100

typology. A value of 1 would be entered for X only for the bicyclist typology of 101

interest and a value of 0 for all other bicyclist typologies. 102

𝛽10 = parameter estimate for regressing outcome of interest on age. This value was

103

multiplied by 30 for all analyses. 104

𝛽₁₁ = parameter estimate for regressing outcome of interest on student-employee status. 105

This value was multiplied by 1 for all analyses. 106

𝛽12 = parameter estimate for regressing outcome of interest on physical activity.

107

𝛾𝑝02 = threshold value for previously described physical activity level of interest.

108

𝛾𝑝₁, 𝛾𝑝2, 𝛾𝑝3, 𝛾𝑝4, 𝛾𝑝5 = parameter estimate for regressing physical activity on each

109

bicyclist typology. A value of 1 would be entered for X only for the bicyclist 110

typology of interest and a value of 0 for all other bicyclist typologies. 111

𝛾𝑝10 = parameter estimate for regressing physical activity on age. This value was

112

multiplied by 30 for all analyses. 113

𝛾𝑝₁₁ = parameter estimate for regressing physical activity on student-employee status. 114

This value was multiplied by 1 for all analyses. 115

𝛽₁₃ = parameter estimate for regressing outcome of interest on commute bicycling. 116

𝛾𝑐₁, 𝛾𝑐2, 𝛾𝑐3, 𝛾𝑐4, 𝛾𝑐5 = parameter estimate for regressing commute bicycling on each

117

bicyclist typology. A value of 1 would be entered for X only for the bicyclist 118

typology of interest and a value of 0 for all other bicyclist typologies. 119

𝛾𝑐₁₀= parameter estimate for regressing commute bicycling on age. This value was 120

multiplied by 30 for all analyses. 121

𝛾𝑐₁₁ = parameter estimate for regressing commute bicycling on student-employee status. 122

This value was multiplied by 1 for all analyses. 123

𝜎₂ = is the variance for the mediator of interest physical activity status. 124

Parameter estimates for each of the other covariates (sex and race/ethnicity dummy variables) 125

were not included as they were multiplied by 0 for all analyses. When presenting the equations 126

for the subsequent conditions, we will exclude the parameter estimates being multiplied by 0. 127

The second condition, Φ(1,0), estimated the z-score of a standard subject of the bicyclist 128

typology of interest who did not perform physical activity three or more times during the past 129

week would have the outcome of interest relative to the Strong and Fearless. The parameters 130

identified below have the same meaning as in the aforementioned equation for the first 131

condition. Parameter estimates for regressing physical activity on each bicyclist typology were 132

Bryan, J. Michael 113

excluded in the equation below as compared to the equation for the first condition as they 133 would be multiplied by 0: 134 𝐶𝐷𝐹(𝐵𝑖𝑐𝑦𝑐𝑙𝑖𝑠𝑡 𝑇𝑦𝑝𝑜𝑙𝑜𝑔𝑦 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡: 1,0) = |𝛽₀₁+ 𝛽₁𝑋₁+ 𝛽₂𝑋₂+ 𝛽₃𝑋₃+ 𝛽₄𝑋₄+ 𝛽₅𝑋₅+ 135 𝛽₁₀𝑋₁₀+ 𝛽₁₁𝑋₁₁+ 𝛽₁₂(𝛾𝑝02+ 𝛾𝑝10𝑋 + 𝛾𝑝11𝑋) + 𝛽13(𝛾𝑐1𝑋1+ 𝛾𝑐2𝑋2+ 𝛾𝑐3𝑋3+ 𝛾𝑐4𝑋4+ 𝛾𝑐5𝑋5+ 136 𝛾𝑐₁₀𝑋₁₀+ 𝛾𝑐₁₁𝑋₁₁)|/√𝛽12𝜎22+ 1 137

The third condition, Φ(0,0), estimated the z-score for the Strong and Fearless who did 138

not perform physical activity three or more times during the past 30 days. 139 𝐶𝐷𝐹(𝐵𝑖𝑐𝑦𝑐𝑙𝑖𝑠𝑡 𝑇𝑦𝑝𝑜𝑙𝑜𝑔𝑦 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡: 0,0) = |𝛽01+ 𝛽1𝑋1+ 𝛽2𝑋2+ 𝛽3𝑋3+ 𝛽4𝑋4+ 𝛽5𝑋5+ 140 𝛽₁₀𝑋₁₀+ 𝛽₁₁𝑋₁₁+ 𝛽₁₂(𝛾𝑝₀₂+ 𝛾𝑝₁₀𝑋 + 𝛾𝑝₁₁𝑋) + 𝛽₁₃(𝛾𝑐₁𝑋₁+ 𝛾𝑐₂𝑋₂+ 𝛾𝑐₃𝑋₃+ 𝛾𝑐₄𝑋₄+ 𝛾𝑐₅𝑋₅+ 141 𝛾𝑐₁₀𝑋₁₀+ 𝛾𝑐₁₁𝑋₁₁)|√𝛽₁2𝜎₂2_{+ 1} 142

The z-score resulting from each CDF for each condition was then converted to a 143

probability in Microsoft Excel using the NORM.S.DIST function. Odds were then calculated 144

from these probabilities. Finally, to calculate the odds ratios, we performed the following two 145

calculations for each bicyclist typology of interest relative to the Strong and Fearless (except in 146

the case of the Supplemental Tables previously identified): 147

1. Indirect Effect through Physical Activity: Φ[probit(1,1)]/(1−Φ[probit(1,1)])_{Φ[probit(1,0)]/(1−Φ[probit(1,0)]} 148

2. Direct Effect: Φ[probit(1,0)]/(1−Φ[probit(1,0)])_{Φ[probit(0,0)]/(1−Φ[probit(0,0)]} 149

In order to compare the odds of worse health through the indirect effect of physical activity for 150

a bicyclist typology of interest relative to the reference typology, the odds produced from 151

calculating Φ(1,1) was divided by the odds produced from calculating Φ(1,0). Similarly, to 152

compare the odds of worse health through the direct effect of the bicyclist typology of interest 153

relative to the reference typology, the odds calculated in Φ(1,0) were divided by the odds 154

calculated from Φ(0,0). Our calculation did assume there was no treatment-mediator 155

interaction. Future studies may find investigating this interaction worthwhile. 156

Bryan, J. Michael 115

8. References

158

1. Centers for Disease Control and Prevention, National Center for Health Statistics, 159

Underlying Cause of Death 1999-2014 on CDC WONDER Database, released 2015. Data are 160

from the Multiple Cause of Death Files, 1999-2014, as compiled from data provided by the 57 161

vital statistics jurisdictions through the Vital Statistics Cooperative Program. 162

2. Hamer, M. and Y. Chida, Active commuting and cardiovascular risk: A meta-analytic review. 163

Preventive Medicine, 2008. 46(1): p. 9-13. 164

3. de Geus, B., et al., Cycling to work: influence on indexes of health in untrained men and women 165

in Flanders. Coronary heart disease and quality of life. Scand J Med Sci Sports, 2008. 18(4): p. 166

498-510. 167

4. United States Census Bureau. 2011-2015 American Community Survey 5-Year Estimates. 168

2017 [cited 2017 January 12, 2017]; Available from: 169

https://factfinder.census.gov/faces/tableservices/jsf/pages/productview.xhtml?pid=ACS_ 170

15_5YR_S0801&prodType=table. 171

5. Dill, J. and N. McNeil Four Types of Cyclists? Examining a Typology to Better Understand 172

Bicycling Behavior and Potential. 2012. 22. 173

6. Falconer, C.L., et al., The tracking of active travel and its relationship with body composition in 174

UK adolescents. Journal of Transport & Health, 2015. 2(4): p. 483-489. 175

7. Andersen, L.B., et al., Cycling to school and cardiovascular risk factors: a longitudinal study. J 176

Phys Act Health, 2011. 8(8): p. 1025-33. 177

8. Tainio, M., et al., Can air pollution negate the health benefits of cycling and walking? Prev 178

Med, 2016. 87: p. 233-6. 179

9. de Hartog, J.J., et al., Do the Health Benefits of Cycling Outweigh the Risks? Ciencia & Saude 180

Coletiva, 2011. 16(12): p. 4731-4744. 181

10. Rojas-Rueda, D., The health risks and benefits of cycling in urban environments compared with 182

car use: health impact assessment study (vol 343, d4521, 2011). British Medical Journal, 2011. 183

343: p. 1. 184

11. Rojas-Rueda, D., et al., Replacing car trips by increasing bike and public transport in the 185

greater Barcelona metropolitan area: a health impact assessment study. Environ Int, 2012. 49: p. 186

100-9. 187

12. Rojas-Rueda, D., et al., Health impact assessment of increasing public transport and cycling use 188

in Barcelona: A morbidity and burden of disease approach. Preventive Medicine, 2013. 57(5): 189

p. 573-579. 190

13. Pucher, J. and R. Buehler, Making cycling irresistible: Lessons from the Netherlands, Denmark 191

and Germany. Transport Reviews, 2008. 28(4): p. 495-528. 192

14. Geller, R. Four Types of Cyclists. 2006. 193

15. Fraser, S.D.S. and K. Lock, Cycling for transport and public health: a systematic review of the 194

effect of the environment on cycling. European Journal of Public Health, 2011. 21(6): p. 738- 195

743. 196

16. Saelens, B.E., J.F. Sallis, and L.D. Frank, Environmental correlates of walking and cycling: 197

Findings from the transportation, urban design, and planning literatures. Annals of Behavioral 198

Medicine, 2003. 25(2): p. 80-91. 199

17. Van Holle, V., et al., Relationship between the physical environment and different domains of 200

physical activity in European adults: a systematic review. BMC Public Health, 2012. 12: p. 201

807. 202

18. Bryan, M., Bicycling for Transportation: Health and Destination, Results of a survey of students 203

and employees from a southern urban university. 2017, Georgia State University. 204

19. Dill, J. and N. McNeil, Revisiting the Four Types of Cyclists. Transportation Research 205

Record: Journal of the Transportation Research Board, 2016. 2587: p. 90-99. 206

20. Dominick, K., et al., Relationship of health-related quality of life to health care utilization and 207

mortality among older adults. Aging Clinical and Experimental Research, 2002. 14(6): p. 208

499-508. 209

21. Colsher, P.L. and R.B. Wallace, Elderly men with histories of heavy drinking: correlates and 210

consequences. Journal of studies on alcohol, 1990. 51(6): p. 528-535. 211

22. Colsher, P.L., et al., Demographic and health characteristics of elderly smokers: results from 212

established populations for epidemiologic studies of the elderly. American journal of 213

preventive medicine, 1990. 6(2): p. 61-70. 214

23. Lamb, K.L., et al., A comparison of selected health-related data from surveys of a general 215

population and a sporting population. Social Science & Medicine, 1991. 33(7): p. 835-839. 216

24. De Forge, B.R., J. Sobal, and J.P. Krick, Relation of Perceived Health with Psychosocial 217

Variables in Elderly Osteoarthritis Patients. Psychological Reports, 1989. 64(1): p. 147-156. 218

25. Permanyer-Miralda, G., et al., Comparison of perceived health status and conventional 219

functional evaluation in stable patients with coronary artery disease. Elsevier, 1991. 44(8): p. 8. 220

26. Wannamethee, G. and A.G. Shaper, Self-assessment of health status and mortality in middle- 221

aged British men. International Journal of Epidemiology, 1991. 20(1): p. 239-245. 222

27. Andresen, E.M., et al., Retest reliability of surveillance questions on health related quality of 223

life. Journal of Epidemiology and Community Health, 2003. 57(5): p. 339-343. 224

28. Toet, J., H. Raat, and E.J. van Ameijden, Validation of the Dutch Version of the CDC Core 225

Healthy Days Measures in a Community Sample. Quality of Life Research, 2006. 15(1): p. 226

179-184. 227

29. Pierannunzi, C., S.S. Hu, and L. Balluz, A systematic review of publications assessing 228

reliability and validity of the Behavioral Risk Factor Surveillance System (BRFSS), 2004-2011. 229

BMC Med Res Methodol, 2013. 13: p. 49. 230

30. Ounpuu, S., et al., Validity of the US Behavioral Risk Factor Surveillance System's health 231

related quality of life survey tool in a group of older Canadians. Chronic diseases in Canada, 232

2001. 22(3-4): p. 93-101. 233

31. Centers for Disease, C., Health-related quality of life and activity limitation - eight states, 1995. 234

Morbidity Mortality Weekly Report, 1998. 47(7): p. 7. 235

32. Centers for Disease, C., Health-related quality of life - Puerto Rico, 1996-2000. Morbidity 236

Mortality Weekly Report, 2002. 51(8): p. 3. 237

33. Andresen, E.M., et al., Performance of health-related quality-of-life instruments in a spinal cord 238

injured population. Archives of Physical Medicine and Rehabilitation, 1999. 80(8): p. 877- 239

884. 240

34. Currey, S.S., et al., Performance of a generic health-related quality of life measure in a clinic 241

population with rheumatic disease. Arthritis Care & Research, 2003. 49(5): p. 658-664. 242

Bryan, J. Michael 117

35. Hayes, A.F. and K.J. Preacher, Statistical mediation analysis with a multicategorical 243

independent variable. British Journal of Mathematical and Statistical Psychology, 2014. 67: 244

p. 20. 245

36. Hemmingsson, E., et al., Increased physical activity in abdominally obese women through 246

support for changed commuting habits: a randomized clinical trial. Int J Obes, 2009. 33(6): p. 247

645-652. 248

37. Oja, P., I. Vuori, and O. Paronen, Daily walking and cycling to work: their utility as health- 249

enhancing physical activity. Patient Educ Couns, 1998. 33(1 Suppl): p. S87-94. 250

38. Saelens, B.E., J.F. Sallis, and L.D. Frank, Environmental correlates of walking and cycling: 251

findings from the transportation, urban design, and planning literatures. Ann Behav Med, 252

2003. 25(2): p. 80-91. 253

39. Sanders, R.L. and J.F. Cooper, Do all roadway users want the same thing? Results from 254

roadway design survey of San Francisco bay area pedestrians, drivers, bicyclists, and transit 255

users. Transportation Research Board: Journal of the Transportation Research Board, 256

2013. 2393: p. 10. 257

40. Fishman, E., et al., Factors influencing bike share membership: An analysis of Melbourne and 258

Brisbane. Transportation Research Part A: Policy and Practice, 2015. 71: p. 14. 259

41. Sener, I.N., N. Eluru, and C.R. Bhat, An analysis of bicycle route choice preferences in Texas, 260

US. Transportation, 2009. 36: p. 29. 261

42. Broach, J., J. Dill, and J. Gliebe, Where do cyclists ride? A route choice model developed with 262

revealed preference GPS data. Transportation Research Part A: Policy and Practice, 2012. 263

46(10): p. 11. 264

43. Kang, L. and J.D. Fricker, Bicyclist commuters' choice of on-street versus off-street route 265

segments. Transportation, 2013. 40: p. 16. 266

44. Winters, M. and K. Teschke, Route preferences among adults in the near market for bicycling: 267

Findings of the cycling in cities study. American Journal of Health Promotion, 2010. 25(1): 268

p. 8. 269

45. Garrard, J., G. Rose, and S.K. Lo, Promoting transportation cycling for women: The role of 270

bicycle infrastructure. Preventive Medicine, 2008. 46(1): p. 5. 271

46. Jackson, M. and E. Ruehr, Let the people be heard: San Diego county bicycle use and attitude 272

survey. Transportation Research Board: Journal of the Transportation Research Board, 273

1998. 1636: p. 5. 274

47. Morckel, V.C., Examining the relationships between perceived neighborhood mobility 275

characteristics, perceived incivilities, travel attitudes, and physical activity amongst university 276

faculty and staff. Journal of Transport & Health, 2016. 3(1): p. 86-95. 277

48. Muthen, B. Applications of Causally Dened Direct and Indirect Effects in Mediation Analysis 278

using SEM in Mplus. 2011. 95. 279

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