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Annotated Output for Binomial Regression SUPPLEMENTAL MATERIAL Appendix 1: Annotated output for logistic regression Appendix 2: Annotated output for binomial regression Appendix 3: Annotated output for proportional odds logistic regression Appendix 4: Annotated output for multinomial regression Appendix 5: Annotated output for Poisson regression Appendix 6: Special cases of Poisson Regression Appendix 1: Annotated output for logistic regression Understanding how to interpret the output R presents can be challenging. The following figures are designed to help demystify the R output for Logistic regression. All data and code can be found here: https://github.com/ejtheobald/BeyondLinearRegression. In addition, a table, akin to what a researcher might publish, is displayed based on the R output. Figure A1.1: Logistic regression annotated output Presenting logistic regression results in manuscripts Table A1.1: Regression coefficients (β), standard error (SE), p-value, and odds-ratio for the logistic regression predicting students reporting being likely to take a mathematical modeling in biology course. The odds of a student reporting being likely to take a mathematical modeling in biology course increased with increasing interest in using math to understand biology and decreased with increasing cost of incorporating math into biology courses. Additionally, the odds of a fourth-year student reporting being likely to take a mathematical modeling course are lower than those of a first-year student. Predictor β SE p-value Odds-ratio Interest 0.71 0.07 < 0.0001 2.04 Utility value 0.11 0.08 0.18 1.12 Cost -0.27 0.06 <0.0001 0.77 Gender (ref: Male) Female -0.22 0.19 0.26 0.81 Year in School (ref: 1st-year) 2nd year -0.16 0.23 0.49 0.85 3rd year -0.19 0.22 0.39 0.82 4th-year -0.52 0.24 0.03 0.60 Highest High School Math Course (ref: Calculus) Pre-calculus -0.03 0.21 0.87 0.97 Algebra/Geometry -0.06 0.38 0.87 0.94 Stats 0.11 0.34 0.75 1.12 Appendix 2: Annotated output for binomial regression Understanding how to interpret the output R presents can be challenging. The following figures are designed to help demystify the R output for binomial regression. All data and code can be found here: https://github.com/ejtheobald/BeyondLinearRegression. In addition, a table, akin to what a researcher might publish, is displayed based on the R output. Figure A2.1: Binomial regression annotated output Presenting binomial regression results in manuscripts Table A2.1: Regression coefficients (β), standard error (SE), p-value, and odds-ratio for the binomial regression predicting the odds of a student completing an optional practice exam. The odds of a student completing a practice exam decreased with increasing GPA. Additionally, the odds of a female student completing a practice exam are higher than those of a male student. Predictor β SE p-value Odds-ratio GPA -1.40 0.09 < 0.0001 0.25 Gender (ref: Male) Female 0.17 0.06 < 0.0001 1.19 First-generation (ref: Continuing-generation) First-generation -0.04 0.09 0.64 0.96 Appendix 3: Annotated output for proportional odds logistic regression Understanding how to interpret the output R presents can be challenging. The following figures are designed to help demystify the R output for proportional odds logistic regression. All data and code can be found here: https://github.com/ejtheobald/BeyondLinearRegression. In addition, a table, akin to what a researcher might publish, is displayed based on the R output. Figure A3.1: Proportional Odds Logistic Regression annotated output Presenting proportional odds results, using model selection, in manuscripts When model selection is conducted to test hypotheses, it is important to indicate the starting model, the best model, and the comparison of the best model to the null model (to show how much better the best model fits; done here with ΔAIC). Here is one way to do that for the hypothesis we tested. Table A3.1: Students were less likely to report a dominator after the interactive activity, compared to the constructive activity. Additionally, students with higher course grades were less likely to report a dominator than students with lower course grades. The table show odds ratios. Outcome Ethnicity1 Activity Type2 Course Grade ΔAIC3 Dominator4 Asian American5 1.67 0.56 0.76 20.207 International 3.35 URM 1.39 1 Reference group is White students 2 Reference group is Constructive activity; effect shown of Interactive activity. 3 In comparison to the null model 4 Outcome was measured on a 5-point Likert scale. Higher numbers indicate more agreement that one person dominated the group. 5 Bold face coefficients show statistically significant relationships but note that interpreting t- values for significance testing is unreliable with a small sample. 6 Units are shown as odds ratios. Appendix 4: Annotated output for multinomial regression Understanding how to interpret the output R presents can be challenging. The following figures are designed to help demystify the R output for multinomial regression. All data and code can be found here: https://github.com/ejtheobald/BeyondLinearRegression. In addition, a table, akin to what a researcher might publish, is displayed based on the R output. Figure A4.1: Multinomial regression annotated R output Presenting multinomial regression results in manuscripts Table A4.1: Model selection using backwards model selection and the likelihood ratio test. In each comparison the bolded term is the one being tested. Degrees Likelihood Models of Ratio p-value Freedom Statistic Gender + Class Standing + University GPA + Gender x Class Standing + Gender x University GPA vs. 8 11.1 0.195 Gender + Class Standing + University GPA + Gender x University GPA Gender + Class Standing + University GPA + Gender x University GPA 4 9.6 0.048 vs. Gender + Class Standing + College GPA Gender + Class Standing + University GPA + Gender x University GPA 8 5.2 0.736 vs. Gender + University GPA + Gender x University GPA Gender + University GPA + (Gender x University GPA) vs. 20 50.0 < 0.0001 Null (no predictors) Best model: Gender + University GPA + (Gender x University GPA) Table A4.2: Regression coefficients from multinomial regression exploring the impact of gender, class standing, and university GPA on the role students assume in groups. Collaborator is the reference level for the outcome variables, so all the other roles are compared to it and the estimates are the log odds of being in that role versus being a collaborator. Table shows estimates, standard error, and p-value from the Wald statistic (in parentheses). Gender x Outcome Gender: Male University GPA Intercept University GPA (ref: Female) at start of class (ref: Female) -1.17 ± 0.272 1.4 ± 0.366 -1.9 ± 0.723 2.7 ± 0.903 Leader vs. Collaborator (< 0.001) (< 0.001) (0.009) (0.002) Listener vs. Collaborator -1.54 ± 0.308 0.12 ± 0.522 -1.2 ± 0.839 0.76 ± 1.13 (< 0.001) (0.812) (0.156) (0.502) Recorder vs. Collaborator -1.75 ± 0.354 -1.0 ± 0.848 0.53 ± 0.955 0.63 ± 2.06 (< 0.001) (0.227) (0.578) (0.760) Other vs. Collaborator -1.29 ± 0.286 -0.12 ± 0.514 -2.0 ± 0.753 1.2 ± 1.03 (< 0.001) (0.820) (0.007) (0.255) Summarizing the output from the effects package verbally example. Figure 2D visually summarizes the output from the effects package, but sometimes researchers may want to explain a particular variable in more detail. Here we present how one might write up the effects package output for the gender x GPA interaction found in our example. Women with a college GPA that is 0.25 points below the mean have a 42% chance of reporting being a collaborator, at the mean they have a 50% chance and 0.5 points above the mean they have a 64% chance (table below). Males with the same range of GPAs do not see the upward shift in percent chance of being a collaborator: at -0.25 points below mean: 37%; at mean: 36%; 0.5 points above mean: 29%). Instead, as GPA increases males become increasingly likely to report preferring to be leaders: -0.25 points below mean: 38%; at mean GPA: 45%; 0.5 points above mean: 57%. The table below shows these probabilities as well as the 95% confidence intervals (shown in parentheses). No other outcome categories see shifts based on gender (as indicated by the 95% confidence interval on all their estimates overlapping). Table A4.3: The probability of preferring to be a collaborator varied with student reported gender and with GPA. Mean GPA Mean GPA Mean GPA - 0.25 pts + 0.5 pts Collaborator 42% 50% 64% Women (31 – 53%) (42 – 60%) (50 – 76%) Men 37% 36% 29% (27 – 50%) (26 – 46%) (18 – 45%) Leader 38% 45% 57% Men (28 – 51%) (34 – 56%) (41 – 71%) Appendix 5: Annotated output for Poisson regression Understanding how to interpret the output R presents can be challenging. The following figures are designed to help demystify the R output for Poisson regression. All data and code can be found here: https://github.com/ejtheobald/BeyondLinearRegression. In addition, a table, akin to what a researcher might publish, is displayed based on the R output. Figure A5.1: Poisson Regression annotated output Presenting Poisson regression results in manuscripts Table A5.1: Regression coefficients (β), standard error (SE), p-value, and the effect of the predictor on the outcome variable (eβ) for the Poisson regression predicting the number of times students raise their hands in class. The number of times a student raises their hand in class increases if they have higher total exam points and if they are a physics major.
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