C.J. 8054 – Problem Set #2 (Fall 2024)

Multilevel Nonlinear Models (25 points)

Instructions

Type answers that do not require math calculations.

Write out math calculations neatly and within the margins of notebook paper, and show all steps involved in deriving each answer.

For each question, include all relevant math calculations and other handwritten work immediately before or after your typed answer to that question (i.e., do not place handwritten work at the end of your document, after your typed answers to all 20 questions).

Present any necessary computer output within your answer to each question instead of appending the output to the end of your document.

Hard copies of your answers are due at the beginning of class on Monday, November 18^th.    

Questions (use ‘restricted maximum likelihood’ estimation for all models)

1. Treating PPROPVIC as a binary dependent variable and CROWDING as a level-2 independent variable, estimate an unconditional model and a means-as-outcome model to determine whether it would be worthwhile to include CROWDING in a multilevel model predicting PPROPVIC. Defend your decision using 3 unique pieces of information provided in or derived from the output. Copy and insert only the information from the output needed to make your decision. (1 point)  

2. Determine whether PROP_OFF and GANG should be included as (a) grand mean-centered or group mean-centered, and (b) random or fixed effects in a multilevel model predicting PPROPVIC. For the decisions of random versus fixed effects, use p < .10 as the alpha level due to the small sample at level-2. Defend all decisions. Copy and insert only the information from the output needed to defend these decisions. (1 point)   

3. Estimate an intercepts-as-outcome model using the variables and information from questions 1 and 2, and add LINEARC as a level-2 independent variable. Compute the proportion of variance explained at level-2, and conduct a hypothesis test of improvement in fit for the full model relative to the null model using LaPlace estimation in HLM (use 20 iterations). Summarize the findings for the model overall and for each independent variable. For the independent variables, interpret the odds ratios at level-1 and the regression coefficients at level-2. Copy and insert only the information from the output needed for your summary. (2 points)

4. Estimate a slopes-as-outcome model with the significantly varying estimate from question 2. Include CROWDING and LINEARC as the independent variables at level-2.  Determine the proportion of variance in the slope that is explained by the level-2 model. Summarize the findings for the model overall and for each independent variable. Copy and insert only the information from the output needed for your summary. (1 point)         

5. Using the level-1 model from question 3, compute the probability of property victimization for four types of prisoners: gang members who engage in property crimes inside prison, gang members who do not engage in property crimes inside prison, non-gang members who engage in property crimes inside prison, and non-gang members who do not engage in property crimes inside prison. Are these probabilities consistent with the directions and statistical significance of the effects of GANG and PROP_OFF on PPROPVIC? Defend your decision. (1 point)

6. Treating CO_LEG3 as a multinomial dependent variable and CO2INMTS as a level-2 independent variable, estimate an unconditional model and a means-as-outcome model to determine whether it would be worthwhile to include CO2INMTS in a multilevel model predicting CO_LEG3. Defend your decision using the same pieces of information used for question 1. Copy and insert only the information from the output needed to make your decision. (1 point)

7. Determine whether AF_AMER should be included as (a) grand mean-centered or group mean-centered, and (b) random or fixed effects in a multilevel model predicting CO_LEG3. For the decisions of random versus fixed effects, use p < .10 as the alpha level due to the small sample at level-2. Defend all decisions. Copy and insert only the information from the output needed to defend these decisions. (1 point) 

8. Estimate an intercepts-as-outcome model using the variables and information from questions 6 and 7, and add GANG as grand mean-centered and fixed at level-1. Compute the proportion of variance explained at level-2. Summarize the findings for the model overall and for each independent variable. For the independent variables, interpret the odds ratios at level-1 and the regression coefficients at level-2. Copy and insert only the information from the output needed for your summary. (2 points) 

9. Estimate a slopes-as-outcome model from the multinomial model above with AF_AMER treated as random and GANG treated as grand mean-centered and fixed at level-1. Include CO2INMTS in the model for β₁₍₁₎ ONLY. Determine the proportion of variance in β₁₍₁₎ that is explained by the level-2 model. Summarize the findings for the model overall and for CO2INMTS. Copy and insert only the information from the output needed for your summary. (1 point)  

10. Using the level-1 model from question 8, compute the odds ratios for four types of prisoners: African American gang members, African-American non-gang members, non-African American gang members, and non-African-American non-gang members. Are these probabilities consistent with the directions and statistical significance of the effects of AF_AMER and GANG on CO_LEG3? Defend your decision. (1 point)

11. Build an ordinal intercepts-as-outcome model using the same dependent and independent variables from questions 6 and 7. Defend all decisions related to the treatment of both dependent and independent variables in the model. Copy and insert only the information from the output needed to defend these decisions. (1 point)

12. Add GANG as grand mean-centered and fixed at level-1 to the model you developed for question 11, and estimate the model. Compute the proportion of variance explained at level-2. Summarize the findings for the model overall and for each independent variable. For the independent variables, interpret the odds ratios at level-1 and the regression coefficients at level-2. Copy and insert only the information from the output needed for your summary. (2 points)

13. Estimate a slopes-as-outcome model from the ordinal model above with AF_AMER treated as random and GANG treated as grand mean-centered and fixed at level-1. Determine the proportion of variance in the estimate of AF_AMER that is explained by the level-2 model. Summarize the findings for the model overall and for CO2INMTS. Copy and insert only the information from the output needed for your summary. (1 point) 

14. Summarize the similarities and differences in substantive findings (not raw coefficient differences) between the analyses of the multinomial and ordinal models above. Explain why each similarity and difference exists. (1 point)

15. Identify one strength and one weakness of a multinomial model relative to an ordinal model with the same dependent and independent variables. Demonstrate the strength and weakness using the models and output for this problem set. (1 point) 

16. Build a Poisson intercepts-as-outcome model with IVIOLVIC as the dependent variable, LNREC and VIOLPRV1 as level-1 independent variables, and CO2INMTS and LINEARC as (possible) level-2 independent variables. Defend all decisions related to the treatment of both dependent and independent variables in the model. Copy and insert only the information from the output needed to defend these decisions. (1 point)

17. Estimate the Poisson intercepts-as-outcome model from question 16. Compute the proportion of variance explained at level-2, and conduct a hypothesis test of improvement in fit for the full model relative to the null model using LaPlace estimation in HLM (use 20 iterations). Summarize the findings for the model overall and for each independent variable. Also interpret the regression coefficients for the independent variables. Copy and insert only the information from the output needed for your summary. (2 points)

18. Repeat question 16 but adjust for over-dispersion in the dependent variable. Defend all decisions related to the treatment of both dependent and independent variables in the model. Copy and insert only the information from the output needed to defend these decisions. (1 point)

19. Estimate the intercepts-as-outcome model from question 18. Compute the proportion of variance explained at level-2, and conduct a hypothesis test of improvement in fit for the full model relative to the null model using LaPlace estimation in HLM (use 20 iterations). Create a table with the statistics from the model for question 17 and from the model for this question placed side-by-side (proportion variance explained at level-2, hypothesis test of improvement in fit for the full model relative to the null model, and the regression coefficients for the independent variables followed by asterisks when statistically significant within your predetermined critical regions. Describe the differences between these two sets of statistics. (2 points)

20. Build and estimate a linear intercepts-as-outcome model using the same variables from question 16. Demonstrate whether or not the Poisson model with the correction for over-dispersion provides substantively different results from the linear model (for the model overall and for each independent variable). (1 point)