---

Homework

1. Make sure you have the files USSCLEV1.SAV and USSCLEV2.SAV. These are data from the U.S. Sentencing Commission, supplied by and used with the permission of Prof. Brian Johnson, University of Maryland. Data are not for redistribution and are not appropriate for publication as they consist only of the first 7,000 cases from a much larger USSC dataset. Prof. Johnson has kindly provided the slides from his ICPSR 2009 Summer Workshop, which uses the full dataset and has additional models. Consulting these slides may help with the assignments below as well.

2. Following the description in Statnotes, use these files to create a file called USSC.MDM. Use the File, Make new MDM file, Stat package input menu choices in HLM. Select HLM2 as the MDM file type. Let DISTRICT be the ID variable at both levels 1 and 2.

3. In HLM, click on Basic Settings and make sure the output and graph file output goes to a valid file folder to which you have read/write access.

4. In HLM, set SENTLENG as the outcome variable and create an intercept-only model, then Run Analysis, then File, View Output. When running, select the "Save and Run" option and save the command file as "USSCnull". Print out the output and mark it "Run 1: HLM Null Model". Be prepared to answer the following questions:
   1. What is the variation in sentence length between individuals?
   2. What is the variation in sentence length across districts?
   3. The intraclass correlation (rho) is the percent of total variation in sentence length at the between-districts level. Compute it by hand - what is this percent?
   4. What is the mean sentence length across districts? Hint: find the intercept coefficient.

5. Select Other Settings, Hypothesis Testing and enter the deviance and number of parameters from Run 1.

6. We now create an unconditional severity-of-sentence model by adding SEVERITY as a level 1 predictor. Add it to the model centered about its grand mean. All error terms should be toggled to the "on" status (not greyed out) EXCEPT the level 2 error term for B1. This makes it a random intercept model (there are no slopes treated as random)

   Run Analysis, then File, View Output. When running, select the "Save and Run" option and save the command file as "USSCuncon". Print out the output and mark it "Run 2: HLM Unconditional Severity Model". Be prepared to answer the following questions:

   1. What is the variation in sentence length among individuals once SEVERITY is controlled? Between districts?
   2. Has district-level variation in sentence length as a percent of total variation gone down? Hint: compute ICC.
   3. What is the slope of SEVERITY? Is it significant? How does one put into an English-language sentence the effect of SEVERITY on sentence length?
   4. What is the standard deviation of the intercept when SEVERITY is controlled? Some 95% of districts have a mean sentence within plus or minus how many months of each other?
   5. Recall that the effect of adding SEVERITY to the model can be assessed by looking at the change in random effect variance components in the current model compared to the null model. The level 1 effect is the change in the R variance component divided by the null model R variance component, for example. Adding SEVERITY to the null model; explains what percent of the individual-level variation in sentence length? What percent of district-level variation?

7. We now add the level 2 variable SOUTH as a predictor, retaining SEVERITY at level 1. Add SOUTH as grand mean centered. Toggle on all error terms (none should be greyed out). This makes it a random coefficient model (not just random intercept model) because the slope of SEVERITY as well as the intercept are borh modeled at level 2.

   Under Other Settings, Hypothesis Testing, enter the Run 2 deviance and number of parameters. then Run Analysis, then File, View Output. When running, select the "Save and Run" option and save the command file as "USSCcon". Print out the output and mark it "Run 3: HLM Conditional Severity Model" because SEVERITY is now modeled as conditional on SOUTH. Be prepared to answer the following questions:

   1. What is the level 1 (individual) variation after SEVERITY and SOUTH are controlled? The level 2 (district level) variation?
   2. What is the ICC for this run? How is it interpreted?
   3. As a result of controlling for SOUTH, how much does the slope of SEVERITY change compared to Run 2?
   4. In Run 2 you should have found considerable residual variation in sentence length at the district level. Recall the U0 level 2 error term models the level 1 B0 intercept, which represents mean sentence length after controls. How much does adding SOUTH to the model here in Run 3 help in explaining district-level variation in sentencing length?
   5. Interpret the likelihood ratio (deviance) test for Run 3.

Explore

• Free student edition of HLM 6 software (SSI)
• Online resource page for HLM software
• Using SAS PROC MIXED to fit multilevel models (Singer)