Analyzing Poisson MLM level 2 residuals for outliers

I am using HLM to analyze school discipline data. My outcome variable here is a count - the number of discipline referrals per student in the academic year - and I'm using binary predictor variables of gender, SES, and race/ethncity (5 dummy variables, White as reference). My model is then

$\log(\lambda_i)=\beta_{0j}+\beta_{1j}SES_{ij}+\beta_{2j}gender_{ij}+\sum_{k=3}^7\beta_{kj}RACE_kj$

$\beta_{0j}=\gamma_{00}+\mu_{0j}$

$\beta_{1j}=\gamma_{10}+\mu_{1j}$

$\beta_{2j}=\gamma_{20}+\mu_{2j}$

$\beta_{(3-7)j}=\gamma_{(3-7)0}+\mu_{(3-7)j}$

My primary question at this point is trying to determine if particular schools are under- or over-disciplining students, adjusting for SES, gender, and race/ethnicity. I'm using HLM, so I have the level 2 residuals, and they vary from -3.8 to +2.5.

I'm still new to multi-level modeling, so I'm struggling with interpreting the actual meaning of those residuals and if they get at what I want to know. My understanding is that they represent the difference between $\beta_{0j}$, the log of the mean rate of referrals for school $j$, and $\gamma_{00}$, the overall mean log rate of referrals, so if I calculate $e^{\mu_{ij}}$ I should get the ratio of the mean referral rate in school $j$ to the overall mean rate of referrals.

The thing is, that's not really what I want. I know the schools are going to have very different mean referral rates, simply because they have very different demographics. What I'm more interested in is the difference or ratio between what we would expect them to have given their demographics and what their actual raw mean discipline rate is.

Is there a way to answer this question? I feel like it's staring at me, but I'm still learning multi-level modeling and HLM and just can't see it.

• How do outliers enter into this question? – Michael Chernick Apr 28 '18 at 4:33
• Good point - I never mentioned that! What I was thinking is looking for outliers in that "expected mean count per school" versus "actual mean count per school" comparison. I'm using "outlier" a little casually here, I guess. – dankernler Apr 28 '18 at 4:36
• – Florian Hartig Apr 28 '18 at 9:41