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Heck et al (2012) give the following model

$Y_{ij} = \gamma_{00} + \gamma_{01}\text{ses_mean}_j + \gamma_{02}\text{pro4yrc}_j + \gamma_{03}\text{public}_j + \gamma_{10}\text{ses}_{ij} + u_{0j} + u_{1j} \text{ses}_{ij} + \epsilon_{ij}$

They report that the estimated variance of the random slope of SES is 1.314, and say that the results "suggest that a model could be developed to explain the variability in the SES-achievement slope across schools"

They build such a model by including a cross-level interaction $\text{public}_j\times\text{SES}_{ij}$:

$Y_{ij} = \gamma_{00} + \gamma_{01}\text{ses_mean}_j + \gamma_{02}\text{pro4yrc}_j + \gamma_{03}\text{public}_j + \gamma_{10}\text{ses}_{ij} + \gamma_{13}\text{public}_j\times\text{SES}_{ij} + u_{0j} + u_{1j} \text{ses}_{ij}+ \epsilon_{ij}$

They then report that cross-level interaction is statistically significant. In their book I can also see that the estimated variance of the random slope is now 1.345. They do not seem to any stage explain how in seeking to explain the slope variability they have actually caused the variability of the slope to actually increase.

I managed to track down their data set, and can only partially replicate their results in both R and SPSS. I can replicate their results from the first model, which really does give an estimate of random slope variance of 1.314. However, for the second model I get an estimate of 1.308, not 1.345.

library(haven)
library(lme4)

d <- read_sav(file ="https://github.com/user1205901/codeforposting/blob/main/ch3multilevel.sav?raw=true")

model1 <- lmer(math ~ 1 + ses + ses_mean + pro4yrc + public + (ses||schcode), data = d)
summary(model1)

model2 <- lmer(math ~ 1 + ses + ses_mean + pro4yrc + public + public*ses + (ses||schcode), data = d)
summary(model2)

I am only slightly interested in the specifics of why the slope variance increased in that particular example in that particular book, i.e. the question of whether it was it just an error in the book. In another thread I already noticed a bit of an issue with their data, inasmuch as certain Level 1 units have discrepant values on Level 2 variables despite being in the same Level 2 unit.

I am more interested in whether this can happen in general, and under what circumstances.

Heck, R. H., Thomas, S. L., & Tabata, L. N. (2012). Multilevel and longitudinal modeling with IBM SPSS. Routledge.

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  • $\begingroup$ You seem to equate "estimate of the random slope" with "the slope variability." Could you clarify what you mean by these phrases? $\endgroup$
    – whuber ♦
    Commented May 23, 2022 at 14:20
  • $\begingroup$ Sorry, I mistyped that sentence, and have fixed it now. $\endgroup$
    – user1205901 - Слава Україні
    Commented May 23, 2022 at 22:41

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