Statistical results
To simultaneously test county- and state-level effects of contextual
factors on the outbreak rate with cross-level interactions, we estimate
a two-level linear model using full maximum likelihood in HLM 7.03
(Figure 3). This accounts for potential similarities in counties within
the same state.24 We center all predictors around the
group mean at level 1 and grand mean at level 2.
We first estimate a one-way random effects ANOVA (unconditional model),
which has an intraclass correlation coefficient (ICC) of 0.243. That is,
more than 24% of the variation in the outbreak rate is between states,
and about 76% is within the states and between their counties. The
variation between states is statistically significant
(u 0 = 4.50E-04, p < 0.001). We
thus deem it prudent to proceed with a multilevel model as follows:
Level 1 (counties):\(\text{Outbreak\ rate}_{\text{ij}}=\beta_{0j}+\beta_{1j}\left[Black\ \&\ African\ American\right]+\ \beta_{2j}\left[\text{Native\ American}\right]+\beta_{3j}\left[\text{Asian\ American}\right]+\beta_{4j}\left[\text{Native\ Hawaiian}\right]+\beta_{5j}\left[\text{Hispanic\ American}\right]+\beta_{6j}\left[\text{Household\ income}\right]+\beta_{7j}\left[\text{Nonproficiency\ in\ English}\right]+\beta_{8j}\left[\text{Math\ grade}\right]+\beta_{9j}\left[Persons\ under\ 18\ years\right]+\beta_{10j}\left[\text{Median\ age}\right]+\beta_{11j}\left[\text{Female\ persons}\right]+\beta_{12j}\left[\text{Social\ associations}\right]+\beta_{13j}\left[\text{Sleep\ deprivation}\right]+\beta_{14j}\left[\text{Preventable\ hospitalization}\right]+\beta_{15j}\left[\text{Obesity}\right]+\beta_{16j}\left[\text{Smoking}\right]+\beta_{17j}\left[\text{Air\ pollution}\right]+\beta_{18j}\left[\text{Rural\ area}\right]+\beta_{19j}\left[\text{Food\ environment}\right]+\beta_{20j}\left[\text{Outbreak\ date}\right]+\beta_{21j}\left[\text{Density}\right]+\beta_{22j}\left[\text{Temperature}\right]+r_{\text{ij}}\)
Level 2 (states):\(\beta_{0j}=\gamma_{00}+\gamma_{01}\left[\text{Party\ control}\right]+\gamma_{02}\left[\text{Gender\ of\ governor}\right]+\gamma_{03}\left[\text{Government\ spending}\right]+\gamma_{04}\left[\text{Collectivism}\right]+u_{0j}\);\(\beta_{1j}=\gamma_{10}+u_{1j}\); \(\beta_{2j}=\gamma_{20}\);\(\beta_{3j}=\gamma_{30}\); \(\beta_{4j}=\gamma_{40}\);\(\beta_{5j}=\gamma_{50}\); \(\beta_{6j}=\gamma_{60}\);\(\beta_{7j}=\gamma_{70}\); \(\beta_{8j}=\gamma_{80}\);\(\beta_{9j}=\gamma_{90}\); \(\beta_{10j}=\gamma_{100}\);\(\beta_{11j}=\gamma_{110}\); \(\beta_{12j}=\gamma_{120}\);\(\beta_{13j}=\gamma_{130}\); \(\beta_{14j}=\gamma_{140}\);\(\beta_{15j}=\gamma_{150}\); \(\beta_{16j}=\gamma_{160}\);\(\beta_{17j}=\gamma_{170}\); \(\beta_{18j}=\gamma_{180}\);\(\beta_{19j}=\gamma_{190}+u_{19j}\);\(\beta_{20j}=\gamma_{200}+u_{20j}\); \(\beta_{21j}=\gamma_{21}\);\(\beta_{22j}=\gamma_{22}\)
We provide the inter-item correlation matrix in Table 2, and the results
of the multilevel model in Table 3. Additionally, we perform several
robustness tests to inform our results.
First, because outbreak rates change over time and their estimation is
somewhat sensitive to the starting figure, we alternatively calculate
them after 10 and 25 cases in the respective unit, finding a high
correlation among the rates. When using the rate after 10 cases, the
outbreak date as a control variable changes its sign and loses
significance (p = 0.065). Notably, the following coefficients
gain significance: government spending (p = 0.064); temperature
(p = 0.011). Conversely, the following coefficients lose
significance: household income (p = 0.989); food environment
(p = 0.144); density (p = 0.709). More importantly, the
signs of the coefficients remain the same. The variable outbreak date
controls for temporality of the outbreak in the original model (1.912,
[1.322; 2.502], p < 0.001).
Second, we iteratively include several other contextual variables and
logged versions to assess the robustness of the results. But because it
is nearly impossible to establish a complete list of confounding
variables, we quantify the potential impact of unobserved confounds
(Table 3; impact threshold).28 For instance, the
necessary impact of such a confound for air pollution would be 0.043,
that is, to invalidate the variable’s inference on the outbreak rate, a
confounding variable would have to be correlated with both the outbreak
rate and air pollution at \(\sqrt{0.043}=0.207\). Next, to alleviate
concerns that some counties are omitted from the analysis because they
are not yet affected by the virus,24 we ask how many
counties would have to be replaced with unobserved cases for which the
null hypothesis is true (i.e., a contextual variables has no influence
on the outbreak rate) in order to invalidate the
inference.28 As Table 3 (confound threshold) shows,
43.962% of the counties would have to be replaced with counties for
which the effect is zero in order to invalidate the influence of air
pollution. In summary, it can be claimed that the influence of the
identified contextual variables on the pandemic is reasonably robust.
Third, a potential omission of relevant variables can lead to
multicollinearity issues, which are generally a serious problem in
epidemiological studies.29 Even though HLM 7.03 checks
for multicollinearity, we conduct several additional diagnostics to
eliminate any potential issues. In the inter-item correlation matrix
(Table 2), the average (absolute) correlation is 0.172, and the highest
correlation is 0.754, which is below the typical cutoff of 0.8. Most
high correlations exist between racial composition and income and
education. Additionally, we conduct a linear regression analysis at
level 1 in IBM SPSS 26 (R 2 = 0.495; without
variable math grade; pairwise exclusion of missing values), and find
that the variable inflation factor (VIF) never exceeds the threshold of
5 (highest being nonproficiency in English, 4.024). The
variance-decomposition matrix also does not show any groups of
predictors with high values.
Fourth, we rerun our model excluding the 23 counties of the New York
metropolitan area. As a COVID-19 hotspot, they could unduly influence
our analysis. All signs remain the same, and the following coefficients
gain significance: household income (p = 0.009); persons under 18
years (p = 0.038).
Fifth, because there is no statistically correct choice for centering
decisions in multi-level models30, we retest our model
with raw values. With the exception of the variable collectivism losing
statistical significance (p = 0.711), the results are fully
consistent with the group- and grand-mean centered predictors in Table
3.
Lastly, we are aware that an accurate estimation and comparison of the
outbreak rate across units depends on similar testing strategies, test
sensitivities, specificities, and reporting of tests performed vs.
individuals tested.10,31 Even within the U.S., some
states report tests performed and others individuals
tested.31 The number of tests administered and the
number of confirmed cases therefore correlates to varying extents across
states.32 By using a multi-level model, we aim to
accommodate such differences between states.