_{1}

^{*}

Assessing geographic variations in health events is one of the major tasks in spatial epidemiologic studies. Geographic variation in a health event can be estimated using the neighborhood-level variance that is derived from a generalized mixed linear model or a Bayesian spatial hierarchical model. Two novel heterogeneity measures, including median odds ratio and interquartile odds ratio, have been developed to quantify the magnitude of geographic variations and facilitate the data interpretation. However, the statistical significance of geographic heterogeneity measures was inaccurately estimated in previous epidemiologic studies that reported two-sided 95% confidence intervals based on standard error of the variance or 95% credible intervals with a range from 2.5
^{th} to 97.5
^{th} percentiles of the Bayesian posterior distribution. Given the mathematical algorithms of heterogeneity measures, the statistical significance of geographic variation should be evaluated using a one-tailed P value. Therefore, previous studies using two-tailed 95% confidence intervals based on a standard error of the variance may have underestimated the geographic variation in events of their interest and those using 95% Bayesian credible intervals may need to re-evaluate the geographic variation of their study outcomes.

Spatial epidemiology is an important methodology to deal with spatial-correlated issues in epidemiologic studies. One of its core tasks is to determine geographic variations and quantify the magnitude of geographic variations in diseases, health behaviors, or environmental exposures [

The generalized linear mixed model and the Bayesian spatial hierarchical model are the most commonly applied to fit the data with a multilevel spatial structure. A geographic variation can be directly quantified as neighborhood-level variance

where ^{th} percentile (0.6745).

where ^{th} and 12.5^{th} percentiles (1.1504, −1.1504), respectively.

Both MOR and IqOR are derived from the variance and are always greater than or equal to one. Larger values of MOR and IqOR denote greater geographic variations in the event of interest. The MOR reflects the average difference of risk when comparing two subjects who have the same individual characteristics and are selected randomly from two different neighborhoods. The IqOR represents the average difference of risk when comparing the first quartile of study subjects residing in neighborhoods with the highest risk to the fourth quartile of study subjects residing in neighborhoods with the lowest risk [

Geographic variations can be qualitatively assessed by using neighborhood-level variance estimation derived from a generalized linear mixed model. The modeling conducted by a commonly used statistical analysis package, such as the SAS, also gives a Z value and a corresponding P value based on an approximately normal distribution of the estimated parameter. With the standard error of the variance from the multilevel model fitting, a 95% CI is able to be computed mathematically. However, one cannot perform a generalized linear mixed analysis to estimate the statistical significance and 95% CIs of the MOR and IqOR because both MOR and IqOR are derived from the variance and do not have their own standard errors.

Alternatively, a Bayesian spatial hierarchical model with a Markov Chain Monte Carlo (MCMC) simulation has been used to estimate geographic heterogeneities. In this setting, the 95% Bayesian credible interval (CrI), defined by the 2.5^{th} and 97.5^{th} percentiles of Bayesian posterior distribution of the geographic heterogeneity measure, has been commonly reported.

In the estimation of a fixed effect of an exposure, its statistical significance can be identified if the 95% confidence/credible interval of its regression coefficient does not cross zero. However, this empirical method conflicts with the nature of geographic heterogeneity measures. Two unreasonable results are usually reported in the studies in which the 95% CI or CrI of geographic heterogeneity measures were used to determine their statistical significance. The 95% CI of the variance could cross zero based on an approximately normal distribution^{th} percentile of the Bayesian posterior distribution of the variance is always greater than 0 and consequently the MOR and IqOR are always greater than one. This leads to the overestimation of geographic disparities.

A simulation analysis was performed to illustrate the issues relevant to the statistical significance of spatial heterogeneity measures. It is assumed that a population of colorectal cancer (CRC) survivors come randomly from 100 neighborhoods, each with 5 - 20 patients, and that the probability of smoking for each patient is 0.2 - 0.5. A random simulation generated a dataset that included 1245 patients and 420 smokers. Multilevel logistic regression is applied to quantify small-area geographic variation in smoking behavior among these CRC patients (Equation (3)).

where

To simplify the explanation, an empty model without neighborhood- and individual-level covariates was fit to estimate the overall geographic heterogeneity of smoking among these CRC patients using the Bayesian hierarchical approach with a MCMC simulation in WinBUGS (Version 1.4.3, MRC, UK). After 50,000 iterations for the convergence, additional 50,000 iterations were run to obtain the posterior estimates of three spatial heterogeneity measures. Because the dataset was simulated randomly, the geographic variation in smoking was expected to be small.

^{th} to their 97.5^{th} percentiles. However, the inconsistent results were observed when comparing the 95% CIs of the variance, MOR and IqOR to their 95% CrIs. The 95% CI of the variance crossed zero and the 95% CIs of both MOR and IqOR crossed 1, suggesting no significant geographic variation in smoking behavior among CRC survivors. In contrast, the 95% CrI of the variance was more than zero and the 95% CrIs of the MOR and IqOR were greater than one, suggesting a significant geographic variation in smoking behavior.

^{th}

Measure | Mean (µ) | SD (σ) | 2.50% | Median | 97.50% | 95% confidence interval | 95% credible interval |
---|---|---|---|---|---|---|---|

VAR^{*} | 0.007 | 0.009 | 0.001 | 0.004 | 0.033 | 0.007 (−0.011, 0.025) | 0.004 (0.001, 0.033) |

MOR^{†} | 1.074 | 0.045 | 1.022 | 1.061 | 1.189 | 1.074 (0.986, 1.162) | 1.061 (1.022, 1.189) |

IqOR^{‡} | 1.190 | 0.125 | 1.054 | 1.155 | 1.517 | 1.190 (0.945, 1.435) | 1.155 (1.054, 1.517) |

^{*}VAR, variance; ^{†}MOR, median odds ratio; ^{‡}IqOR, interquartile odds ratio.

Measure | Range | Null hypothesis (H_{0}) | Alternative hypothesis (H_{1}) |
---|---|---|---|

VAR^{*} | ≥0 | VAR = 0 | VAR > 0 |

MOR^{†} | ≥1 | MOR = 1 | MOR > 1 |

IqOR^{‡} | ≥1 | IqOR = 1 | IqOR > 1 |

^{*}VAR, variance; ^{†}MOR, median odds ratio; ^{‡}IqOR, interquartile odds ratio.

and the 97.5^{th} percentiles of Bayesian posterior distribution (95% CrI) of geographic heterogeneity measures to avoid the misinterpretation of geographic variations. In fact, a one-tailed P value for the variation/heterogeneity estimation has been given from a generalized linear mixed model fitting using common statistical analysis packages, such as the SAS. For the heterogeneity estimation from a Bayesian hierarchical model, one should compute the corresponding statistics, based on the prior distribution of the variance, to obtain their one-tailed P value to determine its statistical significance. In the simulated example, since the Z value for the variance is:

The purpose of this study was to point out an inappropriate method that was used to determine the statistical significance of geographic heterogeneity measures. The simulated data suggested that empirically reporting of the 95% CI/CrI of geographic heterogeneity measures may lead to misunderstanding of the statistical significance of geographic variations of an event.

According to the nature of geographic heterogeneity measures, the statistical inference should be one-tailed (right-tailed). It is inappropriate to report a two-tailed 95% CI/CrI of a heterogeneity measure in spatial epidemiologic studies. It could mislead one in understanding the statistical significances of heterogeneity measures. In the studies using standard errors to obtain two-tailed P values or 95% CIs, geographic variations in the events may be underestimated because a two-tailed test is more conservative than a one-tailed test. In contrast, the studies using the interval between the 2.5^{th} and the 97.5^{th} percentiles of a Bayesian posterior distribution to obtain a 95% CrI may overestimate the statistical significance of geographic variation of the event because a Bayesian 95% CrI never crosses zero for the variance and one for both MOR and IqOR. The issue of statistical significance of geographic heterogeneity measures, which was discussed in this paper, is also extendible to a general multilevel study aiming to investigate the variation(s) in one or multiple event(s) of interest across a non-spatial higher level, such as healthcare providers or medical service facilities.

This work is supported partly by a Career Development Award (K07 CA178331) and a Research Award (R21 CA169807) funded by the National Cancer Institute, and a Research Award (R01 AA021492) funded by the National Institute of Alcohol Abuse and Alcoholism, both at the National Institutes of Health. I also thank the service provided by the Health Behavior, Communication and Outreach Core, which is supported by the National Cancer Institute Cancer Center Support grant (P30 CA091842) awarded to the Alvin J. Siteman Cancer Center at Barnes-Jewish Hospital and Washington University School of Medicine. I declare no conflict of interest.

VAR: variance;

MOR: median odds ratio;

MRR: median rate ratio;

MHR: median hazard ratio;

IqOR: interquartile odds ratio;

IqRR: interquartile rate ratio;

IqHR: interquartile hazard ratio.