# Residential Mobility and Breast Cancer in Marin County, California, USA

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## Abstract

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## 1. Introduction

- H1: The cases and controls do not exhibit substantial residential mobility over the life course.
- Rationale: The breast cancer risk factors found significant in the parent study operate at different points in a woman’s life course, some early in life, some later in life, and others involve long-term behaviors over several years. Substantial residential mobility in the study group suggests residence in Marin County may not be indicative of risk factors that occurred in Marin County.
- H2: There is no statistically significant global clustering of breast cancer cases relative to controls after accounting for known risk factors and residential mobility.
- Rationale: Global clustering might suggest the action of an unidentified risk factor not accounted for in the original case-control study design that impacts risk for most if not all of the cases (a large-scale signal, Global Ǫ statistic).
- H3: There are no time periods when the breast cancer cases, considered as a group, exhibit statistically significant clustering relative to the controls.
- Rationale: Large-scale spatial clustering at specific time periods may indicate past exposures that impacted many if not all of the cases (Global Ǫ
_{t}test). - H4: None of the cases exhibit statistically significant clustering over their life course, such that they tend to have other cases as neighbors.
- Rationale: Clustering over the life course might indicate cases with similar residential histories—they either tend to travel together because of behavioral factors (e.g., seeking treatment, friendship) and/or have lived in areas that have elevated breast cancer risk (Ǫ
_{i}test). - H5: Cases that cluster over the life course (Ǫ
_{i}) are not part of local clusters at specific times (Ǫ_{t}). - Rationale: Such a pattern (excess risk over the life course coupled with local clusters of excess risk) might indicate the action of ephemeral, geographically localized risk factors that were not accounted for in the parent case-control study.

## 2. Data

## 3. Methods

#### 3.1. Space-Time Analysis

**Figure 1.**Residential histories as space-time step functions. The axes x and y define a geographic domain (i.e., longitude and latitude decimal degrees), the t axes represents time (i.e., date). The study extends from time t

_{0}to time t

_{T}. The residential histories for persons i and j are shown as step functions through space-time. For example, person i begins the study residing at location x

_{i}, y

_{i}, t

_{0}. They remain at that geographic coordinate until the instant before time t

_{1}, when they move to x

_{i}, y

_{i}, t

_{1}. The duration of time they reside at this first place of residence is ω

_{0}.

#### 3.2. Ǫ-Statistics

_{0}, t

_{1}…, t

_{T}

_{‒1}(refer to Figure 1). The duration-weighted statistics account for the corresponding duration of time (e.g., ω

_{0,}ω

_{1,}…, ω

_{T}

_{‒1}) over which cases are obtained in a given geographic arrangement.

_{i}and c

_{j}

_{,}that are defined as 1 if a participant i is a case, and 0 otherwise. N is the total number of participants (cases and controls) in a study. The term η

_{i,j,k,t}is a binary spatial proximity metric that is 1 when participant j is a k nearest neighbor at time t of participant i; otherwise it is 0. Since a given individual i may have k unique nearest neighbors, the Ǫ

_{i,k,t}statistic is in the range 0..., k. When i is a control, Ǫ

_{i,k,t}= 0. When i is a case low values indicate cluster avoidance (e.g., a case surrounded by controls) and large values indicate a cluster of cases. When Ǫ

_{i,k,t}= k at time t, all of the k nearest neighbors of case i are cases. This statistic is recalculated for each participant every time there is a change in residence for person i or any of his/her k nearest neighbors. Therefore, Equation (1) reports a value for each residence at each and every time-geography of the residential histories (e.g., at times t

_{0}, t

_{1}…, t

_{T}

_{‒1}as shown in Figure 1). The user must specify the value for k before a statistic is calculated.

_{0}, which refers to the duration of time spent at a residence. The subscript “o” in ω

_{0}denotes the index for the duration ω

_{0}= t

_{0}

_{+1}− t

_{0}. Equation (3) is used to identify which cases are centers of spatial clusters through time; and is the sum, over all time durations, of Equation (2). Equation (4) is a global version and reports whether clustering occurs throughout the entire area at a particular duration (ω

_{0}) in time. It is calculated by summing Equation (2) over all cases at that particular duration in time. Equation (5), gives a measure of global case clustering of residential histories throughout the study area and over the entire study time period. It is calculated by summing Equation (4) over all T−1 durations t

_{0}, t

_{1}…, t

_{T‒}

_{1}. This statistic indicates whether there is global clustering of residential histories when all of the residential histories over the entire study period are considered simultaneously. It is a measure of the persistence of global clustering, and is large when case clustering persists through time.

_{ikt}is used to identify when and where an individual is the center of a local cluster. Ǫ

_{ik}identifies which individuals tend to be centers of clusters over their life-course, but not when those clusters occur. The global Ǫ

_{k}identifies whether global clustering tends to occur over the residential histories, but not when or where the clustering occurs.

#### 3.2.1. A Diagnostic Framework for Ǫ-Statistics in Relation to Disease Processes

_{it}—the local statistic, Ǫ

_{i}—the life-course statistic, and Ǫ

_{t}—the large-scale spatial cluster statistic, are evaluated here. The subscript i is a case identifier, so Ǫ

_{it}and Ǫ

_{i}make statements regarding clustering of individual cases. How might these be used to generate inferences regarding space-time cluster processes?

_{1}that are cases and n

_{0}that are controls. The beginning of the study period is t = 0, the end is t = T. Consider the sets defined as follows:

_{it}that are statistically significant at the type I error level α, Ǫ

_{i}, is the set of all Ǫ

_{i}that are statistically significant at α, and is the set of all Ǫ

_{t}that are statistically significant at α. It turns out that Ǫ

_{t}and Ǫ

_{i}are global statistics that assess case-clustering at specific times (e.g., Ǫ

_{t}) and over the life course of specific cases (e.g., Ǫ

_{i}) such that:

_{it}. This allows for the mapping of sets of the local statistics Ǫ

_{it}to sets of significant statistics and . This mapping is comprised of those Ǫ

_{it}that contribute to the significant (through Equation (9)) and those Ǫ

_{it}that contribute to the significant (through Equation (10)). Understanding this allows for the consideration of the following operations:

_{it}that contribute to the sets of significant global statistics that are the operands of Equations (11)–(14). These operations are represented in Figure 2.

#### 3.2.2. Assessing Overall Significance of Cluster Types

_{it}, the life course statistic Ǫ

_{i}, and the spatial clustering statistic Ǫ

_{t}. The significance at a given α level yields membership in the sets illustrated in Figure 2. The number of local statistics in this case is quite large, and the use of the nominal type I error α will yield false positives. It is therefore necessary to derive statistical tests for evaluating the significance of the cluster types in Figure 2 that are not subject to erroneous inference attributable to multiple testing.

**Figure 2.**Venn diagram illustrating types of space-time clusters that can be identified using Ǫ-statistics. The rectangle represents all Ǫ

_{it}statistics in a study, significant or not. Each circle represents clusters that are found statistically significant locally (e.g., excess of cases about case i at time t, ), over a cases’ life course (e.g., excess of cases about the residential history of case i, ), and globally at a given time t when all cases are considered together (e.g., large-scale spatial clusters at time t, ). These cluster sets and their intersections (Ã, , , ) can provide insights into, and generate hypotheses regarding, disease etiologies. When the underlying Ǫ-statistics have been adjusted for the risk factors and covariates found significant in the parent case-control study these cluster types identify where, when, and to whom to allocate unexplained (e.g., excess) risk (Table 1).

**Table 1.**Description of cluster sets, summary space-time pattern descriptions, and disease etiologies that may give rise to those patterns.

Cluster set | Description | Pattern | Possible etiology |
---|---|---|---|

Cases (i) that at times t have a significant number of nearest neighbors that are cases | Cases (i) that at times t have a significant number of nearest neighbors that are cases. | Infection: Contagious process such that infection spreads from a case to its susceptible neighbors. Vector-borne disease process such that individuals in specific areas have increased risk of infection. Chronic (e.g., cancer): Increased cancer risk for individuals residing in local areas over a defined time period. Duration of elevated risk must be sufficiently long relative to the duration of time individuals live in the affected areas (e.g.,) exposure time must be sufficient to induce disease response. | |

Clustering over the life course | Cases (i) who, over the study, have a significant number of nearest neighbors that are cases. | Infection: The “typhoid Mary” or “super-spreader” process, whereby case (i) (the super-spreader) is infectious over the study period and transmits infections to nearest neighbors. Chronic (e.g., cancer): A process whereby neighbors of case i have increased cancer risk and such risk is elevated over the life course of case i. An example would be behaviors that increase cancer risk for others such as second hand smoke. May also arise when groups with elevated risk tend to move or remain together over their life course (e.g., familial groups with common genetic and/or behavioral risk factors). | |

Temporal case clustering | Large scale spatial clustering of cases at time t. Clustering of cases relative to controls is significant at time t when all cases and controls are considered. | Infection: Infection outbreak such that the infection impacts a large portion of the study population; endemic phase of infection with multiple local outbreaks. Chronic (e.g., cancer): Chronic disease with an underlying infectious etiology (e.g., viral hypothesis of cancer) that impacts a large portion of the study participants; Disease risk mediated by environmental exposures that vary across the study area such that risk is elevated for a large number of study participants. Duration of elevated risk must be sufficiently long relative to the duration of time individuals live in the affected areas (e.g., exposure time must be sufficient to induce disease response). | |

Ã | Locations and time when cases with significant clustering over their life course are members of a geographically localized cluster. Includes both ephemeral and persistent clusters. | Infection: Local foci of infection occurring at times t from which infected and infectious cases move away. Chronic (e.g., cancer): Local areas of persistent elevated risk that are sustained for a sufficient period of time that (1) disease risk is increased for individuals residing in the local area and (2) the duration of residence of cases in the area is of sufficient length to result in a significant Ǫ _{i} statistic. | |

Cases (i) who, over the study, have a significant number of nearest neighbors that are cases. | Infection: Large-scale outbreak at specific times, t, that may be comprised of local pockets of infection. For vector-borne diseases this can arise when large portions of the study area have suitable vector habitat during some parts of the study period. Chronic (e.g., cancer): Large scale exposures that occur at a specific time(s) t. An example would be leukemia in response to the Chernobyl and Hiroshima incidents. | ||

Cases that have clustering over their life course and are part of large-scale spatial clusters at times t. Includes cases whose Ǫ_{it} are not statistically significant, and some whose Ǫ_{it} are statistically significant. | Infection: Large-scale outbreak at times t with at least some of the resulting cases that (1) move together over their life course; and/or (2) remain infectious over their life course and continue to infect their neighbors. For a vector-borne disease this may arise when there is an initial large scale outbreak with some of the resulting cases continuing to be disease reservoirs (e.g., pathogen sources) whose infection can then be transmitted to neighbors. Behavioral: Individuals who have a behavior link that causes them to be at an increased exposure to an environmental factor, pathogen, or vector. The difference from Ǫ _{i} is that here, the exposure factor must temporally “outbreak” in nature in that it either cycles in population like a vector/pathogen can (such as bird flu) or in severity for an environmental factor. For example, imagine a poultry reseller who moves around. He has an elevated risk any time a bird flu epidemic breaks out so will show a Ǫ_{i} cluster and when the epidemics outbreak, there will be Ǫ_{t} clusters. An example would be where a pesticide is applied but because of laws is phased out, but later on people start using it again.Chronic (e.g., cancer): Large scale exposures that occur at a specific time(s) t with some of the resulting cases that (1) move together through life course or (2) continue to reside in the affected area over most of the study period. | ||

Cases that have clustering over their life course, are part of large scale clusters at time t and whose local clusters Ǫ_{it} are all statistically significant. | Etiology is similar to set , but is restricted to include only those individuals that are centers of significant local clustering of cases at times t. For infection, this may be indicative of index cases; for chronic diseases this may indicate individuals who are within local pockets of the largest exposure. |

**Table 2.**Statistics to evaluate the overall significance of the cluster types in a manner that does not involve multiple testing.

Cluster type | Cluster description | Test statistic | Probability of test statistic |
---|---|---|---|

Local case-time | |||

Life course | |||

Temporal case clustering | |||

Ã | |||

_{i}) = n

_{1}, where n

_{2}is the number of cases in the study. Table 3 enumerates n(Ǫ) for the different cluster types. Finally, is the desired type I error of the test, often set to = 0.05.

**Table 3.**Number of possible test statistics including those significant and not significant for each cluster type, using the duration-weighted tests. Here n

_{1t}is the number of cases extant in the study area at time t.

Cluster type | Cluster description | Test statistic | Number of possible elements in each set (n(Ǫ) in Equations (15) and (16)) |
---|---|---|---|

Local case-time | |||

Life course | n_{1T} | ||

Temporal case clustering | T |

#### 3.2.3. Calculating the Empirical Type I Error under Multiple Testing

#### 3.2.4. Adjusting Critical Values of the Test for Different Cluster Types

_{it}. One advantage of defining the cluster types described in Figure 2 is that the underlying tests are based on the number of elements in a set of a given cluster type, such as . Should this statistic prove significant (e.g., ), it is helpful to identify those Ǫ

^{*}

_{i}subsumed within the set that are likewise significant. This is a much smaller number than the maximum number of Ǫ

_{i}that can be calculated, yet there is still a multiple testing issue.

_{i}.

#### 3.3. Analysis Steps

- H1: Evaluate residential mobility by mapping places of residence for the study participants at three spatial scales: Marin County, California, and the continental United States.
- H2: Evaluate global clustering over the entire study using the Global Ǫ-statistic.
- H3: Evaluate whether and when there are times that cases cluster relative to controls when all of the study participants are considered together using the Ǫ
_{t}statistic. - H4: Evaluate whether specific cases tend, over their life course, to have other neighbors as cases using the Ǫ
_{i}statistic. - H5: Using the significance of set A, evaluate whether cases that cluster over their life course (significant Ǫ
_{i}) are part of local clusters at specific times (Ǫ_{it}).

## 4. Results

#### 4.1. Geocoding

#### 4.2. Hypotheses

#### 4.2.1. H1: Cases and Controls Do not Exhibit Substantial Residential Mobility over the Life Course

**Figure 3.**The study population is comprised of “movers” and “stayers”. Prior to 1980, the study population was comprised primarily of long-term residents, illustrated by the bimodal distribution of residence months (RESMONTHS, upper left histogram). The time plot (graph lower left) shows how residence time at each participant’s current residence changes through time. Each line corresponds to a study participant. Lines highlighted in orange are current Marin residents who have lived in the same home for at least 240 months (20 years), when the study was conducted. The map of Marin on 10-9-1977 (right) is comprised almost entirely of long-term residents.

**Figure 4.**Life course place of residence for study participants in Marin County (

**upper left**), California (

**upper right**), and from across the United States (

**bottom**). Cases are denoted as purple circles, and controls as gray crosses.

#### 4.2.2. H2: There Is No Statistically Significant Global Clustering of Breast Cancer Cases Relative to Controls When Accounting for Known Risk Factors and Covariates and for Residential Mobility

**Figure 5.**The sensitivity analysis using the Global Ǫ-statistic found significant global clustering of cases relative to controls after statistical adjustment for covariates and risk factors at k = 2, 3, and 4, with p < 0.01 at k = 4 nearest neighbors. This analysis was conducted at the spatial scale of Marin County.

#### 4.2.3. H3: There Are No Time Periods When the Breast Cancer Cases, Considered as a Group, Exhibit Statistically Significant Clustering Relative to the Controls

_{t}statistic. The identification of large-scale spatial clustering at specific time periods may indicate past exposures that impacted many, if not all, of the cases. This is the specific pattern the Ǫ

_{t}statistic is designed to detect.

_{t}test defined by Equation (4) was performed at k = 4 using 999 randomization runs. This yielded one test statistic and p-value for each of the 1,568 time periods with unique geographic arrangements of cases and controls between 1-1-1960 and 1-1-1999 (Figure 6). Using a type I error of 0.05 it was expected that 1,568 × 0.05 = 78.4 of the global tests would be significant. 122 tests were identified with p-values less than 0.05, an excess of 43.6.

_{t}under the null hypothesis of no clustering? To evaluate this question the probabilities of Ǫ

_{t}were ranked from smallest to highest, and then ordered through time. 3 epochs were identified as having significant clustering of cases relative to controls—February 1967 through February 1968 (Epoch 1); 1 April 1980 through 4 April 1980 (Epoch 2); and July 1989 through February 1990 (Epoch 3). Epoch 1 and epoch 2, which is relatively brief, were the focus of inquiries regarding the specific locations of case-clusters.

**Figure 6.**The probability of the Ǫ

_{t}test for spatial clustering of cases relative to controls through time. This test assesses whether and when there is spatial clustering of cases when all of the cases and controls are considered simultaneously at a given time t. p = 0.05 is shown by the blue horizontal line. The count of the number of observations below this line is = 122 and is highly significant (p = 0.000001625).

#### 4.2.4. H4: Cases Do not Exhibit Statistically Significant Clustering over Their Life Course

_{i}test in Equation (3) to identify individuals with clustering over their residential history, and Equation (7) to evaluate whether the observed number of significant Ǫ

_{i}( ) is itself statistically unusual. This second step has the advantage of not being subject to multiple testing, since only one measure ( ) is evaluated The test for Ǫ

_{i}was performed using k = 5 and 999 randomization runs, while adjusting for the probability of being a case from the logistic regression in order to account for the risk factors and covariates found significant in the parent study by Wrensch et al. [5]. 30 cases were found to have significant clustering over their entire residential history. An evaluation of the probability of = 30 using the binomial expressions in Equations (15) and (16) yielded a probability of observing this outcome of p = 0.000115. 14.25 cases would have been expected to be significant at the nominal type I error of 0.05.

#### 4.2.5. H5: Cases that Cluster over the Life Course (Ǫ_{i}) Are not Part of Local Clusters at Specific Times (Ǫ_{it})

_{i}(life course clustering) were identified. Visual examination of the geographic distribution of these 16 cases was performed focusing on epochs 1 (Figure 7) and 3 (Figure 8).

**Figure 7.**Life course clusters in Epoch 1. Locations of cases with significant clustering over their life course at the end of Epoch 1. Five cases had residential history data recorded on 2-1-1968. One case with significant clustering resided in Marin County (

**upper left**), two were in the Bay area (

**bottom**), and two in the Northeast near Long Island (

**upper right**).

**Figure 8.**Locations of cases with significant clustering over their life course at the end of Epoch 3. At that time 9 cases with significant clustering resided in Marin County (

**upper left**), with other clusters found in the upper Midwest (

**upper right**), southern California and Long Island (

**bottom**).

#### 4.3. Synopsis and Synthesis: Locations of Persistent Life Course Clusters

_{i}versus residence time at current residence during the study was constructed (1-1-1999), and from this individuals were identified that (1) are the most likely life-course clusters; and (2) have been at their current residence for at least 15 and also for 20 years. These cases were then mapped in Marin County (Figure 9).

**Figure 9.**Local clusters of breast cancer cases (large red circles) for twenty year (

**left**) and fifteen year (

**right**) residents of Marin County, California. The observed clustering is statistically significant when residential mobility and significant risk factors and covariates are accounted for.

## 5. Discussion

_{i}, which evaluates the number of nearest neighbors of a breast cancer case that also were breast cancer cases, and not controls. Data on 285 cases were used in this study, of which 30 had probabilities of Ǫ

_{i}that were less than the type I error level of 0.05. This outcome (30 significant of 285 cases) was evaluated using a binomial probability, and it is considered a highly unlikely outcome (p = 0.000115). The authors concluded there is evidence of statistically significant clustering over the life course. Of these 30 significant cases, at the 0.05 level, we would expect to find 14.25 cases significant. The authors therefore continued to analyze only those remaining 16 cases with the smallest p-values, once multiple testing was accounted for.

_{t}statistic, which evaluates clustering of cases relative to controls at specific time points. Of the 1,568 time periods that were defined between 1-1-1960 and 1-1-1999 whenever a woman moved from one residence to another, 122 had significant Ǫ

_{t}statistics. This count (122) is a highly unlikely outcome (p = 0.000001625). Of these 122 periods, one would expect at the 0.05 level to find 78.4 to be statistically significant under the null hypothesis. The remaining 44 most extreme cases defined three time periods (Epochs) when there was significant clustering of cases relative to controls when all of the study participants are considered simultaneously. These are February 1967 through February 1968 (Epoch 1), 1 April 1980 through 4 April 1980 (Epoch 2), and July 1989 through February 1990 (Epoch 3). Epoch 2 is brief and focus was directed the specific locations of case-clusters to Epochs 1 and 2.

_{i}, Ǫ

_{t}, and the cardinality of the results from these methods) were used to identify cluster constituents along the lines illustrated in Figure 2. While the burden of evidence demonstrates the persistent, local clusters identified in this study are statistically unusual, a biologically plausible exposure or risk factor has yet to be identified.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Jacquez, G.M.; Barlow, J.; Rommel, R.; Kaufmann, A.; Rienti, M., Jr.; AvRuskin, G.; Rasul, J.
Residential Mobility and Breast Cancer in Marin County, California, USA. *Int. J. Environ. Res. Public Health* **2014**, *11*, 271-295.
https://doi.org/10.3390/ijerph110100271

**AMA Style**

Jacquez GM, Barlow J, Rommel R, Kaufmann A, Rienti M Jr., AvRuskin G, Rasul J.
Residential Mobility and Breast Cancer in Marin County, California, USA. *International Journal of Environmental Research and Public Health*. 2014; 11(1):271-295.
https://doi.org/10.3390/ijerph110100271

**Chicago/Turabian Style**

Jacquez, Geoffrey M., Janice Barlow, Robert Rommel, Andy Kaufmann, Michael Rienti, Jr., Gillian AvRuskin, and Jawaid Rasul.
2014. "Residential Mobility and Breast Cancer in Marin County, California, USA" *International Journal of Environmental Research and Public Health* 11, no. 1: 271-295.
https://doi.org/10.3390/ijerph110100271