The **smacpod** package implements methods for analyzing
case-control point pattern data. It relies on the
**spatstat** package for many of the internal computations,
and many function arguments are simply the relevant arguments of a
**spatstat** function.

We briefly summarize functionality below using the `grave`

data set. The `grave`

data set provides information related
to 143 medieval grave locations. 30 of the graves are
`"affected"`

by a tooth deformity, while the remaining graves
are `"unaffected"`

by the deformity. The `grave`

data set is a `ppp`

object, which is a class utilized by the
**spatstat** package.

We load and plot the data to start.

The `spdensity`

function computes the spatial density
function of a point pattern. The **spatstat** package
provides a `density`

function, which is actually the
estimated intensity of a point pattern. The `spdensity`

function scales the intensity to integrate to 1. The result is an
`im`

object, a class from the **spatstat**
package, which has an associated `plot`

method. We estimate
and plot the spatial density of the `affected`

grave
locations below. We do not specify the bandwidth or kernel function used
in the estimation process, so the default choices of the
**spatstat** package are used. The heat map of the
estimated spatial density indicates greater density of grave locations
in the lower right part of the study area.

Let \(D\) denote the study area
under consideration. Let \(\lambda_1(s)\) denote the intensity
function at location \(s\) of the group
designated as the `case`

group. Then the spatial density of
the `case`

group can be denoted as \[f(s)=\frac{\lambda_1(s)}{\int_D \lambda_1(s)
ds}.\] Similarly the intensity function and spatial denstiy
function of the `control`

group are denoted as \(\lambda_0(s)\) and \(g(s)\), respectively.

The log relative risk function is defined as \[r(s) =
\ln\left(\frac{f(s)}{g(s)}\right).\] The log relative risk
function can be estimated by \[\hat{r}(s) =
\ln\left(\frac{\hat{f}(s)}{\hat{g}(s)}\right),\] where \(\hat{f}\) and \(\hat{g}\) are the estimates of \(f\) and \(g\) from the `spdensity`

function.

The log relative risk can be estimated using the `logrr`

function. The function takes:

`x`

: a`ppp`

object with`marks`

for the cases and controls`sigma`

: the bandwidth parameter of the`case`

group`sigmacon`

: the bandwidth parameter of the control group`case`

: a character string specifying the name of the case group or the index of the case group in`levels(x$marks)`

. The default is`case = 2`

, the second level of`x$marks`

.

If `sigma`

and `sigmacon`

aren’t specified,
then the `spatstat.explore::bw.relrisk`

function is used for
these arguments.

We estimate the log relative risk below for the `grave`

data, selecting the `affected`

graves as the
`case`

group, and then plot the results. The
`logrr`

returns an object of class `im`

, which has
an associated `plot`

method. We also add contours of \(\hat{r}(s)\) using the `contour`

function. The ares with levels around 1-1.5 have the highest relative
risk of cases to controls. These are the areas most likely to have a
cluster of affected grave locations relative to unaffected grave
locations.

`## affected has been selected as the case group`

```
plot(rhat, main = "log relative risk of affected versus unaffected graves")
contour(rhat, add = TRUE)
```

If the ratio \(f(s)/g(s)\) is
constant, then the relative risk is 1 and \(r(s)=0\). To determine where the log
relative risk differs significantly from zero, we can simulate
`nsim`

new case-control data sets under the random labeling
hypothesis, estimate \(r(s)\) for each
simulated data set, and construct tolerance envelopes for \(r(s)\) at each location. If \(\hat{r}(s)\) is outside the tolerance
envelopes, then the estimated log relative risk is inconsistent with
\(r(s)=0\) under the random labeling
hypothesis. This is the same is believing the spatial densities of the
cases and controls differ from zero at that location. If the spatial
density of the cases is greater than the spatial density of the
controls, then we believe there is a cluster of cases relative to the
controls. If the spatial density of the controls is greater than the
spatial density of the cases, then we believe there is a cluster of
controls relative to the cases. The tolerance envelopes can be
constructed by specifying the following arguments of
`logrr`

:

`nsim`

: the number of randomly labeled data sets to generate.`level`

: the level of the tolerance envelopes. The default is`0.90`

.`alternative`

: the type of envelopes to construct. The default is`two.sided`

.

When `nsim>0`

, the `logrr`

function returns
an object of class `renv`

, which has a `plot`

method that makes it easy to identify the parts of the study area where
the estimated log relative risk is outside the tolerance envelopes. Any
colored area indicates a location where the log relative risk is outside
the tolerance envelopes. Non-colored areas indicate the estimated log
relative risk is inside the tolerance envelopes.

Similarly, there is a `print`

method for the
`renv`

class summarizing information about the object
constructed.

For the `grave`

data, the yellow areas indicate parts of
the study area where the `affected`

grave locations are
clustered relative to the `unaffected`

grave locations beyond
what is expected under the random labeling hypothesis. The purple areas
indicate parts of the study area where the `unaffected`

grave
locations are clustered relative to the `affected`

grave
locations beyond what is expected under the random labeling
hypothesis.

```
# construct tolerance envelopes for logrr
renv = logrr(grave, case = "affected", nsim = 99, level = 0.90)
```

`## affected has been selected as the case group`

```
##
## pixelwise tolerance envelopes for log relative risk, r(s)
##
## r(s) = ln[f(s)/g(s)]
## f(s) = spatial density of cases at location s
## g(s) = spatial density of controls at location s
## case label: affected
## control label: unaffected
## number of simulations: 99
## simulation procedure: random labeling
## envelope level: 0.9
```

The color scale can be manipulated to use different colors, as in the example below.

```
# a better color scale (in my opinion)
# making it easier to distinguish the clusters of cases relative
# to controls (red) and vice versa (blue)
grad = gradient.color.scale(min(renv$v, na.rm = TRUE), max(renv$v, na.rm = TRUE))
plot(renv, col = grad$col, breaks = grad$breaks)
```

Because the tolerance envelopes do not account for multiple comparisons, Kelsall and Diggle (1995) proposed a test of

\(H_0: r(s) = 0\) for all \(s\) in \(D\)

\(H_a: r(s) \neq 0\) for at least one \(s\) in \(D\)

using the test statistic \(\int_D \{\hat{r}(s)\}^2\,ds\).

This test is implemented in the `logrr.test`

function,
which takes an object of class `renv`

.

For the `grave`

data, since the p-value is relative large,
there isn’t convincing evidence that the log relative risk of the
affected and unaffected grave locations differs from 0 for at least one
location in the study area.

```
##
## Kelsall and Diggle (1995) test for log relative risk
##
## r(s) = ln[f(s)/g(s)]
## f(s) = spatial density of cases at location s
## g(s) = spatial density of controls at location s
## case label: affected
## control label: unaffected
##
## null hypothesis: r(s) = 0 for all s in study area
## alternative hypothesis: r(s) != 0 for at least one s in study area
## test statistic: 19478034
## p-value: 0.22
## nsim: 99
## simulation procedure: random labeling
```

Ripley’s K function is often used to assess the variability of the distances between events for point patterns. A difference in K functions can be used to compare the clustering behavior for case-control point data, where \[\widehat{KD}(r) = \widehat{K}_{case}(r) - \widehat{KD}_{control}(r)\] is the estimated difference in K functions for the case and control groups for a specific distance \(r\).

The `kdest`

function can be used to compute \(\widehat{KD}(r)\). The function
requires:

`x`

: a`ppp`

object with`marks`

for the cases and controls`case`

: a character string specifying the name of the case group or the index of the case group in`levels(x$marks)`

. The default is`case = 2`

, the second level of`x$marks`

.

The function relies internally on
`spatstat.explore::Kest`

, and users are directed there for
more details about the estimation process. The `kest`

function produces on object of class `fv`

, defined by the
**spatstat.explore** package, which has an associated
`plot`

method that can be used to plot the results.

We compute the estimated difference in K functions for the case and controls groups below, and then plot the results.

`## affected has been selected as the case group`

To investigate where \(\widehat{KD}(r)\) differs significantly from zero, tolerance envelopes for \(KD(r)\) can be estimated by simulating data under the random labeling hypothesis. We need to specify:

`nsim`

: the number of randomly labeled data sets to generate.`level`

: the level of the tolerance envelopes. The default is`0.90`

.

When `nsim>0`

, the `kdest`

function returns
an object of class `kdenv`

. `print`

,
`summary`

, and `plot`

methods are defined for
`kdenv`

objects. The `print`

method will print
information about the `kdenv`

object, `summary`

will summarize the distances at which \(\widehat{KD}(r)\) is outside the tolerance
envelopes, and `plot`

will plot \(\widehat{KD}(r)\) versus \(r\), along with the average \(\widehat{KD}(r)\) for data sets simulated
under the null hypothesis, the tolerance envelopes, and envelopes of the
minimum and maximum \(\widehat{KD}(r)\)
for the simulated data sets.

We contruct tolerance envelopes for \(KD(r)\) based on 49 data sets simulated
under the random labeling hypothesis at the 0.95 level. Using the output
of the `summary`

function, we see that \(\widehat{KD}(r)\) is outside the tolerance
envelopes for \(r\) in the
(approximate) intervals, 9-48, 353-387, 395-425, 985-1004, 1081-1085,
and at \(r=1090.454\). The
corresponding plot is shown below with a manual legend created to
identify the various parts of the plot.

`## affected has been selected as the case group`

```
## Generating 49 simulations by evaluating expression ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49.
##
## Done.
```

```
##
## Envelopes for difference in estimated K functions
##
## KD(r) = K_case(r) - K_control(r)
## case label: affected
## control label: unaffected
## KD(r) computed for r betwen 0 and 1533.825
## number of simulations: 49
## simulation procedure: random labeling
## envelope level: 0.95
```

```
## KD(r) > upper envelope limit for the following r:
## 8.987257 to 47.93204
## 353.4988 to 386.4521
## 395.4393 to 425.3968
## 985.6025 to 1003.577
## 1081.467 to 1084.462
## 1090.454
```

```
# plot results
plot(kdenv, ylab = "difference in K functions")
legend("topleft", legend = c("obs", "avg", "max/min env", "95% env"),
lty = c(1, 2, 1, 2), col = c("black", "red", "darkgrey", "lightgrey"),
lwd = c(1, 1, 10, 10))
```

The tolerance envelopes do not account for multiple comparisons, so Diggle and Chetwynd (1991) proposed a test of

\(H_0: KD(r) = 0\) for all \(r\) in \([0, r_{max}]\)

\(H_a: KD(r) > 0\) for at least one \(r\) in \([0, r_{max}]\)

using the test statistic \(KD_+=\sum_{i=1}^m \widehat{KD}(r_i)\), where \(r_1,\ldots,r_m\) are the sequence of values for which \(\widehat{KD}(r)\) is estimated.

This test is implemented in the `kdplus.test`

function.

For the `grave`

data, since the p-value is somewhat small
(0.06), there is weak evidence that the affected grave locations are
clustered relative to the unaffected grave locations for some distance
between 0 and 1533 meters.

```
##
## Diggle and Chetwynd (1991) test for difference in K functions
##
## KD(r) = K_case(r) - K_control(r)
## case label: affected
## control label: unaffected
##
## null hypothesis: KD(r) = 0 for all r between 0 and 1533.825
## alternative hypothesis: KD(r) > 0 for at least one r between 0 and 1533.825
## test statistic: 576.2432
## p-value: 0.06
## nsim: 49
## simulation procedure: random labeling
```

Kulldorff (1997) proposed a spatial scan test for case-control point
data. The test scans the study area using circular windows to identify
windows with unusual collections of cases compared to what is outside
the window. The significance of the most likely cluster (window) is
assessed via the random labeling hypothesis. The test is implemented
using the `spscan.test`

function, which takes the
arguments:

`x`

: a`ppp`

object with`marks`

for the cases and controls`case`

: a character string specifying the name of the case group or the index of the case group in`levels(x$marks)`

. The default is`case = 2`

, the second level of`x$marks`

.`nsim`

: the number of randomly labeled data sets to generate.`alpha`

: the significance level of the test`maxd`

: the maximum size of the window. the default is half the maximum inter-event distance.

The method assess whether the most likely cluster of cases is significant under the random labeling hypothesis. It can also be used to identify the most likely clusters of events that do not overlap.

The `print`

function will summarize some basic information
of the test. The `summary`

function will summarize any
identified clusters. The `clusters`

function will extract the
events comprising the significant clusters. The `plot`

function will draw the most likely and any secondary clusters.

For the `grave`

data, we identify one significant cluster.
The cluster is centered at \((10324,
4389)\) with a radius of 1359.7 meters and includes 19 events (11
affected graves). The cluster was found on the lower right part of the
study area. The cluster includes events 126, 124, …, 75.

`## affected has been selected as the case group`

```
## method: circular scan
## distance upperbound: 3689.4017468961
## realizations: 49
```

```
## centroid_x centroid_y radius events cases ex rr stat p
## 1 10324 4389 1359.7 19 11 4 3.8 7.4 0.1
```

```
# plot scan test results
plot(scan, chars = c(1, 20), main = "most likely cluster for grave data",
border = "orange")
```

```
## [[1]]
## [1] 126 124 125 132 120 113 140 141 112 41 136 30 133 108 31 143 110 139 75
```

Cuzick and Edwards (1990) proposed a test to examine whether the
number of cases within the \(Q\)
nearest neighbors of other cases is unusually large relative to the
random labeling hypothesis. This test is implemented in the
`qnn.test`

function, which takes the arguments:

`x`

: a`ppp`

object with`marks`

for the cases and controls`q`

: a sequence of \(Q\) values to consider`case`

: a character string specifying the name of the case group or the index of the case group in`levels(x$marks)`

. The default is`case = 2`

, the second level of`x$marks`

.`nsim`

: the number of randomly labeled data sets to generate.

Note that the statistics for different \(Q\) will be correlated since, for example, the 3 nearest neighbors of a case will be included in the 5 nearest neighbors of a case. Thus, the test also measures the significance of the excess number of cases for different values of \(Q\), i.e., the contrast between the test statistic for different values of \(Q\). If the contrast is not significant, then any clustering observed is caused by the clustering observed for the smaller value of \(Q\).

For the `grave`

data, we examine whether there is excess
clustering of cases among the \(Q\)
nearest neighbors of each case for \(Q=3,5,\ldots,15\), using the
`affected`

group as the `case`

group. Since the
p-value is small for each value of \(Q\), there is evidence of a significant
excess of cases among the \(Q\) nearest
neighbors of each case compared to the random labeling hypothesis for
\(Q=3,5,\ldots,15\). Looking at the
p-values for the contrasts, it appears that the clustering of cases
observed for \(Q=5,7,9,11\) are caused
by the \(Q=3\) nearest neighbors.
However, the clustering observed at \(Q=13,15\) is in excess to the clustering
observed at \(Q=3\). Similarly, the
clustering observed for \(Q=15\) is
caused by the clustering observed for \(Q=13\).

`## affected has been selected as the case group`

```
## Q nearest neighbors test
##
## case label: affected
## control label: unaffected
##
## Summary of observed test statistics
##
## q Tq pvalue
## 3 32 0.02
## 5 45 0.04
## 7 58 0.04
## 9 73 0.04
## 11 91 0.04
## 13 109 0.02
## 15 122 0.06
##
## Summary of observed contrasts between test statistics
##
## contrast Tcontrast pvalue
## T5 - T3 13 0.54
## T7 - T3 26 0.46
## T9 - T3 41 0.26
## T11 - T3 59 0.16
## T13 - T3 77 0.14
## T15 - T3 90 0.12
## T7 - T5 13 0.42
## T9 - T5 28 0.28
## T11 - T5 46 0.14
## T13 - T5 64 0.10
## T15 - T5 77 0.12
## T9 - T7 15 0.20
## T11 - T7 33 0.08
## T13 - T7 51 0.08
## T15 - T7 64 0.10
## T11 - T9 18 0.10
## T13 - T9 36 0.06
## T15 - T9 49 0.10
## T13 - T11 18 0.08
## T15 - T11 31 0.16
## T15 - T13 13 0.46
```

Cuzick, J., and Edwards, R. (1990). Spatial clustering for inhomogeneous populations. Journal of the Royal Statistical Society. Series B (Methodological), 73-104.

Diggle, Peter J., and Amanda G. Chetwynd. “Second-order analysis of spatial clustering for inhomogeneous populations.” Biometrics (1991): 1155-1163.

Kelsall, Julia E., and Peter J. Diggle. “Non-parametric estimation of spatial variation in relative risk.” Statistics in Medicine 14.21-22 (1995): 2335-2342.

Kulldorff, M. (1997) A spatial scan statistic. Communications in Statistics – Theory and Methods 26, 1481-1496.