Case-control studies often collect many exposures but have relatively
small sample sizes in comparison to the number of exposures.
Conventional analysis techniques like multivariate logistic regression
can only accommodate a limited number of exposures. The
`nncc`

package was designed to help address this limitation.
`nncc`

matches a case to its nearest available neighbors
based on a user-defined measure of distance (by default Gower distance)
calculated using all collected confounders for a given exposure. In the
end, `nncc`

creates a matched data set for each exposure of
interest.

Ideally, one prefers to follow the analysis plan pre-defined at study design. Decisions to change your analysis plan to this approach should be considered carefully in light of how they may affect bias and uncertainty in model estimates. In addition, constructing variables are a routine and sometimes necessary part of multifactor studies. However, constructing new outcome variables from primary variables will increase dependencies and the difficulty of interpretation, especially where there is missingness of data.

For illustration, we created a toy example case-control data set
called `anifood`

which only contains 250 cases, 250 controls,
and 11 variables including the one indicating case status,
`case`

, which equals 0 for controls, 1 for cases.

In addition to the data set, `nncc`

needs to know the
exposures of interest, the variables used for matching, and a data set
that contains variables to be excluded from matching for each exposures
(e.g., possible intermediates in the pathway between an exposure and
case status).

```
# exposures of interest. In a real study, this list can be much longer
exp_interest <- c("exp01","exp09", "exp27")
# exposures to be controlled for any exposure of interest
exp_match <- setdiff(names(anifood), "case")
# variables to be excluded from matching for each exposure
# both exp_var and rm_vars are character variables
excl_vars %>% head
#> exp_var rm_vars
#> 1 exp01 exp09
#> 2 exp09 exp24 exp27
#> 3 exp27 exp43 exp45 exp50
```

The first step is to establish a proper distance threshold for matching. If too loose, there will be poor confounding control; if too tight, it will be difficult to find enough controls for the cases due to overmatching.

The `get_threshold`

function uses the maximum bipartite
graph algorithm to find each case’s closest control while ensuring each
control is used no more than once. Then, for each case, a control is
selected randomly.

A logistic regression on whether the control is the closest or the
randomly selected one based on the distance is created. By default, the
threshold used is the distance at which the probability is at least 50%
that the control is the closest vs. a randomly selected one (see the
plot below generated by the function `threshold_model_plot`

).
You can choose a different probability with the `p_threshold`

argument.

The `distance_density_plot`

function plots the density
distribution of Gower distances for the maximum bipartite graph
algorithm matches (solid line), the density distribution of distances
for the random matches (dashed line), and the threshold (red dashed
line). As expected, a maximum bipartite graph algorithm matching results
in closer matching than a random matching.

The function `threshold_model_plot`

plots the probability
of being the maximum bipartite graph algorithm match vs. a random one by
distance. It also plots the threshold for distance corresponding to a
probability specified through the `p_threshold`

argument.

`threshold_model_plot(threshold_results, p_threshold = 0.50) + ggtitle("Example of threshold_model_plot")`

This package can also be used to analyze data from a case-control
study that was originally matched. The
`original_compare_plot`

function makes a density plot of the
distance for originally matched pairs. It also gives the proportion of
originally matched pairs that have a distance greater than the
threshold.

```
# create a variable (i.e., pair) indicating originally matched pairs
anifood_matched <- anifood %>% group_by(case) %>% mutate(pair = seq_along(case)) %>% ungroup
```

In this example, 84% of original matched case-control pairs have a distance greater than the threshold.

The `make_knn_strata`

function calculates Gower distances,
by default, for pairs formed by every case and every control for a given
exposure and creates strata by selecting the closest 250 controls for
each case (by default, can be modified through the `ncntls`

argument). Thus, for that exposure, the number of rows in data frame
equals the number of cases * 251. The data frames for all exposures are
stored in a list with a length equal to
`length(exp_interest)`

.

This step can be computationally intensive depending on the amount of
data. `furrr::future_map()`

can be used so that the
computation can be easily adjusted to use multiple cores on your local
computer or a high-performance computing (HPC). More information is
provided below.

```
library(furrr)
strata1 <- future_map(exp_interest, make_knn_strata, rmvars = excl_vars, matchvars = exp_match, df = anifood) %>%
setNames(exp_interest)
```

```
length(strata1) == length(exp_interest)
#> [1] TRUE
# rows in a matched data set
all.equal(anifood %>% filter(case == 1) %>% NROW %>% `*`(250 + 1), NROW(strata1[[1]]))
#> [1] TRUE
```

In the `make_knn_strata`

function, a data frame should be
supplied to the `rmvars`

argument (see `excl_vars`

above). That data frame should include two variables: *exp_var: a
character variable (only a single exposure in a row), and *rm_vars:
a character variable that contains the names of the variables that are
not adjusted for. Names are separated by a space.

For a given exposure, the `make_analysis_set`

function
selects the closest 20 controls for each case; subset these 20 to those
that fall within the threshold; collapse strata that share the same
controls; and remove strata without any control.

```
strata2 <- future_map(exp_interest, make_analysis_set, stratified_data = strata1, data = anifood, maxdist = threshold_results$threshold) %>% setNames(exp_interest)
#> exp01
#> exp09
#> exp27
```

The `finalize_data`

function ensures that a control
retained in a data frame is used once; removes strata without any case
or any control. In this process, priority is given to the smallest
strata then smallest distance if a control is matched to multiple cases
(i.e., that control exists in multiple strata).

The last step is to exclude exposures, if any, with odds ratios that
are not estimable (e.g., none of cases and controls was exposed). For
example, `exp09`

in this example.

```
# exposures to which neither cases nor controls were exposed
strata3 %>%
lapply(function(dfm) dfm %>%
mutate(case = as.character(case), exp = as.character(exp)) %>%
filter(case != "" & exp != "") %>%
with(table(case, exp))) %>%
lapply(function(x) x==0) %>%
lapply(function(x) {sum = sum(x); length = length(x); cbind(sum, length)}) %>%
do.call(rbind,.) %>%
as.data.frame %>%
mutate(var = names(strata3)) %>%
filter(sum >= 2 | length < 4) %>% select(var) %>%
unclass %>%
unlist -> expvars_invalid
expvars_invalid
#> character(0)
# none exposed and odds ratio cannot be estimated
strata3[["exp09"]] %>% with(table(case, exp))
#> exp
#> case 0 1
#> 0 236 0
#> 1 231 1
data_final <- strata3[setdiff(names(strata3), expvars_invalid)]
```

The `test_mh`

function performs the Mantel–Haenszel test
for data frame with more than one stratum and the Fisher’s exact test
for a data frame with only one stratum.

```
or_mh <- future_map(data_final, function(dfm) with(dfm, test_mh(case = case, exp = exp, strata = strata)))
or_mh[["exp01"]]
#> $p.value
#> [1] 0.727695
#>
#> $conf.int
#> [1] 0.7440193 1.5998105
#> attr(,"conf.level")
#> [1] 0.95
#>
#> $estimate
#> common odds ratio
#> 1.091004
#>
#> $null.value
#> common odds ratio
#> 1
#>
#> $alternative
#> [1] "two.sided"
#>
#> $method
#> [1] "Mantel-Haenszel chi-squared test with continuity correction"
#>
#> $data.name
#> [1] "case and exp and strata"
```

However, when the number of exposed is small, this method may yield an infinite result.

```
data_final[["exp27"]] %>% select(case, exp) %>% table
#> exp
#> case 0 1
#> 0 236 0
#> 1 231 1
or_mh[["exp27"]]
#> $p.value
#> [1] 0.9698768
#>
#> $conf.int
#> [1] NaN NaN
#> attr(,"conf.level")
#> [1] 0.95
#>
#> $estimate
#> common odds ratio
#> Inf
#>
#> $null.value
#> common odds ratio
#> 1
#>
#> $alternative
#> [1] "two.sided"
#>
#> $method
#> [1] "Mantel-Haenszel chi-squared test with continuity correction"
#>
#> $data.name
#> [1] "case and exp and strata"
```

Conditional logistic regression gives results similar to those from Mantel–Haenszel test, but may also give less than ideal results when few participants are exposed.

```
results_clogit <- future_map(data_final, function(dfm){clogit(case ~ exp + strata(strata) , data = dfm)})
results_clogit[["exp27"]] %>% summary() %>% `$`(conf.int)
#> exp(coef) exp(-coef) lower .95 upper .95
#> exp1 11766455 8.498736e-08 0 Inf
```

When combined with multiple imputation, we recommend a Bayesian approach.

```
library(rstanarm)
results_stan_clogit <- future_map(data_final, function(dfm) {
dfm$strata <- factor(dfm$strata)
stan_clogit(case ~ exp,
data = dfm,
strata = strata,
# rstanarm suggests us specifying priors even if they are the
# defaults because they might change in the future
prior = normal(0, 4),
#default: prior = normal(autoscale = TRUE),
prior_covariance = decov(),
iter = 100,
chains = 2,
prior_PD = FALSE)}) # for speed only
```

When the number of participants being exposed is small, Firth’s bias-Reduced penalized-likelihood logistic regression can also be used (1).

```
library(logistf)
# regression coefficients
coef_logistf <- future_map(data_final, function(dfm){
if(length(unique(dfm[["strata"]])) == 1) {
x <- model.matrix(case ~ exp, data = dfm)
plconf <- grep("exp", colnames(x))
o <- logistf(case ~ exp, data = dfm, plconf = plconf) %>%
`[`(c("terms", "coefficients", "ci.lower", "ci.upper", "prob", "call", "loglik", "model"))
} else {
x <- model.matrix(case ~ exp + strata, data = dfm)
plconf <- grep("exp", colnames(x))
o <- logistf(case ~ exp + strata, data = dfm, plconf = plconf) %>%
`[`(c("terms", "coefficients", "ci.lower", "ci.upper", "prob", "call", "loglik", "model"))
}
},
.progress = TRUE)
# odds ratios
or_logistf <- lapply(names(coef_logistf), function(var_name){
coef_logistf[[var_name]] %>%
`[`(c("terms", "coefficients", "ci.lower", "ci.upper", "prob")) %>%
bind_cols %>%
filter(grepl("exp", terms)) %>%
mutate(variable = var_name, or = exp(coefficients), ci.lower = exp(ci.lower), ci.upper = exp(ci.upper)) %>%
select(variable, terms, or, ci.lower, ci.upper, prob)}) %>%
setNames(names(coef_logistf))
# odds ratio for exp27
or_logistf[["exp27"]]
#> # A tibble: 1 × 6
#> variable terms or ci.lower ci.upper prob
#> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 exp27 exp1 3.30 0.172 486. 0.434
```

PAF can be estimated using the method described by Bruzzi and
colleagues (3). We implemented this method in `get_paf`

.

```
# prepare a data frame for calculating PAF
df_or <- bind_rows(or_logistf)
df_or
#> # A tibble: 3 × 6
#> variable terms or ci.lower ci.upper prob
#> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 exp01 exp1 1.09 0.744 1.60 0.656
#> 2 exp09 exp1 3.54 0.182 523. 0.408
#> 3 exp27 exp1 3.30 0.172 486. 0.434
```

```
# point estimate of PAF
paf <- get_paf(df_or = df_or, which_or = or, exp_var = variable, exp_level = terms, df_matched = data_final)
paf
#> # A tibble: 3 × 2
#> paf_or variable
#> <dbl> <chr>
#> 1 0.0549 exp01
#> 2 0.00309 exp09
#> 3 0.00300 exp27
# lower confidence limit of PAF
paf_ci.lower <- get_paf(df_or = df_or, which_or = ci.lower, exp_var = variable, exp_level = terms, df_matched = data_final)
```

When the case-control data contains a large proportion of missingness, multiple imputation may be useful.

For regression coefficients generated by Mantel–Haenszel test or conditional logistic regression, Rubin’s rule may be proper for combining the results (4).

Originally in our paper (2), we used Firth’s bias-Reduced penalized-likelihood logistic regression, but the regression coefficients are not normally distributed. The results from imputed data sets should therefore be combined using penalized likelihood profiles (5). Penalized likelihood profiles are implemented in

`logistf::CLIP.confint()`

, but the function was developed to combine results from imputed data sets that have the same structure. However, if we impute the original case-control data, and then apply nearest-neighbors matching to the imputed data, the final analytic data sets for a given exposure could have inconsistent structures because they could have different numbers of strata.

Thus, we modified the `logistf::CLIP.confint()`

to
accommodate our method. The modified function is included in version
1.0.1 of the `nncc`

package along with a compatible, bundled
version of the `logistf`

package version 1.24, and the
function is called `CLIP.confint.difflevel()`

. Please cite
the original paper (4) and/or the logistf
package if you use this modified function for publication.

- Now we recommend a Bayesian approach, based on drawing samples from the stacked posteriors of the model run on each imputed data set.

`cacheit()`

When the number of exposures of interest or the sample size of the
original case-control study is large, it may be time-consuming for some
analyses (e.g., strata creation). You may not want to have the same code
evaluated again when you run your analysis later. The helper function
`cacheit()`

caches the results for you. Specifically, before
your run `cacheit()`

, create a folder named “cache” in the
working directory of your project. When you run
`cacheit("abc", code)`

for the first time, it saves the
results of your code as abc.rds in the `cache`

folder; next
time, when you run the code, it directly read the saved results in
without evaluating your code.

Considering the intensive computation for this approach, especially when multiple imputation is conducted, a future can be used to accelerate the analysis. Below we briefly introduce the use of futures on your local multi-core computer and HPC.

If you have a multi-core computer, the following code can assign a specified number of cores for the analysis.

```
library(nncc)
library(dplyr)
library(furrr)
library(future.batchtools)
# the workers argument is used to define the number of cores for the analysis. By default, all cores will be used.
plan(multisession, workers = 3)
strata1 <- future_map(exp_interest, make_knn_strata, rmvars = excl_vars, matchvars = exp_match, df = anifood) %>%
setNames(exp_interest)
```

To run futures on HPC, you may need a template file, a job script, and a R script. The template file and the job script are created using a Linux application such as nano and saved as .sh files.

There are some exemplary template files here.
The template file should be saved as `batchtools.sge.tmpl`

in
your current working directory or as `.batchtools.sge.tmpl`

in your home directory on the HPC. Below is an example of a template
file for Sun Grid Engine.

```
#!/bin/bash
## name of the job
#$ -N <%= job.name %>
## tell the queue system to use the current directory as the working directory
## Or else the script may fail as it will execute in your top level home directory /home/username
#$ -cwd
## Use environment variables
#$ -V
## specify a queue
#$ -q <%= resources$queue %>
## free to add other modules if necessary
module load R/3.6.2
Rscript -e 'batchtools::doJobCollection("<%= uri %>")'
exit 0
```

Below is an example of a job script to be submitted to the scheduler through a shell:

```
#!/bin/bash -l
# name of the job is helloR
#$ -N helloR
#$ -cwd
#$ -V
#$ -pe smp 2-12
#$ -q all.q
module load R/3.6.2
# to execute fugure-map.R
Rscript future-map.R
exit 0
```

The R code looks like:

- Firth D. Bias reduction of maximum likelihood estimates. Biometrika 1993;80:27-38.
- Bruzzi P, Green SB, Byar DP, et al. Estimating the population attributable risk for multiple risk factors using case-control data. Am J Epidemiol 1985;122(5):904-14.
- Cui Z, Marder EP, Click ES, Hoekstra RM, Bruce BB. Nearest-neighbors
matching for case-control study analyses: better risk factor
identification from a study of sporadic campylobacteriosis in the United
States. Epidemiology 2022;33(5):633-41.

- Rubin DB. Multiple imputation for nonresponse in surveys. New York: John Wiley and Sons; 1987.
- Heinze G, Ploner M, Beyea J. Confidence intervals after multiple imputation: combining profile likelihood information from logistic regressions. Stat Med 2013;32(29):5062-76.