```
# basic setup
library(smdi)
library(ggplot2)
library(survival)
library(gt)
library(broom)
library(fastDummies)
library(survival)
library(mice)
library(dplyr)
library(gridExtra)
library(tibble)
library(forcats)
```

`MNAR(value)`

As we have seen earlier, if the missingness mechanism of an important
covariate, such as a strong partially observed confounder, truly follows
either a missing completely at random (MCAR) or a missing not at random
*value* (MNAR[value]) scenario, it may be possible to
differentiate these two mechanisms using `smdi_diagnose()`

,
but the distinction between the two can also turn out difficult in
practice. This can have serious consequences since covariates that
follow a true MNAR(value) mechanism may severely bias the effect
estimation of an exposure of interest.

In reality, we will not be able to know 100% what the true
missingness mechanism is and there may always be some residual
uncertainty in case of MNAR(value). On the positive side, however, Leacy
and Moreno-Betancur et al. have proposed a useful procedure to impute
multivariable missing data under MNAR(value) conditions. By definition,
under a MNAR(value) mechanism, the true (but unobserved) values of a
variable systematically differ from the observed values. The idea of the
not at random fully conditional specification (NARFCS) sensitivity
analysis is to specify an imputation model containing a sensitivity
parameter **𝛅** that reflects this systematic departure
from a missing at random (MAR) mechanism.

For example, the `age_num`

of patients below a certain age
cut-off may be systematically less frequently recorded. Despite this,
`age_num`

may still be an important predictor for the
initiation of the treatment of interest and simultaneously an import
prognostic variable for the outcome (time to death due to any reason
[overall survival] in this example).

For simplicity, we don’t assume any other missing covariates for now.
In order to introduce an MNAR(value) mechanism for the
`age_num`

covariate, we:

- Determine a missingess
pattern, in which only
`age_num`

will be set to missing - Create a weight
vector, in which only
`age_num`

itself (by a non-zero value) is the linear predictor for the probability of observations becoming missing - Specify a type of logistic probability distribution for the
missingness weights, so that patients with low weighted sum scores
(i.e., younger
`age_num`

) will have a larger probability of becoming incomplete - Define an overall missingness proportion of ~40%

```
# load complete dataset
smdi_data_complete <- smdi_data_complete %>%
dummy_columns(
select_columns = "ses_cat",
remove_most_frequent_dummy = TRUE,
remove_selected_columns = TRUE
)
# determine missingness pattern
age_col <- which(colnames(smdi_data_complete)=="age_num")
miss_pattern <- rep(1, ncol(smdi_data_complete))
miss_pattern_age <- replace(miss_pattern, age_col, 0)
# weights to compute missingness probability
# covariate itself is only predictor
miss_weights_mnar_v <- rep(0, ncol(smdi_data_complete))
miss_weights_mnar_v <- replace(miss_weights_mnar_v, age_col, 1)
miss_prop_age <- .55
set.seed(42)
smdi_data_mnar_v <- ampute(
data = smdi_data_complete,
prop = miss_prop_age,
mech = "MNAR",
patterns = miss_pattern_age,
weights = miss_weights_mnar_v,
bycases = TRUE,
type = "LEFT"
)$amp
```

If we plot the direct comparison of `age_num`

distributions between the original data and the data with complete cases
after removing those who have a missing value following an MNAR(value)
mechanism, we can observe the systematic difference in the two
datasets.

```
# plot
bind_rows(
smdi_data_complete %>% select(age_num) %>% mutate(dataset = "Complete"),
smdi_data_mnar_v %>% select(age_num) %>% mutate(dataset = "MNAR(value)")
) %>%
ggplot(aes(x = dataset, y = age_num)) +
geom_violin() +
geom_boxplot(width = 0.3) +
stat_summary(
fun = "mean",
geom = "pointrange",
color = "red"
) +
labs(
y = "Age [years]",
x = "Cohort"
) +
theme_bw()
```

`smdi`

diagnosticsNow let’s use `smdi_diagnose`

to investigate the three
group diagnostics from `age_num`

. This may help characterize
the underlying missingness mechanism.

```
smdi_diagnose(
data = smdi_data_mnar_v,
covar = "age_num",
model = "cox",
form_lhs = "Surv(eventtime, status)"
) %>%
smdi_style_gt()
```

Covariate | ASMD (min/max)^{1} |
p Hotelling^{1} |
AUC^{2} |
beta univariate (95% CI)^{3} |
beta (95% CI)^{3} |
---|---|---|---|---|---|

age_num | 0.056 (0.004, 0.360) | <.001 | 0.543 | -0.41 (95% CI -0.50, -0.32) | -0.40 (95% CI -0.49, -0.32) |

p little: <.001, Abbreviations: ASMD = Median absolute standardized mean difference across all covariates, AUC = Area under the curve, beta = beta coefficient, CI = Confidence interval, max = Maximum, min = Minimum | |||||

^{1} Group 1 diagnostic: Differences in patient characteristics between patients with and without covariate |
|||||

^{2} Group 2 diagnostic: Ability to predict missingness |
|||||

^{3} Group 3 diagnostic: Assessment if missingness is associated with the outcome (univariate, adjusted) |

We observe from this diagnostics a median absolute standardized mean
difference (ASMD) of 0.056 with a meaningful spread between the minimum
ASMD (0.004) and the maximum ASMD (0.360), indicating some differences
in patient with and without an observed `age`

value which is
in line with the p-value derived by Hotelling’s test. While the ability
to predict missingness is rather low, group 3 diagnostic tends to show
that even after adjustment, there is a significant difference in the
survival of patients with and without an observed `age`

measurement. In total, these characteristics are reminiscent of the
diagnostic patterns observed in simulations in case of an **MNAR
mechanism**. Hence, we could suspect that
**MNAR(value)** could be the underlying missingness
mechanism, in which case we would want to run some sensitivity analyses
to investigate how much an MNAR(value) mechanism could bias our
results.

Since we know the true treatment effect estimate, we can quickly compare and estimate the bias caused by the 40% of MNAR(value) missing values.

First, we specify a general outcome model.

```
# outcome model (see data generation script)
cox_lhs <- "survival::Surv(eventtime, status)"
covariates <- smdi_data_complete %>%
select(-c(exposure, eventtime, status)) %>%
names()
cox_rhs <- paste(covariates, collapse = " + ")
cox_form <- as.formula(paste(cox_lhs, "~ exposure +", cox_rhs))
cox_form
#> survival::Surv(eventtime, status) ~ exposure + age_num + female_cat +
#> ecog_cat + smoking_cat + physical_cat + egfr_cat + alk_cat +
#> pdl1_num + histology_cat + copd_cat + ses_cat_1_low + ses_cat_2_middle
```

Next, we compute the true Hazard Ratio (HR), the complete case HR and
the standard *mice* multiple imputation HR.

```
# true outcome model
cox_fit_true <- coxph(cox_form, data = smdi_data_complete) %>%
tidy(exponentiate = TRUE, conf.int = TRUE) %>%
filter(term == "exposure") %>%
select(term, estimate, conf.low, conf.high, std.error) %>%
mutate(analysis = "True estimate")
# complete case analysis
cox_fit_cc <- coxph(cox_form, data = smdi_data_mnar_v) %>%
tidy(exponentiate = TRUE, conf.int = TRUE) %>%
filter(term == "exposure") %>%
select(term, estimate, conf.low, conf.high, std.error) %>%
mutate(analysis = "Complete case analysis")
# Multiple imputation (predictive mean matching)
cox_fit_imp <- mice(
data = smdi_data_mnar_v,
seed = 42,
print = FALSE
) %>%
with(
expr = coxph(formula(paste(format(cox_form), collapse = "")))
) %>%
pool() %>%
summary(conf.int = TRUE, exponentiate = TRUE) %>%
filter(term == "exposure") %>%
select(term, estimate, conf.low = `2.5 %`, conf.high = `97.5 %`, std.error) %>%
mutate(analysis = "Multiple imputation")
forest <- bind_rows(cox_fit_true, cox_fit_cc, cox_fit_imp) %>%
mutate(analysis = factor(analysis, levels = c("True estimate", "Complete case analysis", "Multiple imputation"))) %>%
ggplot(aes(y = fct_rev(analysis))) +
geom_point(aes(x = estimate), shape = 15, size = 3) +
geom_errorbar(aes(xmin = conf.low, xmax = conf.high)) +
geom_vline(xintercept = cox_fit_true[["estimate"]], linetype = "dashed") +
labs(x = "Hazard ratio (95% CI)", y= "") +
theme_bw()
table <- bind_rows(cox_fit_true, cox_fit_cc, cox_fit_imp) %>%
select(-term) %>%
relocate(analysis, .before = estimate) %>%
mutate(across(where(is.numeric), ~round(.x, 2)))
grid.arrange(tableGrob(table, rows = NULL), forest)
```

The deviation of the complete case and multiple imputation estimates
aren’t too off which has to do with `age_num`

being a rather
moderate confounder in this simulation. Nevertheless, in reality we
don’t know this for sure and a sensitivity analysis would probably make
sense to test the robustness of our primary analysis.

As mentioned above, under an MNAR(value) mechanism, the true (but
unobserved) values of a variable systematically differ from the observed
values and this difference is reflected in the sensitivity parameter
**𝛅**. More specifically, the interpretation of
**𝛅** would be the difference in the distribution of
missing and observed `age_num`

values conditional on other
covariates. Hence, for continuous covariates (like in our example) it
would be the difference in mean conditional on female_cat + ecog_cat +
smoking_cat + physical_cat + egfr_cat + alk_cat + pdl1_num +
histology_cat + copd_cat + ses_cat_1_low + ses_cat_2_middle.

If in reality, we would know this sensitivity parameter, we could easily plug it into our NARFCS.

But unfortunately, in reality we usually don’t know the value of
**𝛅** and relying on a single **𝛅** may be not
too reassuring. Nevertheless, we want to make sure that whatever
analytic decision we chose for our primary analysis (be it a complete
case approach or an imputation approach), our results would not
drastically change if the true underlying missingness mechanism was
MNAR(value).

This is where the NARFCS sensitivity analysis comes in handy as its
strengths as it is also often used as a **tipping point
analysis**, i.e., we model multiple NARFCS-specified imputation
models over a range of realistic **𝛅** sensitivity
parameters and evaluate if at any pre-specified **𝛅**, our
confidence interval would cross a certain estimate threshold which would
discard the global conclusion of our primary analysis.

```
# initialize method vector
method_vector <- rep("", ncol(smdi_data_mnar_v))
# columns for narfcs imputation with sensitivity parameter
mnar_imp_method <- which(colnames(smdi_data_mnar_v) == "age_num")
# update method vector
method_vector <- replace(x = method_vector, list = c(mnar_imp_method), values = c("mnar.norm"))
# modeled over a range of deltas
narfcs_modeled <- function(i){
# mnar model specification ('i' is delta parameter)
mnar_blot <- list(age_num = list(ums = paste(i)))
narfcs_imp <- mice(
data = smdi_data_mnar_v,
method = method_vector,
blots = mnar_blot,
seed = 42,
print = FALSE
) %>%
with(
expr = coxph(formula(paste(format(cox_form), collapse = "")))
) %>%
pool() %>%
summary(conf.int = TRUE, exponentiate = TRUE) %>%
filter(term == "exposure") %>%
select(term, estimate, conf.low = `2.5 %`, conf.high = `97.5 %`, std.error) %>%
mutate(delta = i)
}
narfcs_range <- lapply(
X = seq(-25, 25, 1),
FUN = narfcs_modeled
)
narfcs_range_df <- do.call(rbind, narfcs_range)
reference_lines <- tibble(
yintercept = c(cox_fit_true[[2]]),
Reference = c("TRUE HR"),
color = c("darkgreen")
)
narfcs_range_df %>%
ggplot(aes(x = delta, y = estimate)) +
geom_point() +
geom_line() +
geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.15) +
labs(x = "Sensitivity parameter δ", y = "Hazard ratio (95% CI)") +
scale_x_continuous(breaks = seq(-25, 25, 2), limits = c(-25, 25)) +
scale_y_continuous(breaks = seq(0.6, 1.2, 0.1), limits = c(0.6, 1.2)) +
geom_hline(aes(yintercept = yintercept, color = Reference), reference_lines) +
scale_colour_manual(values = reference_lines$color) +
theme_bw() +
theme(legend.position="top")
```

Based on this tipping point analysis, the systematic difference
between observed and unobserved `age_num`

values would have
needed to be so drastic that a significant departure of our primary
analysis is rather implausible. This is also supported by the fact that
`age_num`

is a rather weak to moderate confounder and
complete case and non-NARFCS multiple imputation results may have been
different, had `age_num`

been a stronger prognostic
factor.

Now, we can do the same in our *missing*
`smdi_data`

dataset. However, this gets a little tricky since
we have multiple missing covariates, each of which follow different
missingness mechanisms. Here is again a quick overview of the missing
variables using the `smdi_summarize()`

function.

```
smdi_data %>%
smdi_summarize()
#> # A tibble: 3 × 4
#> covariate n_miss prop_miss prop_miss_label
#> <chr> <int> <dbl> <chr>
#> 1 egfr_cat 1015 40.6 40.60%
#> 2 ecog_cat 899 36.0 35.96%
#> 3 pdl1_num 517 20.7 20.68%
```

The NARFCS
procedure that comes with the `mice`

package provides the
flexibility to execute the NARFCS procedure in the presence of other
partially observed covariates which may be assumed missing at
random.

Given the `smdi_diagnose`

analysis, we hypothesize that
`pdl1_num`

may be MNAR(value). This is plausible in the
context that the `pdl1_num`

expression of patients who had
been tested before and showed lower PD-L1 staining levels are less
likely to be tested in the future. Despite this, `pdl1_num`

is still an important predictor for the initiation of the treatment of
interest and simultaneously an import prognostic variable for the
outcome (time to death due to any reason [overall survival].(Khozin et al.
2019) Hence, we decide to run a tipping point sensitivity
analysis by imputing `pdl1_num`

over a range of
**𝛅** sensitivity parameters while imputing
`ecog_cat`

and `egfr_cat`

under a “normal”
imputation model.

```
# we one hot encode the `ses_cat` variable again
# in the smdi_data dataset
smdi_data <- smdi_data %>%
dummy_columns(
select_columns = "ses_cat",
remove_most_frequent_dummy = TRUE,
remove_selected_columns = TRUE
)
# initialize method vector
method_vector <- rep("", ncol(smdi_data))
# specify columns for narfcs and 'normal' imputation
pdl1_mnar_col <- which(colnames(smdi_data) == "pdl1_num")
ecog_mar_col <- which(colnames(smdi_data) == "ecog_cat")
egfr_mnar_col <- which(colnames(smdi_data) == "egfr_cat")
# update method vector
method_vector <- replace(
x = method_vector,
list = c(pdl1_mnar_col, ecog_mar_col, egfr_mnar_col),
values = c("mnar.norm", "logreg", "logreg")
)
# modeled over a range of deltas
narfcs_modeled <- function(i){
# mnar model specification for pdl1_num ('i' is delta parameter)
mnar_blot <- list(pdl1_num = list(ums = paste(i)))
narfcs_imp <- mice(
data = smdi_data,
method = method_vector,
blots = mnar_blot,
seed = 42,
print = FALSE
) %>%
with(
expr = coxph(formula(paste(format(cox_form), collapse = "")))
) %>%
pool() %>%
summary(conf.int = TRUE, exponentiate = TRUE) %>%
filter(term == "exposure") %>%
select(term, estimate, conf.low = `2.5 %`, conf.high = `97.5 %`, std.error) %>%
mutate(delta = i)
}
narfcs_range <- lapply(
X = seq(-25, 25, 1),
FUN = narfcs_modeled
)
narfcs_range_df <- do.call(rbind, narfcs_range)
reference_lines <- tibble(
yintercept = c(cox_fit_true[[2]]),
Reference = c("TRUE HR"),
color = c("darkgreen")
)
narfcs_range_df %>%
ggplot(aes(x = delta, y = estimate)) +
geom_point() +
geom_line() +
geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.15) +
labs(x = "Sensitivity parameter δ", y = "Hazard ratio (95% CI)") +
scale_x_continuous(breaks = seq(-25, 25, 2), limits = c(-25, 25)) +
scale_y_continuous(breaks = seq(0.5, 1.2, 0.1), limits = c(0.5, 1.2)) +
geom_hline(aes(yintercept = yintercept, color = Reference), reference_lines) +
scale_colour_manual(values = reference_lines$color) +
theme_bw() +
theme(legend.position="top")
```

As in the example above, `pdl1_num`

is sensitive to
departures from MAR, but results would only change under unrealistically
strong differences. In fact, the point estimates perfectly align with
the true estimate by sliding **𝛅** a bit to the left of the
range scale. The interpretation would be that patients with lower
`pdl1_num`

values are systematically less likely to be
observed conditional on all other covariates - which is exactly what we
simulated.

For more information and helpful resources regarding sensitivity analyses for missing data, we recommend checking out the following links and manuscripts(Tompsett et al. 2018; Buuren, Boshuizen, and Knook 1999):

https://stefvanbuuren.name/fimd/sec-sensitivity.html

https://amices.org/mice/reference/mice.impute.mnar.html

https://raw.githack.com/moreno-betancur/NARFCS/master/Vignette.html

Buuren, S. van, H. C. Boshuizen, and D. L. Knook. 1999. “Multiple
Imputation of Missing Blood Pressure Covariates in Survival
Analysis.” *Statistics in Medicine* 18 (6): 681–94. https://doi.org/10.1002/(sici)1097-0258(19990330)18:6<681::aid-sim71>3.0.co;2-r.

Khozin, Sean, Rebecca A. Miksad, Johan Adami, Mariel Boyd, Nicholas R.
Brown, Anala Gossai, Irene Kaganman, et al. 2019.
“Real-World Progression, Treatment, and Survival
Outcomes During Rapid Adoption of Immunotherapy for Advanced
Nonsmall Cell Lung Cancer.” *Cancer* 125
(22): 4019–32. https://doi.org/10.1002/cncr.32383.

Tompsett, Daniel Mark, Finbarr Leacy, Margarita Moreno-Betancur, Jon
Heron, and Ian R. White. 2018. “On the Use of the Not-at-Random
Fully Conditional Specification (NARFCS) Procedure in Practice.”
*Statistics in Medicine* 37 (15): 2338–53. https://doi.org/10.1002/sim.7643.