```
library(smdi)
suppressPackageStartupMessages(library(dplyr))
# some global simulation parameters
seed_value <- 42
n <- 2500
```

`smdi`

dataset backgroundTo get acquainted with the functionality and usage of the
`smdi`

package, the package includes a simulated example
dataset. The dataset is an exemplary low-dimensional electronic health
records (EHR) dataset depicting a cohort of 2,500 lung cancer patients.
The dataset follows the general *one-row-per-patient* structure,
in which one row stands for an individual patient and the columns
represent the variables.

Let’s assume that we are interested in studying the comparative
effectiveness of two antineoplastic systemic drug treatment regimens
(`exposure (0/1)`

) on the time to death due to any reason as
the outcome (`overall survival`

). The anticipated time of
follow-up is truncated to 5 years.

The desired strength of effectiveness of the **exposure of
interest** is defined with a hazard ratio (HR) of 1.0, i.e. there
is no difference in overall survival among patients who are treated with
the exposure of interest as compared to the comparator regimen. The
proportional hazards assumption is fulfilled for this dataset.

We further assume that there are a some confounders which we need to specify to estimate our outcome model. Most of the covariates are associated with both the probability of treatment initiation and the outcome but there are also some that are not predictive of the exposure and just the outcome or not associated with any of the exposure or outcome, whatsoever.

Despite the low dimensionality, the dataset is simulated as realistically as possible with varying strengths of associations between covariates and treatment initiation and the outcome.

Most importantly for testing the functionality of this package, some of the above mentioned confounders are just partially observed according to the missingness mechanisms and proportions specified below in the table below.

To get an overview of the dataset, this table provides a summary of the different covariate-exposure-outcome-missingness correlations.

Variable | Description | Associated with exposure/outcome | Missingness [%] |
---|---|---|---|

age_num | Age at baseline (continuous) | Yes/Yes | |

female_cat | Female gender (binary) | Yes/Yes | |

ecog_cat | ECOG performance score 0/1 or >1 (binary) | Yes/Yes | MCAR [35%] |

smoking_cat | Smoker vs. non-smoker at baseline (binary) | Yes/Yes | |

physical_cat | Physically active vs not active (binary) | Yes/Yes | |

egfr_cat | EGFR alteration (binary) | Yes/Yes | MAR [40%] |

alk_cat | ALK translocation (binary) | No/Yes | |

pdl1_num | PD-L1 expression in % (continuous) | Yes/Yes | MNAR(value) [20%] |

histology_cat | Tumor histology squamous vs nonsquamous (binary) | No/Yes | |

ses_cat | Socio-economic status (multi-categorical) | No/No | |

copd_cat | History of COPD (binary) | No/No | Auxiliary to smoking |

```
set.seed(seed_value)
# start with basic dataframe, covariates and their association with exposure
sim_covar <- tibble::tibble(
exposure = rbinom(n = n, size = 1, prob = 0.4),
age_num = rnorm(n, mean = 64 - 7.5*exposure, sd = 13.7),
female_cat = rbinom(n, size = 1, prob = 0.39 - 0.05*exposure),
ecog_cat = rbinom(n, size = 1, prob = 0.63 - 0.04*exposure),
smoking_cat = rbinom(n, size = 1, prob = 0.45 + 0.1*exposure),
physical_cat = rbinom(n, size = 1, prob = 0.35 + 0.02*exposure),
egfr_cat = rbinom(n, size = 1, prob = 0.20 + 0.07*exposure),
alk_cat = rbinom(n, size = 1, prob = 0.03),
pdl1_num = rnorm(n, mean = 40 + 10*exposure, sd = 10.5),
histology_cat = rbinom(n, size = 1, prob = 0.2),
ses_cat = sample(x = c("1_low", "2_middle", "3_high"), size = n, replace = TRUE, prob = c(0.2 , 0.4, 0.4)),
copd_cat = rbinom(n, size = 1, prob = 0.3 + 0.5*smoking_cat)
) %>%
# bring data in right format
dplyr::mutate(across(ends_with("num"), as.numeric)) %>%
dplyr::mutate(across(ends_with("num"), function(x) round(x, digits = 2)))
```

In the first step, we create a dataset with 2,500 patients and 12 variables with the following distributions.

Characteristic |
0, N = 1,502^{1} |
1, N = 998^{1} |
Difference^{2} |
95% CI^{2,3} |
p-value^{2} |
---|---|---|---|---|---|

age_num | 63 (54, 73) | 57 (48, 66) | 6.5 | 5.4, 7.6 | <0.001 |

female_cat | 588 (39%) | 334 (33%) | 5.7% | 1.8%, 9.6% | 0.004 |

ecog_cat | 923 (61%) | 578 (58%) | 3.5% | -0.48%, 7.5% | 0.084 |

smoking_cat | 686 (46%) | 567 (57%) | -11% | -15%, -7.1% | <0.001 |

physical_cat | 502 (33%) | 380 (38%) | -4.7% | -8.6%, -0.73% | 0.019 |

egfr_cat | 286 (19%) | 255 (26%) | -6.5% | -10%, -3.1% | <0.001 |

alk_cat | 39 (2.6%) | 30 (3.0%) | -0.41% | -1.8%, 1.0% | 0.6 |

pdl1_num | 40 (33, 47) | 50 (43, 57) | -10 | -11, -9.2 | <0.001 |

histology_cat | 288 (19%) | 220 (22%) | -2.9% | -6.2%, 0.47% | 0.090 |

ses_cat | 0.05 | -0.03, 0.13 | |||

1_low | 302 (20%) | 222 (22%) | |||

2_middle | 592 (39%) | 377 (38%) | |||

3_high | 608 (40%) | 399 (40%) | |||

copd_cat | 738 (49%) | 600 (60%) | -11% | -15%, -7.0% | <0.001 |

^{1} Median (IQR); n (%) |
|||||

^{2} Welch Two Sample t-test; Two sample test for equality of proportions; Standardized Mean Difference |
|||||

^{3} CI = Confidence Interval |

The following table illustrates the odds of exposure assignment.

```
exposure_form <- as.formula(paste("exposure ~ ", paste(colnames(sim_covar %>% dplyr::select(-exposure)), collapse = " + ")))
exposure_fit <- stats::glm(
exposure_form,
data = sim_covar,
family = "binomial"
)
exposure_fit %>%
gtsummary::tbl_regression(exponentiate = T)
```

Characteristic |
OR^{1} |
95% CI^{1} |
p-value |
---|---|---|---|

age_num | 0.97 | 0.96, 0.97 | <0.001 |

female_cat | 0.73 | 0.60, 0.88 | 0.001 |

ecog_cat | 0.83 | 0.69, 1.00 | 0.054 |

smoking_cat | 1.36 | 1.10, 1.69 | 0.005 |

physical_cat | 1.27 | 1.05, 1.54 | 0.015 |

egfr_cat | 1.47 | 1.18, 1.84 | <0.001 |

alk_cat | 1.36 | 0.77, 2.36 | 0.3 |

pdl1_num | 1.10 | 1.08, 1.11 | <0.001 |

histology_cat | 1.15 | 0.91, 1.44 | 0.2 |

ses_cat | |||

1_low | — | — | |

2_middle | 0.88 | 0.69, 1.13 | 0.3 |

3_high | 0.85 | 0.66, 1.09 | 0.2 |

copd_cat | 1.39 | 1.12, 1.72 | 0.003 |

^{1} OR = Odds Ratio, CI = Confidence Interval |

Fitting a generalized linear model and assessing the probability of treatment assignment, the above constellation of odds results in the following simulated distributions depicting propensities of treatment initiation (aka propensity scores).

```
# compute propensity score
exposure_plot <- sim_covar %>%
dplyr::mutate(ps = fitted(exposure_fit))
# plot density
exposure_plot %>%
ggplot2::ggplot(ggplot2::aes(x = ps, fill = factor(exposure))) +
ggplot2::geom_density(alpha = .5) +
ggplot2::theme_bw() +
ggplot2::labs(
x = "Pr(exposure)",
y = "Density",
fill = "Exposed"
)
```

Next, we simulate a time-to-event outcome for
`overall survival`

. For this, the `simsurv`

package is used with the following assumptions:

- Parametric event times, following an exponential distribution
- Max event times: 5 years of follow-up
- Event times depend on some (time-fixed/baseline) covariate effects as listed in the table above
- beta coefficients for the outcome model as defined in the following

```
betas_os <- c(
exposure = log(1),
age_num = log(1.05),
female_cat = log(0.94),
ecog_cat = log(1.25),
smoking_cat = log(1.3),
physical_cat = log(0.79),
egfr_cat = log(0.5),
alk_cat = log(0.91),
pdl1_num = log(0.98),
histology_cat = log(1.15)
)
betas_os %>%
as.data.frame() %>%
dplyr::transmute(logHR = round(`.`, 2)) %>%
tibble::rownames_to_column(var = "Covariate") %>%
dplyr::mutate(HR = round(exp(logHR), 2)) %>%
gt::gt()
```

Covariate | logHR | HR |
---|---|---|

exposure | 0.00 | 1.00 |

age_num | 0.05 | 1.05 |

female_cat | -0.06 | 0.94 |

ecog_cat | 0.22 | 1.25 |

smoking_cat | 0.26 | 1.30 |

physical_cat | -0.24 | 0.79 |

egfr_cat | -0.69 | 0.50 |

alk_cat | -0.09 | 0.91 |

pdl1_num | -0.02 | 0.98 |

histology_cat | 0.14 | 1.15 |

```
set.seed(seed_value)
sim_df <- sim_covar %>% dplyr::bind_cols(
simsurv::simsurv(
dist = "exponential",
lambdas = 0.05,
betas = betas_os,
x = sim_covar,
maxt = 5
)
) %>%
dplyr::select(-id)
```

The simulation resulted in the following crude Kaplan-Meier estimates for 1-yr and 5-yr survival. Unadjusted estimates belown are shown overall and by exposure for overall survival.

```
km_overall <- survival::survfit(survival::Surv(eventtime, status) ~ 1, data = sim_df)
km_exposure <- survival::survfit(survival::Surv(eventtime, status) ~ exposure, data = sim_df)
gtsummary::tbl_survfit(
list(km_overall, km_exposure),
times = c(1, 5),
label_header = "**{time} Years**"
)
```

Characteristic |
1 Years |
5 Years |
---|---|---|

Overall | 61% (60%, 63%) | 19% (18%, 21%) |

exposure | ||

0 | 56% (54%, 59%) | 15% (13%, 17%) |

1 | 69% (66%, 72%) | 26% (23%, 29%) |

Given, that the true exposure effect is null, the crude model is severely biased as we can see even more clearly in the crude Kaplan-Meier curve.

After adjusting, the simulated data results in the following hazard ratio (HR) estimates.

```
cox_lhs <- "survival::Surv(eventtime, status)"
cox_rhs <- paste(colnames(sim_covar), collapse = " + ")
cox_form = as.formula(paste(cox_lhs, "~ exposure +", cox_rhs))
cox_fit <- survival::coxph(cox_form, data = sim_df)
cox_fit %>%
gtsummary::tbl_regression(exponentiate = T)
```

Characteristic |
HR^{1} |
95% CI^{1} |
p-value |
---|---|---|---|

exposure | 1.01 | 0.91, 1.12 | 0.8 |

age_num | 1.05 | 1.04, 1.05 | <0.001 |

female_cat | 0.92 | 0.84, 1.01 | 0.087 |

ecog_cat | 1.15 | 1.05, 1.26 | 0.002 |

smoking_cat | 1.44 | 1.30, 1.60 | <0.001 |

physical_cat | 0.84 | 0.77, 0.92 | <0.001 |

egfr_cat | 0.52 | 0.46, 0.58 | <0.001 |

alk_cat | 0.88 | 0.66, 1.16 | 0.4 |

pdl1_num | 0.98 | 0.98, 0.98 | <0.001 |

histology_cat | 1.17 | 1.05, 1.30 | 0.004 |

ses_cat | |||

1_low | — | — | |

2_middle | 1.03 | 0.92, 1.16 | 0.6 |

3_high | 1.04 | 0.92, 1.17 | 0.5 |

copd_cat | 0.90 | 0.81, 1.00 | 0.042 |

^{1} HR = Hazard Ratio, CI = Confidence Interval |

`smdi_data_complete`

To provide flexibility to play with the complete data *before*
introducing missingness, we offer two datasets:

`smdi_data_complete`

: complete data`smdi_data`

: data with certain covariates missing (see next chapter)

In many different quantitative disciplines from classic epidemiology to machine and deep learning there is an increasing interest in utilizing electronic health records (EHR) to develop prognostic/predictive models or study the comparative effectiveness and safety of medical interventions. Especially information on variables which are not readily available in other datasets (e.g. administrative claims) are of high interest, including vital signs, biomarkers and lab data. However, these covariates are often just partially observed for various reasons:

- Physician did simply not perform/order a certain test
- Certain measurements are just collected for particularly sick patients
- Information is ‘hiding’ in unstructured records, e.g. clinical notes

To illustrate `smdi's`

main functions using this dataset,
we introduce some missingness to relevant covariates, which are critical
confounders of the causal exposure-outcome relationship in the
`smdi_data`

dataset.

In order to introduce missingness following different missingness
mechanisms, we use the `ampute`

function of the mice package. An excellent
tutorial on this very flexible and elegant function can be found here.

In brief, we introduce missingness by…

- Determining a missingess pattern
- Creating a weight vector, using fitted linear models for the probability of observations becoming missing
- Specifying a type of logistic probability distribution for the missingness weights
- Defining an overall missingness proportion

```
# prepare a placeholder df for missing simulation
# we do not consider ses_cat
tmp <- smdi_data_complete %>%
dplyr::select(-c(ses_cat))
# determine missingness pattern template
miss_pattern <- rep(1, ncol(tmp))
```

`ecog_cat`

^{1} will be set to missing according to the
following specification:

- Missingness pattern: Only
`ecog_cat`

will be set to missing - Weight vector: There are no weights, since the missingness is not dependent on any other observed or unobserved covariates
- Type: Not applicable

```
# specify missingness pattern
# (0 = set to missing, 1 = remains complete)
mcar_col <- which(colnames(tmp)=="ecog_cat")
miss_pattern_mcar <- replace(miss_pattern, mcar_col, 0)
miss_prop_mcar <- .35
set.seed(42)
smdi_data_mcar <- mice::ampute(
data = tmp,
prop = miss_prop_mcar,
mech = "MCAR",
patterns = miss_pattern_mcar,
bycases = TRUE
)$amp %>%
dplyr::select(ecog_cat)
smdi_data_mcar %>%
dplyr::select(ecog_cat) %>%
dplyr::mutate(ecog_cat = forcats::fct_na_value_to_level(factor(ecog_cat), level = "missing")) %>%
gtsummary::tbl_summary()
```

Characteristic |
N = 2,500^{1} |
---|---|

ecog_cat | |

0 | 629 (25%) |

1 | 972 (39%) |

missing | 899 (36%) |

^{1} n (%) |

`egfr_cat`

will be set to missing according to the
following specification:

- Missingness pattern: Only
`egfr_cat`

will be set to missing - Weight vector: all covariates (except
`egfr_cat`

itself and`ses_cat`

) contribute with equal weights as linear predictors for the probability of observations becoming missing - Type: higher weighted sum scores (i.e. higher odds of having an
`egfr_cat`

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

```
# specify missingness pattern
# (0 = set to missing, 1 = remains complete)
mar_col <- which(colnames(tmp)=="egfr_cat")
miss_pattern_mar <- replace(miss_pattern, mar_col, 0)
# weights to compute missingness probability
# by assigning a non-zero value
miss_weights_mar <- rep(1, ncol(tmp))
miss_weights_mar <- replace(miss_weights_mar, mar_col, 0)
miss_prop_mar <- .4
set.seed(42)
smdi_data_mar <- mice::ampute(
data = tmp,
prop = miss_prop_mar,
mech = "MAR",
patterns = miss_pattern_mar,
weights = miss_weights_mar,
bycases = TRUE,
type = "RIGHT"
)$amp
smdi_data_mar %>%
dplyr::select(egfr_cat) %>%
dplyr::mutate(egfr_cat = forcats::fct_na_value_to_level(factor(egfr_cat), level = "missing")) %>%
gtsummary::tbl_summary()
```

Characteristic |
N = 2,500^{1} |
---|---|

egfr_cat | |

0 | 1,175 (47%) |

1 | 310 (12%) |

missing | 1,015 (41%) |

^{1} n (%) |

- Missingness pattern: Only
`pdl1_num`

will be set to missing - Weight vector: Only
`pdl1_num`

itself (by a non-zero value) is the linear predictor for the probability of observations becoming missing - Type: Specify a type of logistic probability distribution for the
missingness weights, so that cases with low weighted sum scores
(i.e. lower
`pdl1_num`

expression) will have a larger probability of becoming incomplete - Define an overall missingness proportion of ~20%

```
# determine missingness pattern
mnar_v_col <- which(colnames(tmp)=="pdl1_num")
miss_pattern <- rep(1, ncol(tmp))
miss_pattern_mnar_v <- replace(miss_pattern, mnar_v_col, 0)
# weights to compute missingness probability
# by assigning a non-zero value
# MNAR_v: covariate itself is only predictor
miss_weights_mnar_v <- rep(0, ncol(tmp))
miss_weights_mnar_v <- replace(miss_weights_mnar_v, mnar_v_col, 1)
miss_prop_mnar_v <- .2
set.seed(42)
smdi_data_mnar_v <- mice::ampute(
data = tmp,
prop = miss_prop_mnar_v,
mech = "MNAR",
patterns = miss_pattern_mnar_v,
weights = miss_weights_mnar_v,
bycases = TRUE,
type = "LEFT"
)$amp
smdi_data_mnar_v %>%
dplyr::select(pdl1_num) %>%
gtsummary::tbl_summary()
```

Characteristic |
N = 2,500^{1} |
---|---|

pdl1_num | 46 (39, 53) |

Unknown | 517 |

^{1} Median (IQR) |

Let’s have a look at the final dataset.

```
smdi_data %>%
gtsummary::tbl_summary(by = "exposure") %>%
gtsummary::add_overall() %>%
gtsummary::add_difference()
```

Characteristic |
Overall, N = 2,500^{1} |
0, N = 1,502^{1} |
1, N = 998^{1} |
Difference^{2} |
95% CI^{2,3} |
p-value^{2} |
---|---|---|---|---|---|---|

age_num | 61 (51, 70) | 63 (54, 73) | 57 (48, 66) | 6.5 | 5.4, 7.6 | <0.001 |

female_cat | 0.12 | 0.04, 0.20 | ||||

0 | 1,578 (63%) | 914 (61%) | 664 (67%) | |||

1 | 922 (37%) | 588 (39%) | 334 (33%) | |||

smoking_cat | 0.22 | 0.14, 0.30 | ||||

0 | 1,247 (50%) | 816 (54%) | 431 (43%) | |||

1 | 1,253 (50%) | 686 (46%) | 567 (57%) | |||

physical_cat | 0.10 | 0.02, 0.18 | ||||

0 | 1,618 (65%) | 1,000 (67%) | 618 (62%) | |||

1 | 882 (35%) | 502 (33%) | 380 (38%) | |||

alk_cat | 0.02 | -0.06, 0.10 | ||||

0 | 2,431 (97%) | 1,463 (97%) | 968 (97%) | |||

1 | 69 (2.8%) | 39 (2.6%) | 30 (3.0%) | |||

histology_cat | 0.07 | -0.01, 0.15 | ||||

0 | 1,992 (80%) | 1,214 (81%) | 778 (78%) | |||

1 | 508 (20%) | 288 (19%) | 220 (22%) | |||

ses_cat | 0.05 | -0.03, 0.13 | ||||

1_low | 524 (21%) | 302 (20%) | 222 (22%) | |||

2_middle | 969 (39%) | 592 (39%) | 377 (38%) | |||

3_high | 1,007 (40%) | 608 (40%) | 399 (40%) | |||

copd_cat | 0.22 | 0.14, 0.30 | ||||

0 | 1,162 (46%) | 764 (51%) | 398 (40%) | |||

1 | 1,338 (54%) | 738 (49%) | 600 (60%) | |||

eventtime | 1.57 (0.52, 3.95) | 1.28 (0.43, 3.13) | 2.18 (0.75, 5.00) | -0.64 | -0.78, -0.49 | <0.001 |

status | 2,017 (81%) | 1,277 (85%) | 740 (74%) | 11% | <0.001 | |

ecog_cat | 0.04 | -0.06, 0.14 | ||||

0 | 629 (39%) | 368 (39%) | 261 (40%) | |||

1 | 972 (61%) | 586 (61%) | 386 (60%) | |||

Unknown | 899 | 548 | 351 | |||

egfr_cat | 0.20 | 0.09, 0.30 | ||||

0 | 1,175 (79%) | 823 (82%) | 352 (74%) | |||

1 | 310 (21%) | 184 (18%) | 126 (26%) | |||

Unknown | 1,015 | 495 | 520 | |||

pdl1_num | 46 (39, 53) | 42 (35, 49) | 51 (45, 58) | -9.1 | -10, -8.2 | <0.001 |

Unknown | 517 | 376 | 141 | |||

^{1} Median (IQR); n (%) |
||||||

^{2} Welch Two Sample t-test; Standardized Mean Difference; 3-sample test for equality of proportions without continuity correction |
||||||

^{3} CI = Confidence Interval |

`smdi_data`

Exporting the data to `data/`

, so it can be used for
educative purposes and for playing around with the package.

The Eastern Cooperative Oncology Group (ECOG) performance score is a clinical measure for how the cancer disease affects the daily living abilities of the patient and is often used as a patient inclusion criterion for clinical trials. The scale ranges from 0 to 5 and typically only patients with 0 and 1 are eligible. Let’s assume for our example, we already have a subset of such a pre-selected clinical trial-like cohort, but we still want to adjust for a baseline ECOG of 0 and 1 (definitions are taken from https://ecog-acrin.org/resources/ecog-performance-status/↩︎