# Multivariate missingness and monotonicity

#### Janick Weberpals

    library(smdi)
    library(gt)
    suppressPackageStartupMessages(library(dplyr))

# Multivariate missing data in `smdi`

In this article, we want to briefly highlight two aspect regarding multivariate missingness:

1. How does `smdi` handle multivariate missingness?

2. What is the link between missing data patterns and missing data mechanisms and how does this affect the behavior and performance of the `smdi` functionality?

## Established taxonomies

In general, there are two classic established missing data taxonomies:

- Mechanisms: Missing completely at random (MCAR), at random (MAR) and not at random (MNAR)

- Patterns: Monotone versus Non-monotone missingness

# How does `smdi` handle multivariate missingness?

In all `smdi` functions, except `smdi_little()`, binary missing indicator variables are created for each partially observed variable (either specified by the analyst using the `covar` parameter or automatically identified via `smdi_check_covar()` if `covar` = NULL) and the columns with the actual variable values are dropped. For the variable importance visualization in `smdi_rf()`, these variables are indicated with a *“_NA”* suffix. Missing values are accordingly indicated with a 1 and complete observations with a 0. This functionality is controlled via the `smdi_na_indicator()` utility function.

    smdi_data %>%
      smdi_na_indicator(
        drop_NA_col = FALSE # usually TRUE, but for demonstration purposes set to FALSE
        ) %>%
      select(
        ecog_cat, ecog_cat_NA,
        egfr_cat, egfr_cat_NA,
        pdl1_num, pdl1_num_NA
        ) %>%
      head() %>%
      gt()

ecog_cat | ecog_cat_NA | egfr_cat | egfr_cat_NA | pdl1_num | pdl1_num_NA
--- | --- | --- | --- | --- | ---
1 | 0 | NA | 1 | 45.03 | 0
NA | 1 | 0 | 0 | NA | 1
0 | 0 | 1 | 0 | 41.74 | 0
1 | 0 | NA | 1 | 45.51 | 0
NA | 1 | 1 | 0 | 31.28 | 0
0 | 0 | NA | 1 | NA | 1

Now, let’s assume we have three partially observed covariates X, Y and Z, which we would like to include in our missingness diagnostics. All `smdi_diagnose()` functions, except `smdi_little()`, create X_NA, Y_NA and Z_NA and X, Y and Z are discarded from the dataset. The functions will then iterate the diagnostics through all X_NA, Y_NA and Z_NA one-by-one. That is, if, for example, X_NA is assessed, Y_NA and Z_NA serve as predictor variables along with all other covariates in the dataset. If Y_NA is assessed, X_NA and Z_NA are included as predictor variables, and so forth.

Important! It is important to notice that this strategy is the default to deal with multivariate missingness in the `smdi` package, however, another possible approach could be to not consider the other partially observed variables in the first place (e.g. by dropping them before applying any `smdi` function) and stacking the diagnostics focusing on one partially observed variable at a time. Such a strategy would be advisable in scenarios of monotone missing data patterns (see next section).

# `smdi` in case of monotone missing data patterns

While in the `smdi` package we mainly focus on missing data mechanisms, missing data patterns always need to be considered, too. Please refer to the routine structural missing data diagnostics article, where we highlight the importance of describing missingness proportions and patterns before running any of the `smdi` diagnostics.

As mentioned in the section before, in case of monotone missing data patterns, the `smdi` functionality may be misleading.

Monotonicity A missing data pattern is said to be monotone if the variables Yj can be ordered such that if Yj is missing then all variables Yk with k > j are also missing (taken from Stef van Buuren 1).

A good example for monotone missing data could be clinical blood laboratory tests (“labs”) which are often tested together in a lab panel. If one lab is missing, typically the other labs of this panel are also missing.

    # we simulatea monotone missingness pattern
    # following an MCAR mechanism
    data_monotone <- smdi_data_complete %>%
      mutate(
        lab1 = rnorm(nrow(smdi_data_complete), mean = 5, sd = 0.5),
        lab2 = rnorm(nrow(smdi_data_complete), mean = 10, sd = 2.25)
        )

    data_monotone[3:503, "lab1"] <- NA
    data_monotone[1:500, "lab2"] <- NA

    smdi::gg_miss_upset(data = data_monotone)

    smdi::md.pattern(data_monotone[, c("lab1", "lab2")], plot = FALSE)
    #>      lab2 lab1
    #> 1997    1    1    0
    #> 3       1    0    1
    #> 2       0    1    1
    #> 498     0    0    2
    #>       500  501 1001

In extreme cases of perfect linearity, this can lead to multiple warnings and errors such as `system is exactly singular` or `-InfWarning: Variable has only NA's in at least one stratum`.

In cases in which monotonicity is still clearly present but not as extreme (like in the example above), `smdi` will prompt a message to the analyst to raise awareness of this issue as the `smdi` output can be highly misleading in such instances.

    diagnostics_jointly <- smdi_diagnose(
      data = data_monotone,
      covar = NULL, # NULL includes all covariates with at least one NA
      model = "cox",
      form_lhs = "Surv(eventtime, status)"
      )
    #> Important note:
    #> AUC for predicting covariate lab1 is very high (>0.9).
    #> Predictor with highest importance: lab2_NA.
    #> Check for potentially underlying monotone missing data pattern.
    #> Important note:
    #> AUC for predicting covariate lab2 is very high (>0.9).
    #> Predictor with highest importance: lab1_NA.
    #> Check for potentially underlying monotone missing data pattern.

    diagnostics_jointly %>%
      smdi_style_gt()

Covariate | ASMD (min/max)1 | p Hotelling1 | AUC2 | beta univariate (95% CI)3 | beta (95% CI)3
--- | --- | --- | --- | --- | ---
lab1 | 0.060 (0.008, 0.730) | <.001 | 0.998 | -0.01 (95% CI -0.12, 0.09) | 1.06 (95% CI 0.01, 2.11)
lab2 | 0.054 (0.003, 0.170) | <.001 | 0.993 | -0.03 (95% CI -0.14, 0.08) | -1.11 (95% CI -2.16, -0.05)
p little: 0.146, 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)

In such cases, it may be advisable to not consider including lab2 in the missingness diagnostics of lab1 and vice versa and stack the diagnostics focusing on one partially observed variable at a time.

## Lab 1 analyzed without Lab 2

    # lab 1
    lab1_diagnostics <- smdi_diagnose(
      data = data_monotone %>% select(-lab2),
      model = "cox",
      form_lhs = "Surv(eventtime, status)"
      )

    lab1_diagnostics %>%
      smdi_style_gt()

Covariate | ASMD (min/max)1 | p Hotelling1 | AUC2 | beta univariate (95% CI)3 | beta (95% CI)3
--- | --- | --- | --- | --- | ---
lab1 | 0.050 (0.008, 0.083) | 0.296 | 0.500 | -0.01 (95% CI -0.12, 0.09) | -0.04 (95% CI -0.15, 0.07)
p little: 0.236, 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)

## Lab 2 analyzed without Lab 1

    # lab 2
    lab2_diagnostics <- smdi_diagnose(
      data = data_monotone %>% select(-lab1),
      model = "cox",
      form_lhs = "Surv(eventtime, status)"
      )

    lab2_diagnostics %>%
      smdi_style_gt()

Covariate | ASMD (min/max)1 | p Hotelling1 | AUC2 | beta univariate (95% CI)3 | beta (95% CI)3
--- | --- | --- | --- | --- | ---
lab2 | 0.051 (0.003, 0.082) | 0.291 | 0.500 | -0.03 (95% CI -0.14, 0.08) | -0.05 (95% CI -0.16, 0.06)
p little: 0.233, 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)

## Presented in one table using `smdi_style_gt()`

We can also combine the output of individually stacked `smdi_diagnose` tables and enhance it with a global Little’s test that takes into account the multivariate missingness of the entire dataset.

    # computing a gloabl p-value for Little's test including both lab1 and lab2
    little_global <- smdi_little(data = data_monotone)

    # combining two individual lab smdi tables and global Little's test
    smdi_style_gt(
      smdi_object = rbind(lab1_diagnostics$smdi_tbl, lab2_diagnostics$smdi_tbl),
      include_little = little_global
      )

Covariate | ASMD (min/max)1 | p Hotelling1 | AUC2 | beta univariate (95% CI)3 | beta (95% CI)3
--- | --- | --- | --- | --- | ---
lab1 | 0.050 (0.008, 0.083) | 0.296 | 0.500 | -0.01 (95% CI -0.12, 0.09) | -0.04 (95% CI -0.15, 0.07)
lab2 | 0.051 (0.003, 0.082) | 0.291 | 0.500 | -0.03 (95% CI -0.14, 0.08) | -0.05 (95% CI -0.16, 0.06)
p little: 0.146, 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)

Since the missingness follows an MCAR mechanism, `smdi_diagnose()` now shows the expected missingness diagnostics patterns one would expect from an MCAR mechanism.

1. For more information on missing data patterns see https://stefvanbuuren.name/fimd/missing-data-pattern.html↩︎