diff --git a/README.Rmd b/README.Rmd index 0ee7ff4..1f10c03 100644 --- a/README.Rmd +++ b/README.Rmd @@ -80,19 +80,19 @@ remotes::install_github("ncn-foreigners/nonprobsvy@dev") ## Basic idea Consider the following setting where two samples are available: -non-probability (denoted as $S_A$ ) and probability (denoted as $S_B$) +non-probability (denoted as $S_A$) and probability (denoted as $S_B$) where set of auxiliary variables (denoted as $\boldsymbol{X}$) is available for both sources while $Y$ and $\boldsymbol{d}$ (or $\boldsymbol{w}$) is present only in probability sample. -| Sample | | Auxiliary variables $\boldsymbol{X}$ | Target variable $Y$ | Design ($\boldsymbol{d}$) or calibrated ($\boldsymbol{w}$) weights | +| Sample | | Auxiliary variables $\bol dsymbol{X}$ | Target variable $Y$ | Design ($\bold symbol{d}$) or calibrated ($\bold symbol{w}$) weights | |---------------|--------------:|:-------------:|:-------------:|:-------------:| -| $S_A$ (non-probability) | 1 | $\checkmark$ | $\checkmark$ | ? | -| | ... | $\checkmark$ | $\checkmark$ | ? | -| | $n_A$ | $\checkmark$ | $\checkmark$ | ? | -| $S_B$ (probability) | $n_A+1$ | $\checkmark$ | ? | $\checkmark$ | -| | ... | $\checkmark$ | ? | $\checkmark$ | -| | $n_A+n_B$ | $\checkmark$ | ? | $\checkmark$ | +| $S_A$ (non-p robability) | 1 | \$ \checkmark\$ | \$ \checkmark\$ | ? | +| | ... | \$ \checkmark\$ | \$ \checkmark\$ | ? | +| | $n_A$ | \$ \checkmark\$ | \$ \checkmark\$ | ? | +| $S_B$ (p robability) | $n_A+1$ | \$ \checkmark\$ | ? | \$ \checkmark\$ | +| | ... | \$ \checkmark\$ | ? | \$ \checkmark\$ | +| | $n_A+n_B$ | \$ \checkmark\$ | ? | \$ \checkmark\$ | ## Basic functionalities @@ -104,12 +104,12 @@ $(y_k, \boldsymbol{x}_k, R_k)$, we can approach this problem with the possible scenarios: - unit-level data is available for the non-probability sample $S_{A}$, - i.e. $(y_{k}, \boldsymbol{x}_{k})$ is available for all units - $k \in S_{A}$, and population-level data is available for - $\boldsymbol{x}_{1}, ..., \boldsymbol{x}_{p}$, denoted as - $\tau_{x_{1}}, \tau_{x_{2}}, ..., \tau_{x_{p}}$ and population size $N$ is - known. We can also consider situations where population data are - estimated (e.g. on the basis of a survey to which we do not have + i.e. \(y_{k}, \boldsymbol{x}_{k}\) is available for all units + \(k \in S_{A}\), and population-level data is available for + \(\boldsymbol{x}_{1}, ..., \boldsymbol{x}_{p}\), denoted as + $\tau_{x_{1}}, \tau_{x_{2}}, ..., \tau_{x_{p}}$ and population size + $N$ is known. We can also consider situations where population data + are estimated (e.g. on the basis of a survey to which we do not have access), - unit-level data is available for the non-probability sample $S_A$ and the probability sample $S_B$, i.e. @@ -120,253 +120,41 @@ possible scenarios: ### When unit-level data is available for non-probability survey only - - - - - - - - - - - - - - - - - - -
Estimator Example code
-Mass imputation based on regression imputation - -```{r, eval = FALSE} -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - pop_totals = c(`(Intercept)`= N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - method_outcome = "glm", - family_outcome = "gaussian" -) -``` -
-Inverse probability weighting - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - pop_totals = c(`(Intercept)` = N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - method_selection = "logit" -) -``` -
-Inverse probability weighting with calibration constraint - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - pop_totals = c(`(Intercept)`= N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - method_selection = "logit", - control_selection = controlSel(est_method_sel = "gee", h = 1) -) -``` -
-Doubly robust estimator - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - outcome = y ~ x1 + x2 + …, + xk, - pop_totals = c(`(Intercept)` = N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian" -) -``` -
+| Estimator | Example code | +|------------------------------------|------------------------------------| +| | | +| Mass imputation based on regression imputation | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + \... + xk, data = nonprob, pop_totals = c(\`(Intercept)\`= N, x1 = tau_x1, x2 = tau_x2, \..., xk = tau_xk), method_outcome = "glm", family_outcome = "gaussian" ) \`\`\` | +| | | +| Inverse probability weighting | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, target = \~ y, data = nonprob, pop_totals = c(\`(Intercept)\` = N, x1 = tau_x1, x2 = tau_x2, \..., xk = tau_xk), method_selection = "logit" ) \`\`\` | +| | | +| Inverse probability weighting with calibration constraint | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, target = \~ y, data = nonprob, pop_totals = c(\`(Intercept)\`= N, x1 = tau_x1, x2 = tau_x2, \..., xk = tau_xk), method_selection = "logit", control_selection = controlSel(est_method_sel = "gee", h = 1) ) \`\`\` | +| | | +| Doubly robust estimator | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, outcome = y \~ x1 + x2 + ..., + xk, pop_totals = c(\`(Intercept)\` = N, x1 = tau_x1, x2 = tau_x2, \..., xk = tau_xk), svydesign = prob, method_outcome = "glm", family_outcome = "gaussian" ) \`\`\` | +| | | ### When unit-level data are available for both surveys - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Estimator Example code
-Mass imputation based on regression imputation - -```{r, eval = FALSE} -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian" -) -``` -
-Mass imputation based on nearest neighbour imputation - -```{r, eval = FALSE} -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "nn", - family_outcome = "gaussian", - control_outcome = controlOutcome(k = 2) -) -``` -
-Mass imputation based on predictive mean matching - -```{r, eval = FALSE} -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "pmm", - family_outcome = "gaussian" -) -``` -
-Mass imputation based on regression imputation with variable selection (LASSO) - -```{r, eval = FALSE} -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "pmm", - family_outcome = "gaussian", - control_outcome = controlOut(penalty = "lasso"), - control_inference = controlInf(vars_selection = TRUE) -) -``` -
-Inverse probability weighting - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - svydesign = prob, - method_selection = "logit" -) -``` -
-Inverse probability weighting with calibration constraint - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - svydesign = prob, - method_selection = "logit", - control_selection = controlSel(est_method_sel = "gee", h = 1) -) -``` -
-Inverse probability weighting with calibration constraint with variable selection (SCAD) - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - svydesign = prob, - method_outcome = "pmm", - family_outcome = "gaussian", - control_inference = controlInf(vars_selection = TRUE) -) -``` -
-Doubly robust estimator - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian" -) -``` -
-Doubly robust estimator with variable selection (SCAD) and bias minimization - -```{r, eval = FALSE} -nonprob( - selection = ~ x1 + x2 + ... + xk, - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian", - control_inference = controlInf( - vars_selection = TRUE, - bias_correction = TRUE - ) -) -``` -
+| Estimator | Example code | +|------------------------------------|------------------------------------| +| | | +| Mass imputation based on regression imputation | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + \... + xk, data = nonprob, svydesign = prob, method_outcome = "glm", family_outcome = "gaussian" ) \`\`\` | +| | | +| Mass imputation based on nearest neighbour imputation | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + \... + xk, data = nonprob, svydesign = prob, method_outcome = "nn", family_outcome = "gaussian", control_outcome = controlOutcome(k = 2) ) \`\`\` | +| | | +| Mass imputation based on predictive mean matching | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + \... + xk, data = nonprob, svydesign = prob, method_outcome = "pmm", family_outcome = "gaussian" ) \`\`\` | +| | | +| Mass imputation based on regression imputation with variable selection (LASSO) | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + \... + xk, data = nonprob, svydesign = prob, method_outcome = "pmm", family_outcome = "gaussian", control_outcome = controlOut(penalty = "lasso"), control_inference = controlInf(vars_selection = TRUE) ) \`\`\` | +| | | +| Inverse probability weighting | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, target = \~ y, data = nonprob, svydesign = prob, method_selection = "logit" ) \`\`\` | +| | | +| Inverse probability weighting with calibration constraint | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, target = \~ y, data = nonprob, svydesign = prob, method_selection = "logit", control_selection = controlSel(est_method_sel = "gee", h = 1) ) \`\`\` | +| | | +| Inverse probability weighting with calibration constraint with variable selection (SCAD) | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, target = \~ y, data = nonprob, svydesign = prob, method_outcome = "pmm", family_outcome = "gaussian", control_inference = controlInf(vars_selection = TRUE) ) \`\`\` | +| | | +| Doubly robust estimator | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, outcome = y \~ x1 + x2 + \... + xk, data = nonprob, svydesign = prob, method_outcome = "glm", family_outcome = "gaussian" ) \`\`\` | +| | | +| Doubly robust estimator with variable selection (SCAD) and bias minimization | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + \... + xk, outcome = y \~ x1 + x2 + \... + xk, data = nonprob, svydesign = prob, method_outcome = "glm", family_outcome = "gaussian", control_inference = controlInf( vars_selection = TRUE, bias_correction = TRUE ) ) \`\`\` | +| | | ## Examples diff --git a/README.md b/README.md index 7e841da..39b2822 100644 --- a/README.md +++ b/README.md @@ -64,19 +64,19 @@ remotes::install_github("ncn-foreigners/nonprobsvy@dev") ## Basic idea Consider the following setting where two samples are available: -non-probability (denoted as $S_A$ ) and probability (denoted as $S_B$) +non-probability (denoted as $S_A$) and probability (denoted as $S_B$) where set of auxiliary variables (denoted as $\boldsymbol{X}$) is available for both sources while $Y$ and $\boldsymbol{d}$ (or $\boldsymbol{w}$) is present only in probability sample. -| Sample | | Auxiliary variables $\boldsymbol{X}$ | Target variable $Y$ | Design ($\boldsymbol{d}$) or calibrated ($\boldsymbol{w}$) weights | -|-------------------------|----------:|:------------------------------------:|:-------------------:|:------------------------------------------------------------------:| -| $S_A$ (non-probability) | 1 | $\checkmark$ | $\checkmark$ | ? | -| | … | $\checkmark$ | $\checkmark$ | ? | -| | $n_A$ | $\checkmark$ | $\checkmark$ | ? | -| $S_B$ (probability) | $n_A+1$ | $\checkmark$ | ? | $\checkmark$ | -| | … | $\checkmark$ | ? | $\checkmark$ | -| | $n_A+n_B$ | $\checkmark$ | ? | $\checkmark$ | +| Sample | | Auxiliary variables $\bol dsymbol{X}$ | Target variable $Y$ | Design ($\bold symbol{d}$) or calibrated ($\bold symbol{w}$) weights | +|--------------------------|----------:|:-------------------------------------:|:-------------------:|:--------------------------------------------------------------------:| +| $S_A$ (non-p robability) | 1 | \$ \$ | \$ \$ | ? | +| | … | \$ \$ | \$ \$ | ? | +| | $n_A$ | \$ \$ | \$ \$ | ? | +| $S_B$ (p robability) | $n_A+1$ | \$ \$ | ? | \$ \$ | +| | … | \$ \$ | ? | \$ \$ | +| | $n_A+n_B$ | \$ \$ | ? | \$ \$ | ## Basic functionalities @@ -88,7 +88,7 @@ $(y_k, \boldsymbol{x}_k, R_k)$, we can approach this problem with the possible scenarios: - unit-level data is available for the non-probability sample $S_{A}$, - i.e. $(y_{k}, \boldsymbol{x}_{k})$ is available for all units + i.e. $y_{k}, \boldsymbol{x}_{k}$ is available for all units $k \in S_{A}$, and population-level data is available for $\boldsymbol{x}_{1}, ..., \boldsymbol{x}_{p}$, denoted as $\tau_{x_{1}}, \tau_{x_{2}}, ..., \tau_{x_{p}}$ and population size @@ -103,296 +103,41 @@ possible scenarios: ### When unit-level data is available for non-probability survey only - - - - - - - - - - - - - - - - - - - - - -
-Estimator - -Example code -
-Mass imputation based on regression imputation - - -``` r -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - pop_totals = c(`(Intercept)`= N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - method_outcome = "glm", - family_outcome = "gaussian" -) -``` - -
-Inverse probability weighting - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - pop_totals = c(`(Intercept)` = N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - method_selection = "logit" -) -``` - -
-Inverse probability weighting with calibration constraint - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - pop_totals = c(`(Intercept)`= N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - method_selection = "logit", - control_selection = controlSel(est_method_sel = "gee", h = 1) -) -``` - -
-Doubly robust estimator - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - outcome = y ~ x1 + x2 + …, + xk, - pop_totals = c(`(Intercept)` = N, - x1 = tau_x1, - x2 = tau_x2, - ..., - xk = tau_xk), - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian" -) -``` - -
+| Estimator | Example code | +|-----------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +| | | +| Mass imputation based on regression imputation | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + ... + xk, data = nonprob, pop_totals = c(\`(Intercept)\`= N, x1 = tau_x1, x2 = tau_x2, ..., xk = tau_xk), method_outcome = “glm”, family_outcome = “gaussian” ) \`\`\` | +| | | +| Inverse probability weighting | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, target = \~ y, data = nonprob, pop_totals = c(\`(Intercept)\` = N, x1 = tau_x1, x2 = tau_x2, ..., xk = tau_xk), method_selection = “logit” ) \`\`\` | +| | | +| Inverse probability weighting with calibration constraint | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, target = \~ y, data = nonprob, pop_totals = c(\`(Intercept)\`= N, x1 = tau_x1, x2 = tau_x2, ..., xk = tau_xk), method_selection = “logit”, control_selection = controlSel(est_method_sel = “gee”, h = 1) ) \`\`\` | +| | | +| Doubly robust estimator | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, outcome = y \~ x1 + x2 + …, + xk, pop_totals = c(\`(Intercept)\` = N, x1 = tau_x1, x2 = tau_x2, ..., xk = tau_xk), svydesign = prob, method_outcome = “glm”, family_outcome = “gaussian” ) \`\`\` | +| | | ### When unit-level data are available for both surveys - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-Estimator - -Example code -
-Mass imputation based on regression imputation - - -``` r -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian" -) -``` - -
-Mass imputation based on nearest neighbour imputation - - -``` r -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "nn", - family_outcome = "gaussian", - control_outcome = controlOutcome(k = 2) -) -``` - -
-Mass imputation based on predictive mean matching - - -``` r -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "pmm", - family_outcome = "gaussian" -) -``` - -
-Mass imputation based on regression imputation with variable selection -(LASSO) - - -``` r -nonprob( - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "pmm", - family_outcome = "gaussian", - control_outcome = controlOut(penalty = "lasso"), - control_inference = controlInf(vars_selection = TRUE) -) -``` - -
-Inverse probability weighting - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - svydesign = prob, - method_selection = "logit" -) -``` - -
-Inverse probability weighting with calibration constraint - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - svydesign = prob, - method_selection = "logit", - control_selection = controlSel(est_method_sel = "gee", h = 1) -) -``` - -
-Inverse probability weighting with calibration constraint with variable -selection (SCAD) - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - target = ~ y, - data = nonprob, - svydesign = prob, - method_outcome = "pmm", - family_outcome = "gaussian", - control_inference = controlInf(vars_selection = TRUE) -) -``` - -
-Doubly robust estimator - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian" -) -``` - -
-Doubly robust estimator with variable selection (SCAD) and bias -minimization - - -``` r -nonprob( - selection = ~ x1 + x2 + ... + xk, - outcome = y ~ x1 + x2 + ... + xk, - data = nonprob, - svydesign = prob, - method_outcome = "glm", - family_outcome = "gaussian", - control_inference = controlInf( - vars_selection = TRUE, - bias_correction = TRUE - ) -) -``` - -
+| Estimator | Example code | +|------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +| | | +| Mass imputation based on regression imputation | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + ... + xk, data = nonprob, svydesign = prob, method_outcome = “glm”, family_outcome = “gaussian” ) \`\`\` | +| | | +| Mass imputation based on nearest neighbour imputation | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + ... + xk, data = nonprob, svydesign = prob, method_outcome = “nn”, family_outcome = “gaussian”, control_outcome = controlOutcome(k = 2) ) \`\`\` | +| | | +| Mass imputation based on predictive mean matching | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + ... + xk, data = nonprob, svydesign = prob, method_outcome = “pmm”, family_outcome = “gaussian” ) \`\`\` | +| | | +| Mass imputation based on regression imputation with variable selection (LASSO) | \`\`\`{r, eval = FALSE} nonprob( outcome = y \~ x1 + x2 + ... + xk, data = nonprob, svydesign = prob, method_outcome = “pmm”, family_outcome = “gaussian”, control_outcome = controlOut(penalty = “lasso”), control_inference = controlInf(vars_selection = TRUE) ) \`\`\` | +| | | +| Inverse probability weighting | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, target = \~ y, data = nonprob, svydesign = prob, method_selection = “logit” ) \`\`\` | +| | | +| Inverse probability weighting with calibration constraint | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, target = \~ y, data = nonprob, svydesign = prob, method_selection = “logit”, control_selection = controlSel(est_method_sel = “gee”, h = 1) ) \`\`\` | +| | | +| Inverse probability weighting with calibration constraint with variable selection (SCAD) | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, target = \~ y, data = nonprob, svydesign = prob, method_outcome = “pmm”, family_outcome = “gaussian”, control_inference = controlInf(vars_selection = TRUE) ) \`\`\` | +| | | +| Doubly robust estimator | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, outcome = y \~ x1 + x2 + ... + xk, data = nonprob, svydesign = prob, method_outcome = “glm”, family_outcome = “gaussian” ) \`\`\` | +| | | +| Doubly robust estimator with variable selection (SCAD) and bias minimization | \`\`\`{r, eval = FALSE} nonprob( selection = \~ x1 + x2 + ... + xk, outcome = y \~ x1 + x2 + ... + xk, data = nonprob, svydesign = prob, method_outcome = “glm”, family_outcome = “gaussian”, control_inference = controlInf( vars_selection = TRUE, bias_correction = TRUE ) ) \`\`\` | +| | | ## Examples @@ -609,8 +354,7 @@ Work on this package is supported by the National Science Centre, OPUS ## References (selected) -
+