Data generation function for various underlying models

Usage

simdat(
  n = c(300, 300, 300),
  effect = 1,
  sigma_Y = 1,
  model = "ols",
  shift = 0,
  scale = 1
)

Arguments

n: Integer vector of size 3 indicating the sample sizes in the training, labeled, and unlabeled data sets, respectively
effect: Regression coefficient for the first variable of interest for inference. Defaults is 1.
sigma_Y: Residual variance for the generated outcome. Defaults is 1.
model: The type of model to be generated. Must be one of "mean", "quantile", "ols", "logistic", or "poisson". Default is "ols".
shift: Scalar shift of the predictions for continuous outcomes (i.e., "mean", "quantile", and "ols"). Defaults to 0.
scale: Scaling factor for the predictions for continuous outcomes (i.e., "mean", "quantile", and "ols"). Defaults to 1.

Value

A data.frame containing n rows and columns corresponding to the labeled outcome (Y), the predicted outcome (f), a character variable (set_label) indicating which data set the observation belongs to (training, labeled, or unlabeled), and four independent, normally distributed predictors (X1, X2, X3, and X4), where applicable.

Details

The simdat function generates three datasets consisting of independent realizations of $Y$ (for model = "mean" or "quantile"), or $\{Y, \boldsymbol{X}\}$ (for model = "ols", "logistic", or "poisson"): a training dataset of size $n_t$, a labeled dataset of size $n_l$, and an unlabeled dataset of size $n_u$. These sizes are specified by the argument n.

NOTE: In the unlabeled data subset, outcome data are still generated to facilitate a benchmark for comparison with an "oracle" model that uses the true $Y^{\mathcal{U}}$ values for estimation and inference.

Generating Data

For "mean" and "quantile", we simulate a continuous outcome, $Y \in \mathbb{R}$, with mean given by the effect argument and error variance given by the sigma_y argument.

For "ols", "logistic", or "poisson" models, predictor data, $\boldsymbol{X} \in \mathbb{R}^4$ are simulated such that the $i$th observation follows a standard multivariate normal distribution with a zero mean vector and identity covariance matrix:

$$ \boldsymbol{X_i} = (X_{i1}, X_{i2}, X_{i3}, X_{i4}) \sim \mathcal{N}_4(\boldsymbol{0}, \boldsymbol{I}). $$

For "ols", a continuous outcome $Y \in \mathbb{R}$ is simulated to depend on $X_1$ through a linear term with the effect size specified by the effect argument, while the other predictors, $\boldsymbol{X} \setminus X_1$, have nonlinear effects:

$$ Y_i = effect \times Z_{i1} + \frac{1}{2} Z_{i2}^2 + \frac{1}{3} Z_{i3}^3 + \frac{1}{4} Z_{i4}^2 + \varepsilon_y, $$

and $\varepsilon_y \sim \mathcal{N}(0, sigma_y)$, where the sigma_y argument specifies the error variance.

For "logistic", we simulate:

$$ \Pr(Y_i = 1 \mid \boldsymbol{X}) = logit^{-1}(effect \times Z_{i1} + \frac{1}{2} Z_{i2}^2 + \frac{1}{3} Z_{i3}^3 + \frac{1}{4} Z_{i4}^2 + \varepsilon_y) $$

and generate:

$$ Y_i \sim Bern[1, \Pr(Y_i = 1 \mid \boldsymbol{X})] $$

where $\varepsilon_y \sim \mathcal{N}(0, sigma\_y)$.

For "poisson", we simulate:

$$ \lambda_Y = exp(effect \times Z_{i1} + \frac{1}{2} Z_{i2}^2 + \frac{1}{3} Z_{i3}^3 + \frac{1}{4} Z_{i4}^2 + \varepsilon_y) $$

and generate:

$$ Y_i \sim Poisson(\lambda_Y) $$

Generating Predictions

To generate predicted outcomes for "mean" and "quantile", we simulate a continuous variable with mean given by the empirical mean of the training data and error variance given by the sigma_y argument.

For "ols", we fit a generalized additive model (GAM) on the simulated training dataset and calculate predictions for the labeled and unlabeled datasets as deterministic functions of $\boldsymbol{X}$. Specifically, we fit the following GAM:

$$ Y^{\mathcal{T}} = s_0 + s_1(X_1^{\mathcal{T}}) + s_2(X_2^{\mathcal{T}}) + s_3(X_3^{\mathcal{T}}) + s_4(X_4^{\mathcal{T}}) + \varepsilon_p, $$

where $\mathcal{T}$ denotes the training dataset, $s_0$ is an intercept term, and $s_1(\cdot)$, $s_2(\cdot)$, $s_3(\cdot)$, and $s_4(\cdot)$ are smoothing spline functions for $X_1$, $X_2$, $X_3$, and $X_4$, respectively, with three target equivalent degrees of freedom. Residual error is modeled as $\varepsilon_p$.

Predictions for labeled and unlabeled datasets are calculated as:

$$ f(\boldsymbol{X}^{\mathcal{L}\cup\mathcal{U}}) = \hat{s}_0 + \hat{s}_1(X_1^{\mathcal{L}\cup\mathcal{U}}) + \hat{s}_2(X_2^{\mathcal{L}\cup\mathcal{U}}) + \hat{s}_3(X_3^{\mathcal{L}\cup\mathcal{U}}) + \hat{s}_4(X_4^{\mathcal{L}\cup\mathcal{U}}), $$

where $\hat{s}_0, \hat{s}_1, \hat{s}_2, \hat{s}_3$, and $\hat{s}_4$ are estimates of $s_0, s_1, s_2, s_3$, and $s_4$, respectively.

NOTE: For continuous outcomes, we provide optional arguments shift and scale to further apply a location shift and scaling factor, respectively, to the predicted outcomes. These default to shift = 0 and scale = 1, i.e., no location shift or scaling.

For "logistic", we train k-nearest neighbors (k-NN) classifiers on the simulated training dataset for values of $k$ ranging from 1 to 10. The optimal $k$ is chosen via cross-validation, minimizing the misclassification error on the validation folds. Predictions for the labeled and unlabeled datasets are obtained by applying the k-NN classifier with the optimal $k$ to $\boldsymbol{X}$.

Specifically, for each observation in the labeled and unlabeled datasets:

$$ \hat{Y} = \operatorname{argmax}_c \sum_{i \in \mathcal{N}_k} I(Y_i = c), $$

where $\mathcal{N}_k$ represents the set of $k$ nearest neighbors in the training dataset, $c$ indexes the possible classes (0 or 1), and $I(\cdot)$ is an indicator function.

For "poisson", we fit a generalized linear model (GLM) with a log link function to the simulated training dataset. The model is of the form:

$$ \log(\mu^{\mathcal{T}}) = \gamma_0 + \gamma_1 X_1^{\mathcal{T}} + \gamma_2 X_2^{\mathcal{T}} + \gamma_3 X_3^{\mathcal{T}} + \gamma_4 X_4^{\mathcal{T}}, $$

where $\mu^{\mathcal{T}}$ is the expected count for the response variable in the training dataset, $\gamma_0$ is the intercept, and $\gamma_1$, $\gamma_2$, $\gamma_3$, and $\gamma_4$ are the regression coefficients for the predictors $X_1$, $X_2$, $X_3$, and $X_4$, respectively.

Predictions for the labeled and unlabeled datasets are calculated as:

$$ \hat{\mu}^{\mathcal{L} \cup \mathcal{U}} = \exp(\hat{\gamma}_0 + \hat{\gamma}_1 X_1^{\mathcal{L} \cup \mathcal{U}} + \hat{\gamma}_2 X_2^{\mathcal{L} \cup \mathcal{U}} + \hat{\gamma}_3 X_3^{\mathcal{L} \cup \mathcal{U}} + \hat{\gamma}_4 X_4^{\mathcal{L} \cup \mathcal{U}}), $$

where $\hat{\gamma}_0$, $\hat{\gamma}_1$, $\hat{\gamma}_2$, $\hat{\gamma}_3$, and $\hat{\gamma}_4$ are the estimated coefficients.

Examples


#-- Mean

dat_mean <- simdat(c(100, 100, 100), effect = 1, sigma_Y = 1,

  model = "mean")

head(dat_mean)
#>           Y  f set_label
#> 1 1.2358347 NA  training
#> 2 0.3104177 NA  training
#> 3 2.2010616 NA  training
#> 4 1.5780407 NA  training
#> 5 2.2034014 NA  training
#> 6 0.3235828 NA  training

#-- Linear Regression

dat_ols <- simdat(c(100, 100, 100), effect = 1, sigma_Y = 1,

  model = "ols")

head(dat_ols)
#>            X1         X2          X3           X4          Y  f set_label
#> 1 -0.01987176 -0.1397668  1.13837121 -0.318232616  1.2596350 NA  training
#> 2  1.38072091  0.6508478 -1.36065237 -0.083392001  0.7739356 NA  training
#> 3 -0.16047478  0.8209136 -0.45652812 -0.027092071  0.6575092 NA  training
#> 4  1.29612624 -1.1946885 -0.28393557 -0.989618989  2.5554902 NA  training
#> 5 -0.86328948 -0.7039109 -0.01353596 -0.639295088 -1.3541555 NA  training
#> 6  0.78215060 -0.1660609 -0.26126609  0.005894084  1.7686558 NA  training