Minimum sample size for continuous‐outcome prediction models

Compute the minimum sample size required to develop a prediction model with a continuous outcome. This wraps the same simulation engine as simulate_binary(), combining bisection search with Gaussian-process learning-curve modelling. From user inputs (maximum achievable performance, target performance, etc.) it constructs a data-generating function, model-fitting function, and metric function, then searches for the smallest $n$ meeting the chosen criterion.

Usage

simulate_continuous(
  signal_parameters,
  noise_parameters = 0,
  complexity = 1,
  data_control = NULL,
  maximum_achievable_rsquared,
  model = c("lm", "lasso", "ridge", "rf", "xgboost"),
  metric = "calibration_slope",
  target_performance,
  n_reps_total = 1000,
  mean_or_assurance = "assurance",
  ...
)

Arguments

signal_parameters

Integer. Number of candidate predictors associated with the outcome (i.e., true signal features).

noise_parameters

Integer. Number of candidate predictors not associated with the outcome (noise features). Default is 0.

complexity

Integer in 1:4 selecting the data-generating signal structure (see Data control). Default 1.

data_control

Optional named list controlling the predictors (see Data control). Default NULL (generator defaults).

maximum_achievable_rsquared

Numeric in (0, 1). Maximum achievable $R^2$ with effectively unlimited data. This is used to calibrate the data-generating mechanism and is not the minimum acceptable threshold.

model

Character string specifying the modelling algorithm. One of "lm" (linear regression), "lasso", "ridge", "rf" (random forest), or "xgboost" (gradient-boosted trees).

metric

Character string naming the performance metric used to assess the sample size; defaults to "calibration_slope". Metric identifiers use one canonical form throughout the package, such as "calibration_slope", "calibration_in_the_large", "auc", "r2", "cindex", and "csse".

"calibration_slope" is the slope from regressing the observed outcome on the model's linear predictor in held-out data; 1 indicates perfect calibration, and values below 1 indicate overfitting. Note that for the machine-learning models ("lasso", "ridge", "rf", "xgboost") this is converted internally to the calibration slope squared error for optimisation and translated back before results are returned; you don't need to do anything, and target_performance is still given on the calibration slope scale. Results derived this way carry a footnote marker in the printed output.

"csse" is the calibration slope squared error, $-(1 - s)^2$ for a calibration slope $s$, so that larger is better and 0 is perfect calibration. It can be requested directly, which is mainly useful for advanced use and for comparison against the internal conversion described above. When requesting it directly you are responsible for supplying target_performance on the CSSE scale: a calibration slope target of 0.9 corresponds to a CSSE target of -0.01. No adjustment is applied on your behalf, and results are reported on the CSSE scale.

target_performance

Numeric. Minimum acceptable value of the selected performance metric $M^*$; the algorithm searches for the smallest $n$ meeting the chosen criterion with respect to this threshold.

n_reps_total

Integer. Total number of simulation replications used by the engine across the search.

mean_or_assurance

Character string, either "mean" or "assurance". Controls whether the minimum $n$ is defined by the mean-based criterion or the assurance-based criterion (with the assurance level $\delta$ controlled by the engine's defaults or additional arguments in ...).

...

Additional options passed to simulate_custom() (e.g., assurance level $\delta$, per-iteration settings).

Value

An object of class "pmsims" containing the estimated minimum sample size and simulation diagnostics (inputs, fitted GP curve, intermediate evaluations, and summary metrics).

Criteria

Two formulations are supported.

Mean-based: find the smallest $n$ such that the expected model performance exceeds the target $M^*$, i.e. $$\min_n \; \mathbb{E}_{D_n}\{ M \mid D_n \} \ge M^*.$$
Assurance-based: find the smallest $n$ such that the probability the performance exceeds $M^*$ is at least $\delta$ (e.g. 0.80), i.e. $$\min_n \; \mathbb{P}_{D_n}\!\left( M \mid D_n \ge M^* \right) \ge \delta.$$

Here, $M$ is the chosen performance metric and the probability/expectation is over repeated samples of training data of size $n$. The assurance criterion explicitly accounts for variability across training sets; models with higher variance typically require larger $n$ to satisfy it.

Data control

complexity selects the signal structure of the data-generating mechanism: 1 purely linear, 2 linear + quadratic, 3 linear + quadratic + interaction, 4 the Friedman function. data_control is an optional list fine-tuning the predictors:

nonlinear_strength: Numeric in [0, 1). Fraction of the signal variance carried by the nonlinear, linearly-inaccessible component. Applies to complexity 2 and 3 only; ignored (with a warning) for 1 and 4. If omitted, the generator's per-complexity default is used.
correlation: Numeric in $[-1, 1]$. Pairwise correlation among the candidate predictors. Default 0.3.
predictor_distribution: One of "normal", "uniform", "binary", "exponential", "lognormal", "t", "laplace". "binary" selects 0/1 predictors and requires binary_predictor_prevalence; any other value selects continuous predictors from that family. Default "normal".
binary_predictor_prevalence: Numeric in (0, 1). Prevalence of the binary predictors; required when predictor_distribution = "binary", ignored (with a warning) otherwise. Note: binary predictors are incompatible with complexity 2/3 because squaring a 0/1 variable returns itself.

Examples

# \donttest{
set.seed(123)
est <- simulate_continuous(
  signal_parameters = 3,
  noise_parameters = 0,
  complexity = 1,
  data_control = list(correlation = 0),
  maximum_achievable_rsquared = 0.50,
  model = "lm",
  metric = "calibration_slope",
  target_performance = 0.9,
  mean_or_assurance = "assurance",
  min_sample_size = 50,
  max_sample_size = 1000,
  n_reps_total = 1000,
  test_n = 30000,
  progress = FALSE
)
#> ℹ Using user-specified min_sample_size and max_sample_size. Adaptive starting values will not be used.
#> ℹ Estimating second stage... (Gaussian process algorithm)
est
#>                     ┌────────────────────────────────────────┐
#>                     │ pmsims: Sample size simulation summary │
#>                     └────────────────────────────────────────┘
#> 
#> ──────────────────────────────────── Inputs ────────────────────────────────────
#> 
#> Data-generating scenario
#>   Outcome                  Continuous
#>   Predictors               3 signal
#>   Predictor distribution   Normal
#>   Predictor correlation    0.00
#>   Signal form              Linear
#> 
#> Model and performance
#>   Model                    Linear regression
#>   Large-sample R²          0.500
#>   Sample-size criterion    Calibration slope ≥ 0.900
#> 
#> Simulation
#>   Mode                     Assurance
#>   Replications             1,000
#> 
#> ──────────────────────────────────── Results ───────────────────────────────────
#> 
#>   Minimum sample size      83
#> 
#>   Performance at N = 83
#>     Calibration slope      0.900    (target ≥ 0.900)
#>     R²                     0.494
#> 
#>   Running time             26 seconds
#> 
#> ────────────────────────────────────────────────────────────────────────────────
#> Assurance mode selects N so that the target is achieved with high probability
#> across repeated datasets.
est$min_n
#> [1] 83
plot(est)

# }