Simulation, Data

Misc

Todo - for the “trtAssign” mess with ratio and the number of ratios
Also see
- Bkmks Data >> Data Simulation
- Pandas-Time Series-Simulation
  - A Gaussian and Standard GARCH time-series that’s frequently encountered in econometrics
Packages
- {simChef} (Vignette) - Rapidly plan, carry out, and summarize statistical simulation studies in a flexible, efficient, and low-code manner.
  - Intuitive tidy grammar of data science simulations
  - Powerful abstractions for distributed simulation processing backed by {future}
  - Automated generation of interactive R Markdown simulation documentation, situating results next to the workflows needed to reproduce them.
- {structmcmc} - A set of tools for performing structural inference for Bayesian Networks using MCMC
Papers
- Generation and analysis of synthetic data via Bayesian networks: a robust approach for uncertainty quantification via Bayesian paradigm

{simstudy}

Misc

Docs

Reference

Available distributions (link)

Probability Distributions
nonrandom: For constants; can be a numeric or a string with a formula that defines a dependency on another variable
clusterSize: For variable cluster sizes but a constant total sample size
- formula: The (fixed) total sample size
- variance: A (non-negative) dispersion measure that represents the variability of size across clusters
  - If the dispersion is set to 0, then cluster sizes are constant

trtAssign: For treatment assignment

formula: Ratio which is separated by semicolons and number of treatments
- e.g. 2 values = 2 groups and “1;2” says group 2 has twice as many units and group 1
variance: Stratification; ratio in formula is used as the stratification ratio (e.g. unbalanced treatment groups → unbalanced stratification)

Example

def <- 
  defData(def, 
          varname = "rx", 
          dist = "trtAssign",
          formula = "1;1;2", 
          variance = "male;over65")

count(studytbl, rx)
#> # A tibble: 3 × 2
#>     rx    n
#>   <int> <int>
#> 1    1    84
#> 2    2    82
#> 3    3  164

count(studytbl, male, rx)
#> # A tibble: 6 × 3
#>   male    rx    n
#>   <int> <int> <int>
#> 1    0    1    40
#> 2    0    2    39
#> 3    0    3    78
#> 4    1    1    44
#> 5    1    2    43
#> 6    1    3    86

count(studytbl, over65, rx)
#> # A tibble: 6 × 3
#>   over65    rx    n
#>   <int> <int> <int>
#> 1      0    1    66
#> 2      0    2    65
#> 3      0    3  130
#> 4      1    1    18
#> 5      1    2    17
#> 6      1    3    34

Functions
- defData(dtDefs = NULL, varname, formula, variance = 0, dist = "normal", link = "identity", id = "id") - Initially creates a data.table or adds a column to a data.table with instructions about creating a variable
  - formula: Numeric constant or string formula for the mean, probability of event (binary), probability of success (binomial), etc.
- defDataAdd(dtDefs = NULL, varname, formula, variance = 0, dist = "normal", link = "identity") - Creates a variable definition like defData but is used to augment a already generated dataset. Used as input to addColumns which will generate the variable data from the instructions in this object and add it as a column to the already generated dataset.
- genCluster(dtClust, cLevelVar, numIndsVar, level1ID, allLevel2 = TRUE) - After generating cluster-level data, this function takes the number of clusters and the sizes of each cluster from that data, and does something like expand.grid to generate an individual-level dataset. Also, adds an id variable.
  - dtClust: Cluster-Level Data
  - cLevelVar: Cluster variable from the cluster-level data
  - numIndsvar: Variable with the number of units per cluster from the cluster-level data
  - level1ID: Name you want for your individual-level ID variable

Variable Dependence

Binary depends on a Binary
- Definitions
```
def <- defData(varname = "male", dist = "binary",
               formula = .5 , id="cid")
def <- defData(def, varname = "over65", dist = "binary",
               formula = "-1.7 + .8*male", link="logit")
```
- What’s happening
```
male <- c(1,1,0,1,0,0,0,1,0,1)
logits <- -1.7 + 0.8 * male
probabilities <- boot::inv.logit(logits)
over65 <- rbinom(n = 10, size = 1, prob = probabilities)
```
  - The formula in the logits line defines the relationship between being male and being over 65yrs old.
  - Males in this sample will have a higher probability (0.2890505) of being over 65yrs old than females (0.1544653)
  - To sample from a Bernoulli distribution, set size = 1
  - over65 is an indicator where each value is determined by a separate probability parameter for a Bernoulli distribution

Clustered with Cluster-Level Random Effect

Example: Fixed Cluster sizes; Balanced

Cluster Definitions
```
d0 <- defData(varname = "n", formula = 20, dist = "nonrandom")
d0 <- defData(d0, varname = "a", formula = 0, variance = 0.33)
d0 <- defData(d0, varname = "rx", formula = "1;1", dist = "trtAssign")
d1 <- defDataAdd(varname = "y", formula = "18 + 1.6 * rx + a",
                 variance = 16, dist = "normal")
```
- n: sample size for the cluster
  - dist = “nonrandom” and formula = 20 says use a constant for the cluster sizer
- rx: treatment indicator
  - dist = “trtAssign” and formula = “1;1” says 2 treatment groups and they’re balanced
- y: the individual-level outcome is a function of the treatment assignment and the cluster effect, as well as random individual-level variation
- a: random individual-level variation (i.e. random effect)
  - Random Effects are sampled from \(\mathcal{N}(0, \sigma)\) where the variance is typically estimated in a Mixed Effects model.

Generate Cluster-Level Data

set.seed(2761)
dc <- genData(10, d0, "site")
dc
##    site  n      a rx
##  1:    1 20 -0.3548  1
##  2:    2 20 -1.1232  1
##  3:    3 20 -0.5963  0
##  4:    4 20 -0.0503  1
##  5:    5 20  0.0894  0
##  6:    6 20  0.5294  1
##  7:    7 20  1.2302  0
##  8:    8 20  0.9663  1
##  9:    9 20  0.0993  0
## 10:  10 20  0.6508  0

Generates 10 clusters labelled as site according to the instructions in d0

Generate Individual Level Data

dd <- genCluster(dc, "site", "n", "id")
dd <- addColumns(d1, dd)
dd
##      site  n      a rx  id    y
##  1:    1 20 -0.355  1  1 17.7
##  2:    1 20 -0.355  1  2 16.2
##  3:    1 20 -0.355  1  3 19.2
##  4:    1 20 -0.355  1  4 20.6
##  5:    1 20 -0.355  1  5 14.7
##  ---                           
## 196:  10 20  0.651  0 196 25.3
## 197:  10 20  0.651  0 197 22.1
## 198:  10 20  0.651  0 198 13.2
## 199:  10 20  0.651  0 199 15.6
## 200:  10 20  0.651  0 200 13.8

genCluster performs an expand.grid to generate an individual-level dataset along with adding an ID variable
addColumns uses individual-level data and outcome variable definition to generate the outcome variable and add it to the dataset.

Example: Varying Cluster Sizes and therefore Varying Sample Size
```
d0 <- defData(varname = "n", formula = 20, dist = "poisson")
genData(10, d0, "site")
##    site  n
##  1:    1 13
##  2:    2 18
##  3:    3 21
##  4:    4 26
##  5:    5 25
##  6:    6 27
##  7:    7 23
##  8:    8 30
##  9:    9 23
## 10:  10 20
```
- Formula sets the poisson distribution parameter, \(\lambda = 20\). So sizes are sampled from poisson distribution with that mean/variance
- To increase the variability between clusters, use the negative binomial distribution
- Most likely leads to an unbalanced design

Example: Varying Cluster Sizes but Constant Sample Size

# moderately varying cluster sizes
d0 <- defData(varname = "n", formula = 200, variance = 0.2, dist = "clusterSize")
genData(10, d0, "site")

##    site  n
##  1:    1 20
##  2:    2 28
##  3:    3 25
##  4:    4 24
##  5:    5 28
##  6:    6 22
##  7:    7  7
##  8:    8 13
##  9:    9 22
## 10:  10 11

# Very highly varying cluster sizes
d0 <- defData(varname = "n", formula = 200, variance = 5, dist = "clusterSize")
genData(10, d0, "site")
##    site  n
##  1:    1  10
##  2:    2  2
##  3:    3  17
##  4:    4  2
##  5:    5  49
##  6:    6 110
##  7:    7  1
##  8:    8  4
##  9:    9  1
## 10:  10  4

Total sample size is fixed at 200 (formula), but individual cluster sizes are allowed to vary.
variance: A dispersion parameter that controls the amount of varying of the cluster sizes