Population values for planning a uSDT study

Describes the population a planned study samples from: how awareness and influence sensitivity (\(d'\)) vary and covary across subjects, and how the response criteria vary. The arguments take the names of usdt_priors(), with numbers instead of priors, and the population can be written in either parameterization of hsdt().

Usage

usdt_population(
  dprime = NULL,
  sd_dprime = NULL,
  cor_dprime = NULL,
  criterion = 0,
  sd_criterion = 0.3,
  cor_criterion = 0,
  sd_signal = 1,
  intercept = NULL,
  from = NULL
)

# S3 method for class 'usdt_population'
print(x, ...)

# S3 method for class 'usdt_population'
update(object, ...)

Arguments

dprime: Mean \(d'\) of each task. Under the latent regression parameterization (when intercept is given), the mean of the awareness task only.
sd_dprime: Between-subject standard deviation of \(d'\) in each task.
cor_dprime: Correlation between awareness and influence \(d'\) across subjects.
criterion: Mean criterion \(c\) of each task, on the classical scale where positive values are conservative.
sd_criterion: Between-subject standard deviation of the criterion in each task. Zero gives every subject the same criterion.
cor_criterion: Correlation between the two criteria across subjects.
sd_signal: Standard deviation of the signal distribution of each task, relative to the noise distribution. The default, 1, is the equal-variance model that hsdt() fits.
intercept: Intercept of the latent regression of influence on awareness \(d'\). Giving it selects the latent regression parameterization, in which it replaces the influence mean.
from: A frequentist hsdt() fit, such as a pilot study, whose estimates give every value not set in the call.
x: A usdt_population object.
...: For update(), the arguments of usdt_population() to change. A task argument may name one task only, which keeps the value of the other.
object: A usdt_population object.

Value

An object of class usdt_population: a data frame with one row per scenario and the columns scenario, mu_A, mu_I, sigma_A, sigma_I, rho, the tested quantities diff, intercept and slope, and criterion_A, criterion_I, sd_criterion_A, sd_criterion_I, cor_criterion, sd_signal_A and sd_signal_I. Its latent attribute is "correlation" or "regression".

Values for each task

dprime, sd_dprime, criterion, sd_criterion and sd_signal take one value for both tasks or one for each, as c(awareness = , influence = ) or list(awareness = , influence = ). The correlations and the intercept take a single value.

The two parameterizations

The correlation parameterization gives the bivariate normal distribution of \(d'\): its means \(\mu_A\) and \(\mu_I\), standard deviations \(\sigma_A\) and \(\sigma_I\) and correlation \(\rho\). The latent regression parameterization gives the intercept \(\beta_0\) instead of \(\mu_I\), which then follows as \(\mu_I = \beta_0 + \rho \, (\sigma_I / \sigma_A) \, \mu_A\), as in hsdt(latent = "regression"). Both describe the same distribution, and the printed population reports the tested quantities of either: the difference \(\mu_I - \mu_A\) (H1), the correlation (H2) and the intercept and slope of the latent regression (H3).

The parameterization decides what stays fixed when a value changes, in a scenario or in update(). Under the correlation parameterization a new correlation keeps both means and moves the intercept (unless \(\mu_A = 0\)). Under the latent regression it keeps the intercept and moves the influence mean, and with it the difference.

Criteria

The criteria matter only through the precision with which each subject's trials measure \(d'\). A task split at the median has its criterion fixed at zero by the split, whatever its value here. The criterion correlation applies when both criteria are estimated.

Scenarios

Several values in an argument give one scenario for each, and several arguments with several values give one scenario for each combination. A value shared by both tasks keeps them equal in every scenario: sd_dprime = c(0.2, 0.3) gives two scenarios, while sd_dprime = list(awareness = c(0.2, 0.3), influence = c(0.2, 0.3)) gives four.

Values from a fitted model

With from, the estimates of a frequentist hsdt() fit give the means, standard deviations and correlations, and the arguments set in the call replace them, so usdt_population(from = m, cor_dprime = 0.3) keeps every estimate of m except the correlation. Setting intercept switches to the latent regression parameterization and drops the estimated influence mean. A criterion fixed at zero by the median split is zero with no variation.

Examples

# The correlation parameterization: means, SDs and correlation of d'
pop <- usdt_population(dprime    = c(awareness = 0.15, influence = 0.30),
                       sd_dprime = c(awareness = 0.35, influence = 0.30),
                       cor_dprime = 0.3)
pop
#> ── Population ─────────────────────────────────────────────────── correlation ── 
#> 
#>   Parameter     Awareness  Influence
#>   mean d'           0.150      0.300
#>   sd(d')            0.350      0.300
#>   cor(d')           0.300           
#>   mean c            0.000      0.000
#>   sd(c)             0.300      0.300
#> 
#>        Tested quantity                 Value
#>   H1   Δd' (influence - awareness)     0.150
#>   H2   correlation                     0.300
#>   H3   intercept                       0.261
#>        slope                           0.257
#> 
#>   A task split at the median has its criterion fixed at 0.

# The latent regression parameterization: the intercept replaces the
# influence mean
usdt_population(dprime = 0.15, intercept = 0.25,
                sd_dprime = c(awareness = 0.35, influence = 0.30),
                cor_dprime = 0.3)
#> ── Population ───────────────────────────────────────────── latent regression ── 
#> 
#>   Parameter     Awareness  Influence
#>   mean d'           0.150     0.289*
#>   sd(d')            0.350      0.300
#>   cor(d')           0.300           
#>   intercept         0.250           
#>   mean c            0.000      0.000
#>   sd(c)             0.300      0.300
#> 
#>        Tested quantity                 Value
#>   H1   Δd' (influence - awareness)     0.139
#>   H2   correlation                     0.300
#>   H3   intercept                       0.250
#>        slope                           0.257
#> 
#>   * The influence mean follows from the intercept: mu_I = b0 + rho (sd_I / sd_A) mu_A.
#>   A task split at the median has its criterion fixed at 0.

# Three scenarios for the correlation
update(pop, cor_dprime = c(0.1, 0.3, 0.5))
#> ── Population ───────────────────────────────────── correlation · 3 scenarios ── 
#> 
#>   Parameter     Awareness  Influence
#>   mean d'           0.150      0.300
#>   sd(d')            0.350      0.300
#>   cor(d')          varies           
#>   mean c            0.000      0.000
#>   sd(c)             0.300      0.300
#> 
#>   Scenario    cor(d')       Δd' intercept     slope
#>   1             0.100     0.150     0.287     0.086
#>   2             0.300     0.150     0.261     0.257
#>   3             0.500     0.150     0.236     0.429
#> 
#>   A task split at the median has its criterion fixed at 0.

# The estimates of a pilot study, with a weaker correlation
d <- usdt_data_tasks(
  awareness = vadillo_awareness,
  influence = vadillo_cuing,
  subject_col      = "subj",
  condition_col    = "condition",
  condition_levels = c(signal = "old", noise = "new"),
  response_col     = list(awareness = "judged.old", influence = "rt"),
  response_levels  = list(awareness = c(signal = 1, noise = 0),
                          influence = c(signal = "faster",
                                        noise = "slower")),
  dichotomize      = list(awareness = FALSE, influence = TRUE)
)
usdt_population(from = hsdt(d), cor_dprime = 0.3)
#> ── Population ─────────────────────────────────────────────────── correlation ── 
#> 
#>   Parameter     Awareness  Influence
#>   mean d'           0.235      0.128
#>   sd(d')            0.122      0.088
#>   cor(d')           0.300           
#>   mean c           -0.036      0.000
#>   sd(c)             0.247      0.000
#> 
#>        Tested quantity                 Value
#>   H1   Δd' (influence - awareness)    -0.107
#>   H2   correlation                     0.300
#>   H3   intercept                       0.077
#>        slope                           0.217
#> 
#>   A task split at the median has its criterion fixed at 0.