Synthesise a dataset from paired-sample t-test summary statistics

makePaired() generates a dataset from paired-sample t-test summary statistics.

makePaired() generates correlated values so the data replicate rating scales taken, for example, in a before and after experimental design.

The function is effectively a wrapper function for lfast() and lcor() with the addition of a t-statistic from which the between-column correlation is inferred.

Paired t-tests apply to observations that are associated with each other. For example: the same people before and after a treatment; the same people rating two different objects; ratings by husband & wife; etc.

The t-test for paired data is given by:

• t = mean(D) / (sd(D) / sqrt(n))

where:

• D = differences in values,

• mean(D) = mean of the differences,

• sd(D) = standard deviation of the differences, where

  • sd(D)^2 = sd(X_before)^2 + sd(X_after)^2 - 2 * cov(X_before, X_after)

A paired-sample t-test thus requires an estimate of the covariance between the two sets of observations. makePaired() rearranges these formulae so that the covariance is inferred from the t-statistic.

Usage

makePaired(
  n,
  means,
  sds,
  t_value,
  lowerbound,
  upperbound,
  items = 1,
  precision = 0
)

Arguments

n

(positive, integer) sample size

means

(real) 1:2 vector of target means for two before/after measures

sds

(real) 1:2 vector of target standard deviations

t_value

(real) desired paired t-statistic

lowerbound

(integer) lower bound (e.g. '1' for a 1-5 rating scale)

upperbound

(integer) upper bound (e.g. '5' for a 1-5 rating scale)

items

(positive, integer) number of items in the rating scale. Default = 1

precision

(positive, real) relaxes the level of accuracy required. Default = 0

Value

a dataframe approximating user-specified conditions.

Note

Larger sample sizes usually result in higher t-statistics, and correspondingly small p-values.

Small sample sizes with relatively large standard deviations and relatively high t-statistics can result in impossible correlation values.

Similarly, large sample sizes with low t-statistics can result in impossible correlations. That is, a correlation outside of the -1:+1 range.

If this happens, the function will fail with an ERROR message. The user should review the input parameters and insert more realistic values.

Examples

n <- 20
pair_m <- c(2.5, 3.0)
pair_s <- c(1.0, 1.5)
lower <- 1
upper <- 5
k <- 6
t <- -2.5

pairedDat <- makePaired(
  n = n, means = pair_m, sds = pair_s,
  t_value = t,
  lowerbound = lower, upperbound = upper, items = k
)
#> Initial data vectors
#> reached maximum of 1024 iterations
#> reached maximum of 1024 iterations
#> Rearrange values to conform with desired t-value
#> Complete!

str(pairedDat)
#> 'data.frame':	20 obs. of  2 variables:
#>  $ X1: num  1.17 2.83 2.17 2.17 1.33 ...
#>  $ X2: num  1 2.33 1.5 2.83 1 ...
cor(pairedDat) |> round(2)
#>      X1   X2
#> X1 1.00 0.82
#> X2 0.82 1.00

t.test(pairedDat$V1, pairedDat$V2, paired = TRUE)
#> Error in t.test.default(pairedDat$V1, pairedDat$V2, paired = TRUE): 'y' is missing for paired test