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

The function 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 paired-samples t-test is defined as:

$$ t = \frac{\mathrm{mean}(D)}{\mathrm{sd}(D) / \sqrt{n}} $$

where:

$D$ = differences in values
$\mathrm{mean}(D)$ = mean of the differences
$\mathrm{sd}(D)$ = standard deviation of the differences, where

$$ \mathrm{sd}(D)^2 = \mathrm{sd}(X_{\text{before}})^2 + \mathrm{sd}(X_{\text{after}})^2 - 2\,\mathrm{cov}(X_{\text{before}}, X_{\text{after}}) $$

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
#> Arranging values to conform with desired t-value
#> Complete!

str(pairedDat)
#> 'data.frame':	20 obs. of  2 variables:
#>  $ X1: num  2.17 4.17 1.33 1.33 2.33 ...
#>  $ X2: num  2.33 4.83 1 1.17 3.17 ...
cor(pairedDat) |> round(2)
#>      X1   X2
#> X1 1.00 0.82
#> X2 0.82 1.00

t.test(pairedDat$X1, pairedDat$X2, paired = TRUE)
#> 
#> 	Paired t-test
#> 
#> data:  pairedDat$X1 and pairedDat$X2
#> t = -2.4863, df = 19, p-value = 0.02238
#> alternative hypothesis: true mean difference is not equal to 0
#> 95 percent confidence interval:
#>  -0.90555937 -0.07777397
#> sample estimates:
#> mean difference 
#>      -0.4916667 
#>