# Confidence Interval Fix

#### Jonathan Bakdash and Laura Marusich

#### 2023-08-09

Source:`vignettes/CI_fix.Rmd`

`CI_fix.Rmd`

### Confidence Interval Fix in 0.5.4

In versions of rmcorr prior to 0.5.4, confidence intervals (CIs) were
incorrectly calculated using the error degrees of freedom instead of
using the effective sample size (error df + 2). Generally, the
difference is slight, with the previous incorrect CIs tending to be a
bit too broad (conservative). However, when *both* the number of
participants *and* repeated measures are small, the prior
incorrect confidence intervals were substantially too wide. In other
words, a difference of 2 in the df has diminishing impact on the width
of CIs as the error df increases.

This fix does not change the effect size, *p*-value, or the
error degrees of freedom. All remain the same as prior versions.
Bootstrapped CIs are also unaffected.

#### Example with Slight Difference in CIs

We use the data from raz2005 to compare the old, incorrect CI
calculation to the current, correct CI. The sample size is *N* =
72 with *k* = 2 repeated measures. Note the difference in CI
bounds is slight: old, incorrect 95% CI [-0.8053581, -0.5637514] and
current, correct 95% CI [ -0.804153, -0.566080].

```
brainvolage.rmc <- rmcorr(participant = Participant, measure1 = Age, measure2 = Volume, dataset = raz2005)
#> Warning in rmcorr(participant = Participant, measure1 = Age, measure2 = Volume,
#> : 'Participant' coerced into a factor
#Old, incorrect CI using error df
psych::r.con(brainvolage.rmc$r, brainvolage.rmc$df, p = 0.95)
#> [1] -0.8053581 -0.5637514
#Current, correct CI using effective sample size
brainvolage.rmc$CI
#> [1] -0.804153 -0.566080
#Manual calculation
psych::r.con(brainvolage.rmc$r, brainvolage.rmc$df + 2, p = 0.95)
#> [1] -0.804153 -0.566080
```

#### Example with Substantial Difference in CIs

For a small sample size (*N* = 5 total participants) with
*k* = 2 repeated measures and *r*_{rm} =
0.88, we show a substantial difference in the old 95% CI [-0.5257087,
0.9974701] compared to the correct 95% CI [0.2394418, 0.9868084].