# Function returns RCI given baseline score SD, reliability and confidence sought

`getRCIfromSDandAlpha.Rd`

Function returns RCI given baseline score SD, reliability and confidence sought

## Arguments

- SD
single number giving the baseline score standard deviation

- rel
reliability of the measure, typically, and sensibly, Cronbach's alpha

- conf
the expectation/confidence interval, almost always .95

## Background

Like the CSC, the RCI comes out of the classic paper Jacobson, Follette & Revenstorf (1984). The thrust of the paper was about trying to bridge the gap between (quantitative) researchers who then, and still, tend to think in terms of aggregated data about change in therapy, and clinicians who tend to think about individual client's change.

The authors came up with two indices: the CSC, and the one here: the RCI or Reliable Change Index.

\[RCI = \varphi(conf)\! \times\! \sqrt{2}\! \times SD\times\sqrt{1 - rel}\]

Where

*SD*is the Standard Deviation (doh!) at baseline/t1,\(\varphi(conf)\) is the value of Gaussian distribution for

*conf*coverage with*conf*, usually .95 and hence \(\varphi(conf)\) = 1.96and

*rel*is the reliability of the measures, usually from Cronbach's alpha.

The logic of the this is that this bit: \[\sqrt{2}\! \times SD\times\sqrt{1 - rel}\]

is the standard error (SE) of a difference between two scores in Classical Test Theory (CTT) In fact the 1984 paper omitted the \(\sqrt(2)\) term
and that was corrected by a subsequent letter from Christensen & Menoza (1986). If we accept the model of CTT in which all these
distributions are Gaussian then multiplying the SE by \(\varphi(conf)\) gives you a range of scores that would include a proportion *conf* of change
that would arise from unreliability of measurement alone.

There is a very widespread misunderstanding that the RCI is a generic benchmark, in fact it's a coverage expectation based on the SD of the baseline scores so it's a calculation to be done for any sample. Computing the RCI for your dataset allows you to designate change as "reliable improvement", "reliable deterioration" or "no reliable change". This is perhaps useful for three particular sets of clients:

those with very high starting scores who might have improvement but not enough to drop below the CSC criterion, so not achieving "clinically significant change", being able to see that some of them improve more than would be expected by unreliability of measurement alone

those who started just above the CSC and crossed the CSC but whose change falls in the "no reliable change" range

those who started below the CSC who of course therefore can't show clinically significant change but may still show reliable improvement.

There's also an issue that Jacobson and his colleagues were looking for criteria with which to categorise individual change, sadly, too often the RCSC paradigm seems to have become just another way aggregate scores but my counting.

## References

Christensen, L., & Mendoza, J. L. (1986). A method of assessing change in a single subject: An alteration of the RC index. Behavior Therapy, 17, 305–308.

Jacobson, N. S., Follette, W. C., & Revenstorf, D. (1984). Psychotherapy outcome research: Methods for reporting variability and evaluating clinical significance. Behavior Therapy, 15, 336–352.

Jacobson, N. S., & Truax, P. (1991). Clinical significance: A statistical approach to defining meaningful change in psychotherapy research. Journal of Consulting and Clinical Psychology, 59(1), 12–19.

Evans, C., Margison, F., & Barkham, M. (1998). The contribution of reliable and clinically significant change methods to evidence-based mental health. Evidence Based Mental Health, 1, 70–72. https://doi.org/0.1136/ebmh.1.3.70

## See also

Other RCSC functions:
`classifyScoresVectorByRCI()`

,
`getBootCICSC()`

,
`getCSC()`