---
title: "(advanced) Alternate ways to decorrelate repeated measures from transformations"
bibliography: "../inst/REFERENCES.bib"
csl: "../inst/apa-6th.csl"
output: 
  html_document:
    toc: TRUE
description: >
  This vignette examines the various techniques that performs within-subject decorrelation
  from transfomations of the raw scores. They can be shown to share many elements.
vignette: >
  %\VignetteIndexEntry{(advanced) Alternate ways to decorrelate repeated measures from transformations}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---


```{r, echo = FALSE, message = FALSE, results = 'hide'}
cat("this will be hidden; use for general initializations.\n")
library(superb)
options("superb.feedback" = c("warnings","design"))

library(ggplot2)
dta = GRD(WSFactors="Moments(3)", SubjectsPerGroup =20, 
    Population = list(mean = 12, stddev = 1, rho = 0.45),
    Effects = list("Moments" = custom(2,3,5) ),
    RenameDV = "Score"
)

```

The methods ``CM`` [@c05; @m08] and ``LM`` [@lm94] can be unified when 
the transformations they required are considered. In the 
Cousineau-Morey method, the raw data must be subject-centered and bias-corrected. 

Subject centering is obtained from

$$
Y_{ij} = X_{ij} - \bar{X}_{i\cdot} + \bar{\bar{X}}
$$


in which $i = 1..n$ and $j=1..C$ where $n$ is the number of participants and $C$ is the number
of repeated measures (sometimes noted with $J$).

Bias-correction is obtained from

$$
Z_{ij} = \sqrt{\frac{C}{C-1}} \left( Y_{ij} - \bar{Y}_{\cdot{}j} \right) + \bar{Y}_{\cdot{}j}
$$

These two operations 
can be performed with two matrix transformations. In comparison, the ``LM`` 
method requires one additional step, that is, pooling standard deviation, 
also achievable with the following transformation.

$$
W_{ij} = \sqrt{\frac{S_p^2}{S_i^2}} \left( Z_{ij} - \bar{Z}_{\cdot{}j} \right) + \bar{Z}_{\cdot{}j}
$$

in which $S_j^2$ is the variance in measurement $j$ and $S_p^2$ is the pooled variance across all $j$ mesurements.

With this approach, we can categorize all the proposals to repeated measure precision as requiring or 
not certain transformations. Table 1 shows these.

**Table 1**. Transformations required to implement one of the repeated-measures
method. Preprocessing must precede post-processing


| Method			| preprocessing	    | postprocessing    |         |  
|-------------------|-------------------|-------------------|---------|
| Stand-alone		| -			        | -                 |         |
| Cousineau, 2005	| Subject-centering	| -			        | -       |
| CM			    | Subject-centering	| Bias-correction	|         |
| NKM			    | Subject-centering	| -			        | pool S  | 
| LM			    | Subject-centering	| Bias-correction	| pool S  |

	 
From that point of view, we see that the Nathoo, Kilshaw and Masson NKM [@nkm18; but see @h19  ] method is missing a 
bias-correction transformation, which explains why these error bars are shorter. The original 
proposal found in Cousineau, 2005, is also missing the bias correction step, which led Morey (2008) 
to supplement this approach. With these four approaches, we have exhausted all the possible combinations 
regarding decorrelation methods based on subject-centering.

We added two arguments in superbPlot to handle this transformation approach, the first is ``preprocessfct`` 
and the second is ``postprocessfct``. 

Assuming a dataset dta with replicated measures stored in say columns called ``Score.1``, ``Score.2`` and
``Score.3``, the command

```{r}
pCM <- superbPlot(dta, WSFactors = "moment(3)",     
  variables = c("Score.1","Score.2","Score.3"), 
  adjustments=list(decorrelation="none"),
  preprocessfct = "subjectCenteringTransform",
  postprocessfct = "biasCorrectionTransform",
  plotStyle = "pointjitter",
  errorbarParams = list(color="red", width= 0.1, position = position_nudge(-0.05) )
)
```

will reproduce the ``CM`` error bars because it decorrelates the data as per
this method. With one additional transformation, 

```{r}
pLM <- superbPlot(dta, WSFactors = "moment(3)", 
  variables = c("Score.1","Score.2","Score.3"), 
  adjustments=list(decorrelation="none"),
  preprocessfct = "subjectCenteringTransform",
  postprocessfct = c("biasCorrectionTransform","poolSDTransform"),
  plotStyle = "line",
  errorbarParams = list(color="orange", width= 0.1, position = position_nudge(-0.0) )
)
```

the ``LM`` method is reproduced. Finally, if the ``biasCorrectionTransform`` is omitted, we get the NKM error bars with:


```{r}
pNKM <- superbPlot(dta, WSFactors = "moment(3)", 
  variables = c("Score.1","Score.2","Score.3"), 
  adjustments=list(decorrelation="none"),
  preprocessfct = "subjectCenteringTransform",
  postprocessfct = c("poolSDTransform"),
  plotStyle = "line",
  errorbarParams = list(color="blue", width= 0.1, position = position_nudge(+0.05) )
)
```

In what follow, I justapose the three plots to see the differences:

```{r, fig.height=4, fig.width=7, fig.cap = "**Figure 1**. Plot of the tree decorrelation methods based on subject transformation."}
tlbl <- paste( "(red)    Subject centering & Bias correction == CM\n",
               "(orange) Subject centering, Bias correction & Pooling SDs == LM\n",
               "(blue)   Subject centering & Pooling SDs == NKM", sep="")

ornate <- list(
    xlab("Group"),
    ylab("Score"),
    labs(   title=tlbl),
    coord_cartesian( ylim = c(12,18) ),
    theme_light(base_size=16)
)

# the plots on top are made transparent
pCM2 <- ggplotGrob(pCM + ornate)
pLM2 <- ggplotGrob(pLM + ornate + makeTransparent() )
pNKM2 <- ggplotGrob(pNKM + ornate + makeTransparent() )

# put the grobs onto an empty ggplot 
ggplot() + 
    annotation_custom(grob=pCM2) + 
    annotation_custom(grob=pLM2) + 
    annotation_custom(grob=pNKM2)
```

The method from Cousineau (2005) missing the bias-correction step
is not shown as it should not be used.



## In summary

All the decorrelation methods based on transformations have (probably) been
explored. An alternative approach using correlation was proposed in @c19. All
these approaches requires sphericity of the data.
Other approaches are required to overcome this sphericity limitations.






# References
