---
title: "Between-Within"
package: mmrm
bibliography: '`r system.file("REFERENCES.bib", package = "mmrm")`'
csl: '`r system.file("jss.csl", package = "mmrm")`'
output:
  rmarkdown::html_vignette:
          toc: true
vignette: |
  %\VignetteIndexEntry{Between-Within}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options:
  chunk_output_type: console
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(mmrm)
```

For determining the degrees of freedom (DF) required for the testing of fixed effects, 
one option is to use the "between-within" method, originally proposed by @schluchter1990small as a small-sample adjustment.

## General definition

Using this method, the DF are determined by the grouping level at which the term is estimated. Generally, assuming $G$ levels of grouping:

$DF_g=N_g-(N_{g-1}+p_g), g=1, ..., G+1$

where $N_g$ is the number of groups at the $g$-th grouping level and $p_g$ is the number of parameters estimated at that level.

$N_0=1$ if the model includes an intercept term and $N_0=0$ otherwise. Note however
that the DF for the intercept term itself (when it is included) are calculated at the $G+1$ level, 
i.e. for the intercept we use $DF_{G+1}$ degrees of freedom.

We note that general contrasts $C\beta$ have not been considered in the literature so far. 
Here we therefore use a pragmatic approach and define that for a general contrast matrix $C$
we take the minimum DF across the involved coefficients as the DF.

## MMRM special case

In our case of an MMRM (with only fixed effect terms), there is only a single grouping level (subject), 
so $G=1$. This means there are 3 potential "levels" of parameters (@galecki2013linear):

* Level 0: The intercept term, assuming the model has been fitted with one. 
  - We use $DF_2$ degrees of freedom as defined below.
* Level 1: Effects that change between subjects, but not across observations within subjects.
  - These are the "between parameters".
  - The corresponding degrees of freedom are $DF_1 = N_1 - (N_0 + p_1)$.
  - In words this can be read as:\
    "Between" DF = "number of subjects" - ("1 if intercept otherwise 0" + "number of between parameters").
* Level 2: Effects that change within subjects.
  - These are the "within parameters".
  - The corresponding degrees of freedom are $DF_2 = N_2 - (N_1 + p_2)$.
  - In words this can be read as:\
    "Within" DF = "number of observations" - ("number of subjects" + "number of within parameters").

## Example

Let's look at a concrete example and what the "between-within" degrees of freedom method gives as results:

```{r}
fit <- mmrm(
  formula = FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  control = mmrm_control(method = "Between-Within")
)
summary(fit)
```

Let's try to calculate the degrees of freedom manually now.

In `fev_data` there are 197 subjects with at least one non-missing `FEV1` observation, and 537 non-missing observations in total. Therefore we obtain the following numbers of groups $N_g$ at the levels $g=1,2$:

* $N_1 = 197$
* $N_2 = 537$

And we note that $N_0 = 1$ because we use an intercept term.

Now let's look at the design matrix:

```{r}
head(model.matrix(fit), 1)
```

Leaving the intercept term aside, we therefore have the following number of parameters for the
corresponding effects:

* `RACE`: 2
* `SEX`: 1
* `ARMCD`: 1
* `AVISIT`: 3
* `ARMCD:AVISIT`: 3

In the model above, `RACE`, `SEX` and `ARMCD` are between-subjects effects and belong to level 1; they do not vary within subject across the repeated observations. 
On the other hand, `AVISIT` is a within-subject effect; it represents study visit, so naturally its value changes over repeated observations for each subject. Similarly, the interaction of `ARMCD` and `AVISIT` also belongs to level 2.

Therefore we obtain the following numbers of parameters $p_g$ at the levels $g=1,2$:

* $p_1 = 2 + 1 + 1 = 4$
* $p_2 = 3 + 3 = 6$

And we obtain therefore the degrees of freedom $DF_g$ at the levels $g=1,2$:

* $DF_1 = N_1 - (N_0 + p_1) = 197 - (1 + 4) = 192$
* $DF_2 = N_2 - (N_1 + p_2) = 537 - (197 + 6) = 334$

So we can finally see that those degrees of freedom are exactly as displayed in the summary table above.

## Differences compared to SAS

The implementation described above is not identical to that of SAS. Differences include:

* In SAS, when using an unstructured covariance matrix, all effects are assigned the between-subjects degrees of freedom.
* In SAS, the within-subjects degrees of freedom are affected by the number of subjects in which the effect takes different values. 
* In SAS, if there are multiple within-subject effects containing classification variables, the within-subject degrees of freedom are partitioned into components corresponding to the subject-by-effect interactions.
* In SAS, the final effect you list in the `CONTRAST`/`ESTIMATE` statement is used to define the DF for general contrasts.

Code contributions for adding the SAS version of between-within degrees of freedom to the `mmrm` package are welcome!

## References
