product.covar.weight {CaseCohortCoxSurvival} | R Documentation |

## product.covar.weight

### Description

Computes the product of joint design weights and joint sampling indicators covariances, needed for the phase-two component of the variance (with design or calibrated weights).

### Usage

```
product.covar.weight(casecohort, stratified = NULL)
```

### Arguments

`casecohort` |
if |

`stratified` |
was the sampling of the case-cohort stratified on |

### Details

`product.covar.weight`

creates the matrix with the products of joint design
weights and joint sampling indicator covariances, for the non-cases in the case
cohort. In other words, it has as many rows and columns as non-cases in the case
cohort, and contains the `w_{i,k,j} \sigma_{i,k,j}`

, with

`w_{i,k,j} = \frac{n^{(j)}(n^{(j)} -1)}{m^{(j)}(m^{(j)} -1)}`

if individuals
`i`

and `k`

in stratum `j`

are both non-cases, and
`w_{i,k,j} = \left( \frac{n^{(j)}}{m^{(j)}} \right)^2`

otherwise,
`i \neq k \in \lbrace 1, \dots, n^{(j)} \rbrace`

,
`j \in \lbrace 1, \dots, J \rbrace`

.

`w_{i,i,j} = \frac{n^{(j)}}{m^{(j)}}`

if individuals `i`

in stratum `j`

is a non-case, `i \in \lbrace 1, \dots, n^{(j)} \rbrace`

,
`j \in \lbrace 1, \dots, J \rbrace`

.

```
\sigma_{i,k,j} = \frac{m^{(j)}(m^{(j)} -1)}{n^{(j)}(n^{(j)} -1)} -
\left( \frac{m^{(j)}}{n^{(j)}} \right)^2
```

if individuals `i`

and
`k`

in stratum `j`

are both non-cases,
`i \neq k \in \lbrace 1, \dots, n^{(j)} \rbrace`

,
`j \in \lbrace 1, \dots, J \rbrace`

.

`\sigma_{i,i,j} = \frac{m^{(j)}}{n^{(j)}} - \left(1 - \frac{m^{(j)}}{n^{(j)}} \right)`

if individuals `i`

in stratum `j`

is a non-case,
`i \in \lbrace 1, \dots, n^{(j)} \rbrace`

,
`j \in \lbrace 1, \dots, J \rbrace`

.

See Section 3.3 in Etievant and Gail (2023).

### Value

`product.covar.weight`

: matrix with the products of joint design weights and
joint sampling indicator covariances, for the non-cases in the case-cohort.

### References

Etievant, L., Gail, M.H. (2023). Cox model inference for relative hazard and pure risk from stratified weight-calibrated case-cohort data. Submitted.

### See Also

`variance`

, that uses `product.covar.weight`

to compute the variance
estimate that follows the complete variance decomposition (superpopulation and
phase-two variance components).

### Examples

```
data(dataexample, package="CaseCohortCoxSurvival")
casecohort <- dataexample$casecohort # a simulated stratified case-cohort
prod.covar.weight <- product.covar.weight(casecohort, stratified = TRUE)
nrow(prod.covar.weight)
ncol(prod.covar.weight)
sum(casecohort$status == 0) # number of non-cases in the case-cohort
```

*CaseCohortCoxSurvival*version 0.0.34 Index]