# Variance estimation for sample surveys by the new stratification

### Description

Computes s2g and the variance estimation by the new stratification.

### Usage

1 |

### Arguments

`Y` |
Variables of interest. Object convertible to |

`H` |
The unit stratum variable. One dimensional object convertible to one-column |

`H2` |
The unit new stratum variable. One dimensional object convertible to one-column |

`w_final` |
Weight variable. One dimensional object convertible to one-column |

`N_h` |
optional; either a |

`N_h2` |
optional; either a |

`period` |
Optional variable for the survey periods. If supplied, the values for each period are computed independently. One dimensional object convertible to one-column |

`dataset` |
Optional survey data object convertible to |

### Details

It is possible to compute population size *M_g* from sampling frame. The standard deviation of *g*-th stratum is

*S_g^2 =1/(M_g-1) ∑ k=1...M_g (y_gk - Ym_g)^2= 1/(M_g-1) ∑ k=1...M_g (y_gk)^2 - M_g/(M_g-1)*(Ym_g)^2*

*∑ k=1...M_g (y_gk)^2* and *Ym_g^2* have to be estimeted to estimate *S_g^2*. Estimate of *∑ k=1...M_g (y_gk)^2* is
*∑ h=1...H N_h/n_h ∑ i=1...n_h (y_gi)^2*z_hi*, where
*z_hi=if(0, h_i notin θ_g; 1, h_i in θ_g)*
, *θ_g* is the index group of successfully surveyed units belonging to *g*-th stratum. Estimate of *(Y_g)^2*
is

*Ym_g^2=(Ym_g)^2- Var(Ym)*

*Ym_g =Ym_g/M_g= 1/M_g ∑ h=1...H N_h/n_h ∑ i=1...n_h y_hi z_hi*

So the estimate of *S_g^2* is

*s_g^2=\1/(M_g-1) ∑ h=1...H N_h/n_h ∑ i=1...n_h (y_hi)^2 * z_hi -*

*-M_g/(M_g-1) (1/M_g ∑ h=1...H N_h/n_h ∑ i=1...n_h y_hi z_hi)^2
*

Two conditions have to realize to estimate *S_g^2: n_h>1, forall g* and *θ_g <> 0, forall g.*

Variance of *Y* is

*Var(Y) = ∑ g=1...G M_g^2 (1/m_g - 1/M_g)*(S_g)^2 *

Estimate of *Var(Y)* is

*Var(Y)= ∑ g=1...G M_g^2 (1/m_g - 1/M_g)*(s_g)^2*

### Value

A list with objects are returned by the function:

`s2g` |
A |

`var_est` |
A |

### References

M. Liberts. (2004) Non-response Analysis and Bias Estimation in a Survey on Transportation of Goods by Road.

### See Also

`domain`

, `lin.ratio`

, `linarpr`

,
`linarpt`

, `lingini`

, `lingini2`

,
`lingpg`

, `linpoormed`

, `linqsr`

,
`linrmpg`

, `residual_est`

, `vardom`

,
`vardom_othstr`

, `vardomh`

, `varpoord`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
period=NULL
dataset=NULL
Y <- data.table(matrix(runif(50) * 5, ncol = 5))
H <- data.table(H = as.integer(trunc(5 * runif(10))))
H2 <- data.table(H2 = as.integer(trunc(3 * runif(10))))
N_h <- data.table(matrix(0 : 4, 5, 1))
setnames(N_h, names(N_h), "H")
N_h[, sk:= 10]
N_h2 <- data.table(matrix(0 : 2, 3, 1))
setnames(N_h2, names(N_h2), "H2")
N_h2[, sk2:= 4]
w_final <- rep(2, 10)
vo <- variance_othstr(Y = Y, H = H, H2 = H2,
w_final = w_final,
N_h = N_h, N_h2 = N_h2,
period = NULL,
dataset = NULL)
vo
``` |