# Variance estimation for sample surveys by the ultimate cluster method

### Description

Computes the variance estimation by the ultimate cluster method.

### Usage

1 2 3 4 |

### Arguments

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

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

`PSU` |
Primary sampling unit variable. One dimensional object convertible to one-column |

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

`N_h` |
optional; either a matrix giving the first column - stratum, but the second column - the total of the population in each stratum. |

`fh_zero` |
by default FALSE; fh is calculated as division of n_h and N_h in each strata, if true, fh value is zero in each strata. |

`PSU_level` |
by default TRUE; if PSU_level is true, in each strata fh is calculated as division of count of PSU in sample (n_h) and count of PSU in frame (N_h). if PSU_level is false, in each strata fh is calculated as division of count of units in sample (n_h) and count of units in frame (N_h), which calculated as sum of weights. |

`PSU_sort` |
optional; if PSU_sort is defined, then variance is calculated for systematic sample. |

`period` |
Optional variable for the survey periods. If supplied, the values for each period are computed independently. Object convertible to |

`dataset` |
an optional name of the individual dataset |

`msg` |
an optional printed text, when function print error. |

### Details

If we assume that *n_h>=2* for all *h*, that is, two or more PSUs
are selected from each stratum, then the variance of *θ*
can be estimated from the variation among the estimated PSU totals of the variable *Z*:

*V(θ)=∑ h=1...H (1-f_h)*n_h/(n_h-1)* ∑ i=1...n_h ( z_hi.- z_h..)^2, *

where

*z_hi.=∑ j=1...m_hi ω_hij * z_hij*

*z_h..=(∑ i=1...n_h z_hi.)/n_h*

*f_h* is the sampling fraction of PSUs within stratum

*h* is the stratum number, with a total of H strata

*i* is the primary sampling unit (PSU) number within stratum *h*, with a total of *n_h* PSUs

*j* is the household number within cluster *i* of stratum *h*, with a total of *m_hi* household

*w_hij* is the sampling weight for household *j* in PSU *i* of stratum *h*

*z_hij* denotes the observed value of the analysis variable *z* for household *j* in PSU *i* of stratum *h*

### Value

a `data.table`

containing the values of the
variance estimation by totals.

### References

Morris H. Hansen, William N. Hurwitz, William G. Madow, (1953), Sample survey methods and theory Volume I Methods and applications, 257-258, Wiley.

Guillaume Osier and Emilio Di Meglio. The linearisation approach implemented by Eurostat for the first wave of EU-SILC: what could be done from the second onwards? 2012

Eurostat Methodologies and Working papers, Standard error estimation for the EU-SILC indicators of poverty and social exclusion, 2013, URL http://ec.europa.eu/eurostat/documents/3859598/5927001/KS-RA-13-029-EN.PDF.

Yves G. Berger, Tim Goedeme, Guillame Osier (2013). Handbook on standard
error estimation and other related sampling issues in EU-SILC,
URL https://ec.europa.eu/eurostat/cros/content/handbook-standard-error-estimation-and-other-related-sampling-issues-ver-29072013_en

Eurostat Methodologies and Working papers, Handbook on precision requirements and variance estimation for ESS household surveys, 2013, URL http://ec.europa.eu/eurostat/documents/3859598/5927001/KS-RA-13-029-EN.PDF.

### See Also

`domain`

, `lin.ratio`

, `linarpr`

,
`linarpt`

, `lingini`

, `lingini2`

,
`lingpg`

, `linpoormed`

, `linqsr`

,
`linrmpg`

, `residual_est`

, `vardom`

,
`vardomh`

, `varpoord`

, `variance_othstr`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
Ys <- rchisq(10, 3)
w <- rep(2, 10)
PSU <- 1 : length(Ys)
H <- rep("Strata_1", 10)
# by default without using fh_zero (finite population correction)
variance_est(Y = Ys, H = H, PSU = PSU, w_final = w)
## Not run:
# without using fh_zero (finite population correction)
variance_est(Y = Ys, H = H, PSU = PSU, w_final = w, fh_zero = FALSE)
# with using fh_zero (finite population correction)
variance_est(Y = Ys, H = H, PSU = PSU, w_final = w, fh_zero = TRUE)
## End(Not run)
``` |