Between- and within-school variance decomposition — variance

Decomposes the variance of achievement into a between-unit (e.g. between-school) and a within-unit component, and reports the intraclass correlation icc – the share of achievement variance that lies between schools. This is a foundational stratification quantity: a high between-school share signals a strongly differentiated, segregated system. OECD reports it as "between-school variance as a percentage of total".

Usage

variance_decomposition(
  data,
  achievement,
  unit,
  weight = NULL,
  repweights = NULL,
  rep_method = c("BRR", "JK2", "JK1"),
  fay = 0.5,
  design = NULL,
  statistics = c("icc", "var_between", "var_within", "var_total"),
  level = 0.95
)

Arguments

data: A data frame of student-level records.
achievement: Character vector of achievement plausible-value columns.
unit: Name of the school / nesting-unit identifier column.
weight: Name of the final student weight column. If NULL, equal weights are used (with a message).
repweights: Optional character vector of replicate-weight columns.
rep_method, fay: Replication design and Fay factor; see rep_factor().
design: Optional lsa_design() bundling weight, repweights, rep_method and fay; when supplied it overrides those arguments.
statistics: Which quantities to return: "icc" (between-unit share), "var_between", "var_within", "var_total".
level: Confidence level for the stored interval (default 0.95).

Value

An object of class "variance_decomposition" / "lsastrat_estimate" (see lsastrat_estimate for methods).

Details

The decomposition is the design-based (weighted) analysis-of-variance decomposition, var_total = var_between + var_within, pooled over plausible values with Rubin's rules and with replicate-weight standard errors when replicate weights are supplied (see pool_pv()).

Examples

data(pisa_mini)
variance_decomposition(pisa_mini, paste0("PV", 1:10, "MATH"),
                       unit = "school_id", weight = "W_FSTUWT",
                       repweights = paste0("W_FSTURWT", 1:64))
#> Between/within-unit variance decomposition
#>   unit: school_id  |  128 units  |  10 plausible value(s)  |  n = 2048
#>   Variance: BRR, 64 replicate weights (+ PV imputation); t reference
#> 
#> 
#>             Estimate Std.Error     t     df        p
#> icc            0.153     0.017  9.15 1259.2  2.3e-19
#> var_between 1110.411   133.168  8.34 1561.8  1.6e-16
#> var_within  6143.320   205.537 29.89  384.4 2.7e-102
#> var_total   7253.731   228.176 31.79  517.7 8.7e-124
#> 
#> icc = between-unit share of achievement variance