Package 'sampleone'

Title: Survey Sampling Methods for Teaching and Practice
Description: Planning, selection, and estimation for surveys under the main probability sampling designs taught in introductory sampling courses: simple random sampling with and without replacement, stratified sampling (proportional, Neyman, and cost-based allocation), systematic sampling, one- and two-stage cluster sampling, and ratio and regression estimators. Functions accept a superset of the inputs a given formula requires and silently ignore arguments that are not needed for the selected computation. Each function returns a structured object with the point estimate, the associated variance and confidence interval, a step-by-step breakdown of the calculation, and a short pedagogical note suggesting related methods to explore, making the package suitable both for applied survey work and for classroom use. Methods follow Cochran (1977, ISBN:0-471-16240-X) and Bolfarine & Bussab (2005, ISBN:8521203675), among others referenced in each function's documentation.
Authors: Miguel Luiz de Almeida Pinto [aut, cre], Mariano Martinez Espinosa [aut] (Author of the source course material (course notes) this package implements)
Maintainer: Miguel Luiz de Almeida Pinto <[email protected]>
License: MIT + file LICENSE
Version: 0.1.0
Built: 2026-10-13 23:52:25 UTC
Source: https://github.com/cran/sampleone

Help Index

Adjust a sample size by a flat percentage buffer ("acrescimo")

Description

Inflates a sample size by a flat percentage buffer, following Example 3.14 of the course notes: n_ajustado = n * (1 + taxa_acrescimo) (there illustrated as a 25% buffer added to protect against losses, refusals, or other contingencies during fieldwork). This is a distinct, simpler convention from the formal non-response adjustment of Section 4.7.2 (dividing by an expected response rate, see the course notes' Eq. 4.25 and the package's stratified-sampling module) – do not confuse the two: dividing by (1 - taxa) targets a given number of completed responses after losses, while multiplying by (1 + taxa) is a simpler flat safety margin, as used here.

Usage

aas_ajustar_perdas(n, taxa_acrescimo, arredondar = c("cima", "proximo"))

Arguments

n

Base (unadjusted) sample size.
taxa_acrescimo

Flat buffer to add, as a proportion (e.g. 0.25 for a 25% increase).
arredondar

Either "cima" (default, always round up) or "proximo" (round to the nearest integer, reproducing Example 3.14 of the course notes exactly: 357 * 1.25 = 446.25, rounded there to 446, not 447).

Value

A sampling_result object with the adjusted sample size.

Examples

aas_ajustar_perdas(357, 0.25) # arredondar="cima" (padrao) -> 447
aas_ajustar_perdas(357, 0.25, arredondar = "proximo") # Exemplo 3.14 -> 446

Compare the "with replacement" and "without replacement" designs (EPA/deff)

Description

Computes the sampling-plan effect (EPA, "efeito do planejamento amostral", a.k.a. design effect) comparing AASs to AASc for the same N and n, following Eq. (3.46)-(3.47) of the course notes.

Usage

aas_comparar_reposicao(N, n)

Arguments

N

Population size.
n

Sample size.

Value

A sampling_result object with the EPA value and its interpretation.

References

Bolfarine, H.; Bussab, W.O. (2005). Elementos de Amostragem. Edgard Blucher.

Examples

aas_comparar_reposicao(N = 2695, n = 336)

Estimate the population mean under simple random sampling (AAS)

Description

Estimates the population mean from a simple random sample, with or without replacement, together with its variance and a confidence interval. Corresponds to Eqs. (3.4)-(3.7) (with replacement, AASc) and (3.25)-(3.28.1) (without replacement, AASs) of the course notes.

Usage

aas_estima_media(x, N = NULL, reposicao = c("sem", "com"), conf = 0.95, ...)

Arguments

x

Numeric vector with the sample data.
N

Optional population size. If NULL or Inf, the population is treated as unknown/very large (the finite-population correction 1 - n/N is dropped, i.e. f = 0), which also makes the "com" ("with replacement") and "sem" ("without replacement") results coincide.
reposicao

Either "sem" (without replacement, AASs – the default and the usual case in practice) or "com" (with replacement, AASc).
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated mean, its variance, coefficient of variation, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley. Bolfarine, H.; Bussab, W.O. (2005). Elementos de Amostragem. Edgard Blucher.

Examples

aas_estima_media(c(12, 30, 18), N = 3, reposicao = "com")

Estimate a population proportion under simple random sampling (AAS)

Description

Estimates a population proportion from a simple random sample, with or without replacement. Corresponds to Eqs. (3.16)-(3.21) (AASc) and (3.39)-(3.44) (AASs) of the course notes.

Usage

aas_estima_proporcao(
  x,
  N = NULL,
  reposicao = c("sem", "com"),
  conf = 0.95,
  ...
)

Arguments

x

Numeric or logical vector coded 0/1 (or FALSE/TRUE).
N

Optional population size. If NULL or Inf, the population is treated as unknown/very large (the finite-population correction 1 - n/N is dropped, i.e. f = 0), which also makes the "com" ("with replacement") and "sem" ("without replacement") results coincide.
reposicao

Either "sem" (without replacement, AASs – the default and the usual case in practice) or "com" (with replacement, AASc).
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated proportion, its variance, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

Examples

aas_estima_proporcao(c(1, 0, 1, 1, 0, 1, 0, 0), N = 500)

Estimate the population total under simple random sampling (AAS)

Description

Estimates the population total τ^=Nxˉ\hat{\tau} = N \bar{x} from a simple random sample without replacement (Eq. 3.28.2-3.28.3 of the course notes). For the "with replacement" plan, the same expressions are used with f = 0.

Usage

aas_estima_total(x, N, reposicao = c("sem", "com"), conf = 0.95, ...)

Arguments

x

Numeric vector with the sample data.
N

Population size (required to scale the total).
reposicao

Either "sem" (without replacement, AASs – the default and the usual case in practice) or "com" (with replacement, AASc).
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated total, its variance, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 2 (extension not explicit in the course notes for AAS, but consistent with Eq. 3.28.3).

Examples

aas_estima_total(c(12, 30, 18), N = 300)

Sample size for AAS: automatic formula selection (umbrella function)

Description

Convenience "umbrella" function that accepts every input that might be relevant to determine an AAS sample size (population variance or standard deviation, expected proportion, population size, expected losses) and automatically dispatches to aas_tamanho_media() or aas_tamanho_proporcao() depending on parametro, silently ignoring whichever inputs are not needed for the chosen calculation.

Usage

aas_tamanho_amostra(
  parametro = c("media", "proporcao"),
  N = NULL,
  sigma2 = NULL,
  sigma = NULL,
  p = NULL,
  d,
  conf = 0.95,
  perda = 0,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

parametro

Either "media" or "proporcao": which population parameter the sample size should be computed for.
N

Optional population size.
sigma2

Population variance (used only when parametro = "media").
sigma

Population standard deviation (alternative to sigma2, used only when parametro = "media").
p

Expected proportion (used only when parametro = "proporcao").
d

Tolerable margin of error.
conf

Confidence level, default 0.95.
perda

Optional expected non-response/loss rate in ⁠[0, 1)⁠.
arredondar

Either "cima" (default) or "proximo" – see aas_tamanho_media().
...

Ignored. Any other variables from a broader problem statement (e.g. sample data, stratum information) may be passed here without causing an error.

Value

A sampling_result object, as produced by aas_tamanho_media() or aas_tamanho_proporcao().

Examples

aas_tamanho_amostra(parametro = "media", sigma = 7, d = 2)
aas_tamanho_amostra(parametro = "proporcao", p = 0.6, d = 0.03, N = 5000)

Sample size to estimate a population mean (AAS)

Description

Computes the sample size needed to estimate a population mean with a given margin of error, choosing automatically between the infinite/unknown-population formula (Eq. 3.15) and the finite-population formula (Eq. 3.37/3.38) depending on whether N is supplied. Extra arguments not needed for the chosen formula are ignored.

Usage

aas_tamanho_media(
  sigma2 = NULL,
  sigma = NULL,
  d,
  conf = 0.95,
  N = NULL,
  perda = 0,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

sigma2

Population variance (or its estimate). Alternatively supply sigma (standard deviation); if both are given, sigma2 takes precedence.
sigma

Optional population standard deviation, used only if sigma2 is not supplied.
d

Tolerable margin of error (same units as the variable).
conf

Confidence level, default 0.95.
N

Optional population size. If NULL or Inf, the infinite/unknown-population formula (3.15) is used; otherwise the finite-population formula (3.37) is used (cross-checked internally against the algebraically equivalent Eq. 3.38).
perda

Optional flat safety-margin proportion (e.g. 0.25 for a 25% buffer), applied via aas_ajustar_perdas() to inflate the raw sample size (n * (1 + perda), following Example 3.14 of the course notes). Not to be confused with a formal response-rate adjustment (dividing by an expected response rate, Section 4.7.2).
arredondar

Either "cima" (the default – always round up, the conventional, conservative practice in sampling theory, e.g. Cochran 1977) or "proximo" (round to the nearest integer). The course notes round to the nearest integer using a 2-decimal critical value (z = 1.96/1.64); use arredondar = "proximo" together with a matching conf to reproduce the worked examples exactly.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the required sample size.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

Examples

aas_tamanho_media(sigma = 7, d = 2, conf = 0.95)
aas_tamanho_media(sigma2 = 1240, d = 7, conf = 0.95, N = 10000)

Sample size to estimate a population proportion (AAS)

Description

Computes the sample size needed to estimate a population proportion with a given margin of error, using the infinite/unknown-population formula (Eq. 3.23, or the conservative Eq. 3.24 when p is unknown) or the finite-population formula (Eq. 3.45) when N is supplied.

Usage

aas_tamanho_proporcao(
  p = NULL,
  d,
  conf = 0.95,
  N = NULL,
  perda = 0,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

p

Expected proportion. If NULL, the conservative value p = 0.5 is used (maximises p(1-p)), following Eq. (3.24).
d

Tolerable margin of error (same units as the variable).
conf

Confidence level, default 0.95.
N

Optional population size. If NULL or Inf, the infinite/unknown-population formula (3.15) is used; otherwise the finite-population formula (3.37) is used (cross-checked internally against the algebraically equivalent Eq. 3.38).
perda

Optional flat safety-margin proportion (e.g. 0.25 for a 25% buffer), applied via aas_ajustar_perdas() to inflate the raw sample size (n * (1 + perda), following Example 3.14 of the course notes). Not to be confused with a formal response-rate adjustment (dividing by an expected response rate, Section 4.7.2).
arredondar

Either "cima" (the default – always round up, the conventional, conservative practice in sampling theory, e.g. Cochran 1977) or "proximo" (round to the nearest integer). The course notes round to the nearest integer using a 2-decimal critical value (z = 1.96/1.64); use arredondar = "proximo" together with a matching conf to reproduce the worked examples exactly.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the required sample size.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

Examples

aas_tamanho_proporcao(p = 0.6, d = 0.03, conf = 0.95)
aas_tamanho_proporcao(d = 0.05, conf = 0.95) # p desconhecido -> conservador

Sample-based estimators (mean, variance, proportion)

Description

Computes the sample mean (Eq. 2.8), the unbiased sample variance (Eq. 2.10, divisor n-1), and, when the data are binary, the sample proportion (Eq. 2.9) – all from a single call. This function is used internally as a building block by most other functions in the package.

Usage

amostra_estimadores(x, ...)

Arguments

x

Numeric vector with the sample data.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the requested sample estimators.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

Examples

amostra_estimadores(c(12, 30, 18))
amostra_estimadores(c(1, 0, 1, 1, 0))

Adjust a sample size for design effect and expected response rate

Description

Inflates a base sample size (computed under a simple-random-sampling assumption) by a design effect (deff, accounting for the extra variability introduced by a more complex design such as clustering or multi-stage sampling) and by an expected response rate, following Eq. (6.6) of the course notes (Espinosa et al., 2019).

Usage

amostragem_ajustar_deff_resposta(n_prime, deff = 1.5, taxa_resposta = 0.85)

Arguments

n_prime

Base sample size (e.g. from aas_tamanho_media(), aas_tamanho_proporcao(), estr_tamanho_media(), or cong1_tamanho_media()), ignoring the design effect and non-response.
deff

Design effect (⁠>= 1⁠), default 1.5 as commonly assumed in the absence of better information. deff = 1 means no loss of efficiency relative to simple random sampling.
taxa_resposta

Expected response rate, default 0.85.

Details

Although this formula is first introduced in the course notes' cluster sampling chapter, it is a general-purpose adjustment: it only needs a base sample size, a design effect, and a response rate – nothing specific to clusters. Course exercises apply it to plain AAS and stratified problems just as often as to cluster problems (e.g. to scale up a prevalence-survey sample size for an anticipated design effect and non-response), so it is exposed here as a cross-cutting utility rather than tucked away under the cong1_ prefix. cong1_ajustar_deff() remains available as an alias, for continuity with the cluster-sampling chapter and its documentation.

Value

A sampling_result object with the adjusted sample size.

References

Espinosa et al. (2019), as cited in the course notes, Eq. (6.6).

Examples

# Cap. 3 (AAS) + deff + resposta -- ex.: planejamento de prevalencia
n_prime <- aas_tamanho_proporcao(p = 0.20, d = 0.05)$estimativa$n
amostragem_ajustar_deff_resposta(n_prime, deff = 1.2, taxa_resposta = 0.85)

Adjust a sample size for a minimum expected response rate

Description

Inflates a sample size to guarantee a target number of completed responses given an expected response rate, following Eqs. (4.24)-(4.25) of the course notes: ⁠n* = n / taxa_resposta⁠. This targets completed responses after losses and is conceptually distinct from the flat safety-margin buffer of aas_ajustar_perdas() (which multiplies by 1 + taxa, see Example 3.14) – do not confuse the two.

Usage

amostragem_ajustar_resposta(n, taxa_resposta)

Arguments

n

Base sample size computed ignoring non-response.
taxa_resposta

Expected response rate, in ⁠(0, 1]⁠ (e.g. 0.85 for an expected 85% response rate).

Value

A sampling_result object with the adjusted sample size.

Examples

amostragem_ajustar_resposta(1525, 0.85)

Compare the variance of different sampling plans for the same problem

Description

Convenience function that lines up the estimated variance obtained under different sampling designs (as returned by sampling_result objects from this package) so they can be compared directly, computing the sampling-plan effect (EPA / design effect) of each one relative to a chosen baseline. This operationalises the comparisons illustrated throughout the course notes (Section 3.7.7, Examples 4.4-4.5, and the design-effect discussion in Section 6.1.5).

Usage

amostragem_comparar_planos(..., referencia = NULL)

Arguments

...

Two or more named sampling_result objects (e.g. aas = aas_estima_media(...), estratificada = estr_estima_media(...)), each with a variancia element containing var_media or var_p.
referencia

Name of the plan to use as the baseline (denominator of the EPA ratio). Defaults to the first plan supplied.

Value

A sampling_result object with a comparison table of variances and EPA (design effect) ratios relative to referencia.

Examples

x <- rnorm(30, 50, 5)
plano_aas <- aas_estima_media(x, N = 300)
# (em um caso real, plano_estrat viria de estr_estima_media() com os
#  mesmos dados organizados em estratos)
amostragem_comparar_planos(aas = plano_aas)

Umbrella estimator for two-stage cluster sampling

Description

Convenience "umbrella" function analogous to aas_tamanho_amostra(): dispatches automatically to cong2_estima_media() (when M is supplied) or cong2_estima_razao_media() (when M is NULL), ignoring whichever inputs are not needed.

Usage

cong_estima(
  dados = NULL,
  Mi = NULL,
  mi = NULL,
  yi_bar = NULL,
  si2 = NULL,
  Nc,
  M = NULL,
  conf = 0.95,
  ...
)

Arguments

dados

A data.frame (or list of data frames / a list with one element per cluster) with, for every sampled element, at least the columns conglomerado (cluster id), Mi (cluster population size), and y (the measured value for that element). Alternatively, supply Mi, mi, yi_bar, and si2 directly (pre-summarised per cluster) – see Details.
Mi

Optional numeric vector with the population size of each sampled cluster (used instead of dados when the data are already summarised per cluster).
mi

Optional numeric vector with the number of sampled elements in each cluster.
yi_bar

Optional numeric vector with the sample mean of each cluster.
si2

Optional numeric vector with the sample variance within each cluster (Eq. 6.15).
Nc

Total number of clusters in the population.
M

Total number of elements in the population.
conf

Confidence level, default 0.95.
...

Ignored. Any other variables from a broader problem statement may be passed here without causing an error.

Value

A sampling_result object, as produced by cong2_estima_media() or cong2_estima_razao_media().

Examples

Mi <- c(50, 65, 45, 48, 52, 58, 42, 66, 40, 56)
mi <- c(10, 13, 9, 10, 10, 12, 8, 13, 8, 11)
yi_bar <- c(5.4, 4.0, 5.6667, 4.8, 4.3, 3.8333, 5.0, 3.8462, 4.875, 5.0)
si2 <- c(11.3778, 10.6667, 16.75, 13.2889, 11.1222, 14.8788, 5.1429,
         4.3077, 6.125, 11.8)
cong_estima(Mi = Mi, mi = mi, yi_bar = yi_bar, si2 = si2, Nc = 90) # M ausente -> razao

Adjust a cluster/element sample size for design effect and response rate

Description

Alias for amostragem_ajustar_deff_resposta(), kept for continuity with the cluster-sampling chapter, since that is where the course notes first introduce Eq. (6.6). The underlying formula is general-purpose (not specific to clusters) – see amostragem_ajustar_deff_resposta() for the full documentation and for use with non-cluster problems (AAS, stratified, etc.).

Usage

cong1_ajustar_deff(n_prime, deff = 1.5, taxa_resposta = 0.85)

Arguments

n_prime

Base sample size (e.g. from aas_tamanho_media(), aas_tamanho_proporcao(), estr_tamanho_media(), or cong1_tamanho_media()), ignoring the design effect and non-response.
deff

Design effect (⁠>= 1⁠), default 1.5 as commonly assumed in the absence of better information. deff = 1 means no loss of efficiency relative to simple random sampling.
taxa_resposta

Expected response rate, default 0.85.

Value

A sampling_result object with the adjusted sample size.

References

Espinosa et al. (2019), as cited in the course notes, Eq. (6.6).

Examples

cong1_ajustar_deff(n_prime = 518, deff = 1.5, taxa_resposta = 0.85)

Estimate the population mean under one-stage cluster sampling

Description

Estimates the population mean from a sample of whole clusters, all of whose elements were observed (one-stage cluster sampling), following Eqs. (6.1)-(6.4) of the course notes.

Usage

cong1_estima_media(yi, Mi, Nc, M = NULL, conf = 0.95, ...)

Arguments

yi

Numeric vector with the cluster totals (y_i) for each sampled cluster.
Mi

Numeric vector with the number of elements in each sampled cluster.
Nc

Total number of clusters in the population.
M

Optional total number of elements in the population (used to compute the population average cluster size M_barra = M/Nc, as required by Eq. 6.2). When M is unknown, M_barra falls back to the sample average of Mi, which is only an approximation and is flagged as such in the result.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated mean, variance, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 9.

Examples

# Exemplo 6.1 (turmas da UFMT)
cong1_estima_media(
  yi = c(1209, 921, 1116, 1433, 1037, 2112, 1737, 2206, 2112, 1870),
  Mi = c(20, 16, 17, 21, 16, 25, 21, 26, 25, 23),
  Nc = 373, M = 8018
)

Estimate the population proportion under one-stage cluster sampling

Description

Analogous to cong1_estima_media() but for a population proportion, following Eqs. (6.7)-(6.10) of the course notes.

Usage

cong1_estima_proporcao(yi, Mi, Nc, M = NULL, conf = 0.95, ...)

Arguments

yi

Numeric vector with the number of "successes" (elements with the characteristic of interest) in each sampled cluster.
Mi

Numeric vector with the number of elements in each sampled cluster.
Nc

Total number of clusters in the population.
M

Optional total number of elements in the population (see cong1_estima_media() for details on M_barra = M/Nc).
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated proportion, variance, and confidence interval.

Examples

# Exemplo 6.3 (proporcao de alunos com peso acima da mediana)
cong1_estima_proporcao(
  yi = c(7, 6, 5, 6, 5, 7, 5, 6, 6, 6),
  Mi = c(20, 16, 17, 21, 16, 25, 21, 26, 25, 23),
  Nc = 373, M = 8018
)

Sample size (number of clusters) to estimate a mean (one-stage cluster sampling)

Description

Computes the number of clusters needed to estimate a population mean with a given margin of error under one-stage cluster sampling (Eq. 6.5 of the course notes).

Usage

cong1_tamanho_media(
  Nc,
  sigmac2,
  M_barra,
  d,
  conf = 0.95,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

Nc

Total number of clusters in the population.
sigmac2

Between-cluster population variance (or its estimate, from a pilot study).
M_barra

Average number of elements per cluster.
d

Tolerable margin of error (in the original units of the variable, not per-cluster).
conf

Confidence level, default 0.95.
arredondar

Either "cima" (default) or "proximo" – see aas_tamanho_media(). Example 6.1(c) of the course notes uses "proximo" (41.87 rounds to 42, same result either way here).
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the required number of clusters.

Examples

cong1_tamanho_media(Nc = 373, sigmac2 = 51054.3854, M_barra = 8018/373, d = 3)

Sample size (number of clusters) to estimate a proportion (one-stage cluster sampling)

Description

Analogous to cong1_tamanho_media() but for a population proportion (Eq. 6.11 of the course notes).

Usage

cong1_tamanho_proporcao(
  Nc,
  sigmacp2,
  M_barra,
  d,
  conf = 0.95,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

Nc

Total number of clusters in the population.
sigmacp2

Between-cluster population variance for the proportion (or its estimate).
M_barra

Average number of elements per cluster.
d

Tolerable margin of error (in the original units of the variable, not per-cluster).
conf

Confidence level, default 0.95.
arredondar

Either "cima" (default) or "proximo" – see aas_tamanho_media(). Example 6.1(c) of the course notes uses "proximo" (41.87 rounds to 42, same result either way here).
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the required number of clusters.

Examples

cong1_tamanho_proporcao(Nc = 373, sigmacp2 = 0.91785840, M_barra = 8018/373, d = 0.013)

Estimate the population mean under two-stage cluster sampling

Description

Estimates the population mean from a two-stage cluster sample (a sample of clusters – UPA – followed by a sample of elements within each sampled cluster – USA), when the total population size M is known, following Eqs. (6.12)-(6.18) of the course notes.

Usage

cong2_estima_media(
  dados = NULL,
  Mi = NULL,
  mi = NULL,
  yi_bar = NULL,
  si2 = NULL,
  Nc,
  M,
  conf = 0.95,
  ...
)

Arguments

dados

A data.frame (or list of data frames / a list with one element per cluster) with, for every sampled element, at least the columns conglomerado (cluster id), Mi (cluster population size), and y (the measured value for that element). Alternatively, supply Mi, mi, yi_bar, and si2 directly (pre-summarised per cluster) – see Details.
Mi

Optional numeric vector with the population size of each sampled cluster (used instead of dados when the data are already summarised per cluster).
mi

Optional numeric vector with the number of sampled elements in each cluster.
yi_bar

Optional numeric vector with the sample mean of each cluster.
si2

Optional numeric vector with the sample variance within each cluster (Eq. 6.15).
Nc

Total number of clusters in the population.
M

Total number of elements in the population.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Details

Two calling styles are supported: (1) pass raw per-element data via dados (a data.frame with one row per sampled element), or (2) pass already-summarised per-cluster vectors Mi, mi, yi_bar, si2 directly. The second style is convenient when a problem statement (as in a textbook exercise) already gives per-cluster summaries instead of raw data – any of dados/Mi/mi/yi_bar/si2 not needed for the chosen style are simply ignored.

Value

A sampling_result object with the estimated mean, its two-part variance decomposition, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 10.

Examples

Mi <- c(50, 65, 45, 48, 52, 58, 42, 66, 40, 56)
mi <- c(10, 13, 9, 10, 10, 12, 8, 13, 8, 11)
yi_bar <- c(5.4, 4.0, 5.6667, 4.8, 4.3, 3.8333, 5.0, 3.8462, 4.875, 5.0)
si2 <- c(11.3778, 10.6667, 16.75, 13.2889, 11.1222, 14.8788, 5.1429,
         4.3077, 6.125, 11.8)
cong2_estima_media(Mi = Mi, mi = mi, yi_bar = yi_bar, si2 = si2,
                    Nc = 90, M = 4500)

Estimate the population proportion under two-stage cluster sampling

Description

Analogous to cong2_estima_media() but for a population proportion, following Eqs. (6.24)-(6.29) of the course notes. Note that, unlike cong2_estima_media(), the proportion estimator is inherently a ratio-type estimator (structurally analogous to cong2_estima_razao_media()) and therefore always uses the sample average cluster size m_barra = mean(Mi) in its variance formula, even when the population totals Nc/M are known – this is exactly how Example 6.6 of the course notes computes it (using m_barra = 52.2 rather than M/Nc = 50, even though both M and Nc are known in that example).

Usage

cong2_estima_proporcao(Mi, mi, pi_hat, Nc, M = NULL, conf = 0.95, ...)

Arguments

Mi

Numeric vector with the population size of each sampled cluster.
mi

Numeric vector with the number of sampled elements in each cluster.
pi_hat

Numeric vector with the sample proportion of each cluster (Eq. 6.24).
Nc

Total number of clusters in the population.
M

Ignored (kept for signature symmetry with cong2_estima_media() and so that extra arguments from a broader problem statement can be passed without error) – see Details.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated proportion, its variance, and confidence interval.

Examples

Mi <- c(50, 65, 45, 48, 52, 58, 42, 66, 40, 56)
mi <- c(10, 13, 9, 10, 10, 12, 8, 13, 8, 11)
pi_hat <- c(0.4, 0.3077, 0.3333, 0.3, 0.3, 0.25, 0.375, 0.3077, 0.375, 0.3636)
cong2_estima_proporcao(Mi = Mi, mi = mi, pi_hat = pi_hat, Nc = 90)

Estimate the population mean via ratio estimator (two-stage cluster sampling, M unknown)

Description

Analogous to cong2_estima_media() but for the case where the total population size M is unknown, using the ratio estimator of Eqs. (6.19)-(6.23) of the course notes, with m_barra (the sample average cluster size) substituted for M_barra = M/Nc.

Usage

cong2_estima_razao_media(
  dados = NULL,
  Mi = NULL,
  mi = NULL,
  yi_bar = NULL,
  si2 = NULL,
  Nc,
  M = NULL,
  conf = 0.95,
  ...
)

Arguments

dados

A data.frame (or list of data frames / a list with one element per cluster) with, for every sampled element, at least the columns conglomerado (cluster id), Mi (cluster population size), and y (the measured value for that element). Alternatively, supply Mi, mi, yi_bar, and si2 directly (pre-summarised per cluster) – see Details.
Mi

Optional numeric vector with the population size of each sampled cluster (used instead of dados when the data are already summarised per cluster).
mi

Optional numeric vector with the number of sampled elements in each cluster.
yi_bar

Optional numeric vector with the sample mean of each cluster.
si2

Optional numeric vector with the sample variance within each cluster (Eq. 6.15).
Nc

Total number of clusters in the population.
M

Ignored for this estimator (kept for signature symmetry with cong2_estima_media() so extra arguments from a broader problem can be passed without error); m_barra is estimated from Mi instead.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the ratio-estimated mean, its variance, and confidence interval.

Examples

Mi <- c(50, 65, 45, 48, 52, 58, 42, 66, 40, 56)
mi <- c(10, 13, 9, 10, 10, 12, 8, 13, 8, 11)
yi_bar <- c(5.4, 4.0, 5.6667, 4.8, 4.3, 3.8333, 5.0, 3.8462, 4.875, 5.0)
si2 <- c(11.3778, 10.6667, 16.75, 13.2889, 11.1222, 14.8788, 5.1429,
         4.3077, 6.125, 11.8)
cong2_estima_razao_media(Mi = Mi, mi = mi, yi_bar = yi_bar, si2 = si2, Nc = 90)

Cost-based optimum allocation of a stratified sample

Description

Allocates strata sample sizes to minimise variance subject to a fixed total budget, when per-unit sampling costs differ across strata (Eqs. 4.7-4.8 for means, Eq. 4.13 for proportions).

Usage

estr_alocacao_custo(n, Nk, sigmak = NULL, pk = NULL, ck, ...)

Arguments

n

Total sample size to allocate.
Nk

Numeric vector with the population size of each stratum.
sigmak

Numeric vector with the population standard deviation of each stratum (for a mean problem). Ignored if pk is supplied.
pk

Numeric vector with the population proportion of each stratum (for a proportion problem); if supplied, sigmak is derived as sqrt(pk*(1-pk)) and any sigmak argument is ignored.
ck

Numeric vector with the sampling cost per unit in each stratum.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the per-stratum sample sizes under cost-based optimum allocation.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 5.

Examples

estr_alocacao_custo(n = 491, Nk = c(4000, 3000, 2000), sigmak = c(3, 3, 2),
                     ck = c(10, 10, 20))

Neyman (optimum) allocation of a stratified sample

Description

Allocates a total sample size across strata to minimise variance for a fixed total sample size, assuming equal per-unit cost across strata (Eqs. 4.18-4.20 of the course notes). Works for both mean (sigmak known) and proportion (pk known, substituting sigmak = sqrt(pk*(1-pk))) problems.

Usage

estr_alocacao_neyman(n, Nk, sigmak = NULL, pk = NULL, ...)

Arguments

n

Total sample size to allocate.
Nk

Numeric vector with the population size of each stratum.
sigmak

Numeric vector with the population standard deviation of each stratum (for a mean problem). Ignored if pk is supplied.
pk

Numeric vector with the population proportion of each stratum (for a proportion problem); if supplied, sigmak is derived as sqrt(pk*(1-pk)) and any sigmak argument is ignored.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the per-stratum sample sizes under Neyman allocation.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 5.

Examples

estr_alocacao_neyman(n = 261, Nk = c(1540, 770, 385), sigmak = c(2.6, 3, 3.3))

Proportional allocation of a stratified sample

Description

Allocates a total sample size across strata proportionally to stratum size, nk = n * Wk (Eq. 4.6, or Eq. 4.14-4.15 for proportions).

Usage

estr_alocacao_proporcional(n, Nk, ...)

Arguments

n

Total sample size to allocate.
Nk

Numeric vector with the population size of each stratum.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the per-stratum sample sizes.

Examples

estr_alocacao_proporcional(n = 120, Nk = c(1540, 770, 385))

Estimate the population mean under stratified sampling

Description

Estimates the population mean from data collected under stratified random sampling, given per-stratum sample means and sizes (Eqs. 4.1-4.4 of the course notes).

Usage

estr_estima_media(Nk, xk_bar, sk2, nk, conf = 0.95, ...)

Arguments

Nk

Numeric vector with the population size of each stratum.
xk_bar

Numeric vector with the sample mean of each stratum.
sk2

Numeric vector with the sample variance of each stratum.
nk

Numeric vector with the sample size of each stratum.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated stratified mean, its variance, coefficient of variation, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 5.

Examples

estr_estima_media(Nk = c(1540, 770, 385), xk_bar = c(120, 118, 125),
                   sk2 = c(7, 9, 11), nk = c(69, 34, 17))

Estimate the population proportion under stratified sampling

Description

Estimates a population proportion from data collected under stratified random sampling, given per-stratum sample proportions and sizes (Eqs. 4.9-4.11 of the course notes).

Usage

estr_estima_proporcao(Nk, pk, nk, conf = 0.95, ...)

Arguments

Nk

Numeric vector with the population size of each stratum.
pk

Numeric vector with the sample proportion of each stratum.
nk

Numeric vector with the sample size of each stratum.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated stratified proportion, its variance, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 5.

Examples

estr_estima_proporcao(Nk = c(1540, 770, 385), pk = c(0.75, 0.50, 0.25),
                       nk = c(161, 81, 40))

Sample size for stratified sampling: automatic formula selection (umbrella)

Description

Convenience "umbrella" function analogous to aas_tamanho_amostra(): accepts every input potentially relevant to a stratified sample-size problem and dispatches to estr_tamanho_media() or estr_tamanho_proporcao() depending on parametro, ignoring whichever inputs are not needed.

Usage

estr_tamanho_amostra(
  parametro = c("media", "proporcao"),
  Nk,
  sigmak2 = NULL,
  pk = NULL,
  d,
  conf = 0.95,
  ck = NULL,
  alocacao = c("proporcional", "neyman", "custo", "nenhuma"),
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

parametro

Either "media" or "proporcao".
Nk

Numeric vector with the population size of each stratum.
sigmak2

Numeric vector with the population variance of each stratum.
pk

Numeric vector with the population proportion of each stratum (used only when parametro = "proporcao").
d

Tolerable margin of error.
conf

Confidence level, default 0.95.
ck

Optional numeric vector with the sampling cost per unit in each stratum. If supplied, the cost-based formula (Eq. 4.8) is used; otherwise the standard formula (Eq. 4.5) is used.
alocacao

How to allocate n across strata once computed: "proporcional" (default), "neyman", "custo" (requires ck), or "nenhuma" (only the total n is returned).
arredondar

Either "cima" (default, always round up – the conventional, conservative practice) or "proximo" (round to the nearest integer, reproducing the course notes' worked examples, which use a 2-decimal critical value and round to the nearest integer).
...

Ignored. Any other variables from a broader problem statement may be passed here without causing an error.

Value

A sampling_result object, as produced by estr_tamanho_media() or estr_tamanho_proporcao().

Examples

estr_tamanho_amostra(parametro = "media", Nk = c(1540, 770, 385),
                      sigmak2 = c(7, 9, 11), d = 0.5)

Sample size for stratified sampling: mean (proportional / Neyman / cost)

Description

Computes the total sample size and, when alocacao is not "nenhuma", the per-stratum allocation, to estimate a population mean under stratified sampling, dispatching automatically between the proportional/Neyman formula (Eq. 4.5) and the cost-based formula (Eq. 4.8) depending on whether ck is supplied. Extra arguments not needed for the chosen formula are ignored.

Usage

estr_tamanho_media(
  Nk,
  sigmak2,
  d,
  conf = 0.95,
  ck = NULL,
  alocacao = c("proporcional", "neyman", "custo", "nenhuma"),
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

Nk

Numeric vector with the population size of each stratum.
sigmak2

Numeric vector with the population variance of each stratum.
d

Tolerable margin of error.
conf

Confidence level, default 0.95.
ck

Optional numeric vector with the sampling cost per unit in each stratum. If supplied, the cost-based formula (Eq. 4.8) is used; otherwise the standard formula (Eq. 4.5) is used.
alocacao

How to allocate n across strata once computed: "proporcional" (default), "neyman", "custo" (requires ck), or "nenhuma" (only the total n is returned).
arredondar

Either "cima" (default, always round up – the conventional, conservative practice) or "proximo" (round to the nearest integer, reproducing the course notes' worked examples, which use a 2-decimal critical value and round to the nearest integer).
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the total sample size and, unless alocacao = "nenhuma", the per-stratum allocation.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 5.

Examples

estr_tamanho_media(Nk = c(1540, 770, 385), sigmak2 = c(7, 9, 11), d = 0.5, conf = 0.95)

Sample size for stratified sampling: proportion (proportional / Neyman / cost)

Description

Analogous to estr_tamanho_media() but for a population proportion, using Eq. (4.12), with per-stratum proportions substituting for variances (sigmak2 = pk*(1-pk)).

Usage

estr_tamanho_proporcao(
  Nk,
  pk,
  d,
  conf = 0.95,
  ck = NULL,
  alocacao = c("proporcional", "neyman", "custo", "nenhuma"),
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

Nk

Numeric vector with the population size of each stratum.
pk

Numeric vector with the population proportion of each stratum.
d

Tolerable margin of error.
conf

Confidence level, default 0.95.
ck

Optional numeric vector with the sampling cost per unit in each stratum (Eq. 4.13).
alocacao

How to allocate n across strata: "proporcional" (default), "neyman", "custo" (requires ck), or "nenhuma".
arredondar

Either "cima" (default) or "proximo" – see estr_tamanho_media().
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the total sample size and, unless alocacao = "nenhuma", the per-stratum allocation.

Examples

estr_tamanho_proporcao(Nk = c(1540, 770, 385), pk = c(0.75, 0.50, 0.25), d = 0.05)

Population mean

Description

Computes the population mean μ=∑Xi/N\mu = \sum X_i / N (Eq. 2.3) and, when requested, the population variance (Eq. 2.5).

Usage

pop_media(x, ...)

Arguments

x

Numeric vector with the full population values.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the population mean and variance.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

Examples

pop_media(c(44, 45, 48, 42, 46))

Population proportion

Description

Computes the population proportion P=∑Xi/NP = \sum X_i / N for a binary (0/1) characteristic (Eq. 2.4) and its variance P(1−P)P(1-P) (Eq. 2.5.1).

Usage

pop_proporcao(x, ...)

Arguments

x

Numeric or logical vector coded as 0/1 (or FALSE/TRUE).
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the population proportion.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

Examples

pop_proporcao(c(1, 0, 0, 1, 1))

Population total

Description

Computes the population total XT=∑i=1NXiX_T = \sum_{i=1}^{N} X_i (course notes, Eq. 2.2).

Usage

pop_total(x, ...)

Arguments

x

Numeric vector with the full population values.
...

Ignored. Present so that extra, unrelated arguments can be passed without causing an error (see package overview).

Value

A sampling_result object with the population total.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

Examples

pop_total(c(1, 2, 3, 4, 5))

Print method for sampling_result objects

Description

Print method for sampling_result objects

Usage

## S3 method for class 'sampling_result'
print(x, ...)

Arguments

x

A sampling_result object.
...

Further arguments (unused, kept for S3 consistency).

Value

x, invisibly.

Ratio estimator of the population mean

Description

Estimates the population mean using the ratio estimator (Eq. 7.36-7.38 of the course notes). Unlike razao_estima_R(), this does not require knowing X (the population total of the auxiliary variable) – only N.

Usage

razao_estima_media(
  x = NULL,
  y = NULL,
  n = NULL,
  sum_x = NULL,
  sum_y = NULL,
  sum_x2 = NULL,
  sum_y2 = NULL,
  sum_xy = NULL,
  N,
  X = NULL,
  conf = 0.95,
  ...
)

Arguments

x, y

Optional raw numeric vectors (sample data).
n, sum_x, sum_y, sum_x2, sum_y2, sum_xy

Optional pre-aggregated summary statistics, used when raw data are not available (e.g. a problem statement that only gives totals).
N

Population size.
X

Optional population total of the auxiliary variable; not needed to compute the mean itself (only N is required), but accepted here so that a full problem statement (which usually supplies X) can be passed without error. Ignored if NULL.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated mean, its variance, and confidence interval.

Examples

x <- c(1, 30, 44, 20, 0, 10, 15, 5, 2, 50, 35, 25)
y <- c(2, 35, 50, 27, 1, 15, 17, 7, 0, 53, 35, 30)
razao_estima_media(x = x, y = y, N = 1500)

Ratio estimator of the population ratio R

Description

Estimates the population ratio R = Y/X (Eq. 7.27-7.30 of the course notes) and its variance (Eq. 7.31), from either raw sample data (x, y) or pre-aggregated sums – see .razao_resumo().

Usage

razao_estima_R(
  x = NULL,
  y = NULL,
  n = NULL,
  sum_x = NULL,
  sum_y = NULL,
  sum_x2 = NULL,
  sum_y2 = NULL,
  sum_xy = NULL,
  N,
  X,
  conf = 0.95,
  ...
)

Arguments

x, y

Optional raw numeric vectors (sample data).
n, sum_x, sum_y, sum_x2, sum_y2, sum_xy

Optional pre-aggregated summary statistics, used when raw data are not available (e.g. a problem statement that only gives totals).
N

Population size.
X

Population total of the auxiliary variable (used to derive X_barra = X/N, required by Eq. 7.31).
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated ratio, its variance, and confidence interval.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 6.

Examples

x <- c(1, 30, 44, 20, 0, 10, 15, 5, 2, 50, 35, 25)
y <- c(2, 35, 50, 27, 1, 15, 17, 7, 0, 53, 35, 30)
razao_estima_R(x = x, y = y, N = 1500, X = 15000)

Ratio estimator of the population total

Description

Estimates the population total using the ratio estimator (Eq. 7.33-7.35 of the course notes).

Usage

razao_estima_total(
  x = NULL,
  y = NULL,
  n = NULL,
  sum_x = NULL,
  sum_y = NULL,
  sum_x2 = NULL,
  sum_y2 = NULL,
  sum_xy = NULL,
  N,
  X,
  conf = 0.95,
  ...
)

Arguments

x, y

Optional raw numeric vectors (sample data).
n, sum_x, sum_y, sum_x2, sum_y2, sum_xy

Optional pre-aggregated summary statistics, used when raw data are not available (e.g. a problem statement that only gives totals).
N

Population size.
X

Population total of the auxiliary variable (used to derive X_barra = X/N, required by Eq. 7.31).
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the estimated total, its variance, and confidence interval.

Examples

x <- c(1, 30, 44, 20, 0, 10, 15, 5, 2, 50, 35, 25)
y <- c(2, 35, 50, 27, 1, 15, 17, 7, 0, 53, 35, 30)
razao_estima_total(x = x, y = y, N = 1500, X = 15000)

Sample size to estimate the population mean via the ratio estimator

Description

Analogous to razao_tamanho_R() but for the population mean (Eq. 7.44 of the course notes). Unlike razao_tamanho_R(), this does not require X_barra.

Usage

razao_tamanho_media(
  sr2,
  N,
  d,
  conf = 0.95,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

sr2

Variance of the ratio estimator (population value, or a pilot-sample estimate from razao_variancia_piloto()).
N

Population size.
d

Tolerable margin of error.
conf

Confidence level, default 0.95.
arredondar

Either "cima" (default) or "proximo".
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the required sample size.

Examples

razao_tamanho_media(sr2 = 2.2462, N = 1000, d = 0.75)

Sample size to estimate the population ratio R

Description

Computes the sample size needed to estimate the population ratio R with a given margin of error (Eq. 7.41 of the course notes).

Usage

razao_tamanho_R(
  sr2,
  N,
  X_barra,
  d,
  conf = 0.95,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

sr2

Variance of the ratio estimator (population value, or a pilot-sample estimate from razao_variancia_piloto()).
N

Population size.
X_barra

Population mean of the auxiliary variable (X/N).
d

Tolerable margin of error.
conf

Confidence level, default 0.95.
arredondar

Either "cima" (default) or "proximo" – see aas_tamanho_media().
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the required sample size.

Examples

razao_tamanho_R(sr2 = 7.7853, N = 1500, X_barra = 10, d = 0.05)

Sample size to estimate the population total via the ratio estimator

Description

Analogous to razao_tamanho_R() but for the population total (Eq. 7.43 of the course notes).

Usage

razao_tamanho_total(
  sr2,
  N,
  d,
  conf = 0.95,
  arredondar = c("cima", "proximo"),
  ...
)

Arguments

sr2

Variance of the ratio estimator (population value, or a pilot-sample estimate from razao_variancia_piloto()).
N

Population size.
d

Tolerable margin of error.
conf

Confidence level, default 0.95.
arredondar

Either "cima" (default) or "proximo" – see aas_tamanho_media().
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the required sample size.

Examples

razao_tamanho_total(sr2 = 7.7853, N = 1500, d = 500)

Pilot-sample variance estimate for the ratio estimator

Description

Computes s_r^2, the pilot-sample estimate of the population variance of the ratio estimator (Eq. 7.42 of the course notes), used as an input to razao_tamanho_R(), razao_tamanho_total(), and razao_tamanho_media() when no prior estimate of the variance is available.

Usage

razao_variancia_piloto(
  x = NULL,
  y = NULL,
  n = NULL,
  sum_x = NULL,
  sum_y = NULL,
  sum_x2 = NULL,
  sum_y2 = NULL,
  sum_xy = NULL,
  N = NULL,
  X = NULL,
  ...
)

Arguments

x, y

Optional raw numeric vectors (sample data).
n, sum_x, sum_y, sum_x2, sum_y2, sum_xy

Optional pre-aggregated summary statistics, used when raw data are not available (e.g. a problem statement that only gives totals).
N, X

Ignored (kept for signature symmetry with razao_estima_R() so a full problem statement can be passed without error) – s_r^2 itself does not require them.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with s_r^2.

Examples

x <- c(1, 30, 44, 20, 0, 10, 15, 5, 2, 50, 35, 25)
y <- c(2, 35, 50, 27, 1, 15, 17, 7, 0, 53, 35, 30)
razao_variancia_piloto(x = x, y = y)

Ratio vs. regression: full side-by-side comparison (umbrella function)

Description

Convenience "umbrella" function that fits the regression model, tests whether the line passes through the origin, and computes both the ratio and the regression estimators of the population mean side by side, clearly flagging which one the origin test recommends – following the full workflow of Examples 7.1-7.2 of the course notes. This lets students and instructors compare both estimators even when only one is formally "correct" for the data at hand.

Usage

razao_vs_regressao_decidir(x, y, X_barra, N, conf = 0.95, ...)

Arguments

x

Numeric vector with the auxiliary variable.
y

Numeric vector with the variable of interest.
X_barra

Known population mean of the auxiliary variable (required for the regression estimator; not required for the ratio estimator, whose mean form only needs N).
N

Population size.
conf

Confidence level, default 0.95.
...

Ignored. Any other variables from a broader problem statement may be passed here without causing an error.

Details

The function itself does not write anything to the console; print the returned object (or just type its name) to see a side-by-side summary.

Value

An object of class razao_vs_regressao: a named list with elements teste (from reg_teste_origem()), razao (from razao_estima_media()), and regressao (from reg_estima_media()), plus a recomendado character scalar. It has a print() method.

Examples

x <- c(1, 30, 44, 20, 0, 10, 15, 5, 2, 50, 35, 25)
y <- c(2, 35, 50, 27, 1, 15, 17, 7, 0, 53, 35, 30)
res <- razao_vs_regressao_decidir(x, y, X_barra = 10, N = 1500)
res$recomendado
print(res)

Fit a simple linear regression (ratio/regression estimator groundwork)

Description

Fits the simple linear regression of y on x by least squares (Eqs. 7.9-7.10 of the course notes), and computes the residual variance, the standard error of the intercept, and the t-statistic used to test whether the line passes through the origin (Eqs. 7.18-7.20) – the basis for deciding between the ratio and the regression estimator (see reg_teste_origem()).

Usage

reg_ajustar(x, y, ...)

Arguments

x

Numeric vector with the auxiliary variable.
y

Numeric vector with the variable of interest.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the fitted coefficients, residual variance, and the intercept's standard error and t-statistic.

References

Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley, cap. 7.

Examples

x <- c(39, 43, 21, 64, 57, 47, 28, 75, 34, 52)
y <- c(65, 78, 52, 82, 92, 89, 73, 98, 56, 75)
reg_ajustar(x, y)

Regression estimator of the population mean

Description

Estimates the population mean using the regression estimator (Eq. 7.22-7.25 of the course notes), which uses a known population mean X_barra of an auxiliary variable correlated with y.

Usage

reg_estima_media(x, y, X_barra, N = NULL, conf = 0.95, ...)

Arguments

x

Numeric vector with the auxiliary variable.
y

Numeric vector with the variable of interest.
X_barra

Known population mean of the auxiliary variable.
N

Optional population size, used for the finite-population correction f = n/N. If NULL, f = 0 is assumed.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the regression-estimated mean, its variance, and confidence interval.

Examples

x <- c(39, 43, 21, 64, 57, 47, 28, 75, 34, 52)
y <- c(65, 78, 52, 82, 92, 89, 73, 98, 56, 75)
reg_estima_media(x, y, X_barra = 52, N = 486)

Decide between the ratio and regression estimators

Description

Tests whether the regression line of y on x passes through the origin (H0: a0 = 0), using both the t-statistic (Eq. 7.20) and the confidence interval for a0 (Eq. 7.21), and recommends the ratio estimator if the line passes through the origin (fails to reject H0) or the regression estimator otherwise – following the logic of Examples 7.1-7.2 of the course notes.

Usage

reg_teste_origem(x, y, conf = 0.95, ...)

Arguments

x

Numeric vector with the auxiliary variable.
y

Numeric vector with the variable of interest.
conf

Confidence level, default 0.95.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object stating the recommended estimator, the test statistic, the confidence interval for a0, and both estimators computed side by side for comparison.

Examples

x <- c(1, 30, 44, 20, 0, 10, 15, 5, 2, 50, 35, 25)
y <- c(2, 35, 50, 27, 1, 15, 17, 7, 0, 53, 35, 30)
reg_teste_origem(x, y)

Sampling interval for systematic sampling

Description

Computes the sampling interval k = N/n used to select a systematic sample, following Section 5.1-5.2 of the course notes. When N/n is not an integer, k is first rounded up; if k*n > N, k is then decremented by 1, and the correction constant ⁠k* = N - k*n⁠ is computed (Example 5.2 of the course notes).

Usage

sist_intervalo(N, n, ...)

Arguments

N

Population size.
n

Desired sample size.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the sampling interval (and, when N/n is not an integer, the correction constant ⁠k*⁠ needed by sist_selecionar_amostra()).

Examples

sist_intervalo(N = 2000, n = 80)
sist_intervalo(N = 2000, n = 75)

Select a systematic sample

Description

Selects a systematic sample of size n from a population of size N, handling both the integer-interval case and the non-integer-interval case (with the correction constant ⁠k*⁠), following the selection rules of Section 5.2 of the course notes (Examples 5.1-5.2). In the non-integer case, the first selected element is r; from the second element onward, ⁠k*⁠ is added on top of the regular step k (i.e. element i for i >= 2 equals ⁠(i-1)*k + k*⁠ plus r), exactly as illustrated in Example 5.2.

Usage

sist_selecionar_amostra(N, n, r = NULL, semente = NULL, ...)

Arguments

N

Population size.
n

Desired sample size.
r

Optional random start (an integer between 1 and k, inclusive). If NULL, one is drawn uniformly at random.
semente

Optional integer seed for reproducibility. The user's global random state is saved and restored, so calling this function does not affect subsequent unrelated random draws.
...

Ignored (see package overview on tolerated extra arguments).

Value

A sampling_result object with the selected element positions (1:N).

Examples

sist_selecionar_amostra(N = 2000, n = 80, r = 8)
sist_selecionar_amostra(N = 2000, n = 75, r = 26)

Summary method for sampling_result objects

Description

Prints the print method output plus the full step-by-step calculation trace and bibliographic references.

Usage

## S3 method for class 'sampling_result'
summary(object, ...)

Arguments

object

A sampling_result object.
...

Further arguments (unused, kept for S3 consistency).

Value

object, invisibly.