Package 'samplesize4surveys'

Statistical power for a hyphotesis testing on a double difference of means.

Description

This function computes the power for a (right tail) test of double difference of means

Usage

b4ddm(
  N,
  n,
  mu1,
  mu2,
  mu3,
  mu4,
  sigma1,
  sigma2,
  sigma3,
  sigma4,
  D,
  DEFF = 1,
  conf = 0.95,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The population size.
n	The sample size.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
mu3	The value of the estimated mean of the variable of interes for the third population.
mu4	The value of the estimated mean of the variable of interes for the fourth population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
sigma3	The value of the estimated variance of the variable of interes for the third population.
sigma4	The value of the estimated mean of a variable of interes for the fourth population.
D	The value of the null effect.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the power achieved for an specific sample size.

Details

We note that the power is defined as:

$1-\Phi(Z_{1-\alpha} - \frac{(D - [(\mu_1 - \mu_2) - (\mu_3 - \mu_4)])}{\sqrt{\frac{1}{n}(1-\frac{n}{N})S^2}})$

where

$S^2 = DEFF (\sigma_1^2 + \sigma_2^2 + \sigma_3^2 + \sigma_4^2$

Value

The power of the test.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

b4ddm(N = 100000, n = 400, mu1=50, mu2=55, mu3=50, mu4=55, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, D = 7)
b4ddm(N = 100000, n = 400, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, D = 12, plot = TRUE)
b4ddm(N = 100000, n = 4000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, D = 11, DEFF = 2, conf = 0.99, plot = TRUE)

---

Statistical power for a hyphotesis testing on a difference of proportions

Description

This function computes the power for a (right tail) test of difference of proportions.

Usage

b4ddp(N, n, P1, P2, P3, P4, D, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
P1	The value of the first estimated proportion.
P2	The value of the second estimated proportion.
P3	The value of the third estimated proportion.
P4	The value of the fourth estimated proportion.
D	The value of the null effect.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the power achieved for an specific sample size.

Details

We note that the power is defined as:

$1-\Phi(Z_{1-\alpha} - \frac{(D - [(P_1 - P_2) - (P_3 - P_4)])}{\sqrt{\frac{DEFF}{n}(1-\frac{n}{N})(P_1Q_1+P_2Q_2+P_3Q_3+P_4Q_4)}})$

Value

The power of the test.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

b4ddp(N = 10000, n = 400, P1 = 0.5, P2 = 0.5, P3 = 0.5, P4 = 0.5, D = 0.03)
b4ddp(N = 10000, n = 400, P1 = 0.5, P2 = 0.5, P3 = 0.5, P4 = 0.5, D = 0.03, plot = TRUE)
b4ddp(N = 10000, n = 4000, P1 = 0.5, P2 = 0.5, P3 = 0.5, P4 = 0.5, 
D = 0.05, DEFF = 2, conf = 0.99, plot = TRUE)

---

Statistical power for a hyphotesis testing on a difference of means.

Description

This function computes the power for a (right tail) test of difference of means

Usage

b4dm(N, n, mu1, mu2, sigma1, sigma2, D, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
D	The value of the null effect.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the power achieved for an specific sample size.

Details

We note that the power is defined as:

$1-\Phi(Z_{1-\alpha} - \frac{ (D - (\mu_1 - \mu_2))}{\sqrt{\frac{1}{n}(1-\frac{n}{N})S^2}})$

where

$S^2 = DEFF (\sigma_1^2 + \sigma_2^2$

Value

The power of the test.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

b4dm(N = 100000, n = 400, mu1 = 5, mu2 = 5, sigma1 = 10, sigma2 = 15, D = 5)
b4dm(N = 100000, n = 400, mu1 = 5, mu2 = 5, sigma1 = 10, sigma2 = 15, D = 0.03, plot = TRUE)
b4dm(N = 100000, n = 4000, mu1 = 5, mu2 = 5, sigma1 = 10, sigma2 = 15, 
D = 0.05, DEFF = 2, conf = 0.99, plot = TRUE)

---

Statistical power for a hyphotesis testing on a difference of proportions

Description

This function computes the power for a (right tail) test of difference of proportions.

Usage

b4dp(N, n, P1, P2, D, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
P1	The value of the first estimated proportion.
P2	The value of the second estimated proportion.
D	The value of the null effect.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the power achieved for an specific sample size.

Details

We note that the power is defined as:

$1-\Phi(Z_{1-\alpha} - \frac{(D - (P_1 - P_2))}{\sqrt{\frac{DEFF}{n}(1-\frac{n}{N})(P_1Q_1+P_2Q_2)}})$

Value

The power of the test.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

b4dp(N = 100000, n = 400, P1 = 0.5, P2 = 0.5, D = 0.03)
b4dp(N = 100000, n = 400, P1 = 0.5, P2 = 0.5, D = 0.03, plot = TRUE)
b4dp(N = 100000, n = 4000, P1 = 0.5, P2 = 0.5, D = 0.05, DEFF = 2, conf = 0.99, plot = TRUE)

---

Statistical power for a hyphotesis testing on a single mean

Description

This function computes the power for a (right tail) test of means.

Usage

b4m(N, n, mu, sigma, D, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
mu	The value of the estimated mean of the variable of interest.
sigma	The value of the standard deviation of the variable of interest.
D	The value of the null effect. Note that D must be strictly greater than mu.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the power achieved for an specific sample size.

Details

We note that the power is defined as:

$1-\Phi(Z_{1-\alpha} - \frac{(D - \mu)}{\sqrt{\frac{1}{n}(1-\frac{n}{N})S^2}})$

where

$S^2 = DEFF \sigma^2$

Value

The power of the test.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

b4m(N = 100000, n = 400, mu = 3, sigma = 1, D = 3.1)
b4m(N = 100000, n = 400, mu = 5, sigma = 10, D = 7, plot = TRUE)
b4m(N = 100000, n = 400, mu = 50, sigma = 100, D = 100, DEFF = 3.4, conf = 0.99, plot = TRUE)

---

Statistical power for a hyphotesis testing on a single proportion

Description

This function computes the power for a (right tail) test of proportions.

Usage

b4p(N, n, P, D, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
P	The value of the first estimated proportion.
D	The value of the null effect. Note that D must be strictly greater than P.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the power achieved for an specific sample size.

Details

We note that the power is defined as:

$1-\Phi(Z_{1-\alpha} - \frac{(D-P)}{\sqrt{\frac{DEFF}{n}(1-\frac{n}{N})(P (1-P))}})$

Value

The power of the test.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

b4p(N = 100000, n = 400, P = 0.5, D = 0.55)
b4p(N = 100000, n = 400, P = 0.5, D = 0.9, plot = TRUE)
b4p(N = 100000, n = 4000, P = 0.5, D = 0.55, DEFF = 2, conf = 0.99, plot = TRUE)

---

Statistical power for a hyphotesis testing on a single variance

Description

This function computes the power for a (right tail) test of variance

Usage

b4S2(N, n, S2, S20, K = 0, DEFF = 1, conf = 0.95, power = 0.8, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
S2	The value of the first estimated proportion.
S20	The value of the null effect. Note that S2 must be strictly smaller than S2.
K	The excess kurtosis of the variable in the population.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
power	The statistical power. By default power = 0.80.
plot	Optionally plot the power achieved for an specific sample size.

Details

We note that the power is defined as:

$1-\Phi(Z_{1-\alpha} - \frac{(D-P)}{\sqrt{\frac{DEFF}{n}(1-\frac{n}{N})(P (1-P))}})$

Value

The power of the test.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

b4S2(N = 100000, n = 400, S2 = 120, S20 = 100, K = 0, DEFF = 1)
b4S2(N = 100000, n = 400, S2 = 120, S20 = 100, K = 2, DEFF = 1)
b4S2(N = 100000, n = 400, S2 = 120, S20 = 100, K = 2, DEFF = 2.5, plot = TRUE)

---

Some Business Population Database for two periods of time

Description

This data set corresponds to a random sample of BigLucy. It contains some financial variables of 85296 industrial companies of a city in a particular fiscal year.

Usage

BigLucyT0T1

Format

ID

The identifier of the company. It correspond to an alphanumeric sequence (two letters and three digits)

Ubication

The address of the principal office of the company in the city

Level

The industrial companies are discrimitnated according to the Taxes declared. There are small, medium and big companies

Zone

The city is divided by geoghrafical zones. A company is classified in a particular zone according to its address

Income

The total ammount of a company's earnings (or profit) in the previuos fiscal year. It is calculated by taking revenues and adjusting for the cost of doing business

Employees

The total number of persons working for the company in the previuos fiscal year

Taxes

The total ammount of a company's income Tax

SPAM

Indicates if the company uses the Internet and WEBmail options in order to make self-propaganda.

Segments

The cartographic divisions.

Outgoing

Expenses per year.

Years

Age of the company.

ISO

Indicates whether the company is quality-certified.

ISOYears

Indicates the time company has been certified.

CountyP

Indicates wheter the county is participating in the intervention. That is if the county contains companies that have been certified by ISO

Time

Refers to the time of observation.

Author(s)

Hugo Andres Gutierrez Rojas [email protected]

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas.

Examples

data(Lucy)
attach(Lucy)
# The variables of interest are: Income, Employees and Taxes
# This information is stored in a data frame called estima
estima <- data.frame(Income, Employees, Taxes)
# The population totals
colSums(estima)
# Some parameters of interest
table(SPAM,Level)
xtabs(Income ~ Level+SPAM)
# Correlations among characteristics of interest
cor(estima)
# Some useful histograms
hist(Income)
hist(Taxes)
hist(Employees)
# Some useful plots
boxplot(Income ~ Level)
barplot(table(Level))
pie(table(SPAM))

---

Estimated sample Effects of Design (DEFF)

Description

This function returns the estimated design effects for a set of inclusion probabilities and the variables of interest.

Usage

DEFF(y, pik)

Arguments

y	Vector, matrix or data frame containing the recollected information of the variables of interest for every unit in the selected sample.
pik	Vector of inclusion probabilities for each unit in the selected sample.

Details

The design effect is somehow defined to be the ratio between the variance of a complex design and the variance of a simple design. When the design is stratified and the allocation is proportional, this measures reduces to

$DEFF_{Kish} = 1 + CV(w)$

where w is the set of weights (defined as the inverse of the inclusion probabilities) along the sample, and CV refers to the classical coefficient of variation. Although this measure is #' motivated by a stratified sampling design, it is commonly applied to any kind of survey where sampling weight are unequal. On the other hand, the Spencer's DEFF is motivated by the idea that a set of weights may be efficent even when they vary, and is defined by:

$DEFF_{Spencer} = (1 - R^2) * DEFF_{Kish} + \frac{\hat{a}^2}{\hat{\sigma}^2_y} * (DEFF_{Kish} - 1)$

where

$\hat{\sigma}^2_y = \frac{\sum_s w_k (y_k - \bar{y}_w)^2}{\sum_s w_k}$

and $\hat{a}$ is the estimation of the intercept in the following model

$y_k = a + b * p_k + e_k$

with $p_k = \pi_k / n$ is an standardized sampling weight. Finnaly, $R^2$ is the R-squared of this model.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas. Valliant, R, et. al. (2013), Practical tools for Design and Weighting Survey Samples. Springer

Examples

#############################
# Example with BigLucy data #
#############################
data(BigLucy)
attach(BigLucy)

# The sample size
n <- 400
res <- S.piPS(n, Income)
sam <- res[,1]
# The information about the units in the sample is stored in an object called data
data <- BigLucy[sam,]
attach(data)
names(data)
# Pik.s is the inclusion probability of every single unit in the selected sample
pik <- res[,2]
# The variables of interest are: Income, Employees and Taxes
# This information is stored in a data frame called estima
estima <- data.frame(Income, Employees, Taxes)
E.piPS(estima,pik)
DEFF(estima,pik)

---

Statistical errors for the estimation of a double difference of means

Description

This function computes the cofficient of variation and the standard error when estimating a double difference of means under a complex sample design.

Usage

e4ddm(
  N,
  n,
  mu1,
  mu2,
  mu3,
  mu4,
  sigma1,
  sigma2,
  sigma3,
  sigma4,
  DEFF = 1,
  conf = 0.95,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The population size.
n	The sample size.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
mu3	The value of the estimated mean of the variable of interes for the third population.
mu4	The value of the estimated mean of the variable of interes for the fourth population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
sigma3	The value of the estimated variance of the variable of interes for the third population.
sigma4	The value of the estimated mean of a variable of interes for the fourth population.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

We note that the coefficent of variation is defined as:

$cve = \frac{\sqrt{Var((\bar{y}_1 - \bar{y}_2)-(\bar{y}_1 - \bar{y}_2))}}{(\bar{y}_1 - \bar{y}_2)-(\bar{y}_3 - \bar{y}_4)}$

Also, note that the magin of error is defined as:

$\varepsilon = z_{1-\frac{\alpha}{2}}\sqrt{Var((\bar{y}_1 - \bar{y}_2)-(\bar{y}_3 - \bar{y}_4))}$

Value

The coefficient of variation and the margin of error for a predefined sample size.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

e4ddm(N=10000, n=400, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12)
e4ddm(N=10000, n=400, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, plot=TRUE)
e4ddm(N=10000, n=400, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, DEFF=3.45, conf=0.99, plot=TRUE)

---

Statistical errors for the estimation of a double difference of proportions

Description

This function computes the cofficient of variation and the standard error when estimating a double difference of proportions under a complex sample design.

Usage

e4ddp(N, n, P1, P2, P3, P4, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
P1	The value of the first estimated proportion.
P2	The value of the second estimated proportion.
P3	The value of the third estimated proportion.
P4	The value of the fouth estimated proportion.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

We note that the margin of error is defined as:

$cve = \frac{\sqrt{Var((\hat{P}_1 - \hat{P}_2) - (\hat{P}_3 - \hat{P}_4) ) }}{(\hat{P}_1 - \hat{P}_2) - (\hat{P}_3 - \hat{P}_4)}$

Also, note that the magin of error is defined as:

$\varepsilon = z_{1-\frac{\alpha}{2}}\sqrt{Var((\hat{P}_1 - \hat{P}_2) - (\hat{P}_3 - \hat{P}_4) )}$

Value

The coefficient of variation and the margin of error for a predefined sample size.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

e4ddp(N=10000, n=400, P1=0.5, P2=0.6, P3=0.5, P4=0.7)
e4ddp(N=10000, n=400, P1=0.5, P2=0.6, P3=0.5, P4=0.7, plot=TRUE)
e4ddp(N=10000, n=400, P1=0.5, P2=0.6, P3=0.5, P4=0.7, DEFF=3.45, conf=0.99, plot=TRUE)

---

Statistical errors for the estimation of a difference of means

Description

This function computes the cofficient of variation and the standard error when estimating a difference of means under a complex sample design.

Usage

e4dm(N, n, mu1, mu2, sigma1, sigma2, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

We note that the coefficent of variation is defined as:

$cve = \frac{\sqrt{Var(\bar{y}_1 - \bar{y}_2)}}{\bar{y}_1 - \bar{y}_2}$

Also, note that the magin of error is defined as:

$\varepsilon = z_{1-\frac{\alpha}{2}}\sqrt{Var(\bar{y}_1 - \bar{y}_2)}$

Value

The coefficient of variation and the margin of error for a predefined sample size.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

e4dm(N=10000, n=400, mu1 = 100, mu2 = 12, sigma1 = 10, sigma2=8)
e4dm(N=10000, n=400, mu1 = 100, mu2 = 12, sigma1 = 10, sigma2=8, plot=TRUE)
e4dm(N=10000, n=400, mu1 = 100, mu2 = 12, sigma1 = 10, sigma2=8, DEFF=3.45, conf=0.99, plot=TRUE)

---

Statistical errors for the estimation of a difference of proportions

Description

This function computes the cofficient of variation and the standard error when estimating a difference of proportions under a complex sample design.

Usage

e4dp(N, n, P1, P2, DEFF = 1, T = 0, R = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
P1	The value of the first estimated proportion.
P2	The value of the second estimated proportion.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

We note that the margin of error is defined as:

$cve = \frac{\sqrt{Var(\hat{P}_1 - \hat{P}_2) }}{\hat{P}_1 - \hat{P}_2}$

Also, note that the magin of error is defined as:

$\varepsilon = z_{1-\frac{\alpha}{2}}\sqrt{Var(\hat{P}_1 - \hat{P}_2)}$

Value

The coefficient of variation and the margin of error for a predefined sample size.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

e4dp(N=10000, n=400, P1=0.5, P2=0.6)
e4dp(N=10000, n=400, P1=0.5, P2=0.6, plot=TRUE)
e4dp(N=10000, n=400, P1=0.5, P2=0.6, DEFF=3.45, conf=0.99, plot=TRUE)
e4dp(N=10000, n=400, P1=0.5, P2=0.6, T=0.5, R=0.5, DEFF=3.45, conf=0.99, plot=TRUE)

---

Statistical errors for the estimation of a single mean

Description

This function computes the cofficient of variation and the standard error when estimating a single mean under a complex sample design.

Usage

e4m(N, n, mu, sigma, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
mu	The value of the estimated mean of the variable of interest.
sigma	The value of the standard deviation of the variable of interest.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

We note that the coefficent of variation is defined as:

$cve = \frac{\sqrt{Var(\bar{y}_S)}}{\bar{y}_S}$

Also, note that the magin of error is defined as:

$\varepsilon = z_{1-\frac{\alpha}{2}}\sqrt{Var(\bar{y}_S)}$

Value

The coefficient of variation and the margin of error for a predefined sample size.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

e4m(N=10000, n=400, mu = 10, sigma = 10)
e4m(N=10000, n=400, mu = 10, sigma = 10, plot=TRUE)
e4m(N=10000, n=400, mu = 10, sigma = 10, DEFF=3.45, conf=0.99, plot=TRUE)

---

Statistical errors for the estimation of a single proportion

Description

This function computes the cofficient of variation and the standard error when estimating a single proportion under a sample design.

Usage

e4p(N, n, P, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
P	The value of the estimated proportion.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

We note that the coefficent of variation is defined as:

$cve = \frac{\sqrt{Var(\hat{p})}}{\hat{p}}$

Also, note that the magin of error is defined as:

$\varepsilon = z_{1-\frac{\alpha}{2}}\sqrt{Var(\hat{p})}$

Value

The coefficient of variation, the margin of error and the relative margin of error for a predefined sample size.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

e4p(N=10000, n=400, P=0.5)
e4p(N=10000, n=400, P=0.5, plot=TRUE)
e4p(N=10000, n=400, P=0.01, DEFF=3.45, conf=0.99, plot=TRUE)

---

Statistical errors for the estimation of a single variance

Description

This function computes the cofficient of variation and the margin of error when estimating a single variance under a sample design.

Usage

e4S2(N, n, K = 0, DEFF = 1, conf = 0.95, plot = FALSE)

Arguments

N	The population size.
n	The sample size.
K	The excess kurtosis of the variable in the population.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

We note that the coefficient of variation is defined as:

$cve = \frac{\sqrt{Var(\hat{S^2})}}{\hat{S^2}}$

Also, note that the magin of error is defined as:

$\varepsilon = z_{1-\frac{\alpha}{2}}\sqrt{Var(\hat{S^2})}$

Value

The coefficient of variation and the margin of error for a predefined sample size.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

e4S2(N=10000, n=400, K = 0)
e4S2(N=10000, n=400, K = 1, DEFF = 2, conf = 0.99)
e4S2(N=10000, n=400, K = 2, DEFF = 2, conf = 0.99, plot=TRUE)

---

Intraclass Correlation Coefficient

Description

This function computes the intraclass correlation coefficient.

Usage

ICC(y, cl)

Arguments

y	The variable of interest.
cl	The variable indicating the membership of each element to a specific cluster.

Details

The intraclass correlation coefficient is defined as:

$\rho = 1- \frac{m}{m-1} \frac{WSS}{TSS}$

Where $m$ is the average sample sie of units selected inside each sampled cluster.

Value

The total sum of squares (TSS), the between sum of squqres (BSS), the within sum of squares (WSS) and the intraclass correlation coefficient.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4p

Examples

##########################################
# Almost same mean in each cluster       #
#                                        #
# - Heterogeneity within clusters        #
# - Homogeinity between clusters         #
##########################################

# Population size
N <- 100000
# Number of clusters in the population
NI <- 1000
# Number of elements per cluster
N/NI

# The variable of interest
y <- c(1:N)
# The clustering factor
cl <- rep(1:NI, length.out=N)

table(cl)
tapply(y, cl, FUN=mean)
boxplot(y~cl)
rho = ICC(y,cl)$ICC
rho


##########################################
# Very different means per cluster       #
#                                        #
# - Heterogeneity between clusters       #
# - Homogeinity within clusters          #
##########################################

# Population size
N <- 100000
# Number of clusters in the population
NI <- 1000
# Number of elements per cluster
N/NI

# The variable of interest
y <- c(1:N)
# The clustering factor
cl <- kronecker(c(1:NI),rep(1,N/NI))

table(cl)
tapply(y, cl, FUN=mean)
boxplot(y~cl)
rho = ICC(y,cl)$ICC
rho

############################
# Example 1 with Lucy data #
############################

data(Lucy)
attach(Lucy)
N <- nrow(Lucy)
y <- Income
cl <- Zone
ICC(y,cl)

############################
# Example 2 with Lucy data #
############################

data(Lucy)
attach(Lucy)
N <- nrow(Lucy)
y <- as.double(SPAM)
cl <- Zone
ICC(y,cl)

---

Sample Sizes in Two-Stage sampling Designs for Estimating Signle Means

Description

This function computes a grid of possible sample sizes for estimating single means under two-stage sampling designs.

Usage

ss2s4m(N, mu, sigma, conf = 0.95, delta = 0.03, M, to = 20, rho)

Arguments

N	The population size.
mu	The value of the estimated mean of a variable of interest.
sigma	The value of the estimated standard deviation of a variable of interest.
conf	The statistical confidence. By default conf = 0.95. By default conf = 0.95.
delta	The maximun relative margin of error that can be allowed for the estimation.
M	Number of clusters in the population.
to	(integer) maximum number of final units to be selected per cluster. By default to = 20.
rho	The Intraclass Correlation Coefficient.

Details

In two-stage (2S) sampling, the design effect is defined by

$DEFF = 1 + (m-1)\rho$

Where $\rho$ is defined as the intraclass correlation coefficient, m is the average sample size of units selected inside each cluster. The relationship of the full sample size of the two stage design (2S) with the simple random sample (SI) design is given by

$n_{2S} = n_{SI}*DEFF$

Value

This function returns a grid of possible sample sizes. The first column represent the design effect, the second column is the number of clusters to be selected, the third column is the number of units to be selected inside the clusters, and finally, the last column indicates the full sample size induced by this particular strategy.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ICC

Examples

ss2s4m(N=100000, mu=10, sigma=2, conf=0.95, delta=0.03, M=50, rho=0.01)
ss2s4m(N=100000, mu=10, sigma=2, conf=0.95, delta=0.03, M=50, to=40, rho=0.1)
ss2s4m(N=100000, mu=10, sigma=2, conf=0.95, delta=0.03, M=50, to=40, rho=0.2)
ss2s4m(N=100000, mu=10, sigma=2, conf=0.95, delta=0.05, M=50, to=40, rho=0.3)

##########################################
# Almost same mean in each cluster       #
#                                        #
# - Heterogeneity within clusters        #
# - Homogeinity between clusters         #
#                                        #
#  Decision rule:                        #
#    * Select a lot of units per cluster #
#    * Select a few of clusters          #
##########################################

# Population size
N <- 1000000
# Number of clusters in the population
M <- 1000
# Number of elements per cluster
N/M

# The variable of interest
y <- c(1:N)
# The clustering factor
cl <- rep(1:M, length.out=N)

rho = ICC(y,cl)$ICC
rho

ss2s4m(N, mu=mean(y), sigma=sd(y), conf=0.95, delta=0.03, M=M, rho=rho)


##########################################
# Very different means per cluster       #
#                                        #
# - Heterogeneity between clusters       #
# - Homogeinity within clusters          #
#                                        #
#  Decision rule:                        #
#    * Select a few of units per cluster #
#    * Select a lot of clusters          #
##########################################

# Population size
N <- 1000000
# Number of clusters in the population
M <- 1000
# Number of elements per cluster
N/M

# The variable of interest
y <- c(1:N)
# The clustering factor
cl <- kronecker(c(1:M),rep(1,N/M))

rho = ICC(y,cl)$ICC
rho

ss2s4m(N, mu=mean(y), sigma=sd(y), conf=0.95, delta=0.03, M=M, rho=rho)

##########################
# Example with Lucy data #
##########################

data(BigLucy)
attach(BigLucy)
N <- nrow(BigLucy)
P <- prop.table(table(SPAM))[1]
y <- Income
cl <- Segments

rho <- ICC(y,cl)$ICC
M <- length(levels(Segments))

ss2s4m(N, mu=mean(y), sigma=sd(y), conf=0.95, delta=0.03, M=M, rho=rho)

##########################
# Example with Lucy data #
##########################

data(BigLucy)
attach(BigLucy)
N <- nrow(BigLucy)
P <- prop.table(table(SPAM))[1]
y <- Years
cl <- Segments

rho <- ICC(y,cl)$ICC
M <- length(levels(Segments))

ss2s4m(N, mu=mean(y), sigma=sd(y), conf=0.95, delta=0.03, M=M, rho=rho)

---

Sample Sizes in Two-Stage sampling Designs for Estimating Signle Proportions

Description

This function computes a grid of possible sample sizes for estimating single proportions under two-stage sampling designs.

Usage

ss2s4p(N, P, conf = 0.95, delta = 0.03, M, to = 20, rho)

Arguments

N	The population size.
P	The value of the estimated proportion.
conf	The statistical confidence. By default conf = 0.95.
delta	The maximun margin of error that can be allowed for the estimation.
M	Number of clusters in the population.
to	(integer) maximum number of final units to be selected per cluster. By default to = 20.
rho	The Intraclass Correlation Coefficient.

Details

In two-stage (2S) sampling, the design effect is defined by

$DEFF = 1 + (\bar{m}-1)\rho$

Where $\rho$ is defined as the intraclass correlation coefficient, $\bar{m}$ is the average sample size of units selected inside each cluster. The relationship of the full sample size of the two stage design (2S) with the simple random sample (SI) design is given by

$n_{2S} = n_{SI}*DEFF$

Value

This function returns a grid of possible sample sizes. The first column represent the design effect, the second column is the number of clusters to be selected, the third column is the number of units to be selected inside the clusters, and finally, the last column indicates the full sample size induced by this particular strategy.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ICC

Examples

ss2s4p(N=100000, P=0.5, delta=0.05, M=50, rho=0.01)
ss2s4p(N=100000, P=0.5, delta=0.05, M=500, to=40, rho=0.1)
ss2s4p(N=100000, P=0.5, delta=0.03, M=1000, to=100, rho=0.2) 

############################
# Example 2 with Lucy data #
############################

data(BigLucy)
attach(BigLucy)
N <- nrow(BigLucy)
P <- prop.table(table(SPAM))[1]
y <- Domains(SPAM)[, 1]
cl <- Segments

rho <- ICC(y,cl)$ICC
M <- length(levels(Segments))
ss2s4p(N, P, conf=0.95, delta = 0.03, M=M, to=30, rho=rho)

---

The required sample size for estimating a double difference of means

Description

This function returns the minimum sample size required for estimating a double difference of means subjecto to predefined errors.

Usage

ss4ddm(
  N,
  mu1,
  mu2,
  mu3,
  mu4,
  sigma1,
  sigma2,
  sigma3,
  sigma4,
  DEFF = 1,
  conf = 0.95,
  cve = 0.05,
  rme = 0.03,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The maximun population size between the groups (strata) that we want to compare.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
mu3	The value of the estimated mean of the variable of interes for the third population.
mu4	The value of the estimated mean of the variable of interes for the fourth population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
sigma3	The value of the estimated variance of the variable of interes for the third population.
sigma4	The value of the estimated mean of a variable of interes for the fourth population.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95. By default conf = 0.95.
cve	The maximun coeficient of variation that can be allowed for the estimation.
rme	The maximun relative margin of error that can be allowed for the estimation.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

Note that the minimun sample size to achieve a relative margin of error $\varepsilon$ is defined by:

$n = \frac{n_0}{1+\frac{n_0}{N}}$

Where

$n_0=\frac{z^2_{1-\frac{alpha}{2}}S^2}{\varepsilon^2 \mu^2}$

and $S^2=(\sigma_1^2 + \sigma_2^2 + \sigma_3^2 + \sigma_4^2) * (1 - (T * R)) * DEFF$ Also note that the minimun sample size to achieve a coefficient of variation $cve$ is defined by:

$n = \frac{S^2}{|(\bar{y}_1-\bar{y}_2) - (\bar{y}_3-\bar{y}_4) |^2 cve^2 + \frac{S^2}{N}}$

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

e4p

Examples

ss4ddm(N=100000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, cve=0.05, rme=0.03)
ss4ddm(N=100000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, cve=0.05, rme=0.03, plot=TRUE)
ss4ddm(N=100000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, DEFF=3.45, conf=0.99, cve=0.03, 
     rme=0.03, plot=TRUE)

#############################
# Example with BigLucy data #
#############################
data(BigLucyT0T1)
attach(BigLucyT0T1)

BigLucyT0 <- BigLucyT0T1[Time == 0,]
BigLucyT1 <- BigLucyT0T1[Time == 1,]
N1 <- table(BigLucyT0$ISO)[1]
N2 <- table(BigLucyT0$ISO)[2]
N <- max(N1,N2)

BigLucyT0.yes <- subset(BigLucyT0, ISO == "yes")
BigLucyT0.no <- subset(BigLucyT0, ISO == "no")
BigLucyT1.yes <- subset(BigLucyT1, ISO == "yes")
BigLucyT1.no <- subset(BigLucyT1, ISO == "no")
mu1 <- mean(BigLucyT0.yes$Income)
mu2 <- mean(BigLucyT0.no$Income)
mu3 <- mean(BigLucyT1.yes$Income)
mu4 <- mean(BigLucyT1.no$Income)
sigma1 <- sd(BigLucyT0.yes$Income)
sigma2 <- sd(BigLucyT0.no$Income)
sigma3 <- sd(BigLucyT1.yes$Income)
sigma4 <- sd(BigLucyT1.no$Income)

# The minimum sample size for simple random sampling
ss4ddm(N, mu1, mu2, mu3, mu4, sigma1, sigma2, sigma3, sigma4, 
DEFF=1, conf=0.95, cve=0.001, rme=0.001, plot=TRUE)
# The minimum sample size for a complex sampling design
ss4ddm(N, mu1, mu2, mu3, mu4, sigma1, sigma2, sigma3, sigma4, 
DEFF=3.45, conf=0.99, cve=0.03, rme=0.03, plot=TRUE)

---

The required sample size for testing a null hyphotesis for a double difference of proportions

Description

This function returns the minimum sample size required for testing a null hyphotesis regarding a double difference of proportions.

Usage

ss4ddmH(
  N,
  mu1,
  mu2,
  mu3,
  mu4,
  sigma1,
  sigma2,
  sigma3,
  sigma4,
  D,
  DEFF = 1,
  conf = 0.95,
  power = 0.8,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The maximun population size between the groups (strata) that we want to compare.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
mu3	The value of the estimated mean of the variable of interes for the third population.
mu4	The value of the estimated mean of the variable of interes for the fourth population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
sigma3	The value of the estimated variance of the variable of interes for the third population.
sigma4	The value of the estimated mean of a variable of interes for the fourth population.
D	The minimun effect to test.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
power	The statistical power. By default power = 0.80.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the effect against the sample size.

Details

We assume that it is of interest to test the following set of hyphotesis:

$H_0: (mu_1 - mu_2) - (mu_3 - mu_4) = 0 \ \ \ \ vs. \ \ \ \ H_a: (mu_1 - mu_2) - (mu_3 - mu_4) = D \neq 0$

Note that the minimun sample size, restricted to the predefined power $\beta$ and confidence $1-\alpha$, is defined by:

$n = \frac{S^2}{\frac{D^2}{(z_{1-\alpha} + z_{\beta})^2}+\frac{S^2}{N}}$

where $S^2=(\sigma_1^2 + \sigma_2^2 + \sigma_3^2 + \sigma_4^2) * (1 - (T * R)) * DEFF$

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4pH

Examples

ss4ddmH(N = 100000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, D=3)
ss4ddmH(N = 100000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, D=1, plot=TRUE)
ss4ddmH(N = 100000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, D=0.5, DEFF = 2, plot=TRUE)
ss4ddmH(N = 100000, mu1=50, mu2=55, mu3=50, mu4=65, 
sigma1 = 10, sigma2 = 12, sigma3 = 10, sigma4 = 12, D=0.5, DEFF = 2, conf = 0.99, 
       power = 0.9, plot=TRUE)

#############################
# Example with BigLucy data #
#############################
data(BigLucyT0T1)
attach(BigLucyT0T1)

BigLucyT0 <- BigLucyT0T1[Time == 0,]
BigLucyT1 <- BigLucyT0T1[Time == 1,]
N1 <- table(BigLucyT0$ISO)[1]
N2 <- table(BigLucyT0$ISO)[2]
N <- max(N1,N2)

BigLucyT0.yes <- subset(BigLucyT0, ISO == 'yes')
BigLucyT0.no <- subset(BigLucyT0, ISO == 'no')
BigLucyT1.yes <- subset(BigLucyT1, ISO == 'yes')
BigLucyT1.no <- subset(BigLucyT1, ISO == 'no')
mu1 <- mean(BigLucyT0.yes$Income)
mu2 <- mean(BigLucyT0.no$Income)
mu3 <- mean(BigLucyT1.yes$Income)
mu4 <- mean(BigLucyT1.no$Income)
sigma1 <- sd(BigLucyT0.yes$Income)
sigma2 <- sd(BigLucyT0.no$Income)
sigma3 <- sd(BigLucyT1.yes$Income)
sigma4 <- sd(BigLucyT1.no$Income)

# The minimum sample size for testing 
# H_0: (mu_1 - mu_2) - (mu_3 - mu_4) = 0   vs.   
# H_a: (mu_1 - mu_2) - (mu_3 - mu_4) = D = 3

ss4ddmH(N, mu1, mu2, mu3, mu4, sigma1, sigma2, sigma3, sigma4,
 D = 3, conf = 0.99, power = 0.9, DEFF = 3.45, plot=TRUE)

---

The required sample size for estimating a double difference of proportions

Description

This function returns the minimum sample size required for estimating a double difference of proportion subjecto to predefined errors.

Usage

ss4ddp(
  N,
  P1,
  P2,
  P3,
  P4,
  DEFF = 1,
  conf = 0.95,
  cve = 0.05,
  me = 0.03,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The population size.
P1	The value of the first estimated proportion at first wave.
P2	The value of the second estimated proportion at first wave.
P3	The value of the first estimated proportion at second wave.
P4	The value of the second estimated proportion at second wave.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95. By default conf = 0.95.
cve	The maximun coeficient of variation that can be allowed for the estimation.
me	The maximun margin of error that can be allowed for the estimation.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

Note that the minimun sample size (for each group at each wave) to achieve a particular margin of error $\varepsilon$ is defined by:

$n = \frac{n_0}{1+\frac{n_0}{N}}$

Where

$n_0=\frac{z^2_{1-\frac{\alpha}{2}}S^2}{\varepsilon^2}$

and

$S^2 = (P1 * Q1 + P2 * Q2 + P3 * Q3 + P4 * Q4) * (1 - (T * R)) * DEFF$

Also note that the minimun sample size to achieve a particular coefficient of variation $cve$ is defined by:

$n = \frac{S^2}{(ddp)^2cve^2+\frac{S^2}{N}}$

And $ddp$ is the expected estimate of the double difference of proportions.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4dp

Examples

ss4ddp(N=100000, P1=0.05, P2=0.55, P3= 0.5, P4= 0.6, cve=0.05, me=0.03)
ss4ddp(N=100000, P1=0.05, P2=0.55, P3= 0.5, P4= 0.6, cve=0.05, me=0.03, plot=TRUE)
ss4ddp(N=100000, P1=0.05, P2=0.55, P3= 0.5, P4= 0.6, DEFF=3.45, conf=0.99, 
cve=0.03, me=0.03, plot=TRUE)
ss4ddp(N=100000, P1=0.05, P2=0.55, P3= 0.5, P4= 0.6, DEFF=3.45, conf=0.99,
 cve=0.03, me=0.03, T = 0.5, R = 0.9, plot=TRUE)

#################################
# Example with BigLucyT0T1 data #
#################################
data(BigLucyT0T1)
attach(BigLucyT0T1)

BigLucyT0 <- BigLucyT0T1[Time == 0,]
BigLucyT1 <- BigLucyT0T1[Time == 1,]
N1 <- table(BigLucyT0$SPAM)[1]
N2 <- table(BigLucyT1$SPAM)[1]
N <- max(N1,N2)
P1 <- prop.table(table(BigLucyT0$ISO))[1]
P2 <- prop.table(table(BigLucyT1$ISO))[1]
P3 <- prop.table(table(BigLucyT0$ISO))[2]
P4 <- prop.table(table(BigLucyT1$ISO))[2]
# The minimum sample size for simple random sampling
ss4ddp(N, P1, P2, P3, P4, conf=0.95, cve=0.05, me=0.03, plot=TRUE)
# The minimum sample size for a complex sampling design
ss4ddp(N, P1, P2, P3, P4, T = 0.5, R = 0.5, conf=0.95, cve=0.05, me=0.03, plot=TRUE)

---

The required sample size for testing a null hyphotesis for a double difference of proportions

Description

This function returns the minimum sample size required for testing a null hyphotesis regarding a double difference of proportion.

Usage

ss4ddpH(
  N,
  P1,
  P2,
  P3,
  P4,
  D,
  DEFF = 1,
  conf = 0.95,
  power = 0.8,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The maximun population size between the groups (strata) that we want to compare.
P1	The value of the first estimated proportion.
P2	The value of the second estimated proportion.
P3	The value of the thrid estimated proportion.
P4	The value of the fourth estimated proportion.
D	The minimun effect to test.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
power	The statistical power. By default power = 0.80.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the effect against the sample size.

Details

We assume that it is of interest to test the following set of hyphotesis:

$H_0: (P_1 - P_2) - (P_3 - P_4) = 0 \ \ \ \ vs. \ \ \ \ H_a: (P_1 - P_2) - (P_3 - P_4) = D \neq 0$

Note that the minimun sample size, restricted to the predefined power $\beta$ and confidence $1-\alpha$, is defined by:

$n = \frac{S^2}{\frac{D^2}{(z_{1-\alpha} + z_{\beta})^2}+\frac{S^2}{N}}$

Where $S^2 = (P1 * Q1 + P2 * Q2 + P3 * Q3 + P4 * Q4) * (1 - (T * R)) * DEFF$ and $Q_i=1-P_i$ for $i=1,2, 3, 4$.

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4pH

Examples

ss4ddpH(N = 100000, P1 = 0.5, P2 = 0.5, P3 = 0.5, P4 = 0.5, D=0.03)
ss4ddpH(N = 100000, P1 = 0.5, P2 = 0.5, P3 = 0.5, P4 = 0.5, D=0.03, plot=TRUE)
ss4ddpH(N = 100000, P1 = 0.5, P2 = 0.5, P3 = 0.5, P4 = 0.5, D=0.03, DEFF = 2, plot=TRUE)
ss4ddpH(N = 100000, P1 = 0.5, P2 = 0.5, P3 = 0.5, P4 = 0.5, 
D=0.03, conf = 0.99, power = 0.9, DEFF = 2, plot=TRUE)

#################################
# Example with BigLucyT0T1 data #
#################################
data(BigLucyT0T1)
attach(BigLucyT0T1)

BigLucyT0 <- BigLucyT0T1[Time == 0,]
BigLucyT1 <- BigLucyT0T1[Time == 1,]
N1 <- table(BigLucyT0$SPAM)[1]
N2 <- table(BigLucyT1$SPAM)[1]
N <- max(N1,N2)
P1 <- prop.table(table(BigLucyT0$ISO))[1]
P2 <- prop.table(table(BigLucyT1$ISO))[1]
P3 <- prop.table(table(BigLucyT0$ISO))[2]
P4 <- prop.table(table(BigLucyT1$ISO))[2]
# The minimum sample size for simple random sampling
ss4ddpH(N, P1, P2, P3, P4, D = 0.05, plot=TRUE)
# The minimum sample size for a complex sampling design
ss4ddpH(N, P1, P2, P3, P4, D = 0.05, DEFF = 2, T = 0.5, R = 0.5, conf=0.95, plot=TRUE)

---

The required sample size for estimating a single difference of proportions

Description

This function returns the minimum sample size required for estimating a single proportion subjecto to predefined errors.

Usage

ss4dm(
  N,
  mu1,
  mu2,
  sigma1,
  sigma2,
  DEFF = 1,
  conf = 0.95,
  cve = 0.05,
  rme = 0.03,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The maximun population size between the groups (strata) that we want to compare.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95. By default conf = 0.95.
cve	The maximun coeficient of variation that can be allowed for the estimation.
rme	The maximun relative margin of error that can be allowed for the estimation.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

Note that the minimun sample size to achieve a relative margin of error $\varepsilon$ is defined by:

$n = \frac{n_0}{1+\frac{n_0}{N}}$

Where

$n_0=\frac{z^2_{1-\frac{alpha}{2}}S^2}{\varepsilon^2 (\mu_1 - \mu_2)^2}$

and $S^2=(\sigma_1^2 + \sigma_2^2) * (1 - (T * R)) * DEFF$ Also note that the minimun sample size to achieve a coefficient of variation $cve$ is defined by:

$n = \frac{S^2}{|\bar{y}_1-\bar{y}_2|^2 cve^2 + \frac{S^2}{N}}$

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

e4p

Examples

ss4dm(N=100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, cve=0.05, rme=0.03)
ss4dm(N=100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, cve=0.05, rme=0.03, plot=TRUE)
ss4dm(N=100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, DEFF=3.45, conf=0.99, cve=0.03, 
     rme=0.03, plot=TRUE)

#############################
# Example with BigLucy data #
#############################
data(BigLucy)
attach(BigLucy)

N1 <- table(SPAM)[1]
N2 <- table(SPAM)[2]
N <- max(N1,N2)

BigLucy.yes <- subset(BigLucy, SPAM == 'yes')
BigLucy.no <- subset(BigLucy, SPAM == 'no')
mu1 <- mean(BigLucy.yes$Income)
mu2 <- mean(BigLucy.no$Income)
sigma1 <- sd(BigLucy.yes$Income)
sigma2 <- sd(BigLucy.no$Income)

# The minimum sample size for simple random sampling
ss4dm(N, mu1, mu2, sigma1, sigma2, DEFF=1, conf=0.99, cve=0.03, rme=0.03, plot=TRUE)
# The minimum sample size for a complex sampling design
ss4dm(N, mu1, mu2, sigma1, sigma2, DEFF=3.45, conf=0.99, cve=0.03, rme=0.03, plot=TRUE)

---

The required sample size for testing a null hyphotesis for a single difference of proportions

Description

This function returns the minimum sample size required for testing a null hyphotesis regarding a single difference of proportions.

Usage

ss4dmH(
  N,
  mu1,
  mu2,
  sigma1,
  sigma2,
  D,
  DEFF = 1,
  conf = 0.95,
  power = 0.8,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The maximun population size between the groups (strata) that we want to compare.
mu1	The value of the estimated mean of the variable of interes for the first population.
mu2	The value of the estimated mean of the variable of interes for the second population.
sigma1	The value of the estimated variance of the variable of interes for the first population.
sigma2	The value of the estimated mean of a variable of interes for the second population.
D	The minimun effect to test.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95.
power	The statistical power. By default power = 0.80.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the effect against the sample size.

Details

We assume that it is of interest to test the following set of hyphotesis:

$H_0: mu_1 - mu_2 = 0 \ \ \ \ vs. \ \ \ \ H_a: mu_1 - mu_2 = D \neq 0$

Note that the minimun sample size, restricted to the predefined power $\beta$ and confidence $1-\alpha$, is defined by:

$n = \frac{S^2}{\frac{D^2}{(z_{1-\alpha} + z_{\beta})^2}+\frac{S^2}{N}}$

where $S^2=(\sigma_1^2 + \sigma_2^2) * (1 - (T * R)) * DEFF$

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

ss4pH

Examples

ss4dmH(N = 100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, D=3)
ss4dmH(N = 100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, D=1, plot=TRUE)
ss4dmH(N = 100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, D=0.5, DEFF = 2, plot=TRUE)
ss4dmH(N = 100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, D=0.5, DEFF = 2, conf = 0.99, 
       power = 0.9, plot=TRUE)

#############################
# Example with BigLucy data #
#############################
data(BigLucy)
attach(BigLucy)

N1 <- table(SPAM)[1]
N2 <- table(SPAM)[2]
N <- max(N1,N2)

BigLucy.yes <- subset(BigLucy, SPAM == 'yes')
BigLucy.no <- subset(BigLucy, SPAM == 'no')
mu1 <- mean(BigLucy.yes$Income)
mu2 <- mean(BigLucy.no$Income)
sigma1 <- sd(BigLucy.yes$Income)
sigma2 <- sd(BigLucy.no$Income)

# The minimum sample size for testing 
# H_0: mu_1 - mu_2 = 0   vs.   H_a: mu_1 - mu_2 = D = 3
D = 3
ss4dmH(N, mu1, mu2, sigma1, sigma2, D, DEFF = 2, plot=TRUE)

# The minimum sample size for testing 
# H_0: mu_1 - mu_2 = 0   vs.   H_a: mu_1 - mu_2 = D = 3
D = 3
ss4dmH(N, mu1, mu2, sigma1, sigma2, D, conf = 0.99, power = 0.9, DEFF = 3.45, plot=TRUE)

---

The required sample size for estimating a single difference of proportions

Description

This function returns the minimum sample size required for estimating a single proportion subjecto to predefined errors.

Usage

ss4dp(
  N,
  P1,
  P2,
  DEFF = 1,
  conf = 0.95,
  cve = 0.05,
  me = 0.03,
  T = 0,
  R = 1,
  plot = FALSE
)

Arguments

N	The maximun population size between the groups (strata) that we want to compare.
P1	The value of the first estimated proportion.
P2	The value of the second estimated proportion.
DEFF	The design effect of the sample design. By default DEFF = 1, which corresponds to a simple random sampling design.
conf	The statistical confidence. By default conf = 0.95. By default conf = 0.95.
cve	The maximun coeficient of variation that can be allowed for the estimation.
me	The maximun margin of error that can be allowed for the estimation.
T	The overlap between waves. By default T = 0.
R	The correlation between waves. By default R = 1.
plot	Optionally plot the errors (cve and margin of error) against the sample size.

Details

Note that the minimun sample size to achieve a particular margin of error $\varepsilon$ is defined by:

$n = \frac{n_0}{1+\frac{n_0}{N}}$

Where

$n_0=\frac{z^2_{1-\frac{\alpha}{2}}S^2}{\varepsilon^2}$

and

$S^2 = (P1 * Q1 + P2 * Q2) * (1 - (T * R)) * DEFF$

Also note that the minimun sample size to achieve a particular coefficient of variation $cve$ is defined by:

$n = \frac{S^2}{p^2cve^2+\frac{S^2}{N}}$

Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

References

Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

See Also

e4p

Examples

ss4dp(N=100000, P1=0.5, P2=0.55, cve=0.05, me=0.03)
ss4dp(N=100000, P1=0.5, P2=0.55, cve=0.05, me=0.03, plot=TRUE)
ss4dp(N=100000, P1=0.5, P2=0.55, DEFF=3.45, conf=0.99, cve=0.03, me=0.03, plot=TRUE)
ss4dp(N=100000, P1=0.5, P2=0.55, DEFF=3.45, T=0.5, R=0.5, conf=0.99, cve=0.03, me=0.03, plot=TRUE)

#############################
# Example with BigLucy data #
#############################
data(BigLucy)
attach(BigLucy)

N1 <- table(SPAM)[1]
N2 <- table(SPAM)[2]
N <- max(N1,N2)
P1 <- prop.table(table(SPAM))[1]
P2 <- prop.table(table(SPAM))[2]
# The minimum sample size for simple random sampling
ss4dp(N, P1, P2, DEFF=1, conf=0.99, cve=0.03, me=0.03, plot=TRUE)
# The minimum sample size for a complex sampling design
ss4dp(N, P1, P2, DEFF=3.45, conf=0.99, cve=0.03, me=0.03, plot=TRUE)

---