# The required sample size for testing a null hyphotesis for a double difference of proportions This function returns the minimum sample size required for testing a null hyphotesis regarding a double difference of proportions. ```r 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 $S^2=(\sigma_1^2 + \sigma_2^2 + \sigma_3^2 + \sigma_4^2) * (1 - (T * R)) * DEFF$ ## Examples ```r 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) ``` ## References Gutierrez, H. A. (2009), **Estrategias de muestreo: Diseno de encuestas y estimacion de parametros**. Editorial Universidad Santo Tomas ## See Also `ss4pH` ## Author(s) Hugo Andres Gutierrez Rojas

ss4ddmH function