DanielBiostatistics10th (version 0.2.6)

Gosset_Welch: Student's and Welch–Satterthwaite Equation

Description

To determine the degree of freedom, as well as the standard error, of two-sample \(t\)-statistic, with or without the equal-variance assumption.

Usage

Gosset_Welch(
  s1,
  s0,
  c1 = 1,
  c0 = 1,
  v1 = s1^2,
  v0 = s0^2,
  n1,
  n0,
  var.equal = FALSE
)

Value

Function Gosset_Welch returns a numeric scalar or vector

of the degree of freedom, with attributes,

attr(., 'stderr')

numeric scalar or vector, standard error \(s_{\bar\Delta}\);

attr(., 'stderr2')

numeric scalar or vector, standard error squared \(s^2_{\bar\Delta}\), included for downstream compute-intensive functions.

Arguments

s1, s0

(optional) double scalars or vectors, sample standard deviations \(s_1\) and \(s_0\) of the treatment and control sample, respectively

c1, c0

double scalars or vectors, multipliers \(c_1\) and \(c_0\) of the treatment and control sample, respectively, to test the hypothesis \(H_0: c_1\mu_1 - c_0\mu_0 = 0\). Default \(c_1=c_0=1\)

v1, v0

double scalars or vectors, sample variances of the treatment and control sample, respectively. Default \(v_1=s_1^2\), \(v_0=s_0^2\).

n1, n0

integer scalars or vectors, sample sizes \(n_1\) and \(n_0\) of the treatment and control sample, respectively

var.equal

logical scalar, whether to assume \(v_1=v_0\), default FALSE (as in function t.test.default).

Details

If \(v_1=v_0\) is assumed, the standard error of two-sample \(t\)-statistic from William Sealy Gosset (a.k.a., Student) satisfies that \(s^2_{\bar\Delta}=s^2_p(1/n_1+1/n_0)\), with degree-of-freedom \(\text{df} = n_1+n_0-2\), where \(s_p\) is the pooled standard deviation, $$s^2_p=\dfrac{(n_1-1)v_1+(n_0-1)v_0}{n_1+n_0-2}$$

If \(v_1\neq v_0\), the standard error of two-sample \(t\)-statistic satisfies that \(s^2_{\bar\Delta}=v_1/n_1 + v_0/n_0\), with degree of freedom (Welch–Satterthwaite equation), $$\text{df} = \dfrac{\left(\dfrac{v_1}{n_1}+\dfrac{v_0}{n_0}\right)^2}{\dfrac{(v_1/n_1)^2}{n_1-1}+\dfrac{(v_0/n_0)^2}{n_0-1}}$$

References

Student's \(t\)-test by William Sealy Gosset, tools:::Rd_expr_doi("10.1093/biomet/6.1.1").

Welch–Satterthwaite equation by Bernard Lewis Welch tools:::Rd_expr_doi("10.1093/biomet/34.1-2.28") and F. E. Satterthwaite tools:::Rd_expr_doi("10.2307/3002019").

Examples

x = rnorm(32L, sd = 1.6); y = rnorm(57L, sd = 2.1)
vx = var(x); vy = var(y); nx = length(x); ny = length(y)
t.test(x, y, var.equal = FALSE)[c('parameter', 'stderr')]
Gosset_Welch(v1 = vx, v0 = vy, n1 = nx, n0 = ny, var.equal = FALSE)
t.test(x, y, var.equal = TRUE)[c('parameter', 'stderr')]
Gosset_Welch(v1 = vx, v0 = vy, n1 = nx, n0 = ny, var.equal = TRUE)