# Subset pivotality interpretation?

Is interpretation of subset pivotality correct that it is violated when there is corellation between data variables which will undergo hypothesis testing?

The subset pivotality condition states that the distribution of test statistics is the same under any combination of true null hypotheses. That is, the test statistic distribution is invariant to whether all null hypotheses are indeed true ($$H^C_0$$ ) or a partial set of null hypotheses are true.

Let us consider the simulation of three variables for the subsequent hypothesis testing:

$$H_0^1: \mu_1 = 0; H_A^1: \mu_1 \ne 0$$

$$H_0^2: \mu_2 = 0; H_A^2: \mu_2 \ne 0$$

$$H_0^3: \mu_3 = 0; H_A^3: \mu_3 \ne 0$$

The data generation mechanism is below, namely it is generation of 1000 values from normal distribution from popupulations with $$\mu = 0$$ (null) and $$\mu = 100$$ (alternative):

set.seed(223)
x10 <- rnorm(1000) # sample from the Population 1; mu = 0
x20 <- rnorm(1000) # sample from the Population 20: mu = 0
x2a <- 100 + rnorm(1000) # sample from the Population 2A: mu = 100
x30_w_x20 <- rnorm(1000) + x20 # Sample from Population 3 correlated with Sample 20
x30_w_x2a <- rnorm(1000) + x2a # Sample from Population 3 correlated with Sample 2A


I am making the series of t-tests:

t.test(x1)$$p.value # p = 0.6071707 t.test(x20)$$p.value
# p = 0.183821
t.test(x2a)$$p.value # p = 0 t.test(x30_w_x20)$$p.value
# p = 0.45501
t.test(x30_w_x2a)\$p.value
# p = 0


And observe that the falsity of $$H^2_0$$ implies the falsity of $$H^3_0$$, hence falsity of one subset implies the falsity of another hence there is no subset pivotality.