Duplication of data and effects on various statistics I recently had a question about the following based on duplication of data:
Say you have $x_1, x_2, x_3, ..., x_n$. The t-statistic calculated from this dataset is $t_1$. Now, assume the data was duplicated 3 times, i.e., $x_1, x_1, x_1, x_1, x_2, x_2, x_2, x_2, x_3, x_3, x_3, x_3, ..., x_n, x_n, x_n, x_n$. (thus, there are 4 copies of each data point). Now, the t-statistic calculated from this duplicated data is $t_2$. What is the relationship between $t_1$ and $t_2$?
I want to quantify the relationship as an inequality. For example, is $t_2/3$ and $t_1$ somehow related? I know that the mean would stay the same, but the standard deviation would decrease. Thus, it must be that $t_2$ is greater.
In addition, what would be the relationship in a much more generalized case where the data is duplicated $m$ times? I want to know if other similar effects can be observed and quantified for z-statistics (z-scores) or other types of test or sample statistics.
 A: Following is what we get based on calculation:
\begin{align}
t_1&=\frac{\bar x - \mu}{s_1/\sqrt n} \\
\end{align}
where, $s_1=\sqrt{\frac{\sum\limits_{i=1}^n (x_i-\bar x)^2}{(n-1)}}$
For $t_2$,
\begin{align}
s_2 &=\sqrt{\frac{\sum\limits_{i=1}^{4n} (x_i-\bar x)^2}{(4n-1)}}  \\
&=\sqrt{\frac{4\sum\limits_{i=1}^{n} (x_i-\bar x)^2}{(4n-1)}} \\
&=s_1 \cdot \sqrt{\frac{4n-4}{4n-1}} \\
&\approx s_1 \tag{for large n}
\end{align}
So, we get:
\begin{align}
t_2&=\frac{\bar x - \mu}{s_2/\sqrt {4n}} \\
&=2t_1\sqrt{\frac{4n-1}{4n-4}} \\
&\approx 2t_1 \tag{for large n}
\end{align}
For duplicating $m$ times, we can use the approximation $t_2=\sqrt m t_1$ as long as $n>>m$.
Here's an R simulation for $m=4$:
y=rnorm(100)
i=seq(from=5, to=100,by=5)
ylist=lapply(i, function(x) {y[1:x]})
ylist_rep=lapply(i, function(x) {rep(y[1:x],4)})
t1=unlist(purrr::map(ylist,function(x) {t.test(x)$statistic}))
t2=unlist(purrr::map(ylist_rep,function(x) {t.test(x)$statistic}))
plot(i,t2/t1,ylim = c(1.9,2.2),xlab = 'sample size')
abline(2,0)


