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Let's say I have a point $P_1$ on a 2D scatter plot. The coordinates of $P_1$ are $(x_1, y_1)$. $x_1$ has a standard deviation of $\pm u_1$ and $y_1$ has a standard deviation of $\pm v_1$.

I also have a point $P_2$. The coordinates of $P_2$ are $(x_2, y_2)$, with standard deviations of $\pm u_2$ and $\pm v_2$ respectively.

How can I decide if $P_1$ and $P_2$ are different in a statistically significant manner?

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  • $\begingroup$ A P-value is computed given a null distribution for the data. What is the null distribution in this case? $\endgroup$ May 3, 2014 at 23:54
  • $\begingroup$ Is this all of your information or are P1 and P2 actually means of some sample of points? $\endgroup$
    – John
    May 4, 2014 at 1:43
  • $\begingroup$ @John They are means of multiple samples. $\endgroup$
    – Superbest
    May 4, 2014 at 2:09
  • $\begingroup$ Are the x&y independent? $\endgroup$
    – Glen_b
    May 4, 2014 at 8:51
  • $\begingroup$ @Glen_b Yes (filler) $\endgroup$
    – Superbest
    Feb 1, 2018 at 20:54

2 Answers 2

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You have two continuous variables $Y_1, Y_2$, and one nominal grouping variable $X$ and you wish to check if groups defined by X are statistically different in terms of $Y_1, Y_2$. If you have the real data then if $Y_1, Y_2$ are not highly correlated (Pearson r > 0.9) and similar in their meaning then MANOVA would be an option as this method will give you statistical result on difference of mean of Y1, Y2 among groups defined by X. If instead $Y_1, Y_2$ are uncorrelated (r < 0.4) then Linear Discriminant Analysis would be an option, giving you statistical result on the discriminative power of $Y_1, Y_2$ on groups defined by X. If X defines just 2 groups then instead of LDA you may consider Logistic Regression with X as dependent and $Y_1, Y_2$ as independent.

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The question is interesting.

We have assumptions and basic statistics for any test of difference between two means, or two statistics.

How can I decide if P1 and P2 are different in a statistical significant manner?

The answer is, given such statistics, you can not have any conclusion.

I will use counterexample to illustrate.

You have

$P_1=(X_1,Y_1)$, $S^2_1=((S^2_{X1X1},S^2_{X1Y1})'(S^2_{Y1X1},S^2_{Y1Y1})')$

$P_2=(X_2,Y_2)$, $S^2_2=((S^2_{X2X2},S^2_{X2Y2})'(S^2_{Y2X2},S^2_{Y2Y2})')$

You have no information regarding the distribution of both $P_1$ and $P_2$.

If $P_1$ is the mean value from i.i.d. samples, we can use t-distribution as approximation. The same for $P_2$.

If we have no information for the distribution, we can always find distribution $f(P_1)$ that $P_2$ is beyond the $95\%$ CI. And at the same time we can always find another distribution $g(P_1)$ that $P_2$ is within the $95\%$ CI. Thus the same for $P_2$

Thus we can conclude neither $H_0:P_1=P_2$ nor $H_1:P_1\neq P_2$.

Since you always have $f_1(P_1)$ with $f_2(P_2)$ to accept $H_1$.

And you always have $g_1(P_1)$ with $g_2(P_2)$ to accept $H_0$.

Without further statistics, your statement is inconclusive.

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