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What is the difference between having something statistically significant (such as a difference between two samples) and stating if a group of numbers are independent or dependent.

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Significance in an independent-samples t test just means that the probability (if the null were true) of sampling a mean difference as extreme as the mean difference you actually sampled is less than .05.

This is totally unrelated to dependent/independent. "Dependent" means the distribution of some individual observations is connected to the distribution of others, for example A) they are the same person taking the same test a second time, B) people in each group are matched on some pre-test variable, C) people in the two groups are related (i.e. family). "Independent" means there is no such connection.

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    $\begingroup$ Noting also that p=0.05 is a somewhat arbitrary threshold. If you think that 1:20 is too high a chance of a false positive, then your p should be lower. $\endgroup$
    – naught101
    Mar 24 '12 at 9:30
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Why stop at $t$-tests?

You can think of two variables being uncorrelated as two orthogonal vectors, exactly like the $x$ and $y$ axes in a two dimensional Cartesian coordinate system.

When either of two vectors, let's say $\mathbf{x}$ and $\mathbf{y}$ is correlated with the other, there will be a certain part of x that can be projected onto y and vice versa. With that in mind, it's fairly easy to see that since,

$$ \begin{align*} \left<\mathbf{x},\mathbf{y}\right>&=\|x\|\|y\|\cos\left(\theta\right)\\ \frac{\left<\mathbf{x},\mathbf{y}\right>}{\|x\|\|y\|}&=\cos\left(\theta\right)=r \end{align*} $$

Where $r$ is Pearson's correlation coefficient and $\left<\cdot,\cdot\right>$ is the inner product of the arguments. When I learned this I was totally blown away by how geometrically simple the idea of correlation is. And this is definitely not the only way to measure the correlation between two (or more) variables.

Significance testing is a different ball game. Often we want to know by how much two (or more) groups differ on some outcome variable as a result of some manipulation that was performed on said groups. Like Brian said, you want to know if the two groups come from the same distribution, thus you compute the probability of sampling the mean difference (scaled by the standard error of the mean) that you obtained from your experiment, given that the null hypothesis (there's no significant difference in the means) is true. In behavioral research (and often elsewhere) if this probability is less 0.05, you can conclude that the difference in the two (or more) means is likely due to your manipulation.

EDIT: Dilip Sarwate pointed out that two uncorrelated variables can be statistically dependent, so I took out the first part. Thanks for that.

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    $\begingroup$ Wow, my maths background is far more advanced than my stats background. I find that a really intuitive way of understanding Pearson's r. This answer is really helpful, thanks! $\endgroup$
    – naught101
    Mar 24 '12 at 9:40
  • $\begingroup$ Especially the concept that covariance is just an inner product! $\endgroup$
    – naught101
    Mar 24 '12 at 10:34
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    $\begingroup$ -1 for "You can think of two variables being independent (also called uncorrelated sometimes)" Independence is not the same as being uncorrelated; uncorrelated random variables can be very much dependent. $\endgroup$ Mar 24 '12 at 13:56
  • $\begingroup$ OK, Thanks for fixing the problem. I am reversing my down vote. $\endgroup$ Mar 25 '12 at 2:43

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