Hello Sundaram Ramaiyer Karimassery ,

I think a 0.23 correation level has a meaning , we say that a weak relationship exist between the two variables.

I have others remarks :

1- What about the heterogenous effect in your sample ?

2- You deal with a linear correlation, how about a non-lineare correlation?

3-You judged the statistical significance of this relationship based on p-value? how about the bayesian p-value?

4- try to plot the line between the x values and y values , and see the geometric relation between them.

Best wihes

The statistical significance (of your observed data!) is just like a "signal-to-noise ratio" that can be used to judge if the information from the data regarding the parameter (pr property) of interest large enough so that the "signal" is clearly higher than the "noise", indicating that you could at least confidently distinguish a positive from a negative signal using your data.

For r, any sample will give you some value between -1 and +1. A different sample will (very likely) give you a different sample value. You only have that one value, and it happend to be 0.23, say. If you test the data under H0: r=0, the resulting p-value can help you to judge if the data you have is sufficient to at least interpret the sign of this sample value (it' spositive in this example). If the p-value is close to zero, you would reject H0 and thus claim that there is enough information in your data to conclude a positive correlation, and not a negative. If the p-value is not close to zero, you wouldn't reject H0 and thus state that the data you have is not enough to discern a positive from a negative correlation with sufficient confidence. So in this case, the data is not conclusive w.r.t. the sign of r - you woudln't interpret the sign of your sample r.

This is independent of the actual sample value or r. It's only about r > H0 vs. r < H0: is there enough data so be able to discern these cases?

You can have a sample r of 0.8, and if you would take another sample, the other sample values could be -0.1, or +0.03 or whatever. You don't know that, and the test does not tell you that. The large p-value (on H0: r=0) just tells you can not even interpret the sign of this sample value. Not even the sign! Let alone the actual size of that value! If the information from tha sample data is not enough to clearly see the sign, how can you think it is enough to see the (more or less precise) size?

To interpret the size you will need to ask: "how likely is r > +0.5 (or some other value you deem relevant), given the data?".* This is not obtainable from the data alone. You have to make a (prior) judgement about how likely you think that r is in some interval. Only then you can use the information from the data to adjust this judgement (to get a posterior judgement). This is a Bayesian strategy, and this has little (actually nothing) to do with the concept of p-values from significance tests. If it is, on a priori grounds, very unlikely that r > 0.5, a small dataset will not change this judgement much, even though the sample value would be 0.8 or 0.9. If the on a priori grounds p is expected to be very likely > 0.5, than the small sample data with a sample r of 0.8 will confirm that and even slightly shift the posterior judgement more towards higher values.

Again: data alone does not provide the information about the size of r. Significance tests put the data in relation to a fixed hypothesis to see if the information in the data is enough to conclude "at which side of the hypothesis" the "truth" is. If you want an estimate (what size/value has r?) , you would need a Bayesian approach, where you need to put the data in context to some prior knowledge about that size. In large samples, the information from the data is usually overwhelming and the prior is almost irrelevant. But just for small samples, the prior makes a difference, because the information from the data about the size is so little.

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* Note: the test asks the inverse question: "how likely is the data 'more extreme' than the observed data, given that r=0"

I agree.But, the problem is more importance is given to the statistical significance rather than indicating only a weak correlation

... but undo importance shouldn't be placed on statistical significance. For any kind of meaningful interpretation of the results, the size of the effect has to be considered, and the practical implications (cost, and so on) have to be considered.

In this case, maybe calculating a confidence interval around r in each case would be illustrative. In the example of the r of 0.8 for n =10, there is not good evidence that the correlation for the populations from which the samples were drawn is different from 0, looking at either the confidence interval or the p value. (I assume you mean 5 pairs of observations).

x = c(1,2,3,4,5)

y = c(1,3,2,5,4)

cor.test(~ x + y)

### Pearson's product-moment correlation

###

### t = 2.3094, df = 3, p-value = 0.1041

###

### 95 percent confidence interval:

### -0.2796400 0.9861962

###

### cor

### 0.8

Hello,

I add to what my dear professors had written above, in statistics, generally one prefers (and one wishes) to work with large samples (so that they are representative of our population), especially if this population is heterogeneous. therefore, a small sample could give you a high correlation coefficient. but by adding other observations, this coefficient (based on the principles of statistical convergences) converges to its true value.

In the literature, this problem of statistical significance versus practical significance is very popular (especially in the clinical field), and I see that a better study and preliminary analysis of our data before making a sophistical statistical analysis is recommended.

Best Wishes.

The hypothesis is important. In both cases i suspect the null hypothesis is H_0: r=0. Which does not seem appropriate for the majority of tests in applied fields.

The problem is the difference between statistically significant and clinical relevance. In your first example it is not clinical relevent to you, but you have enough information to say that the correlation is statistically significantly different from 0.

In your other example you have a highly clinically relevant estimate but not enough information to state that the estimate is statistically significant from 0.

It all boils down to information and estimates.

If you are in the latter case, you need more observation to state that it also is statistically significant. And I would advice in an applied setting to choose a appropriate null hypothesis.

A correlation of .23 could be very meaningful depending on context. A correlation of 0.8 and n = 10 is not really hugely meaningful because of noise (though it is statistically significant). Remember that the sample statistic is not the population correlation because of sampling error, range restriction (or its inverse), small sample bias, attenuation due to unreliability and so forth.

The 95% CIs here are [0.09426709, 0.35733820] and [0.3432884, 0.9507384]. The main reason the n = 10 correlation is not very trustworthy however is publication or selection bias. In an n = 10 sample only a huge correlation is statistically significant so the chances are that reported correlation is one of the 'lucky' ones that is is larger than typical by chance.

Also r = .23 is about 4 or 5% of the variance in the sample. In many contexts larger effects would be implausible. For instance a personality variable that predicted compliance in taking medication would not plausible account for 50% of variation so is unlikely to be above say .4 or .5 (and indeed 0.2 is more plausible).

I agree with Chellai Fatih 100%.

1) p-values are influenced by sample size for every statistical;

2) correlation does not mean causation and correlation does not take inconsideration the nonlinear relationships.

Therefore, your prior knowledge is the main point to go through every problem.