Ahmad, I think the "acceptability" of the magnitude of r (together with a test result that is significant) will depend on the kind of variables you are correlating. For example a significant relationship of 0.2 between the volume of substance A and the number of infant mortality will not have the same acceptability as the a significant value of 0.2 between the volume of substance A and weight gain, considering the critical difference between mortality and weight gain.

  As a rule of thumb, for absolute value of r:

0.00-0.19: very weak

0.20-0.39: weak

0.40-0.59: moderate 

0.60-0.79: strong

0.80-1.00: very strong.

You should keep in mind that these classes are arbitrary and they may differ for different context.

Statistical significance of r value depends on the number of observation. 

http://faculty.quinnipiac.edu/libarts/polsci/Statistics.html

http://faculty.washington.edu/ddbrewer/s231/s231regr.htm

Hey Ahmad,

there is a lot of information about effect sizes in this short book:

https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CEcQFjAE&url=https%3A%2F%2Fbitbucket.org%2Funikum%2Fbibliography%2Fdownloads%2FEllis.pdf&ei=rZnAVMPZIoTqOMuxgYgO&usg=AFQjCNFKwNTMML16kwNkYuVyBc51QcmFTA&bvm=bv.83829542,d.ZWU&cad=rja

Thank you very much Hormoz,

In the context of relationship between cognitive test (ie. grammar test) and psycho-motor test (ie. speaking test), what is the best classes (scales) to be use?

Thank you very much Timo for the very useful book,

I think the effect size of correlation coefficient is the same as the r value, isn't it?

While effect size of ANOVA is different from F value.

Yes, pearson's r = correlation coefficient = r.

You are welcome Ahmad.

The F Value is a measure of error used in evaluating the choice to accept or reject the null hypothesis and included in reporting the results of the research. The larger the F Value, the smaller the potential for error calculated from the data. Running a G-Power analysis will give the critical F value for accepting or rejecting the null hypothesis based upon the a priori choice of a particular effect size.

The ANOVA represents the relationship of correlation between means two means. The effect size table gives the rule of thumb for interpreting the statistical value and using in reporting whether the relationship is weak, moderate, or strong.

Kind regards,

Janet

http://gpower.hhu.de/

http://www.edanzediting.com/blog/statistics_anova_explained#.VMENnSdl_go

Thank you very much Janet.

The correlation coefficient is an effect size, which can not in and of itself prove the existence between two tests.  The statistical significance, or p-value, is what you will be wanted to look at.  The generally accepted p-value to prove the existence between two variables is .05 or lower.

Are the tests which you are testing the same test, and are you using the same sample group or different groups for each test?  This is important as using the same test on the same group is a test-retest and the individuals retaking the test most likely need to be correlated separate from the individuals taking the test the first time.

Thank you very much Joshua.

Ahmad, I think the "acceptability" of the magnitude of r (together with a test result that is significant) will depend on the kind of variables you are correlating. For example a significant relationship of 0.2 between the volume of substance A and the number of infant mortality will not have the same acceptability as the a significant value of 0.2 between the volume of substance A and weight gain, considering the critical difference between mortality and weight gain.

Ahmad, I’m totally agree with Eddie Seva response, you must have a clear idea about what coefficient of correlation could be of interest depending on the variables and the practical interpretation. I suggest that is important that previously (in the design of the research) you must to define this and then you will not have problem to interpret the results; and something more, the null hypothesis of the significance test can be changed depending on the coefficient that you want to prove, most of the statistical software tests the null hypothesis that your coefficient is equal to cero, but you can change this hypothesis to test that your coefficient is equal to a given value (Rhoo):

Traditional test:                                                 Ho: Rho = 0.0

On the basis of an expected Rho (r) value:      Ho: Rho = Rhoo

In this point you can perform a one tailed t-test depending the direction (sign) of the expected correlation.

Good luck.

Thank you very much Eddie

Thank you very much Federico for your informative answer.

Could you suggest me some paper or research that implement what have you described in your answer?

Especially educational or behavioral research.

Ahmad: I'm sorry, the hypothesis test for a given r value is not a t-test, is Fisher's Z transformation, and I let you some links of it:

For concepts:

http://pages.uoregon.edu/stevensj/MRA/correlat.pdf

Online calculator to transform r to Z (Fisher's Z transformation):

http://onlinestatbook.com/calculators/fisher_z.html

Online correlation calculus (including test hypothesis for any expected Rho):

http://home.ubalt.edu/ntsbarsh/business-stat/otherapplets/correlation.htm

A personal comentary: In about twenty five years to be statistical teacher and consultant, I never found this application in scientific papers or journals (it must have, but most of researchers prefer the traditional hypothesis test or apply some a priori criteria).  I will dedicate some time to do a literature review about that point, because your question is valid and aroused in me some curiosity about that.

Thank you very much.. The information is very useful for me. I appreciate it.

Dear colleagues:

In order to contribute with some evidence about the subjective and discretionary criteria to evaluate the Pearson’s correlation coefficient, I did a random search of papers that used this statistical technique (n=30 for convenience, maybe later I will dedicate more time to increase the sample size).

Of the journals cited, 20 of them have impact factor between 1.21 to 54.42 and the rest hadn't (I did this mini-research in Scholar Google, but maybe I will continue in more confinable databases).

The results are attached.

Something that is important to appoint is that the t-test (Ho: Rho = 0.0), the most used to evaluate the correlation coefficient, is sensible to the number of samples (n), and when this number increase, it’s possible to detect spurious correlations, or significant p-values when r is very low, like the example attached at last (the underlined is mine).

I let this data for a further discussion.

Best regards

Dear Ahmad, unfortunately the interpretation of correlation values usually disregard completely the influence of sample size, and like in statistical significance, size matters. Regarding correlation In Statistics for business and economics, 5e ( 2003) Newbold, P., Carlson, W. L., & Thorne, B. M. (2003) suggested the the threshold r>=2/sqrt( n) as the limit of r indicating a linear relationship. I suggest you to read the work entitled "Correlation Coefficient Rule of Thumb" by Timothy C. Krehbiel, where you can find the statistical justification for such threshold

First look at the level of significant. If it is less than 0.05, then take a look at the amount of r.

0.2 is considered to be weak.

https://www.dummies.com/education/math/statistics/how-to-interpret-a-correlation-coefficient-r/

Nice quires!

1. The first question is, what is the minimum of correlation coefficient for meaningfulness or significant?

It depends strongly on the degree of freedom (df). For instance, in a two-tailed test ( P=0.05 ) with 22 subjects (df=20), the minimum r for significant is 0.423. But when your subjects increase, for example, 122 subjects (df=120), the r=0.178 is significant. That is, it can be determined without knowing the p value whether it is significant or not.

With increasing the number of subjects (degree of freedom) the require correlation coefficient for significant decrease.

2. You asked about weak, moderate or strong correlation.

As mentioned @Shahnaz_Shahrbanian , usually:

Small= 0.1 to 0.3 and -0.1 to -0.3

Medium= 0.3 to 0.5 and -0.3 to -0.5

Large= 0.5 to 1.0 and -0.5 to -1.0

Hello Ahmed

Really, It is not easy to answer this question shortly. But it may be useful to explain each part of the question as below:

1. If you have two variables, the correlation coefficient is computed and interpreted easily.

2. The care should be paid to the sample size and the assumptions of the measures and tests.

3. Correlation is used to test relationships between quantitative variables or categorical variables. In other words, it’s a measure of how things are related. The study of how variables are correlated is called correlation analysis.

4. A correlation coefficient gives a numerical summary of the degree of association between two variables - e,g, to what degree do high values of one variable go with high values of the other one? Correlation coefficients vary from -1 to +1, with positive values indicating an increasing relationship and negative values indicating a decreasing relationship. A “0” means there is no relationship between the variables at all, while -1 or 1 means that there is a perfect negative or positive correlation (negative or positive correlation here refers to the type of graph the relationship will produce).

5. Spearman's rho is the Pearson correlation coefficient applied to the scores after they have been ranked from the smallest to the largest on the two variables separately. It is used when the basic assumptions of the Pearson correlation coefficient have not been met by the data – that is especially when the scores are markedly asymmetrical (skewed) on a variable.

6.Since correlation coefficients are usually based on samples of data, it is usual to include a statement of the statistical significance of the correlation coefficient. Statistical significance is a statement of the likelihood of obtaining a particular correlation coefficient for a sample of data if there is no correlation (i.e. a correlation of 0.00) in the population from which the sample was drawn. SPSS can give statistical significance as an exact value or as one of the conventional critical significance levels (e.g. 0.05 and 0.01).

7. Mathematically the multiple correlation coefficient could be calculated using multivariate normal distribution.See:

. Richard Johnson and Dearn Wichern (2007). Applied Multivariate Statistical Analysis. Chapter 4, pages: 149-208.

.Joseph Hair, Wuilliam Black, Barry Babin, Rolph Anderson and Ronald Tatham (2006).Multivariate Data Analysis. Chapter 4, pages: 237-243.

Regards,

Zuhair

It’s still remain the question: What is the minimum value of correlation coefficient to prove the existence of accepted relationship between two variables? You can have a Pearson’s correlation coeficent almost zero (r=0.07) but significant (p=0.001), with a large sample size (n=2160). It is significant but is spurious, it doesn’t have any practical application. You must to prove the hypothesis that the correlation coefficient is at least equal at one practical and logical given value, different to zero (Rhoo). Other alternative is to apply some standardized criteria about the strength of the association, like the Evans criteria (very weak, weak, moderate, strong or very strong).

Best regards

A hIgh correlation coefficient just mean that the model that was adopted fits well the data you have. Sometimes this model comes from a physical relationship, sometimes this model is just a mathematical function. Therefore, you can obtain a low correlation coefficient, depending on the quality of your data, for a physical derived model and have a high correlatIon coefficient for a mathematical model you’ve hypotetically conceived. In this sense correlation coefficient is meaningless. You can obtain a high correlation coefficient for completely disconnected variables. But, being straight in the answer of your question, for cartesians, a high correlation coefficient, as close as to the unity, is sought. For a natural/social/economics science student, a correlation coefficient higher than 0.6 is enough.