I liked the answer by David E. Drehmer. But I would like to go back to the question of which hypothesis are at work. Using correlation has a basic, often not recognized, assumption: the variables to be correlated must be real. This means that their sample space is the real line for one variable, the plane for two variables, and so on. That means that the variables are free of attaining any positive, null or negative value. Moreover, the sum is the basic operation and the differences are computed using the ordinary subtraction, that is the scale of the variables is assumed absolute. Under these assumptions, the covariance is the inner product of two real vectors, the samples of the two variables, frequently taken as the two columns in the data matrix. The correlation coefficient is this inner product divided by the norms of the two column vectors, so that correlation can be interpreted as the cosinus of the angle they form. Note that, up to now, there is no statistical hypothesis about the distribution (normality, homoscedasticity,...). Consequently, Pearson correlation can be used to explore dependence between two real variables whatever is their joint distribution.

The situation changes when correlation is used in connection to regression analysis where testing hypothesis about the model are going to be use. Then, the previous answers to the initial question hold. However, again the linear regression models require to decide what means "linear" and in which sample space. The automatic, but not always realistic assumption, is that the sample space is real and the variables are real. For instance, residuals are computed as differences between observations and predicted values, i.e. using ordinary subtraction (opposite to sum). In many cases, the variables are not real, for instance, if they are strictly positive and the scale can be considered relative; or they are proportions which cannot attain zero or one and have logistic scale. In these cases, dealing with non real variables, they should be transformed. But transformation should be understood as a way of transforming the variables into real variables so that standard correlation and regression (parametric and non parametric) techniques apply. I am saying that using transformations to maintain assumptions required by some tests is a bad idea; the sensible point is that the transformed variables can be supposed to be real.

Only a last comment on non-parametric tests. Although they allow to test hypothesis under mild or even weak assumptions, their power is low compared with parametric tests, that is they have larger probability of not rejecting false hypothesis. They should be used carefully, as much carefully as the assumptions of parametric tests should also be carefully checked.

if these assumptions are not fulfilled one has to apply non-parametric tests e.g. chi-square in place of correlation. I have mentioned the non parametric tests that can be used in place parametric test when the assumptions are not fulfilled:

1-Sample Sign: Performs a 1-sample sign test of the median and calculates the corresponding point estimate and confidence interval. Nonparametric alternative to 1-sample Z and 1-sample t-tests.

1-Sample Wilcoxon: Performs a 1-sample Wilcoxon signed rank test of the median and calculates the corresponding point estimate and confidence interval. Nonparametric alternative to 1-sample Z and 1-sample t-tests.

Mann-Whitney: Performs a hypothesis test of the equality of two population medians and calculates the corresponding point estimate and confidence interval. Nonparametric alternative to the 2-sample t-test.

Kruskal-Wallis: Performs a hypothesis test of the equality of population medians for a one-way design (two or more populations). Nonparametric alternative to the one-way analysis of variance.

Mood's Median Test: Performs a hypothesis test of the equality of population medians in a one-way design. Mood's median test, like the Kruskal-Wallis test, is a nonparametric alternative to the usual one-way analysis of variance.

Friedman: Performs a nonparametric analysis of a randomized block experiment . Nonparametric alternative to the two-way analysis of variance.

If the relationship is nonlinear then use a transformation to make it linear. The choice that works often is a log transformation. If the transformation works, then all the standard assumptions apply. If the relationship cannot be transformed to be linear then a nonlinear analysis is needed. Nonlinear analysis is a different course, and you probably should consult a statistician.

A good transformation can sometimes be used to solve the other problems. However, nonparametric tests may still be needed.

The real problem: how many replicates do you have in the treatment with the fewest number of replicates? If the answer is 4 or 5, then you should really wonder how accurate a nonlinear or nonparametric analysis is with such a small data set.

If you are thinking about simple linear regression and correlation, then the nonparametric approaches are useful.

These tests do not make assumptions of about the distribution of the data.

For correlation, there is Kendall correlation, and Spearman correlation.

For linear regression, there is Kendall-Theil-Siegel regression, and quantile regression.

The nonparametric correlations do not assume that the data are linearly related. E.g. https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Spearman_fig1.svg/450px-Spearman_fig1.svg.png

Kendall-Theil-Siegel regression fits a best fit line between two variables. It is robust to outliers in the y variable. You can determine the slope and intercept of the line, the p-value, but not anything like an r-squared.

Quantile regression is a very flexible approach, and can be used for a linear regression. It is robust to outliers in the y variable. You can determine the slope and intercept of the line, the p-value, and a pseudo-r-squared value.

Lets assume that you mean continuous interval level data for the two variables, X and Y.  Lets also assume that in a linear regression problem, the regression line follows the means of the conditional distributions of Y|X.  In that case the central limit theorem saves us because the sampling distribution of means will almost always be approximately normal.  It really doesn’t matter much what the skewness, kurtosis, of the sample looks like.  Therefore the issue of normality may be considered a weak assumption for all practical purposes.

The assumption of linearity for well behaved continuous data can usually be achieved by transformation of either dependent or independent variable to create a more or less linear relationship.  If the data is decidedly discontinuous, such as only nominal or ordinal scaling in the dependent variable, then linear or curvilinear regression may not be warranted, and some other non-linear method may be more appropriate. 

Homoscedasticity is either a strong or weak assumption depending on what you are trying to do.  If you just want to estimate the regression line, then homoscedasticity doesn’t matter.  On the other hand, if you want to estimate how good the fit is, i.e., a correlation coefficient or some other proportional reduction in error statistic (PRE), then homoscedasticity is very important.  The assumption of homoscedasticity means that we assume that the variance of the conditional distributions of Y given X, var(Y|X), is the same everywhere.  Then the PRE measure, like R2 can be defined as (var(Y) – var(Y|X)) / var(Y). 

In a  heteroscedastic relationship, the conditional variances are not assumed to be the same, and pooling makes little sense.  In other words var(Y|Xi) depends on which i.  Therefore, the strength of the relationship depends on what level of the independent variable you are looking at.  No single number is likely able to summarize this strength or PRE.  (var(Y) – var(Y|Xi)) / var(Y) cannot be known until a particular i is defined. Think of a curve.  It is a line where the slope is not constant, it changes depending on where one is on the curve.  In many respects, you might think of the relationship strength like a calculus derivative, not a constant.  That is a discussion that is probably beyond the scope of this forum. 

The thing that distinguishes one PRE measure from another is the method used to determine location (center) and the method for assessing variability (loss function).  Check out “proportional reduction in error statistics” in google for many examples and references.

With regard to putative causation, this is a design issue, not a statistical issue. Correlation and other measures of association do not imply causation.  Even structural equation modeling, such as Lisrel, does not prove cause and effect.  It is just a way to construct a model from your theory to determine whether that model could produce the data you have observed.  It can be helpful in disproving you theory, however. 

I hope that you find this helpful in your defining what you need to do. 

Best wishes.

David Drehmer   

I liked the answer by David E. Drehmer. But I would like to go back to the question of which hypothesis are at work. Using correlation has a basic, often not recognized, assumption: the variables to be correlated must be real. This means that their sample space is the real line for one variable, the plane for two variables, and so on. That means that the variables are free of attaining any positive, null or negative value. Moreover, the sum is the basic operation and the differences are computed using the ordinary subtraction, that is the scale of the variables is assumed absolute. Under these assumptions, the covariance is the inner product of two real vectors, the samples of the two variables, frequently taken as the two columns in the data matrix. The correlation coefficient is this inner product divided by the norms of the two column vectors, so that correlation can be interpreted as the cosinus of the angle they form. Note that, up to now, there is no statistical hypothesis about the distribution (normality, homoscedasticity,...). Consequently, Pearson correlation can be used to explore dependence between two real variables whatever is their joint distribution.

The situation changes when correlation is used in connection to regression analysis where testing hypothesis about the model are going to be use. Then, the previous answers to the initial question hold. However, again the linear regression models require to decide what means "linear" and in which sample space. The automatic, but not always realistic assumption, is that the sample space is real and the variables are real. For instance, residuals are computed as differences between observations and predicted values, i.e. using ordinary subtraction (opposite to sum). In many cases, the variables are not real, for instance, if they are strictly positive and the scale can be considered relative; or they are proportions which cannot attain zero or one and have logistic scale. In these cases, dealing with non real variables, they should be transformed. But transformation should be understood as a way of transforming the variables into real variables so that standard correlation and regression (parametric and non parametric) techniques apply. I am saying that using transformations to maintain assumptions required by some tests is a bad idea; the sensible point is that the transformed variables can be supposed to be real.

Only a last comment on non-parametric tests. Although they allow to test hypothesis under mild or even weak assumptions, their power is low compared with parametric tests, that is they have larger probability of not rejecting false hypothesis. They should be used carefully, as much carefully as the assumptions of parametric tests should also be carefully checked.

"If any of the assumptions are not met the method of finding association between two quantitative variables" is you then find the appropriate method.

Statistics researchers have developed dozens of methods that can deal with hundreds of problems in various academic fields, especially with sample size problems and data collection procedures that violate the assumptions of the methods selected to analyze the gathered data.

There is no reason to use an inappropriate method when a viable method exists. 

For example, when statistical assumptions for regression cannot be met (fulfilled by the researcher) pick a different method. Regression requires its dependent variable to be at least least interval or ratio data. If the dependent is not this level of data, the researcher violating this assumption would have spurious results. Instead, the researcher would merely, if the dependent is nominal or ordinal, use logistic regression or multinomial logistic regression as the appropriate method. 

This is why the researcher should read literature in the field, formulate the hypothesis (or hypotheses) first before collecting data. How the hypotheses will be tested is the indication as to how the data should be analyzed and the methods, therefore, that should be used.

Causal findings are more experimental designs, where environmental controls help avoid spurious findings because cause can be uniquely attributed to a treatment variable, and no other explanation is better than the change in variable "A" was a result of its interacting with variable "B."