Other than ANOVA (and maybe Bayesian interactive model, I don't know about those) they're all appropriate, they just show different thing. Linear regression and Pearson correlation are more or less formally equivalent (indeed, in a simple case with only two variables, they should give you the same significance), they just describe the relationship in different ways, either one of which may be appropriate depending on what you're trying to do; pearson correlation describes the strength of the relationship in abstract units, whereas regression predicts one variable based on the other (telling you specifically how much change in Y is expected for a one-unit change in X). Kendall and Spearman correlation are similar to Pearson correlation but appropriate for datasets with different kinds of distributions or different kinds of relationships than what Pearson is appropriate for (if you look up any tutorial on these correlations, it will explain what kind of data they are appropriate for).

Other than ANOVA (and maybe Bayesian interactive model, I don't know about those) they're all appropriate, they just show different thing. Linear regression and Pearson correlation are more or less formally equivalent (indeed, in a simple case with only two variables, they should give you the same significance), they just describe the relationship in different ways, either one of which may be appropriate depending on what you're trying to do; pearson correlation describes the strength of the relationship in abstract units, whereas regression predicts one variable based on the other (telling you specifically how much change in Y is expected for a one-unit change in X). Kendall and Spearman correlation are similar to Pearson correlation but appropriate for datasets with different kinds of distributions or different kinds of relationships than what Pearson is appropriate for (if you look up any tutorial on these correlations, it will explain what kind of data they are appropriate for).

Linear regression and ANOVA are similar. I am not sure if you could apply ANOVA for your case though. Simple correlations are more crude than any other method you listed. PCA seems not related with your aim. I have no idea about Bayesian Interactive Model.

So you had better go with regression.

It depends on

1) the features of your data (different methods are based on different assumptions)

2) the hypotheses you want to test

3) the need for further modelling in order to solve some task (such as prediction based on the temperature)

If you just want to check if there's a dependency or not , I would recommend to use the Spearman's correlation coefficient as it is easy to use and should be based on realistic assumptions about the data. But still, you'll need to take into account some assumptions of this method - in particular, you need to assume that the relationship (if it exists) is monotonic.

More info here:

https://statistics.laerd.com/statistical-guides/spearmans-rank-order-correlation-statistical-guide.php

Correlation provides the relationship between two variables while simple linear regression allows predicting one variable from another. ANOVA, on the other hand, shows significant effects of independent variables over the response, if any. For more details, refer to the WWW.

Hi,

Considering it seems you have an output variable(s) (disease incident data), multiple linear regression might be indicated here. Because it allows you : first to take into account several explanatory variables (temperature and rain records, else...) ; second to test the null hypothesis-(es) for one or several (at the same time) beta-coefficients via t-test and Fischer-test. 

If you just want to present descriptive statistics between your output variable(s) and your explanatory variables, you can perfectly use rho, spearman, kendall coefficients. Moreover, you can also use tests based on these quantities. (see : https://stat.ethz.ch/R-manual/R-patched/library/stats/html/cor.test.html). Note that, on the contrary to MLR tests, these tests are good for one versus one comparisons.

Finally, if you are interested in using PCA to study your problem, I suggest you have a look at its supervised version : see for example Hastie et al. at https://web.stanford.edu/~hastie/Papers/spca_JASA.pdf

I hope this helps ! 

You might consider using rank correlations as well as looking at partial correlations.

You may consider Principal Component Analysis or correlation based analysis.

As suggested by some , for identifying bivariate relation you can use correlation coefficient,  however as this assumes linearity a more generic metric can be mutual information.

Once you go for PCA, you start loosing the original features. So this method should be applied accordingly.

Hi

Answer for your question depends on your interest, scope and also the characteristics of your response variable disease incidence. The methods you list are similar but not equal. for instance ANOVA and regression have similar ideas but in different concepts. In regression you can test the type/direction of relationship(negetive or positive), you can predict the values of response based on predictors, you can check the change on response caused by a unit change on predictors. by Pearson,collerelation you can test the strength and direction of relationship between response and predictor  whereas ANOVA is the extension of two independent sample t-test and you can determine and compare the effect of different categorical predictors on response variable instead checking the relationship between predictors and response. ANOVA means it is Analysis of Variance and as its name indicates you can spilit the variances in two part in ANOVA i.e.; variance due to model and due to errors and you call Mean Square error and Mean square of treatment. Both Regression and ANOVA are only applicable if your response is continuous and error term is normaly distributed with mean zero and constant variance. But if your response variable Disease incidence is Dichothous valued variable you can use Binary logistic regression rather linear regression but if it is also ordenal or nominal valued you can use ordenal or nominal logistic regression in place of linear regression. in place of one way ANOVA you can use the nonparametric analog of it called Kruskal–Wallis test. Spearman's correlation if for testing of degree of agreement of different opinions and it uses ranking method rather original data.Kendal is similar to Spearman's correlation. I have no idea about Iterated baysian.

Mohsen -

If you have weather-related regressor data for predicting disease, that seems very interesting, but I suspect you would also have omitted variable bias in a regression, as it seems hard to believe that weather-related independent variables are all you need. 

It seems you need to concentrate on regression, and the kind depends upon your type of data, best variables to use, and whether or not there may be substantial nonlinearity.   Also, you don't want to overfit or underfit.  Further, heteroscedasticity is often a natural and important part of the error structure as well.

I suggest you consider your subject matter, and perhaps graphical analyses, and from these considerations develop candidate models.  Then you could leave out part of your data for testing, and see which model performs best. 

It may be interesting to see individual regressor variables used to check correlations or regressions, but graphs may show nonlinearity between continuous variables, or other phenomema may be found graphically, and the interaction of regressor variables may be important, so if I understand your situation - and I'm sure I don't understand all of it - I worked mostly with simple one regressor continuous variables - it appears best if you were to concentrate on comparing candidate-model performances. 

Cheers  - Jim

PS -  If you have count data you might want to look at some discussion such as that at the beginning of the document found at the following link, and consider from there where you want to go:

 http://cameron.econ.ucdavis.edu/racd/simplepoisson.pdf