Added part C: Use the R code to calculate the Spearman correlation as well. Reca

Question

Added part C: Use the R code to calculate the Spearman correlation as well. Recall that the Spearman correlation is used to for non-linear relationships. Which correlation measure best reflects the strength and direction of the relationship between SRD and DMS, and why?

I have part (a) how can I find part (b) and for (C) the correlations are:

Pearson: 0.968531317479152

Spearman: 0.782142857142857

4.30 Sulfur, the ocean, and the sun. Sulfur in the atmosphere 18 19 20 20 ffects climate by influencing for mation of clouds. The main natural source of sulfur is dimethyl sulfide (DMS) produced by small organisms in the upper layers of the oceans. DMS production is in turn influenced by the amount of energy the upper ocean receives from sunlight. Here are monthly data on solar radiation dose (SRD, in watts per square meter) and surface DMS concentration (in nanomo- 46 60 46 38 lars) for a region in the Mediterrane FUR 14.34 19.72 21.52 22.41 7.65 48.41 1.744 1.062 0.682 1.517 0.736 0.720 09.38 157.79 262.67 268.96 289.23 2.692 5.134 8.038 7.280 8.872 (a) Make a scatterplot that shows how DMS responds to SRD (b) Describe the overall pattern of the data. Find the corre- lation r between DMS and SRD. Because SRD changes with the seasons of the year, the close relationship between SRD and DMS helps explain other seasonal patterns.

Explanation / Answer

a & b) Use the plot() function in R. plot(DMS,SRD) will give desired result. As the plot is given in the question , it can be inferred from the scatterplot that the relationship can be assumed linear.Again it is based on very small sample size so it may the misinterpretation of the linearity because last four points make it look like linear where if you consider the first few points then it is random. When you calculate the Pearson r it is coming to be 0.969. It indicates the relationship is almost perfectly linear. Use cor(x,y) function to calculate correlation.

c) The Pearson correlation evaluates the linear relationship between two continuous variables where Speaman correlation evaluates the monotonic relationship between two continuos or ordinal variables. Actually spearman correlation is the ranked version of the pearson correlation.Use cor(x,y, method="spearman") to calculate spearman correlation. Pearson correlation has value 0.969 where Spearman has 0.78. Both the measure indicate that there is strong linear and monotonic relationship. To choose between them first check the linearity assumption actually holds. The pearson correlation is very good again non-parametric version spearman can be also used because both indicates strong positive relationship.

Here first apply the linear regression and check the normality assumption by shapiro wilk test and normal probability plot and also the residual plot to check the homoscedasticity of the variances of the residuals. If both satiesfied then go for pearson correlation. If not apply some transformation.After that if it does not work go for the Spearman correlation. As most of the hypothesis testing cases we like to go for the assumtion of normality so parametric version correlation that is pearson correlation is helpful where nonparametric version spearman correlation can be used if that does not hold.  

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