Worksheet Correlation and RegressionSolve the Statistical question proQuestion 2Janssenet al. (2007) studied the relationships between a variety of abiotic factors and benthic invertebrate abundance at sites on beaches along the Dutch coast. One of these abiotic factors was the relative height of the site in relationship to the average sea level of the area (NAP). Positive values of NAP indicate sites that are higher than the average sea level, whereas negative values indicate sites that are below the average sea level.The data are in the file sle251dutch.csv and the relevant variables are the response variable, richness (richness of invertebrate species), and the predictor variable, NAP (relative height of the site in relationship to the average sea level of the area).Format of sle251dutch.csv data fileSite NAP richness1 0.045 112 -1.036 103 -1.336 134 0.616 115 -0.684 10.. .. ..Site The number of the site where the samples were collectedNAP Relative height of the site in relationship to the average sea level of the areaPredictor variablerichness Richness of invertebrate speciesResponse variablea) Janssenet al. (1996) were interested in modeling the linear relationship between invertebrate richness (response) and the relative height of the site in relationship to the average sea level (predictor). List the following:The biological inference of interestThe biological null hypothesis derived from aboveThe statistical null hypothesis (H0) derived from aboveb) Draw a scatterplot of NAP against richness. Draw boxplots for each variable as well. Any evidence of skewness in the distributions or nonlinearity?To create scatterplot in RGraphsScatterplotSelect x-variable (NAP) and y-variable (richness)Check Marginal boxplots and Least-squares lineUnselect Smooth line and show spreadOKc) Fit the regression model richness = intercept + slope x NAP.To fit linear regression and create an ANOVA table in RStatisticsFit modelsLinear regression…You can enter a name for the results object (Enter name for model:) but its simplest to just use the name that R provides.Select richnessfrom Response variablelistSelect NAPfrom Explanatory variables list.OKModelsHypothesis testsANOVA tableSelect Partial, ignoring marginality (“Type III”).OKExamine the regression output and identify and interpret the following:Sample y-interceptValue (estimate in the R output):Interpretation:Slope of regression line (NAP)Value(estimate in the R output):Interpretation:t statistic for main H0 (regression slope equals zero)Value:Interpretation:P-value for main H0 (regression slope equals zero)Value:Interpretation:r2 value (multiple R-squared)Value:Interpretation:d) Complete the following ANOVA table from the regression analysisSource of variation SS df MS F ratioRegressionResidualTotal44Note: To get the MS values from the output – remember to divide the SS value by the df.e) What conclusions would you draw from the regression analysis (statistical and biological)?f) What invertebrate richness would you predict for a new site with an NAP of -2? Simply plug -2 into your regression equation and calculate predicted richness.
