Relationship between two variables Two quantitative variables: correlation and regression methods Two qualitative variables: contingency table methods One quantitative, one qualitative : two-sample methods already considered… We’ll begin with two quantitative variables, continuous measurement variables usually, X and Y. There are usually two situations giving rise to X and Y: 1.bivariate sampling: assume we select pairs at random from a bivariate population 2.fixed-X sampling: an experiment is performed where the X’s are fixed in advance and we observe the values of Y that correspond to those X’s. In either of these cases we may use the correlation coefficient and regression to look for the association between X and Y. Think of Y as the response and X the explanatory variable (though in the first case above, we may not have an explanatory/response situation…)

we define the population correlation coefficient as the value of  ranges from -1 to +1 and measures the strength of the linear relationship between X and Y. -1 corresponds to perfect negative correlation, +1 to perfect positive correlation. If rho = 0 then there is no linear association between X and Y and we say they are uncorrelated variables.. The usual parametric statistic to test the hypothesis that rho=0 is the Pearson product- moment correlation coefficient, r, given by the formula below: The t-statistic (n-2 df) below can be used to test the null hypothesis that rho=0:

Of course, the parametric assumptions are that the (X,Y)-pairs are from a bivariate normal distribution… We may also consider the slope of the so-called regression line relating Y to X: We estimate the unknown slope and intercept via least-squares (the important formulas are given on pages ) and we use the statistic given in the middle of p.147 to test the null hypothesis that    As with our statistic to test 0 correlation, this statistic has n-2 df and assumes the errors are normally distributed. Let’s use SAS and R to show how they implement the correlation and regression tests… Go over Example on page 149… we’ll do the permutation test next, but first, try SAS and R: –in SAS, the procedures of interest here are PROC PLOT (always look at your data), PROC CORR (to get correlations and to test rho=0), and PROC REG (to get estimates for the slope and intercept and to test hypotheses about them). PROC REG also will give plots… –in R use the lm function (fit linear models) and look at the various components

Let’s try to reproduce the empirical distribution of the correlation coefficient, r, as given in Figure on page 153 – use R (see R#9, page 2). Note the correspondence between the transformed r (Z=r(sqrt(n-1))) percentiles and the standard normal percentiles (see Table on page 152). HW: Read Chapter 5, through page 153. Do problems #1, 3 and 4 on page 189 Then for next time, we’ll begin our discussion of the Spearman’s rank correlation coefficient…