Relationship between two variables

1
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
• Well begin with two quantitative variables,
  continuous measurement variables usually, X and
  Y. There are usually two situations giving rise
  to X and Y
• bivariate sampling assume we select pairs at
  random from a bivariate population
• fixed-X sampling an experiment is performed
  where the Xs are fixed in advance and we observe
  the values of Y that correspond to those Xs.
• 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)

2

• we define the population correlation coefficient
  as
• the value of r 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 rho0 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 rho0

3

• 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 146-147) and we use the statistic given
  in the middle of p.147 to test the null
  hypothesis that b1 0. As with our statistic
  to test 0 correlation, this statistic has n-2 df
  and assumes the errors are normally distributed.
• Lets use SAS and R to show how they implement
  the correlation and regression tests Go over
  Example 5.1.1 on page 149 well 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 rho0), 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

4

• Lets try to reproduce the empirical distribution
  of the correlation coefficient, r, as given in
  Figure 5.1.3 on page 153 use R (see R9, page
  2). Note the correspondence between the
  transformed r (Zr(sqrt(n-1))) percentiles and
  the standard normal percentiles (see Table 5.1.3
  on page 152).
• HW Read Chapter 5, through page 153. Do
  problems 1, 3 and 4 on page 189
• Then for next time, well begin our
  discussion of the Spearmans rank correlation
  coefficient