10.8. Measures of Association 63110.7.6.Consider the sequence of local alternatives given by the hypothesesH 0 :β=0versusH 1 n:β=βn=√β^1 n,whereβ 1 >0. Letγ(β) be the power function discussed in Exercise 10.7.5 for an
asymptotic levelαtest based on the test statisticTφ. Using the mean value theorem
to approximateμT(βn), sketch a proof of the limit
lim
n→∞
γ(βn)=1−Φ(zα−cTβ 1 ). (10.7.17)10.8MeasuresofAssociation
In the last section, we discussed the simple linear regression model in which the
random variables,Ys, were the responses or dependent variables, while thexswere
the independent variables and were thought of as fixed. Regression models occur in
several ways. In an experimental design, the values of the independent variables are
prespecified and the responses are observed. Bioassays (dose–response experiments)
are examples. The doses are fixed and the responses are observed. If the experimen-
tal design is performed in a controlled environment (for example, all other variables
are controlled), it may be possible to establish cause and effect betweenxandY.
On the other hand, in observational studies both thexsandYs are observed. In the
regression setting, we are still interested in predictingY in terms ofx, but usually
cause and effect betweenxandY are precluded in such studies (other variables
besidesxmay be changing).
In this section, we focus on observational studies but are interested in the
strength of the association betweenY andx.SobothXandY are treated as
random variables in this section and the underlying distribution of interest is the
bivariate distribution of the pair (X, Y). We assume that this bivariate distribution
is continuous with cdfF(x, y)andpdff(x, y).
Hence, let (X, Y) be a pair of random variables. A natural null model (baseline
model) is that there is no relationship betweenXandY; that is, the null hypothesis
is given byH 0 :XandY are independent. Alternatives, though, depend on which
measure of association is of interest. For example, if we are interested in the cor-
relation betweenXandY, we use the correlation coefficientρ(Section 9.7) as our
measure of the association. A two-sided alternative in this case isH 1 :ρ =0. Re-
call that independence betweenXandYimplies thatρ= 0, but that the converse
is not true. However, the contrapositive is true; that is,ρ = 0 implies thatXand
Y are dependent. So, in rejectingH 0 , we conclude thatXandY are dependent.
Furthermore, the size ofρindicates the strength of the correlation betweenXand
Y.
10.8.1 Kendall’sτ
The first measure of association that we consider in this section is a measure of the
monotonicitybetweenXandY. Monotonicity is an easily understood association
betweenXandY.Let(X 1 ,Y 1 )and(X 2 ,Y 2 ) be independent pairs with the same