Statistical Methods in Medical Research 4ed

where b T ˆ rb S. Equations (20.11) and (20.12) represent two straight lines with
the same intercept, a, on the vertical axis and with slopes in the ratio 1 : r. Hence
the term slope ratio. The intercept a is the expected response at zero dose,
whether of T or of S. The position is shown in Fig. 20.3.
In the analysis of results from a slope-ratio assay the observed responses at
various doses XS of S and XT of T must be fitted by two lines of the form
and
Y ˆ a ‡ bSXS
…20:13†
Y ˆ a ‡ bTXT, …20:14†
which are, like the true regression lines, constrained to pass through the same
intercept on the vertical axis (Fig. 20.3). This is a form of regression analysis not
previously considered in this book. The problem can conveniently be regarded as
one of multiple regression. For each observation we define three variables, y, XS
and XT, of which y is the dependent variable and XS and XT are predictor
variables. For any observation on S, XS is non-zero and XT ˆ 0; for an observation
on T, XS ˆ 0 and XT is non-zero. The assay may include control observations
(so-called blanks) without either S or T; for these, XS ˆ XT ˆ 0. The true
regression equations (20.11) and (20.12) can now be combined into one multiple
regression equation:
E…y† ˆa ‡ b SX S ‡ b TX T, …20:15†
and the estimated regressions (20.13) and (20.14) are combined in the estimated
multiple regression
Response, y
True regression lines
Slope
β T = ρβ S
Dose, X
Y ˆ a ‡ bSXS ‡ bTXT: …20:16†
T
Slope β S
S
Response, y
20.3 Slope-ratio assays 725
Fitted regression lines
Dose, X
Fig. 20.3 Slope-ratio assay. True and fitted regression lines for standard and test preparations.
T
S