The Isotonic Regression Framework

Categorizing continuous variables arises as an important task in
statistical analysis, especially in analyzing dose-response
relationships. Creating meaningful groups of the predictor
variables regarding the outcome variable is desirable in many
settings, especially if the form of the relationship is unknown.
However it is not always obvious how many groups should be build
and where the cutpoints should be placed. Usually more than one
explanatory variable has to be included in the analysis, and
therefore one has to apply an appropriate statistical model. For
this purpose we need a simple approach to model the data without
many requirements. Another important issue in statistical analysis
and especially in toxicology studies is proving a dose response
relationship: increasing response probability with increasing
predictor variable. This theses deals with cases where
categorization of numerical or categorical predictor variables
results as an effect of the dose-response relationship. Isotonic
regression is an alternative proposal when one wishes to establish
a dose-response relationship, categorize continuous variables and
estimate threshold values. The only assumption for this approach is
the monotonicity in the response variable. The isotonic regression
summarizes the description of n observations to l categories (level
sets or solution blocks) by automatically splitting the predictor
in constant risk groups. The result is always a step function, and
therefore the isotonic regression can be used to fit a changepoint
model. The Pooled Adjacent Violators Algorithm (PAVA) is used to
fit the data. In relation to model fitting and testing, some
problems arise when the response is binary, and in the present work
the difficulties are highlighted and some proposals to solve them
are given. Regarding isotonic regression and binary response, the
isotonic test for trend, the reduced isotonic model,
multidimensional isotonic models and methods to assess threshold
limit values are discussed. The isotonic framework provides a
reliable test for trend which unlike other widely used tests (the
Cochran-Armitage test for example) is independent of any monotonic
transformation of the dose variable and does not assume a linear
shape. However the proposed large sample approximation (a weighted
chi-square distribution) does not hold when the overall response
probability is less than 5\% and thus exact methods are proposed in
order to assess the correct p-value. In a simulation study it has
been shown that the isotonic likelihood ratio test is more powerful
than the Cochran-Armitage test, the Wilcoxon test and the
Iso-chi-squared test. The model resulting from PAVA can become more
parsimonious if the level sets which correspond to a non
significant change for the response variable are eliminated. This
model is called reduced isotonic regression. That can be
accomplished by two means: a sequence of Fisher tests for the
adjacent 2x2 tables or the application of a variation of a "closed
testing" procedure. The correction for multiple comparisons is made
for the first method by an a-priori estimation of the overall
significance level in a permutation procedure. In the second method
the control for the expense of the type I error is effected by the
closure principal. To select between full isotonic and reduced
model, a procedure based on parametric bootstrap is proposed. A
simulation study proved that when the maximal coefficient of
determination for the analyzed data set is at least 50% and the
data can be represented by a step function, the reduced monotonic
regression controls successfully the trade off between model
complexity and goodness of fit. When more than one predictor is to
be taken into account an additive isotonic model can be applied.
Alternatively, an isotonic-surfaces model is proposed. This can be
estimated by an iterative version of the Pooled Adjacent Violators
Algorithm. The result is a sequence of surfaces which is monotonic
in every dimension. This approach models interaction and
categorizes the predictors in "multivariate" groups by combining
them regarding restrictions to the outcome variable. This approach
is very useful since, unlike the additive model, it can be easily
combined with the reducing procedures to give a simple and
interpretable model. However, for practical reasons a maximum of
three predictors can be taken into account. A special aspect in
analyzing dose-response relationships for a compound known to have
harmful health effects, is to estimate a threshold limit value
(TVL). On this regard a "hockey stick" threshold model is usually
used. As alternative the use of a step function model by fitting
the data using isotonic regression is proposed. A set of candidate
threshold values is returned, and some threshold value estimation
procedures are studied here. One of them starts from the isotonic
model and applies the likelihood ratio test to detect the threshold
value (method 1). Method 2 is based on the reduced isotonic
regression. The performance of these two approaches is outlined in
a simulation study under different scenarios and their properties
are explored with categorical predictors. It is concluded that
these methods possess a satisfactory power to reject the constant
risk assumption, when a dose-response relationship exists as well
as to estimate the actual threshold. Some limitations regarding the
sample size and the force of trend are also discussed. A third
method has also been presented. This modifies the closed testing
procedure for the special case of thresholds, by setting one end of
the regression line conditional to the other. All three threshold
value estimation methods can be combined with the isotonic-surfaces
model to provide thresholds, taking into account interactions
between the predictor variables. The use of isotonic regression and
its reduced version can also be extended to other settings. The
capability of isotonic regression to be implemented in several
models is outlined by describing how isotonic regression can model
and test time-varying effects in Cox regression. The monotonic
variation in the impact of a predictor included in the model during
an observational period can be represented by a step function. An
estimation of the time-dependent effect in the extended Cox model
is presented based on isotonic regression framework. Smoothing the
Schoenfeld residuals plotted against time applying PAVA, can reveal
the changepoints without any a priori information about their
location. The corresponding step function is then introduced in the
model. The power of the Grambsch and Therneau test (which tests for
time-variation in the effect of the predictors) can be improved if
the isotonic transformation for the Schoenfeld residuals is used.
Although this test appears to increase the type I error, its power
is higher compared to conventional Grambsch and Therneau test and
tests based on fractional polynomials. In summary it arises that
isotonic framework is characterized by simplicity and stability.
The main drawback underlying its application is the lack of
asymptotic support in testing. This can make the use of isotonic
models cumbersome since exact or bootstrap methods need to be used.