In statistics, hypothesis testing is some times known as
decision theory or simply testing. The key result around whichall decision theory revolves is the likelihood ratio test.
The likelihood ratio test
In a binary hypothesis testing problem, four possible outcomes
can result. Model
${}_{0}$ did in fact represent the best model for the data
and the decision rule said it was (a correct decision) or saidit wasn't (an erroneous decision). The other two outcomes
arise when model
${}_{1}$ was in fact true with either a correct or incorrect
decision made. The decision process operates by segmentingthe range of observation values into two disjoint
decision regions${}_{0}$ and
${}_{1}$ . All values of
$r$ fall into either
${}_{0}$ or
${}_{1}$ . If a given
$r$ lies in
${}_{0}$ , for example, we will announce our decision
"model
${}_{0}$ was true"
; if in
${}_{1}$ , model
${}_{1}$ would be proclaimed. To derive a rational method of
deciding which model best describes the observations, we needa criterion to assess the quality of the decision process.
Optimizing this criterion will specify the decision regions.
The
Bayes' decision criterion seeks to minimize a
cost function associated with making a decision. Let
${C}_{ij}$ be the cost of mistaking model
$j$ for model
$i$ (
$i\neq j$ ) and
${C}_{ii}$ the presumably smaller cost of correctly choosing
model
$i$ :
${C}_{ij}> {C}_{ii}$ ,
$i\neq j$ . Let
${}_{i}$ be the
a priori probability of model
$i$ . The so-called
Bayes' cost$\langle C\rangle $ is the average cost of making a decision.
$p(r, {}_{i}, r)$ is the conditional probability density function of
the observed data
$r$ given that model
${}_{i}$ was true. To minimize this expression with respect
to the decision regions
${}_{0}$ and
${}_{1}$ , ponder which integral would yield the smallest
value if its integration domain included a specificobservation vector. This selection process defines the
decision regions; for example, we choose
${}_{0}$ for those values of
$r$ which yield a smaller value for the first integral.
$${}_{0}{C}_{00}p(r, {}_{0}, r)+{}_{1}{C}_{01}p(r, {}_{1}, r)< {}_{0}{C}_{10}p(r, {}_{0}, r)+{}_{1}{C}_{11}p(r, {}_{1}, r)$$ We choose
${}_{1}$ when the inequality is reversed. This expression is
easily manipulated to obtain the decision rule known as the
likelihood ratio test .
The comparison relation means selecting model
${}_{1}$ if the left-hand ratio exceeds the value on the right;
otherwise,
${}_{0}$ is selected. Thus, the
likelihood
ratio$\frac{p(r, {}_{1}, r)}{p(r, {}_{0}, r)}$ symbolically represented by
$(r)$ , is computed from the observed value of
$r$ and then compared with a
threshold$$ equaling
$\frac{{}_{0}({C}_{10}-{C}_{00})}{{}_{1}({C}_{01}-{C}_{11})}$ . Thus, when two models are hypothesized, the
likelihood ratio test can be succinctly expressed as thecomparison of the likelihood ratio with a threshold.
$(r)\underset{{}_{0}}{\overset{{}_{1}}{}}$
The data processing operations are captured entirely by the
likelihood ratio
$\frac{p(r, {}_{1}, r)}{p(r, {}_{0}, r)}$ . Furthermore, note that only the value of the
likelihood ratio
relative to the
threshold matters; to simplify the computation of thelikelihood ratio, we can perform
any positively monotonic operations simultaneously on the
likelihood ratio and the threshold without affecting thecomparison. We can multiply the ratio by a positive constant,
add any constant, or apply a monotonically increasing functionwhich simplifies the expressions. We single one such
function, the logarithm, because it simplifies likelihoodratios that commonly occur in signal processing
applications. Known as the log-likelihood, we explicitlyexpress the likelihood ratio test with it as
Useful simplifying transformations are problem-dependent; by
laying bare that aspect of the observations essential to themodel testing problem, we reveal the
sufficient
statistic$(r)$ : the scalar quantity which best summarizes the data
(
Lehmann, pp. 18-22 ). The
likelihood ratio test is best expressed in terms of thesufficient statistic.
$(r)\underset{{}_{0}}{\overset{{}_{1}}{}}$
We will denote the threshold value by
$$ when the sufficient statistic is used or by
$$ when the likelihood ratio appears prior to its
reduction to a sufficient statistic.
As we shall see, if we use a different criterion other than
the Bayes' criterion, the decision rule often involves thelikelihood ratio. The likelihood ratio is comprised of the
quantities
$p(r, {}_{i}, r)$ , termed the
likelihood function , which
is also important in estimation theory. It is thisconditional density that portrays the probabilistic model
describing data generation. The likelihood functioncompletely characterizes the kind of "world" assumed by each
model; for each model, we must specify the likelihood functionso that we can solve the hypothesis testing problem.
A complication, which arises in some cases, is that the
sufficient statistic may not be monotonic. If monotonic, thedecision regions
${}_{0}$ and
${}_{1}$ are simply connected (all portions of a region can
be reached without crossing into the other region). If not,the regions are not simply connected and decision region
islands are created (see
this problem ). Such regions usually
complicate calculations of decision performance. Monotonic ornot, the decision rule proceeds as described: the sufficient
statistic is computed for each observation vector and comparedto a threshold.
An instructor in a course in detection theory wants to
determine if a particular student studied for his last test.The observed quantity is the student's grade, which we
denote by
$r$ . Failure may not indicate studiousness:
conscientious students may fail the test. Define the modelsas
${}_{0}$ :did not study
${}_{1}$ :did study
The conditional densities of the grade are shown in
.
Based on knowledge of student behavior, the instructor
assigns
a priori probabilities of
${}_{0}=1/4$ and
${}_{1}=3/4$ . The costs
${C}_{ij}$ are chosen to reflect the instructor's sensitivity
to student feelings:
${C}_{01}=1={C}_{10}$ (an erroneous decision either way is given the
same cost) and
${C}_{00}=0={C}_{11}$ . The likelihood ratio is plotted in
and the threshold value
$$ , which is computed from the
a
priori probabilities and the costs to be
$1/3$ , is indicated. The calculations of this
comparison can be simplified in an obvious way.
$$\frac{r}{50}\underset{{}_{0}}{\overset{{}_{1}}{}}\times 1/3$$ or
$$r\underset{{}_{0}}{\overset{{}_{1}}{}}\times 50/3=16.7$$ The multiplication by the factor of 50 is a simple
illustration of the reduction of the likelihood ratio to asufficient statistic. Based on the assigned costs and
a priori probabilities, the optimum
decision rule says the instructor must assume that thestudent did not study if the student's grade is less than
16.7; if greater, the student is assumed to have studieddespite receiving an abysmally low grade such as 20. Note
that as the densities given by each model overlap entirely:the possibility of making the wrong interpretation
always haunts the instructor. However,
no other procedure will be better!
Questions & Answers
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.