King M.R., Mody N.A. Numerical and Statistical Methods for Bioengineering: Applications in MATLAB

Подождите немного. Документ загружается.

Now prove that, for a linear equation in one independent variable, i.e.

y ¼ c

x þ c

, the variance associated with the prediction of the mean,

y,atx

¼ σ





xðÞ





xðÞ

What can you say about the effect of the magnitude of x

on the magnitude of the

variance, s

3.16. Medina et al.(2004) studied the effect that estrogen replacement therapy has on

deoxyglucose (DOG) transport in endometrial cancer cells. The data are presented

in Table 3.9, showing the transport (i.e. “reaction”) rate as a function of DOG

concentration, either with or without 10 nM progesterone treatment.

Note that the control data set contains two fewer points. Perform a linear regres-

sion on these data using the normal equations to determine the Michaelis constant and

max

. Report each of your results with a 68% (1σ) conﬁdence interval, and the units of

measure. Plot the transformed data, your best-ﬁt model, and curves that represent

±1σ of your model prediction on the same set of axes. Use a legend to identify the

different curves.

Are the Michaelis constants, with and without progesterone treatment, signiﬁ-

cantly different? In other words, do the 95% conﬁdence intervals of the two param-

eters overlap? (If there is overlap of the two intervals, than we cannot conclude that

they are signiﬁcantly different.)

Table 3.9.

DOG conc. (mM)

Reaction rate (nmol/(cells min))

With progesterone Control case

00 0

5 3.25 2.4

10 2.4 3.75

20 5.1 –

25 6.0 4.05

30 7.2 5.4

40 7.7 6.5

50 9.1 –

60 9.05 7.8

Table 3.10.

G0 (mM) G10 (mM)

10 5

8 3.1

2 0.5

207

3.9 Problems

3.17. A certain enzyme acts on glucose. In a laboratory reaction (37 ºC, pH 7), measure-

ments were made of the glucose concentration at time zero (G0) and afte r ten

minutes (G10), and are given in Table 3.10.

First, convert the concentration measurements to reaction rates by subtracting

G10 from G0 and dividing by 10 min. Estimate K

and R

max

by performing a linear

regression of the Michaelis–Menten equati on using the normal equations. Report

the 68% conﬁdence intervals and units. Plot the transformed data, your best-ﬁt

model, and curves that represent ±1σ of your model prediction. Use a legend to

identify the different curves.

References

Goldberger, A. L., Bhargava, V., West, B. J., and Mandell, A. J. (1985) On a Mechanism of

Cardiac Electrical Stability. The Fractal Hypothesis. Biophys. J., 48, 525–8.

Goldberger, A. L., Amaral, L. A., Hausdorff, J. M., Ivanov, P., Peng, C. K., and

Stanley, H. E. (2002) Fractal Dynamics in Physiology: Alterations with Disease and

Aging. Proc. Natl. Acad. Sci. USA, 99 (Suppl 1), 2466–72.

Lee, J. S., FitzGibbon, E., Butman, J. A., Dufresne, C. R., Kushner, H., Wientroub, S.,

Robey, P. G. and Collins, M. T. (2002) Normal Vision Despite Narrowing of the Optic

Canal in Fibrous Dysplasia. N. Engl. J. Med., 347, 1670–6.

Mazzocca, A. D., Spang, J. T., Rodriguez, R. R., Rios, C. G., Shea, K. P., Romeo, A. A., and

Arciero, R. A. (2008) Biomechanical and Radiographic Analysis of Partial

Coracoclavicular Ligament Injuries. Am. J. Sports. Med., 36, 1397–402.

Medina, R. A., Meneses, A. M., Vera, J. C. (2004) Differential Regulation of Glucose

Transporter Expression by Estrogen and Progesterone in Ishikawa Endometrial

Cancer Cells. J. Endocrinol., 182, 467–78.

Meyer, C., Robson, D., Rackovsky, N., Nadkarni, V., and Gerich, J. (1997) Role of the

Kidney in Human Leptin Metabolism. Am. J. Physiol., 273, E903–7.

Mody, N. A. and King, M. R. (2007) Inﬂuence of Brownian Motion on Blood Platelet Flow

Behavior and Adhesive Dynamics near a Plane Wall. Langmuir, 23, 6321–8.

Nader, P. R., Bradley, R. H., Houts, R. M., McRitchie, S. L., and O’Brien, M. (2008)

Moderate-to-Vigorous Physical Activity from Ages 9 to 15 Years. Jama, 300, 295–305.

Nelson, D. E., Mowery, P., Asman, K., Pederson, L. L., O’Malley, P. M., Malarcher, A.,

Maibach, E. W., and Pechacek, T. F. (2008) Long-Term Trends in Adolescent and

Young Adult Smoking in the United States: Metapatterns and Implications. Am.

J. Public Health, 98, 905–15.

Reichenbach, H. (1951) The Rise of Scientiﬁc Philosophy (Berkeley, CA: University of

California Press).

Roursgaard, M., Poulsen, S. S., Kepley, C. L., Hammer, M., Nielsen, G. D., and Larsen, S. T.

(2008) Polyhydroxylated C(60) Fullerene (Fullerenol) Attenuates Neutrophilic Lung

Inﬂammation in Mice. Basic Clin. Pharmacol. Toxicol., 103, 386–8.

Tees, D. F., Waugh, R. E., and Hammer, D. A. (2001) A Microcantilever Device to Assess the

Effect of Force on the Lifetime of Selectin-Carbohydrate Bonds. Biophys. J., 80, 668–82.

Waugh, R. E., Narla, M., Jackson, C. W., Mueller, T. J., Suzuki, T., and Dale, G. L. (1992)

Rheologic Properties of Senescent Erythrocytes: Loss of Surface Area and Volume with

Red Blood Cell Age. Blood, 79, 1351–8.

West, B. J. (2006) Where Medicine Went Wrong: Rediscovering the Path to Complexity

(Singapore: World Scientiﬁc Publishing Co.).

208

Probability and statistics

4 Hypothesis testing

4.1 Introduction

Often, a researcher, engineer, or clinician will formulate a statement or a belief about

a certain method, treatment, or behavior with respect to one or more populations

of people, animals, or objects. For example, a bioengineer might be interested in

determining if a new batch of drug-coated stents has a drug coat thickness that is

different from a previous batch of stents. The engineer makes several measurements

on a small number of stents chosen at random. A substantial deviation in drug coating

thickness from the standard range of acceptable values indicates that the process

conditions have unexpectedly changed. The stents from the new batch may have either

a thicker or a thinner drug coating compared to those of the previous batches,

depending on the direction of deviation in coat thickness. Or, it is possible that the

drug coat thickness widely ﬂuctuates in the new batches of stents beyond the range

speciﬁed by design criteria. The engineer formulates a hypothesis based on her

preliminary survey. She will need to perform statistical tests to conﬁrm her hypothesis.

Suppose a biomaterials researcher is testing a group of new promising materials

that enhance the durability or lifetime of a contact lens. The researcher poses a

number of hypotheses regarding the performance qualities of different lens materi-

als. She must use statistical techniques to test her hypotheses and draw appropriate

conclusions. As another example, a clinician may test a number of variations of a

therapeutic procedure to determine which one is most effective in treating patients.

He randomly assigns patients to the treatment groups and then records patient

outcomes with respect to time. A comparison is drawn among the treatment groups

using statistical tests, which yield a probability value that indica tes whether the

clinical evidence is in favor (or not in favor) of the clinician’s hypothesis.

Hypothesis testing is a critical part of the method of scientiﬁc investigation.

The investigator develops a hypothesis on the basis of previous experi ence, data,

or simply intuiti ve reasoning. Experiments are performed or surveys conducted to

generate data that characterize the process or phenomenon for which the hypothesis

is made. The data are inspected for consistency with the hypothesis. If the data are

consistent with the hypothesis, they offer evidence in support of the hypothesis.

Because such studies are conducted using samples from their respective popula-

tions, and different samples from the same popula tion exhibit variability in their

characteristics, sample data suffer from sa mpling error. If we were to r epeat the

study, t he new data set obtained woul d not be exactly the same as the data set

from the previous study. If sampling variability in our data is unavoidable, how do we

know if any discrepancies between our study ﬁndings and our hypothesis are simply

due to sampling error or arise because the hypothesis is incorrect? Here, inferential

statistics plays an important role. Hypothesis tests are statistical and probabilistic

methods that help the researcher or engineer make a decision regarding the hypothesis

based on objectivity and consistency rather than subjective opinions.

In Chapter 3, one application of statistical inference was demonstrated: estim-

ation of population parameters. Probabilistic methods were used to generate 100

(1 – α)% conﬁdence intervals, where α is the level of signiﬁcance. Conﬁdence inter-

vals characterize the precision with which one can estimate a population parameter

such as the population mean. In this chapter, we show how inferential statistics can

be used to perform an even more powerful role: one of guiding decision -making.

Using the concepts of statistics and probability in Chapter 3, such as probability

distributions, error propagation formulas, the standard normal distribution, and the

central-limit theorem, we develop statistical tests of signiﬁcance to perform hypoth-

esis testing, i.e. to retain or reject a hypothesis.

We begin this chap ter by co nsidering how one should formulate a hypothesis

(Section 4.2). Once a hypothesis is made and worded in such a way that a statistical

test can be applied to test it, how would one choose the most appropriate statistical

test to use from the bewildering array of test procedures? Each statistical test is

appropriate only for certain situations, such as when the observed variable is dis-

tributed normally within the population, or when multiple populations being com-

pared have equal variances. A plethora of statistical tests have been designed by

statisticians but their suitability for use depends on the nature of the sample data

(e.g. cardinal, ordinal, or categor ical), the nature of distribution of the random

variable within the population (e.g. normal, binomial, Poisson, or non-normal),

the number of populations to be co mpared, equality of variances among the pop-

ulations being compared, and the sample sizes. Section 4.3 demonstrates how a

statistical test is performed, how the probability value that is derived from the

statistical test should be interpreted, and the likelihood of making an error when

using a statistical test. The variety of statistical tests can be broadly categorized into

parametric and non-param etric tests. Methods to assess whether the population

distribution is normal are discussed in Section 4.4. Sections 4.5–4.7 cover parametric

tests for hypotheses concerning a single population or comparison between two

populations. A parametric test called ANOVA for comparing multiple samples is

discussed in Section 4.8. Several important non-parametric tests are considered in

Sections 4.9 and 4.10.

4.2 Formulating a hypothesis

The scientiﬁc exercise of formulating and testing hypotheses is the backbone of

the scientiﬁc method of conducting and interpreting experimental work. Contrary

to popular notion, hypothesis testing is not restricted to the scientiﬁc research

community. Engineers, doctors, nurses, public health ofﬁcials, environmentalists,

meteorologists, economists, and statisticians often need to make an intuitive guess

or prediction related to a treatment, process, or situation. The best course of action is

determined after analyzing and interpreting available data. The data can either

support the hypothesis or provide evidence against it. First, let’s familiarize our-

selves with what a hypothesis is. A hypothesis is a formal scientiﬁc statement that

a person makes based on a conjecture or educated guess. For example, one may

hypothesize from prior observation,

“Drinking tea reduces the incidence of cancer.”

There may be no concrete evidence to back the above hypothetical statement.

Proof must be sought to conﬁrm or disprove the statement. One must perform a

210

Hypothesis testing

thorough scientiﬁc investigation to decide whether to reject (disprove) or retain a

hypothesis. The invest igation is scientiﬁc because it follows the scientiﬁc principles

of objective exploration. A hypothesi s (or a set of hypotheses) that is consis tently

retained time and again and has received overwhelming evidence to support it

becomes a theory.

4.2.1 Designing a scientific study

The scientiﬁc study that is performed to uncover evidence that either supports or

negates the hypothesis can be of two types: an experimental study or an observa-

tional study.

Experimental study

The investigator of an experimental study actively manipulates one or more con-

ditions (or variables) keeping all other variables as constant as possible. The results

of an experimental study can be used to infer a cause–effect relationship between the

experimental variables (manipulated variables) and the response variable (observed

outcome of interest).

In any experimental study that is performed to investigate the validity of a

hypothesis, it is important to create a control group that mimics the treated group

in every way except in receiving the treatment itself. If any marked deviation is

observed in a patient’s health status after treatment, the cause of change may not

necessarily be the result of the treatment or drug itself, but is possibly due to several

other factors involved in the treatment procedure. More than one control group

may have to be created to test each of these factors. Often a control group may be

included that does not experience any form of treatment but is simply observed for

changes in the response variable with time. Several studies have uncovered the

powerful psychological effect that a placebo has on treating a patient’s ailments.

It has been shown in several cases that a medication offered to a group of people

worked just as well as a placebo, as long as the individuals were not aware of which

type of treatment was being offered. The only effect of the treatment was to give

the individual a psychological boost, convincing him or her that “help is on the

way.” If the control group had not been included in the experiment, the “placebo

effect” would not have been discovered.

Bias in an experimental result is reduced by minimizing the differences between

the treatment groups (which includes the control group) prior to administering

treatments. A single sample of experimental subjects s hould be chosen initially that

represents the population. Subjects from this sample should then be randomly

(not haphazardly) allocated to all study groups. In this way, any inter-individual

differences will be uniformly dispersed among all study groups, and the s tudy

groups will appear identical with respect to all other variables except the treatment

conditions. This method of experimental design is called completely randomized

design.

Observational study

The investigator of an observational study sim ply observes an existing situation to

determine an association or pattern between a risk factor and an outcome. The risk

factor(s) whose effect is studied is called the exposure variable(s). Examples of risk

211

4.2 Formulating a hypothesis

factors are substance abuse, exposure to UV rays, chemicals, x-rays, diet, body

weight, and physical activity. The outcome is measured in terms of the outcome

variable or response variable. Examples of an outcome variable are change in

physical or mental health status, mortality, birth rate, birth weight, preterm labor,

and obesity. Often, researchers are interested in ﬁnding a link between some type of

exposure (risk factor) and disease (respo nse variable). In an observational study,

no attempt is made to manipulate the variables of interest or the subjects of the

study. The disadvantage of this type of study is that a positive or negative correlati on

between two observed variables does not necessarily indicate a cause–effect rela-

tionship between the two variables. The observed correlation may be due to

an extraneous variable not considered in the study that is associated with the

Box 4.1A Use of cyclosporin to reduce reperfusion injury during acute myocardial

infarction

Blockage of blood flow in coronary arteries due to thrombosis or lodging of emboli severely reduces

oxygen supply to the cardiac muscle tissue. Deprived of oxygen, myocardial tissue will die and form an

infarct. This condition is called acute myocardial infarction. Interventions such as stenting allow

reperfusion of the affected tissue by opening the passageways in the occluded artery. Ironically,

reperfusion of tissue can cause fur ther myocardial injury. The sudden availability of oxygen to the

cardiac tissue produces an overload of reactive oxygen species (ROS) within the cells. The restoration of

blood flow brings an abundance of white blood cells, which, upon contact with the damaged tissue,

release pro-inflammatory factors and free radicals. Sudden excessive amounts of ROS invokes trauma

in the ischemic cells and may trigger the apoptotic cascade, possibly through a route involving

permeabilization of mitochondria within the myocardial cells.

Cyclosporin is a well-known and widely used immunosuppressant for allogeneic organ transplantation.

It is a fungal peptide, which also blocks the opening of mitochondrial permeability-transition pores. The

formation of these pores (or holes) in the mitochondrial membrane is responsible for causing mitchondrial

collapse and signaling end-stage apoptosis.

An experimental single-blinded pilot study (Piot et al., 2008) was performed in France to determine

whether administration of cyclosporin along with stenting, when treating a patient suffering from acute

myocardial infarction, would limit infarct size and thereby reduce reperfusion injury. Two study groups

were created. In the treatment group, the patients were given a single intravenous bolus injection of

cyclosporin dissolved in saline, while the control group of patients received an equivalent amount of

saline alone. Patients suffering from chest pain, who underwent coronary angiography and were deemed

eligible for the stenting procedure, were randomly assigned to either the treatment group or to the

control group. Each patient was given a sealed envelope that contained a number revealing the study

group that he or she was assigned to. The experimental trial was single-blinded because the patients

were aware of the study group they were assigned to, but the researchers who analyzed the data were

unaware of the assignments. If the trial had been double-blinded, then the patients themselves would

not have known which study group they were being assigned to. The two study groups were found to be

similar with respect to control variables such as age, gender ratio, body-mass index, physical health

conditions such as diabetes, hypertension, and smoking, area of tissue at risk for infarction, and other

baseline characteristics.

Creatine kinase and troponin I co ncentrations in bl ood were measured repeatedly over a period of

72 hours follow ing the stenting procedure to assess the size of the infarct. The size of the inf arct was

measured using MRI 5 days following reperfusion. A comparison of the creatine kinase concen-

trations as well as the infarct size in the two study groups led to the conclus ion that administering

cyclosporin at the time of reperfusion resulted in reduced infarct size compared to that observed with

placebo. Later in this chapter, we show how this conclusion was obtained using the Wilcoxon rank-

sum test.

212

Hypothesis testing

exposure variable and may be causally related to the outcome variable. Or, the

extraneous variable may have a cause–effect relationship with both the exposure and

outcome variables.

While examining results obtained from a scientiﬁc study, one must be aware of the

possibility that extraneous variables can inﬂuence the outcome. When searching for

the nature and extent of association between the exposure variable and the response

variable, these extraneous variables confound or obscure the true relationship

sought. Response of individuals to exercise, diet, and prescription drugs can depend

on age, gender, body weight, smoking status, daily alcohol consumption, race, and

other factors. Suppose we are interested in determining the type of association that

exists between smoking and body weight. Does heavy smoking have a tendency to

increase body weight? Or do overweight people have a higher tenden cy to smoke?

If the smoking population on average has a higher alcohol consumption, or rela-

tively unhealthy eating habits, then the amount of alcohol consumption or type of

diet may inﬂuence body weight more than smoking. Alcohol consumption and diet

are called confounding variables, and may be linked either to smoking status, to body

weight status, or to both categorical variables.

Box 4.2A Treatment of skeletal muscle injury using angiotensin II receptor blocker

Skeletal muscle injuries are particularly common among athletes. Recovery from muscle strains often

require a lengthy rehabilitation period and time away from being on the field or courts. Importantly, injury

to skeletal muscle tissue predisposes orthopedic patients to recurring injuries because of weakening of

the tissue. Following injury, regeneration of muscle tissue is compromised by simultaneous formation

of fibrous (connective) tissue within the regenerated tissues. Patients who undergo surgical procedures

that involve incisions made in skeletal muscle tissue also face the same problem of compromised

healing and longer recovery times due to fibrosis occurring in the healing tissue.

Angiotensin II is a vasoconstricting peptide. Several antagonists have been developed as drugs

(e.g. Cozaar (losartan) by Merck & Co.) to block angiotensin II receptors as a means to relieve

hypertension. Losartan inhibits angiotensin II binding to its receptor and prevents the activation of

precise signaling pathways that induce vasoconstriction. It has recently been found that angiotensin II

upregulates the expression of transforming growth factor-beta1 (TGF–β1) in fibroblasts; TGF–β1 induces

fibroblast proliferation and formation of fibrotic (collagenous) tissue.

Bedair and co-workers from the University of Pittsburgh (Bedair et al., 2008) hypothesized that

treating animals suffering from skeletal muscle injury with an angiotensin receptor blocker (ARB) may

reduce fibrosis in the regenerating tissue. In order to test their hypothesis, they formed a study group of

40 mice on which lacerations to the gastrocnemius muscle were made to simulate muscle strain injury.

The mice were randomly assigned to either

(1) a control group that received tap water for drinking over a period of 3 weeks and 5 weeks;

(2) a low-dose ARB treatment group that received tap water containing dissolved losartan at a

concentration of 0.05 g/l, or

(3) a high-dose ARB treatment group that received tap water containing dissolved losartan at a

concentration of 0.5 g/l.

The mice were allowed to drink water or ARB solutions at amounts and frequencies of their

own discretion. The mice were sacrificed at the end of 3 weeks or 5 w eeks and the number

of regenerating muscle fibers and extent of fib rous tissue development were determined by

histological analysis.

In later sections of this chapter we will perform t tests to identify significant differences, if any, among

the extents of muscle fiber regeneration and fibrosis occurring in each of the three study groups.

213

4.2 Formulating a hypothesis

Box 4.3 Fluoridation of water supply linked to cancer

A controversy spanning several decades continues over the health effects of fluoridation of the US water

supply. It is well established that fluoride helps to prevent tooth decay caused by plaque – a sticky

bacterial film that forms on teeth. When breaking down sugars in food particles, bacteria secrete acids

that damage the hard enamel surface of teeth and expose the delicate inner layers of the teeth (dentin,

pulp) and thereby cause cavities. Fluoride interferes with the action of bacterial acids. It not only

prevents acid demineralization or dissolution of the enamel, but can reverse low levels of tooth decay by

promoting repair or remineralization of teeth. Moreover, in children under the age of six, fluoride that is

ingested is incorporated into developing permanent teeth, thereby increasing resistance of teeth to

demineralization. Accordingly, fluoride was added to the public water supply to aid in the prevention of

tooth decay.

Much skepticism has arisen recently over the benefit of adding fluoride to the water supply. It is

believed that the protective action of fluoride results primarily from topical application as opposed to

ingestion. More importantly, animal and insect studies have shown that low levels of fluoride have

mutagenic effects on chromosomal DNA and may therefore act as cancer-causing agents. Several

epidemiological studies have been conducted to assess the useful/harmful effects of fluoridation of the

water supply with respect to

(1) improving dental health (reducing tooth decay), and

(2) carcinogenic effects of fluoride.

The first of such epidemiological (observational) studies to pinpoint a relationship between fluo-

ridation of water supply and death due to cancer was carried out by Yiamouyiannis and Burk (1977).

They chose to compare the cancer incidence rates observed in ten fluoridated cities (treatment

group) and ten non-fluoridated cities (control group) across the USA. Prior to commencement of

fluoridating the water supply in any of these cities, cancer death rates (obtained from the National

Center for Health Statistics) showed similar trends in both groups – fluoridated cities and non-

fluoridated cities – between 1940 and 1950. Fluoridation of the water supply was initiated in the

treatment group from 1952 to 1956. In 1969, the cancer death rates in the fluoridated cities were

significantly higher than those observed in the non-fluoridated cities. Cancer death rate (CDR) is a

function of many variables and strongly depends on age, ethnicity, and exposure to carcinogens

(toxins, pollutants, chemicals, sunlight). Yiamouyiannis and Burk (1977) divided the population in

each of these cities into four age groups: 0–24, 25–44, 45–64, and 65+. The changing trends of

cancer death rates with time from 1950 to 1970 were studied in each of these four age groups. It was

found that cancer death rate decreased significantly in the first two age groups from 1950 to 1970,

while the cancer death rate increased significantly in the third age group in both fluoridated and

non-fluoridated cities. An increase in cance r death rate for the 65+ population was only found in

fluoridated cities and not in the control group. Their most impor tant finding was a significantly

greater rise i n cancer death rate in the two age groups 45–64 and 65+ in the fl uoridated ci ties

compared to the control g roup, while there was no difference in the CDR within the 0–44

population of the treatment and control groups. Table 4.1 illustrates the method by which CDRs

in t he treatment and control groups were compared.

No differences in CDR of fluoridated and non-fluoridated cities were found on the basis of ethnicity

differences within the two groups.

While Yiamouyiannis and Burk (1977) controlled for age and ethnicity in their study, they did not

recognize or draw attention to any other confounding variables that could be responsible for the CDR

differences observed between the treated and control groups. Thus, a major drawback of this study is

that the authors assumed that the cities were all similar in every way except for fluoridation status, age,

and ethnicity. If the cities were truly similar, then how does one explain the observed differences in CDR

of the fluoridated and non-fluoridated cities in 1952 (prior to fluoridation), especially for the 45–64 and

65+ age groups?

The authors directly compared the values of CDRs from both groups. However, can cancer death

rates from different cities be compared directly to assess for carcinogenicity of a chemical? Many other

214

Hypothesis testing

An observational study can be of three types:

(1) Prospective study or cohort study In this study, two or more groups of disease-free

individuals that differ based on exposure to certain risk factors are selected using

random sampling procedures. These individuals are followed up until some point

in the future. During this time development of the disease (response variable) in

the individuals is noted. The individuals under study are collectively referred to as a

cohort.

(2) Retrospective study or case-control study Two or more groups of individuals are

identiﬁed from the population who either have one or more conditions (diseases) or

who do not. A study is performed to determine which of these individuals were

exposed to one or more risk factors in the past. The “case” denotes the existence of

the condition and is represented by the group of individuals that have developed that

condition. The “control” group does not show signs of disease. The disadvantages of

a case-control study are that (i) the past must be reconstructed from memory

Table 4.1. Difference in cancer death rates (CDR) of ﬂuoridated and

non-ﬂuoridated cities in the USA

Age group

Difference in CDR of

ﬂuoridated cities and

non-ﬂuoridated cities

Change in difference

in CDR of ﬂuoridated

and non-ﬂuoridated

cities from 1952 to 1969 Calculated p value1952 1969

0–24 −0.1929 +0.2157 +0.41 not signiﬁcant

25–44 +2.212 +2.370 +0.16 not signiﬁcant

45–64 +12.98 +28.13 +15.2 <0.02

65+ +57.36 +92.75 +35.4 <0.05

extraneous variables exist that can confound the result. It is possible that the fluoridated cities

had something else in common, e.g. exposure to pollutants. Perhaps the treated cities were more

industrialized, because of which the public came in contact with pollutants to a greater degree than

inhabitants of cities of the control group. In the absence of identification of and control for the additional

extraneous factors, it is difficult to interpret the results.

Doll and Kinlen (1977) repeated this study and found that the average death rate (actually the ratio of

observed death rate to expected death rate), adjusted for age, gender, and ethnicity, slightly declined in

the fluoridated cities from 1950 to 1970, while they remained the same in the non-fluoridated cities over

this duration. In this study, the decreasing or increasing trend in cancer death rates of each group of

cities with time was studied without having calculating differences between dissimilar groups of cities in

time. A study designed in this fashion, in which confounders are better controlled, yields a more

convincing result.

Doll and Kinlen’s findings do not rule out the possibility that fluoridation of our water supply may be

harmful. Perhaps the amount of data available to them was insufficient to prove that fluoridation has

negative effects. However, it is very important when analyzing data and drawing statistical inferences to

take into account as many confounding variables as possible and to interpret results responsibly. It is

critical that one does not try to extract more information out of one’s result than what is scientifically

justifiable.

215

4.2 Formulating a hypothesis

(authentic baseline data are not available) and (ii) the case group is often a sample of

convenience and therefore not necessarily a random sample.

(3) Cross-sectional study A study group is identiﬁed and the variables of interest are

measured at the same point in time to determine if the variables are associated. The

study group is not followed in time into the future or the past. The purpose of this

study is to ascertain prevalence of disease as opposed to the incidence of disease;

the latter is the goal of prospective and retrospective studies.

Because sociodemographic factors such as age, education, income, race/ethnicity,

marital status, and employment status inﬂuence people in many ways, it is important

to control for these variables in any broad scientiﬁc study. The experiment must be

Box 4.4 Smoking and cognitive decline

A prospective cohort study (1995–2007) was performed in Doetinchem, Netherlands, to determine

the effect of smoking on decline of cognitive functions in middle-aged men and women over a 5-year

period (Nooyens et al., 2008). Reduced cognitive functioning occurs naturally with aging and is also a

symptomatic precursor to neurodegenerative disorders such as dementia and Alzheimer’s disease. If

smoking causes accelerated cognitive decline in the elderly, then intervention programs to prevent or

end the smoking habit must be implemented early on. In this prospective study, participants between

the ages of 43 and 70 years underwent cognitive testing, once at baseline, i.e. when they signed up for

participation, and once at follow-up, i.e. at the end of the 5-year period after entering the study. The

study group (n = 1904) was divided into four categories that described their smoking status:

(1) those who had never smoked (34.9%),

(2) ex-smokers (those who quit smoking before baseline) (43.8%),

(3) recent quitters (those who quit smoking in between baseline and follow-up) (6.3%), and

(4) consistent smokers (15.1%).

The four areas of cognitive testing were

(1) memory,

(2) speed of cognitive processes,

(3) cognitive flexibility, and

(4) global cognitive function.

The distributions of the scores from different cognitive tests were unimodal and skewed to the right. The

results of the tests were transformed to a normal distribution and then converted to standardized z scores

using the means and standard deviations of the respective test scores at baseline. The standard

deviations of the test score distributions at follow-up were the same as those at baseline testing.

Cognitive decline (or reduction in cognitive test scores) was observed in all four areas of testing over

the 5-year period irrespective of the smoking status due to the aging-related decline among middle-

aged people. It was found that smokers were significantly worse off at baseline than those who had never

smoked in the areas of cognitive speed, flexibility, and global function. At follow-up, the decline in all

areas of cognitive functioning over 5 years was found to be more rapid among smokers than among

those who had never smoked. The ex-smokers and recent quitters exhibited cognitive declines

intermediate to that of those who had never smoked and consistent smokers. Variables recognized as

confounders were age, gender, education level, body-mass index, hypertension, cardiovascular

disease, HDL cholesterol levels, physical activity, alcohol consumption, energy intake, fat intake, and

antioxidant intake. These variables exhibited an association either with smoking status or with cognitive

function decline. A multivariable linear regression analysis was performed to adjust (standardize) the

test results based on differences within the study groups with respect to the confounding variables.

It was concluded that smoking impacts cognitive decline, and that the areas of cognitive functioning may

be impacted differently by the negative effects of smoking in a time-dependent, order-specific fashion.

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Hypothesis testing