Article

On a Calculus-based Statistics Course for Life ScienceStudentsJoseph C. Watkins

Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089

Submitted March 17, 2010; Revised June 7, 2010; Accepted June 21, 2010Monitoring Editor: John Jungck

The choice of pedagogy in statistics should take advantage of the quantitative capabilities andscientific background of the students. In this article, we propose a model for a statistics coursethat assumes student competency in calculus and a broadening knowledge in biology. Weillustrate our methods and practices through examples from the curriculum.

INTRODUCTION

When considering quantitative training for the next generationof scholars, the most persistently requested advanced skillsexpressed by the research-active life scientists at the Universityof Arizona are the use of statistics and comfort with calculus.Consequently, one centerpiece for the curricular activities atthe University of Arizona is the development of three coremathematics courses–a two-semester-long sequence that inte-grates calculus and differential equations and, based upon thatknowledge, a one-semester-long course in statistics.

Benefiting from changes in approach in the calculus anddifferential equations course, students enrolled in the sta-tistics course are acquainted and have some facility withopen-ended questions. In addition, because these students aretypically juniors and seniors, they bring a much broaderknowledge base in the life sciences than they had when theyentered the calculus classroom for the first time.

Most life science students, presumed to be proficient incollege algebra, are taught a variety of procedures and stan-dard tests under a well-developed pedagogy. This approachis sufficiently refined so that students have a good intuitiveunderstanding of the underlying principles presented in thecourse. However, if the needs presented by the science falloutside the battery of methods presented in the standard

curriculum, then students are typically at a loss to adjust theprocedures to accommodate the additional demand.

On the other hand, mathematics students frequently havea course in the theory of statistics as a part of their under-graduate program of study. In this case, the standard cur-riculum repeatedly finds itself close to the very practicallyminded subject that statistics is. However, the demands ofthe syllabus provide very little time to explore these appli-cations with any sustained attention.

Our goal for life science students at the University of Ari-zona is to find a middle ground. In their overview “Under-graduate Statistics Education and the National Science Foun-dation,” Hall and Rowell (2008) note that statistics educationreformers have until recently overlooked the issues associatedwith an introductory postcalculus statistics curriculum. Theirnotable exceptions are the data-oriented active-learning ap-proach of Rossman and Chance (2004), Virtual Laboratories inStatistics by Siegrist (2004), and case studies–based approachesto teach mathematical statistics (Nolan and Speed, 1999, 2000).

Despite the fact that calculus is a routine tool in the devel-opment of statistics, the benefits to students who have learnedcalculus are very rarely used in the statistics curriculum forundergraduate biology students. Our objective is to meet thisneed with a one-semester course in statistics that moves for-ward in recognition of the coherent body of knowledge pro-vided by statistical theory having an eye consistently on theapplication of the subject. Even though such a course may notbe able to achieve the same degree of completeness now pre-sented by the two more standard courses described above, thequestion is whether it leaves the student capable of under-standing what statistical thinking is, understanding how tointegrate this with scientific procedures and quantitative mod-eling, how to ask statistics experts productive questions, andhow to implement their ideas using statistical software and

DOI: 10.1187/cbe.10–03–0035Address correspondence to: Joseph C. Watkins (jwatkins@math.arizona.edu).

© 2010 J. C. Watkins CBE—Life Sciences Education © 2010 TheAmerican Society for Cell Biology under license from the au-thor(s). It is available to the public under Attribution–Noncom-mercial–Share Alike 3.0 Unported Creative Commons License(http://creativecommons.org/licenses/by-nc-sa/3.0).

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other computational tools. For a thoughtful essay on theseissues, see Nolan and Lang (2009).

The efforts described above, despite having similar goals,arrive at very different approaches. In this article, we shallintroduce the course at the University of Arizona with anannotated syllabus and through classroom examples, takehome assignments, and end-of-the-semester projects.

AN ANNOTATED SYLLABUS

The four parts of the course—organizing and collectingdata, an introduction to probability, estimation procedures,and hypothesis testing—are the standard building blocks ofmany statistics courses. This section highlights some of thefeatures of a calculus-based course.

Organizing and Collecting DataMuch of this is standard and essential—organizing categor-ical and quantitative data, appropriately displayed as barcharts, histograms, boxplots, time plots, and scatterplots,and summarized using medians, quartiles, means, weightedmeans, trimmed means, standard deviations, and regressionlines. We use this as an opportunity to introduce students tothe statistical software package R (R Development CoreTeam, 2009) and to add additional summaries like the em-pirical cumulative distribution function and the empiricalsurvival function. One example integrating the use of this isthe comparison of the lifetimes of wild-type and transgenicmosquitoes and a discussion of the best strategy to displayand summarize data if the goal is to examine the differencesin these two genotypes of mosquitoes in their ability to carryand spread malaria. A bit later, we will do an integration byparts exercise to show that the mean lifetime of the mosqui-toes is the area under the survival function.

Collecting data under a good design is introduced early inthe course, and discussion of the underlying principles ofexperimental design is an abiding issue throughout thiscourse. With each new mathematical or statistical conceptcomes an enhanced understanding of what an experimentmight uncover through a more sophisticated design thanwhat was previously thought possible. The students aregiven readings on design of experiment and examples usingR to create simple and stratified random samples. The rec-ommended lecture is to tell a story that illustrates how theseissues appear in personal research activities.

Introduction to ProbabilityProbability theory is the analysis of random phenomena. Itis built on the axioms of probability and is explored, forexample, through the introduction of random variables. Thegoal of probability theory is to uncover properties arisingfrom the phenomena under study. Statistics is devoted to theanalysis of data. The goal of statistical theory is to articulateas well as possible what model of random phenomena un-derlies the production of the data. The focus of this sectionof the course is to develop those probabilistic ideas thatrelate most directly to the needs of statistics.

Thus, we must study the axioms of probability to theextent that the students understand conditional probabilityand independence. Conditional probability is necessary to

develop Bayes formula, which we will later use to give ataste of the Bayesian approach to statistics. Independencewill be needed to describe the likelihood function in the caseof an experimental design that is based on independentobservations. Densities for continuous random variablesand mass function for discrete random variables are neces-sary to write these likelihood functions explicitly. Expecta-tion will be used to standardize a sample sum or samplemean and to perform method of moments estimates.

Random variables are developed for a variety of reasons.Some, like the Poisson random variable or the gamma randomvariable, arise from considerations based on Bernoulli trials orexponential waiting. The hypergeometric random variablehelps us understand the difference between sampling with andwithout replacement. The F, t, and �2 random variables willlater become test statistics. Uniform random variables are theones simulated by random number generators. Because of thecentral limit theorem, the normal family is the most importantamong the list of parametric families of random variables.

The flavor of the course returns to becoming more authen-tically statistical with the law of large numbers and thecentral limit theorem. These are largely developed usingsimulation explorations and first applied to simple MonteCarlo techniques and importance sampling to evaluate inte-grals. One cautionary tale is an example of the failure ofthese simulation techniques when applied without carefulanalysis. If one uses, for example, Cauchy random variablesin the evaluation of some quantity, then the simulated sam-ple means can appear to be converging only to experience anabrupt and unpredictable jump. The lack of convergence ofan improper integral reveals the difficulty.

The central object of study is, of course, the central limittheorem. It is developed both in terms of sample sums andsample means and used in relatively standard ways to esti-mate probabilities. However, in this course, we can intro-duce the delta method, which adds ideas associated to thecentral limit theorem to the context of propagation of error.

Estimation ProceduresIn the simplest possible terms, the goal of estimation theoryis to answer the question: What is that number? An estimatoris a statistic (i.e., a function of the data). We look to two typesof estimation techniques—method of moments and maxi-mum likelihood and several criteria for an estimator (e.g.,bias and variance). Several examples including capture-re-capture and the distribution of fitness effects are developedfor both types of estimators. The variance is estimated usingthe delta method for method of moments estimators and usingFisher information for maximum likelihood estimators. Ananalysis of bias is based on quadratic Taylor series approxima-tions and the properties of expectations. R is routinely used insimulations to gain insight into the quality of estimators.

The point estimation techniques are followed by intervalestimation and, notably, by confidence intervals. This bringsus to the familiar one- and two-sample t intervals for pop-ulation means and one- and two-sample z intervals for pop-ulation proportions. In addition, we can return to the deltamethod and the observed Fisher information to constructconfidence intervals associated, respectively, with method ofmoment estimators and maximum likelihood estimators.

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Hypothesis TestingFor hypothesis testing, we begin with the central issues—null and alternative hypotheses, type I and type II errors,test statistics and critical regions, significance, and power.We begin with the ideas of likelihood ratio tests as best testsfor a simple hypothesis. This is motivated by a game. Ex-tensions of this result, known as the Neyman Pearsonlemma, form the basis for the t test for means, the �2 test forgoodness of fit, and the F test for analysis of variance. Theseresults follow from the application of optimization tech-niques from calculus, including Langrange multiplier tech-niques to develop goodness of fit tests.

The desire of a powerful test is articulated in a variety ofways. In engineering terms, power is called sensitivity. Weillustrate this with a radon detector. An insensitive instru-ment is a risky purchase. This can be either because theinstrument is substandard in the detection of fluctuations orpoor in the statistical test that results in an algorithm toannounce a change in radon level. An insensitive detectorhas the undesirable property of not sounding its alarm whenthe radon level has indeed risen.

The course ends by looking at the logic of hypothesestesting and the results of different likelihood ratio analysesapplied to a variety of experimental designs. The deltamethod allows us to extend the resulting test statistics tomultivariate nonlinear transformations of the data.

EXAMPLES FROM THE CURRICULUM

In this section, we describe three examples from the Univer-sity of Arizona course. These examples are presented tohighlight how a calculus-based course differs from an alge-bra-based course. These abbreviated descriptions cannotbring the same sense of background preparation or thebreadth of issues under consideration that a student experi-ences in the classroom. Consequently, more detailed notesare available from the author upon request.

For the first two examples, we begin with the presentationseen in a typical algebra-based statistics course and thenprovide extensions of these ideas that are possible for thosestudents who have good calculus skills. The third exampledescribes a strategy to introduce the concept of likelihood. Insubsequent classes, the students will use this as motivationfor the rationale for the optimization problems in the devel-opment of likelihood ratio tests.

Extensions on RegressionGiven observations (x1, y1), (x2, y2), …(xn, yn), ordinary linearregression has as its goal to find the best linear fit g(x��, �) �� � �x to the data. The least squares criterion refers to theminimization of the sum of squares

SS��, �� � �i � 1

n

�yi � g �xi � �, ���2

over all real parameter values � and �. Calling � and � thevalues that achieve this minimum, we obtain the regressionline

yi � � � �xi

fit to the response �i. Ordinary linear regression is a staple ofany introductory statistics course [see, e.g., Moore et al. (2007)or Agresti and Franklin (2008)]. Students compute regressionlines using software or a formula and learn to check the ap-propriateness of the regression line by examining the residuals(i.e., the differences between the data and the fit). In-formally, a good fit would show no structure in the residualplot. Two departures from this are common. Either

• the size of the residuals depends on x, or• the sign of the residuals depends on x.

The goal here is to have the ability to extend the use ofregression beyond the formulas and qualitative reasoning.To address the first case above, we can modify the leastsquares criterion and solve a weighted least squares regres-sion with weight function w,

SSw��, �� � �i � 1

n

w�xi��yi � g�xi � �, ���2.

Later in the course, after the students have learned aboutmaximum likelihood estimation, they learn that the weightsshould often be chosen to be inversely proportional to thevariance of the residual.

Here, we focus on the second case in which the scatter-plots display a curved relationship. With the tools of collegealgebra, this relationship can be transformed to be suitablefor linear regression by guessing the relationship and apply-ing the inverse transformation to the response variable. Thisis seen most frequently in scatterplots of a quantity overtime exhibiting exponential growth or decay. Thus, we takethe logarithm of the response variable and proceed.

This strategy is sometimes too superficial to be successful.For students who have been acquainted with differentialequations, we can consider a common chemical reaction,

E � S^k � 1

k1

ES3k2

E � P

Here E is an enzyme, S is the substrate, ES is the substrate-bound enzyme, and P is the product. The production rate,V � d[P]/dt, of the product is measured for five differentconcentrations [S] of the substrate to obtain the followingdata: (See Figure 1.)

Naive attempts to give a transformation that yields a linearrelationship are unlikely to succeed. Despite intense specu-lation, students have never been able to guess correctly. Amodel-based approach is needed. With this in mind, webegin with the law of mass action to obtain differentialequations for the concentrations

d�ES�

dt� k1�E��S� � �k � 1 � k2��ES� and

d�P�

dt� k2�ES�.

The strategy used by Michaelis and Menten applies to situ-ations in which the concentration of the substrate-bound

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enzyme (and hence also the unbound enzyme) change muchmore slowly than those of the product and substrate. If, bymeasuring the change of [ES] directly over time or by argu-ing that if the concentration of enzyme [E] is small comparedwith the concentration of substrate [S], we find that

0 �d�ES�

dtand, thus, �E��S� � Km�ES� where Km �

k � 1 � k2

k1,

the Michaelis constant. Equipped with these ideas, we findafter some algebraic manipulation, the well-known Line-weaver-Burk double reciprocal plot

1V

�Km � �S�

Vmax�S��

Km

Vmax

1�S�

�1

Vmax.

In this way, we have found that the desired linear relation-ship is between 1/V and 1/[S]. We can now perform ordi-nary least squares on the transformed data and estimateVmax, the maximum rate of production, and Km. [See, forexample, Nelson and Cox (2005), pp. 204–210.]

However, this classical approach to estimation is nolonger considered satisfactory (Piegorsch and Bailer, 2005).Examination of residual plots reveals the drawback—theresiduals measured as the reciprocal of the concentrationbecome magnified for very small concentrations. Conse-quently, the preferred method is to use a nonlinear leastsquares criterion. In this case, given data (V1, [S]1), (V2, [S]2),… (Vn, [S]n), find the values of Vmax and [S] that minimize

SS�Vmax, Km� � �j � 1

n

�Vj � g��S�j � Vmax, Km��2

where

g��S� � Vmax, Km� � Vmax

�S�

Km � �S�.

This minimization problem does not yield explicit equationsfor the estimators and Rather than embarking

on a minicourse on numerical techniques for optimization,we use this opportunity to discuss the intricacies of thenonlinear regression analysis and strategies to engage anexpert in statistical science on the science, the experimentalprotocol, and the data.

One of the aspects of any new course is the understand-able uncertainty by the students that the approaches pre-sented in class have applicability to questions of their ownconcern. Thus, in the problem sets, we ask the students touse the ideas presented in lecture and make substantial useof them in another biological context. In this case, the ideason nonlinear transformations are further explored by thestudents in the following question addressed by Wiehe andStephan (1993):

Due to selection, neutral sites near genes are hitchhikedalong with the gene to higher levels in the population untila recombination event separates the neutral site from theselected gene. This reduces the diversity of the genome neargenes. The nucleotide diversity � versus recombination rate is given for 17 gene regions in Drosophila melanoganster.After a short amount of reasoning based on the nature ofrecombination and selection, the students learn the relation-ship

� ��

� � .

This expression is also amenable to a Lineweaver-Burk dou-ble reciprocal plot. Thus, we rewrite this expression as

1�

��

�

1

�1�

,

a linear expression in the variables 1/� and 1/. As thescatterplot of these reciprocal variables in Figure 2 shows,we can see that the correlation is not very high. In this case,strength of selection is a good candidate for a hidden orlurking variable. In this data set, we are likely looking atgenes that range from small selection that will not reducediversity much to large selection that will. In the same waythat estimated value of Vmax gives the reaction rate at highconcentration of the substrate, the estimated value of � givesthe nucleotide diversity at recombination distances faraway from genes. This value gives a sense of the intrinsicdiversity of the Drosophila genome due solely to the effects ofneutral mutation and demographic history.

The Central Limit Theorem, Propagation of Error,and the Delta MethodGeorge Polya (1920) used the phrase the central limit theo-rem because of its centrality to the theory of probability. Insimple terms, the central limit theorem states that, irrespec-tive of the distribution of the observations from a simplerandom sample, its sample mean has, for sufficiently

many observations, approximately a normal distribution.Many of the commonly used hypotheses rely on a teststatistic that is based on the normal distribution. Thus, theuse of the t test, the �2 test, and the analysis of variance F testdepends on the appropriateness of the normal distribution

Figure 1. Michaelis–Menten kinetics. Measurements of productionrate V � d[P]/dt versus [S], substrate concentration.

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as an approximation for the distribution of sample meanscomputed from the data. So, understanding the use of thesestatistics depends in part on grasping the logic behind thecentral limit theorem.

Unfortunately, the best proofs of the central limit theoremrely on indirect and sophisticated methods. Moreover, theseproofs yield very little insight into the emergence, with anincreasing number of observations, of the bell curve for thedistribution of Consequently, rather than proving the

central limit theorem in an introductory statistics course atthis level, the typical pedagogical choice is to use graphicaland other empirical methods to convey the key ideas (seeRoss (2009) or Pitman (1999) for a proof).

Most elementary courses dedicate a week to the under-standing of the central limit theorem. This developmentmight culminate with a graph like that seen in Figure 3. Thisshows the standardization of random variables and relates it

to the test statistics that the students will later encounter inthe study of hypothesis testing. This can then be used toestimate probabilities for X� by evaluating z scores.

The extension we can have in a calculus-based course ismotivated by interest in nonlinear transformations of themeasured quantities. Some physics, chemistry, and engi-neering students have seen a bit of this in the study ofpropagation of error (see, e.g., Meyer [1975], Bevington andRobinson [2002]). Having investigated the properties of vari-ance and covariance in the context of the independence ofrandom variables, the goal is to combine the benefits of thecentral limit theorem and the propagation of error analyses.

After investigating some one-dimensional examples likethose seen in Figure 4, the students are prepared to examinea more sophisticated example. For this, we consider a sug-gestion by Powell (2007) to questions in avian biology. Tointroduce one of his examples, define the fecundity B as themean number of female fledglings per year. Then, B is aproduct of three quantities,

B � F � p � N,

where F equals the mean number of female fledglings persuccessful nest, p equals nest survival probability, and Nequals the mean number of nests built per female per year.Let’s collect measurements on n1 nests to count female fledg-lings in a successful nest, check n2 nests for survival proba-bility, and follow n3 females to count the number of success-ful nests per year. To begin, assume that the experimentaldesign is structured so that measurements are independent.

A critical value for B is 1. If B is greater than 1, then thepopulation grows over time. If it is less than 1, then thepopulation is headed toward extinction. So, for example, ifour estimate B � 1.03, then we may or may not be confidentthat the actual value of B is greater than 1 depending on theway the estimator B distributes its values.

The central limit theorem can directly determine appropriatenormal distributions to approximate F� , the sample mean of thenumber of female fledglings per successful nest, p, the sampleproportion of surviving nests, and N� , the sample mean numberof nests built per female per year. If our estimate B of thefecundity is the product of these three numbers, then what is agood approximation for the distribution of this statistic?

Propagation of error analysis suggests that we find alinear approximation to B. Because the measurements for F,

Figure 2. Double reciprocal plot for genetic hitchhiking. 1/�versus 1/. The regression line y intercept is 277.8, giving anestimate � 1/ 277.8 � 0.0036.

Figure 3. Displaying the central limit theo-rem graphically. Density of the standardizedversion of the sum of n independent exponen-tial random variables for n � 2 (dark blue), 4(green), 8 (red), 16 (light blue), and 32 (ma-genta). Note how the skewness of the exponen-tial distribution slowly gives way to the bellcurve shape of the normal distribution.

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p, and N are independent, then B2, the variance of B is easy

to approximate. This solution give us the size of the error,but we do not yet know its distribution.

The delta method extends the propagation of error analysisby noting that, by the central limit theorem, the random quan-tity given by the linear approximation can be approximated bya normal distribution. Returning to the question of estimatingfecundity, the delta method tells us that B can be approximated bya normal distribution. The variance can be determined from F

2,the variance in the fecundity measurement and N

2, the variance inthe number of nests built per adult female per year. By combiningthis information, we derive in the class an expression that high-lights in the contribution to the variance in the estimate of fecun-dity originating from each of the three measurements:

B2

B2 �1n1

F2

F2 �1n2

1 � pp

�1n3

N2

N2.

This formula now plays an essential role in settling on a datacollecting protocol. With a goal to bring the standard devi-ation B down as much as possible given available resources,the choices of n1, n2, and n3 are under control of the biolo-gists. In addition, if independence of observations between theestimation of F, p, and N is an issue, we can extend the prop-agation of error and with it the delta method by including thecovariances between the pairs of the sets of observations.

This application of the calculus greatly extends the appli-cability of the central limit theorem in the practical applica-tion of statistics. This idea is explored by the students whoapply the delta method to describe the estimator for the focallength f of a convex lens based on repeated measurement ofthe distance, s1, to an object and the distance, s2, to its imageusing the thin lens formula,

1f

�1s1

�1s2

.

(See Figure 5.)The appearances of the delta method do not end here. We

shall use this technique to determine the variance for amethod of moments estimator. Later, we can use these ideasto construct test statistics for hypothesis tests beyond thosegenerally encountered in more elementary courses.

Likelihood RatiosThe study of statistical hypotheses begins with a shot ofjargon that will take a student some time to absorb. Under-standing the terminology is important because it codifies the

Figure 4. Illustrating the delta method. Here the mean � � EX �1.5 and the blue curve h(x) � x2. Thus, is approximately normal

with mean close to 2.25 and . The bell curve on the y axis

is the reflection of the bell curve on the x axis about the (black)tangent line y � h(�) � h�(�)(x �).

Figure 5. Simulating a sampling distribution for the estimate offocal length. In this example, the distance from a convex lens to anobject s1 � 12 cm and to its image s2 � 15 cm. The standarddeviation of the measurement is 0.1 cm for s1 and 0.5 cm for s2.Based on 25 measurements each for s1 and s2, the delta method gives0.0207 for the standard deviation of the estimate for f. This agrees tothree decimal places with the value based on the 1000 simulationssummarized in the histogram above. The bell curve shape shows thequality of an approximation to a normal random variable.

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relationship between hypothesis testing and scientific advance-ment. Under a simple hypothesis, we write the test as:

H0 : data are from distribution 0

versus H1 : data are from distribution 1.

H0 is called the null hypothesis. H1 is called the alternativehypothesis. The possible actions are:

• Reject the hypothesis. Rejecting the hypothesis when it istrue is called a type I error or a false positive. Its proba-bility � is called the size of the test or the significance level.

• Fail to reject the hypothesis. Failing to reject the hypoth-esis when it is false is called a type II error or a falsenegative. If the false negative probability is �, then thepower of the test is 1 �.

The rejection of the hypothesis is based on whether or notthe data X land in a critical region C. Thus,

reject H0 if and only if X � C.

Given a choice � for the size of the test, a critical region C iscalled best or most powerful if it has the lowest probabilityof a type II error among all regions that have size � (Table 1).

� � PX � C � H1 is true}

To see whether the students can discover on their own thebest possible test, we play a game. The goal here is toestablish a foundation for the fundamental role of likelihoodratios. This will form the rationale behind the optimizationproblems that must be solved in order to derive likelihoodratio tests. Both the null hypothesis (our side) and the alter-native hypothesis (our opponent) are given 100 pointsamong the values for the data X that run from 11 to 11.These are our likelihood functions. These can be created anddisplayed quickly in R using the commands:

� x�-c(11:11)� L0�-c(0,0:10,9:0,0)� L1�-sample(L0,length(L0))� data.frame(x,L0,L1)Thus L1 is a random rearrangement of the values in L0.

Here is the output from one simulation.The goal of this game is to pick values x so that your

accumulated points increase as quickly as possible from

your likelihood L0, keeping your opponent’s points from L1as low as possible. The natural start is to pick values of x sothat L1 (x) � 0. Then, the points you collect begin to add upwithout your opponent gaining anything.

Being ahead by a score of 23–0 can be translated into a bestcritical region in the following way. If we take as our criticalregion C all the values for x except 2, 3, 5, and 7, then, the sizeof the test � � 0.77 and the power of the test 1 � � 1.00because there is no chance of type II error with this criticalregion.

Understanding the next choice is crucial. Candidatesare

x � 4, with L0�4� � 6 against L1�4� � 1 and

x � 1, with L0�1� � 9 against L1�1� � 2.

After some discussion, the class will come to the conclusionthat having a high ratio is more valuable than having a highdifference. The choice 6 against 1 is better than 9 against 2because choosing 6 against 1 twice will put us in a betterplace than the single choice of 9 against 2. Now we can pickthe next few candidates, keeping track of the size and thepower of the test with the choice of critical region being thevalues of x not yet chosen (see Table 2).

From this exercise we see how the likelihood ratio test isthe choice for a most powerful test. This is more carefullydescribed in the Neyman-Pearson lemma [See Hoel et al.(1972) and Hogg and Tanis (2009)].

Neyman Pearson LemmaLet L0 denote the likelihood function for the random vari-able X corresponding to H0 and L1 denote the likelihoodfunction for the random variable X corresponding to H1. Ifthere exists a critical region C of size � and a nonnegativeconstant k such that

L1�x�

L0�x�� k for x � C and

L1�x�

L0�x� k for x�C,

then C is the most powerful critical region of size �.Using R, we can complete the table for L0 total and L1 total.� o�-order(L1/L0)� sumL0�-cumsum(L0[o])� sumL1�-cumsum(L1[o])� alpha�-1-sumL0/100� beta�-sumL1/100� data.frame(x[o],L0[o],L1[o],� sumL0,sumL1,alpha,1-beta)

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Completing the curve, known as the receiver operatorcharacteristic (ROC), is shown in Figure 6. The ROC can beused to discuss the inevitable trade-offs between type I andtype II errors. For example, by the mere fact that the graphis increasing, we can see that by setting a more rigorous testachieved by lowering the level of significance (decreasingthe value on the horizontal axis) necessarily reduces thepower (decreasing the value on the vertical axis) [see, e.g.,Zweig and Campbell (1993)].

Now the students are prepared to understand the rea-soning behind the use of critical values for the t statistic,the �2 statistic, or the F statistic as related to the criticalvalue k in extensions of the ideas of likelihood ratiosand can extend the likelihood ratio tests to more novelsituations.

These examples have been presented here in such a way asto emphasize transitions from the use of algebra to the use ofcalculus in the learning of concepts in statistics. However,students do not notice such a shift in the sense that theydo not see calculus as a special tool. Limits need to beevaluated, rates change, areas under curves need to bedetermined, functions need to be maximized, and theirconcavity needs to be assessed. Moreover, calculus is

more than a powerful computational tool: it provides away of thinking that enhances the students’ view of whatis possible. I asked a student how much is calculus used inthe course. The answer she gave, “So much we don’t thinkabout it,“ is a goal for this course.

For many years, quantitative subjects as diverse as physicsand economics have developed distinct pedagogies for analgebra-based and for a calculus-based course. For example,Newton’s second law applied to projectile motion or to themotion of springs gives, for the calculus student, simpledifferential equations from which the basic algebraic or trig-onometric relationships for the variables position, velocity,acceleration, and time can be derived. For the algebra stu-dent, these relationships have to be taken on faith. In asimilar way, students can receive a more comprehensive andelegant understanding of the fundamental principles of sta-tistical science if they are equipped with a working knowl-edge of calculus.

PROJECT DESCRIPTIONS

One important feature of the course is the end-of-the-semes-ter project. Typically for this assignment, students work inpairs. All of the suggested projects are based on researchactivities that have taken place at the University of Arizona,and all of the projects require statistical analyses beyond themethods presented in the course. This gives the students theopportunity to work as a team on a project and to work withone of the scientists (typically a graduate student or post-doctoral fellow) who was involved in the original research.This way, the students obtain a first-hand account of thefundamental questions that are meant to be addressed bythe research and learn how to use the ideas from theirstatistics course and apply them to new situations. Theycertainly see the nature of open-ended questions that are apart of a research scientist’s daily life.

We describe two projects to give a sense of the breadth ofthe choices from the point of view of both biology andstatistics.

Language and Genes in SumbaHuman populations and the languages that they speakchange over time. The movement of people and the inno-vations they make in their languages are difficult to ob-serve and quantify over short periods and impossible towitness over long periods. Consequently, researchershave been forced to undertake indirect approaches toinfer associations between human languages and humangenes. Many well-known studies focus their questions onthe movement of people on continental scales. These stud-

Figure 6. Receiver operator characteristic. The graph of� PX � C � H0 is true} (significance) versus 1 � � P{X �

C � H1 is true} (power) in the example. The horizontal axis � is alsocalled the false positive fraction (FPF). The vertical axis 1 � is alsocalled the true positive fraction (TPF).

Table 2: Results for Neyman-Peason game.

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ies led Diamond and Bellwood (2003) to suggest thatmany of the correlations that we presently see in lan-guages and genes result from the movement of prehistoricfarmers from the places in which these agricultural tech-niques first arose.

Using the Indonesian island of Sumba as a test case,Lansing et al. (2007) look to see the degree in which thesecorrelations can be seen at much smaller scales. Both geneticand archeological evidence place the first migration of ana-tomically modern humans in Southeast Asia and Oceanabetween 40,000 and 45,000 years ago. Further archeologicalevidence places the transition on Sumba from the originalhunter-gatherer technology to the neolithic technology be-tween 3500 and 4000 years ago. At that time, a small numberof farmers speaking an Austronesian language likely cameinto contact with resident foragers speaking presumably aPapuan language.

To investigate the paternal histories of the Sumbanese,Lansing et al. (2007) obtained genetic samples from 352 meninhabiting eight villages. Their genetic information is de-rived from the Y chromosome. Based on the Y ChromosomeConsortium worldwide genealogical tree of paternal ances-try, haplogroup O appears to be associated with the expan-sion of Austronesian societies from southeast Asia to Indo-nesia and Oceana.

The linguistic data consist of 29 200-word Swadesh listsfrom sites well distributed throughout the island. Swadeshlists are built from words ascribed to meanings that are basicto everyday life. Using the methodology of comparativelinguistics, some words from different language lists can betraced to a common ancestral word. These techniques havebeen used previously to construct a Proto-Austronesian(PAn) language. In addition, a phylogenetic tree built fromthese 29 word lists branches to form five major language

subgroups and leads us to the conclusion that the presentday Sumbanese all speak a language derived from a singlecommon ancestral Proto-Sumbanese. From this we cancount the number of words on the Swadesh word lists thatare derived from the PAn language. This information issummarized in Figure 7 (Lansing et al., 2007).

Statistical ProceduresThe Mantel test (Sokal and Rohlf, 1994) computes the sig-nificance of the correlation between two positive symmet-ric n n matrices. The ij entry in a matrix is meant to givea distance between sites i and j. Distance matrices M andN are square and the calculations for the test are carriedout on the entries above the diagonal. The computationyields a statistic:

Z � �i � 1

n �j � i � 1

n

Mij Nij

similar to the correlation statistic well known to the stu-dents. The null hypothesis is that the observed relationshipbetween the two distance matrices could have been obtainedby any random arrangement of the observations.

In this example, we have n � 8 villages and threemeasures of distance— geographic, linguistic, and genetic.In previous correlation computations, changing one ob-servation was not at all prohibited by the structure of theproblem. However, distance matrices are highly con-strained. For example, if we change one entry in thematrix by altering distance from one village to the next,then we must also change other entries in the matrix tocompensate. These constraints force us to look for a re-finement in the standard hypothesis testing procedure forcorrelation.

Figure 7. Phylogenetic and geographic distri-bution of languages and Y chromosome haplo-groups.

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CBE—Life Sciences Education306

To see whether the value of Z is bigger than what wemight find by chance, we perform a permutation on one ofthe matrices. For example, if we were to keep the labels onthe villages geographic distances in M and rearrange thedistances in the N matrix with a permutation �, then wehave a new statistic:

Z� � �i � 1

n �j � i � 1

n

Mij N��i��� j�.

If the alternative hypothesis holds, then Z is likely to bebigger than most of the Z�. Indeed, the P value of the test isa fraction of the Z� that are larger than Z. The results of thesetests are summarized in Table 3.

We next test the null assumption of no correlation be-tween the fraction of O haplogroup men and retainedSwadesh list PAn cognates among the eight villages. Thealternative is that these two quantities are positively corre-lated. A naive strategy to analyze the correlation in Figure 8is to perform simple linear regression using software andreport the P value for a test of zero slope. This methodwastes much of the effort in collecting the substantialamount of genetic and linguistic information necessary toplot each of the points in Figure 8. We can tell that the

method is not most powerful in that if we were to double theamount of data and obtain the same regression line, then theP value remains the same.

An exact or even an approximate computation of thedistribution of correlation under the null hypothesis is dif-ficult. Consequently a bootstrap analysis (Efron and Tib-shirani, 1994) is used. Let p be the fraction of men through-out all of Sumba typed as O haplogroup and let q be thefraction of neutral words throughout all of Sumba that arePAn cognates. Under the null hypothesis, the data from eachof the villages are independent Bernoulli trials based onthese parameters. The students know that, under these con-ditions, the distribution of O haplogroup men and PAncognate words from each village will have binomial distri-butions and can easily simulate them using R. What isunknown are the actual values of p and q. The bootstrapsuggests that these two parameters should be estimatedfrom the actual data, estimations that the students have seenin simpler contexts. Next, pseudodata should be simulatedrepeatedly under the null hypothesis.

For each of the resulting bootstrap samples, the studentscan compute the correlation. Repeat this procedure manytimes to create the bootstrap distribution of correlation un-der the null hypothesis. The bootstrapped P value, 0.047,for the test of no correlation is simply the fraction ofbootstrap correlations greater than 0.627, the observedvalue of correlation.

Force-dependent Kinetic Models for Single MoleculesOne of the most widely used relations in chemistry is theArrhenius relation,

k � exp ��G‡

kBT.

Here k is a reaction rate, kB is Boltzmann’s constant, T is theabsolute temperature, and �G‡ is the free energy (see, Nel-son and Cox [2005], pp. 489–506.) Svante Arrhenius couldnot have imagined the development of micromanipulationtechniques that allow measurement and control of piconew-ton size forces and nanometer size displacements with timeresolution as short as a millisecond (see Figure 9). Suchmodern techniques have made it possible to probe, under

Figure 8. Scatterplot of PAn cognates versus the percentage ofsample from haplogroup O. The correlation is 0.627 (Lansing et al.,2007).

Figure 9. Single molecule experiments. Set-up for kinesin attachedto a bead, translocating along a microtubule subject to optical twee-zers pulling on bead. (Walton, 2002).

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mechanical load, the kinetics of biomolecular processes atthe single-molecule level. The goal of this research is toexamine the Arrhenius relationship at this level.

Taking displacement to be our reaction coordinate, thefree energy �G‡ may be split into the sum of the unloadedfree energy, and the work done against or assisted by

the external force, F,

�G‡ � �G0‡ � F�

where � is the displacement to the transition state. The signof � depends on whether the applied force helps or hindersthe reaction. Setting k0 to be the zero-force rate, we have

k � k0eF�/kBT

for the form of the force-dependent rates in the system.At the level of a single interaction, then for a simple reaction,

bound3 unbound,

describing an irreversible detachment of two molecules withno substeps, the physics leads us to two assumptions:

• The dwell time is based on thermal fluctuations and thuspossesses the memorylessness property. In other words,the dwell times are random and follow an exponentialdistribution (see Ross [2009]). Thus, the density of dwelltimes takes the form

f�t � �� � � exp���t�

for some rate � � 0. The mean of this random variable is 1/�.

• The Arrenhius relation holds at the level of a single mol-ecule, i.e., the mean dwell time equals

�0e � F�/kBT, where �0 �1k0

.

Consequently, � � eF�/kBT/�0 and the density of dwell timesis

f�t � �0, �, F� �1�0

eF�/kBTexp ��t�0

eF�/kBT�.

The force F is under the control of the experimentalist. Theparameters �0 and � need to be estimated from data. More-over, because we have no way of estimating either �0 or �separately, they need to be estimated simultaneously.

Previously published methods used seat of the pants boot-strapping techniques. The graduate student who broughtthis problem to my attention performed some numericalexperimentation, and it appears that the estimators do noteven converge as the number of observations increases.However, students can apply the concepts in the course todesign most powerful tests and with it asymptotically nar-rowest possible confidence intervals.

Statistical ProceduresIf our data are from an experiment with independentlymeasured forces F � (F1, F2, …, Fn) and correspondingindependent dwell times t � (t1, t2, …, tn), we have anexplicit expression for the likelihood function L(�0,��t, F).The equations for the maximum likelihood estimators,

�

��L��0, � � t, F� � 0 and

�

��0L��0, � � t, F� � 0,

do not separate to produce closed forms for the maximumlikelihood estimators and However, they can be re-

arranged to obtain two algebraic relationships.

�0 �1n�

i � 1

n

tieFi � ⁄ kBT and �0 �1

nF� �i � 1

n

tiFieFi� ⁄ kBT.

This facilitates a simple graphical interpretation for and

as the intersection point of the curves determined by

these two relationships.This experiment has not yet been performed (Kalafut,

Liang, Watkins, and Visscher, unpublished results), so weask the students to simulate data and find the maximumlikelihood estimates. This gives us the opportunity to dis-cuss the value of generating data via simulation and test theinference methods on these data before moving to the actualdata. To keep the situation simple, we take � � 1 and �0 � 1.We also take the forces uniformly distributed between 0 andkbT. As can be seen from the simulated data, any relationshipin Figure 10 between the dwell times and the force would bedifficult to discern by inspection.

To simulate the data with n � 1000:

� F�-runif(1000); t�-rep(0,1000)� for(k in 1:1000){t[k]�-rexp(1,exp(F[k]))}� plot(F,t)

If we solve numerically for the estimates in this simula-tion, we obtain and for the intersection of the two

curves defined above. This is displayed graphically in Fig-ure 11.

Figure 10. Force versus dwell time for the simulated data. TheArrhenius relationship is not easy to visualize in the scatterplot of1000 observations.

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CBE—Life Sciences Education308

The covariance matrix for � 1.008 and � 1.011 can

be approximated by the inverse of the Fisher informationmatrix. In this case, we can compute this matrix explicitlyand take its inverse.

I��0, �� �1 �1

nvar�F�� �02F� 2 �0kBTF�

�0kBTF� �kBT�2 �.

This information matrix is the fundamental object that di-rects the design of an experiment. For example, we canreduce the variance of our estimators and and hence

the length of confidence intervals by making the variance ofF as large as possible. This is accomplished by having theforce be as small as the laser trap allows and as large aspossible maintaining the stability for the single molecule. Onthe other hand the variance depends on the product nvar(F).Thus, the ability to perform many measurements may be aneasier experiment to perform to achieve a desired length forconfidence intervals.

The remaining three projects are drawn from questions onbacterial growth and division, on otoacoustic emissions inthe human ear, and on flour beetle population dynamics.The expectation is that if the student has a genuine interestin the life sciences, then at least one of these projects will beenticing.

ASSESSMENT AND IMPACT

At the University of Arizona, we are seeing a transformationin the view that mathematics plays in the education of lifescientist students. To see this evolution reflected in the sta-tistics class, we saw six students choose to add a mathemat-ics minor during the first time the course was given. For thesecond and third time, the students entered the class asmathematics minors. The course, offered for a fourth time

this fall, is full and the capacity is being increased to accom-modate demand.

The head of our math center is now writing to studentswho have completed the first two years of mathematicscourse work and is encouraging them to take the calculus-based statistics course. In addition, more than a fourth of thestudents (26 of 98) in the Undergraduate Biology ResearchProgram are either mathematics majors or minors. Severalresearchers who have had statistics students in their lab arerecommending it to other students.

From the course evaluations, the students especially liked“the applicability of statistics to multiple areas of interest,”that “the homework sets were challenging and engaging,”“relating course work to software used in research,” having“taken stats classes before I feel like I actually learned someof the basics of stats in this one,” “learn(ing) about the realworld,” “forc(ing) me to learn a subject I really didn’t enjoy(I like stats now),” that “the study examples were particu-larly effective as motivation to learn the material,” “using Rand relating everything to the real world,” and that “thehomework was nicely challenging.” Thus, we can see thatstudents are favorably impressed by the applicability of thecourse and the ability of theoretical considerations to lead tostrategies for addressing practical problems. After somegentle reminding, they recognize that the material in someof the more mathy courses in their past were essential intheir ability to address life science questions with theirpresent level of understanding. Students are showing morefacility with software as they continue to use it and havinga general sense that mathematical and statistical issues areaspects of the challenges that come with exciting research.

The students, especially those who are working in a lab-oratory, were invited to select their own topic of interest.About a third of the students picked this option. Yeastgenetics, earthquake prediction, rat behavior, fruit flybehavior, strength of material, educational value of cer-tain curricula and evaluation methods, land subsidence,malaria prevalence, and the effects of monetary policywere among the chosen topics. Most of these student-generated projects resulted in a presentation in their re-search group’s lab meeting.

To get a sense of the source and motivation of this trans-formation, we are now systematically gathering informationon changes in student and educator attitude as a conse-quence of a variety of experiences, including this course.Analysis of these data will bring further insights into thetype of attitudes that students who choose this course haveand how their attitude is impacted by the experiences of thecourse.

DISCUSSION

The National Research Council Committee on Undergradu-ate Biology Education to Prepare Research Scientists for the21st Century (2003) produced BIO2010: Transforming Under-graduate Education for Future Research Biologists for the Na-tional Research Council of the National Academies. Theyremark that “Mathematics teaching presents a special case.Most biology majors take no more than one year of calculus,although some also take an additional semester of statistics.Very few are exposed to discrete mathematics, linear alge

Figure 11. Maximum likelihood estimation. The two curves are thetwo expressions for as a function of for the simulated data.

Here � 1.008 and � 1.011. The actual values, �0 � 1 and � � 1,

are indicated by the red .

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bra, probability, and modeling topics, which could greatlyenhance their future research careers.” The issues of thesynergies of these mathematics subjects are left largely un-explored in BIO2010. Some connections are well known andhave become standard in the curriculum. Knowledge oflinear algebra is certainly essential in learning both linearmodels in statistics and multidimensional differential equa-tions. Students with a solid background in discrete mathe-matics have an advantage in the early stages of a course inprobability. Any additional course will bring its comple-ment of tools for a modeling course. Here we explore thehow the combined effect of the unfolding understanding ofbiology and proficiency in calculus impact the students’ability to gain a firm foundation in statistical science.

As the course described in this article settles on an ap-proach and a curriculum, we can now see that the uses ofcalculus in these contexts turn out to be neither particularlydifficult nor novel for a student comfortable with the subject.In addition, after a couple of frustrating weeks with syntax,students continue acquiring skills in the use of statisticalsoftware. What is difficult for the students is the increasedmaturity necessary to apprehend the breadth of applicabilitythat statistics brings to any carefully conceived data-richexploration. In the end, having the tools of algebra andcalculus, with probability as a frame of reference and readyaccess to computational software, students can kindle excite-ment in their new discoveries in the life science and bringthemselves to a level of understanding that is not accessibleabsent the combined use of these quantitative tools.

Despite the aspiration inherent in the title BIO2010, westill have much work to do to create the type of curriculumin the quantitative sciences that suits the demands of thenext generation of life science researchers. This effort towarda calculus-based statistics course as a small contributionamong many other efforts seems to be bringing us closer tothat goal.

ACKNOWLEDGMENTSI thank the Howard Hughes Medical Institute (HHMI) BiomathCommittee at the University of Arizona for many thought-provok-ing discussions. Student work and their feedback were essential insetting and refining the pedagogical approaches in the course. Aspecial note of thanks goes to Christopher Bergevin, who wasco-instructor the first time the course was taught, and to CarolBender, the Director of the Undergraduate Biology Research Pro-gram, whose efforts have provided many undergraduate studentswith the opportunity to experience an authentic research experi-ence. The activities described in this article were supported in partby a grant to the University of Arizona from the HHMI (52005889).

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