Much of this is standard and essential—organizing categorical and quantitative data, appropriately displayed as bar charts, histograms, boxplots, time plots, and scatterplots, and summarized using medians, quartiles, means, weighted means, trimmed means, standard deviations, and regression lines. We use this as an opportunity to introduce students to the statistical software package R (R Development Core Team, 2009) and to add additional summaries like the empirical cumulative distribution function and the empirical survival function. One example integrating the use of this is the comparison of the lifetimes of wild-type and transgenic mosquitoes and a discussion of the best strategy to display and summarize data if the goal is to examine the differences in these two genotypes of mosquitoes in their ability to carry and spread malaria. A bit later, we will do an integration by parts exercise to show that the mean lifetime of the mosquitoes is the area under the survival function.

Collecting data under a good design is introduced early in the course, and discussion of the underlying principles of experimental design is an abiding issue throughout this course. With each new mathematical or statistical concept comes an enhanced understanding of what an experiment might uncover through a more sophisticated design than what was previously thought possible. The students are given readings on design of experiment and examples using R to create simple and stratified random samples. The recommended lecture is to tell a story that illustrates how these issues appear in personal research activities.

Probability theory is the analysis of random phenomena. It is built on the axioms of probability and is explored, for example, through the introduction of random variables. The goal of probability theory is to uncover properties arising from the phenomena under study. Statistics is devoted to the analysis of data. The goal of statistical theory is to articulate as well as possible what model of random phenomena underlies the production of the data. The focus of this section of the course is to develop those probabilistic ideas that relate most directly to the needs of statistics.

Thus, we must study the axioms of probability to the extent that the students understand conditional probability and independence. Conditional probability is necessary to develop Bayes formula, which we will later use to give a taste of the Bayesian approach to statistics. Independence will be needed to describe the likelihood function in the case of an experimental design that is based on independent observations. Densities for continuous random variables and mass function for discrete random variables are necessary to write these likelihood functions explicitly. Expectation will be used to standardize a sample sum or sample mean 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 random variable, arise from considerations based on Bernoulli trials or exponential waiting. The hypergeometric random variable helps us understand the difference between sampling with and without replacement. The F, t, and χ² random variables will later become test statistics. Uniform random variables are the ones simulated by random number generators. Because of the central limit theorem, the normal family is the most important among the list of parametric families of random variables.

The flavor of the course returns to becoming more authentically statistical with the law of large numbers and the central limit theorem. These are largely developed using simulation explorations and first applied to simple Monte Carlo techniques and importance sampling to evaluate integrals. One cautionary tale is an example of the failure of these simulation techniques when applied without careful analysis. If one uses, for example, Cauchy random variables in the evaluation of some quantity, then the simulated sample means can appear to be converging only to experience an abrupt and unpredictable jump. The lack of convergence of an improper integral reveals the difficulty.

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

In the simplest possible terms, the goal of estimation theory is to answer the question: What is that number? An estimator is a statistic (i.e., a function of the data). We look to two types of estimation techniques—method of moments and maximum likelihood and several criteria for an estimator (e.g., bias and variance). Several examples including capture-recapture and the distribution of fitness effects are developed for both types of estimators. The variance is estimated using the delta method for method of moments estimators and using Fisher information for maximum likelihood estimators. An analysis of bias is based on quadratic Taylor series approximations and the properties of expectations. R is routinely used in simulations to gain insight into the quality of estimators.

The point estimation techniques are followed by interval estimation and, notably, by confidence intervals. This brings us to the familiar one- and two-sample t intervals for population means and one- and two-sample z intervals for population proportions. In addition, we can return to the delta method and the observed Fisher information to construct confidence intervals associated, respectively, with method of moment estimators and maximum likelihood estimators.

For 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 tests for a simple hypothesis. This is motivated by a game. Extensions of this result, known as the Neyman Pearson lemma, form the basis for the t test for means, the χ² test for goodness of fit, and the F test for analysis of variance. These results follow from the application of optimization techniques from calculus, including Langrange multiplier techniques to develop goodness of fit tests.

The desire of a powerful test is articulated in a variety of ways. In engineering terms, power is called sensitivity. We illustrate this with a radon detector. An insensitive instrument is a risky purchase. This can be either because the instrument is substandard in the detection of fluctuations or poor in the statistical test that results in an algorithm to announce a change in radon level. An insensitive detector has the undesirable property of not sounding its alarm when the radon level has indeed risen.

The course ends by looking at the logic of hypotheses testing and the results of different likelihood ratio analyses applied to a variety of experimental designs. The delta method allows us to extend the resulting test statistics to multivariate nonlinear transformations of the data.

Given observations (x₁, y₁), (x₂, y₂), …(x_n, y_n), ordinary linear regression has as its goal to find the best linear fit g(x|α, β) = α + βx to the data. The least squares criterion refers to the minimization of the sum of squares over all real parameter values α and β. Calling α̂ and β̂ the values that achieve this minimum, we obtain the regression line fit to the response γ_i. Ordinary linear regression is a staple of any introductory statistics course [see, e.g., Moore et al. (2007) or Agresti and Franklin (2008)]. Students compute regression lines using software or a formula and learn to check the appropriateness of the regression line by examining the residuals (i.e., the differences between the data and the fit). Informally, a good fit would show no structure in the residual plot. Two departures from this are common. Either

the size of the residuals depends on x, or

Here, we focus on the second case in which the scatterplots display a curved relationship. With the tools of college algebra, this relationship can be transformed to be suitable for linear regression by guessing the relationship and applying the inverse transformation to the response variable. This is seen most frequently in scatterplots of a quantity over time exhibiting exponential growth or decay. Thus, we take the logarithm of the response variable and proceed.

This strategy is sometimes too superficial to be successful. For students who have been acquainted with differential equations, we can consider a common chemical reaction, 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 different concentrations [S] of the substrate to obtain the following data: (See Figure 1.) Naive attempts to give a transformation that yields a linear relationship are unlikely to succeed. Despite intense speculation, students have never been able to guess correctly. A model-based approach is needed. With this in mind, we begin with the law of mass action to obtain differential equations for the concentrations The strategy used by Michaelis and Menten applies to situations in which the concentration of the substrate-bound enzyme (and hence also the unbound enzyme) change much more slowly than those of the product and substrate. If, by measuring the change of [ES] directly over time or by arguing that if the concentration of enzyme [E] is small compared with the concentration of substrate [S], we find that the Michaelis constant. Equipped with these ideas, we find after some algebraic manipulation, the well-known Lineweaver-Burk double reciprocal plot In this way, we have found that the desired linear relationship is between 1/V and 1/[S]. We can now perform ordinary least squares on the transformed data and estimate V_max, the maximum rate of production, and K_m. [See, for example, Nelson and Cox (2005), pp. 204–210.]

However, this classical approach to estimation is no longer considered satisfactory (Piegorsch and Bailer, 2005). Examination of residual plots reveals the drawback—the residuals measured as the reciprocal of the concentration become magnified for very small concentrations. Consequently, the preferred method is to use a nonlinear least squares criterion. In this case, given data (V₁, [S]₁), (V₂, [S]₂), … (V_n, [S]_n), find the values of V_max and [S] that minimize where This minimization problem does not yield explicit equations for the estimators and Rather than embarking on a minicourse on numerical techniques for optimization, we use this opportunity to discuss the intricacies of the nonlinear regression analysis and strategies to engage an expert in statistical science on the science, the experimental protocol, and the data.

One of the aspects of any new course is the understandable uncertainty by the students that the approaches presented in class have applicability to questions of their own concern. Thus, in the problem sets, we ask the students to use the ideas presented in lecture and make substantial use of them in another biological context. In this case, the ideas on nonlinear transformations are further explored by the students in the following question addressed by Wiehe and Stephan (1993):

Due to selection, neutral sites near genes are hitchhiked along with the gene to higher levels in the population until a recombination event separates the neutral site from the selected gene. This reduces the diversity of the genome near genes. The nucleotide diversity π versus recombination rate ρ is given for 17 gene regions in Drosophila melanoganster. After a short amount of reasoning based on the nature of recombination and selection, the students learn the relationship This expression is also amenable to a Lineweaver-Burk double reciprocal plot. Thus, we rewrite this expression as a linear expression in the variables 1/π and 1/ρ. As the scatterplot 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 or lurking variable. In this data set, we are likely looking at genes that range from small selection that will not reduce diversity much to large selection that will. In the same way that estimated value of V_max gives the reaction rate at high concentration of the substrate, the estimated value of α gives the nucleotide diversity at recombination distances ρ far away from genes. This value gives a sense of the intrinsic diversity of the Drosophila genome due solely to the effects of neutral mutation and demographic history.

George Polya (1920) used the phrase the central limit theorem because of its centrality to the theory of probability. In simple terms, the central limit theorem states that, irrespective of the distribution of the observations from a simple random sample, its sample mean has, for sufficiently many observations, approximately a normal distribution. Many of the commonly used hypotheses rely on a test statistic that is based on the normal distribution. Thus, the use of the t test, the χ² test, and the analysis of variance F test depends on the appropriateness of the normal distribution as an approximation for the distribution of sample means computed from the data. So, understanding the use of these statistics depends in part on grasping the logic behind the central limit theorem.

Unfortunately, the best proofs of the central limit theorem rely on indirect and sophisticated methods. Moreover, these proofs yield very little insight into the emergence, with an increasing number of observations, of the bell curve for the distribution of Consequently, rather than proving the central limit theorem in an introductory statistics course at this level, the typical pedagogical choice is to use graphical and other empirical methods to convey the key ideas (see Ross (2009) or Pitman (1999) for a proof).

Most elementary courses dedicate a week to the understanding of the central limit theorem. This development might culminate with a graph like that seen in Figure 3. This shows the standardization of random variables and relates it to the test statistics that the students will later encounter in the study of hypothesis testing. This can then be used to estimate probabilities for X̄ by evaluating z scores.

The extension we can have in a calculus-based course is motivated by interest in nonlinear transformations of the measured quantities. Some physics, chemistry, and engineering students have seen a bit of this in the study of propagation of error (see, e.g., Meyer [1975], Bevington and Robinson [2002]). Having investigated the properties of variance and covariance in the context of the independence of random variables, the goal is to combine the benefits of the central limit theorem and the propagation of error analyses.

After investigating some one-dimensional examples like those seen in Figure 4, the students are prepared to examine a more sophisticated example. For this, we consider a suggestion by Powell (2007) to questions in avian biology. To introduce one of his examples, define the fecundity B as the mean number of female fledglings per year. Then, B is a product of three quantities, where F equals the mean number of female fledglings per successful nest, p equals nest survival probability, and N equals the mean number of nests built per female per year. Let's collect measurements on n₁ nests to count female fledglings in a successful nest, check n₂ nests for survival probability, and follow n₃ females to count the number of successful nests per year. To begin, assume that the experimental design is structured so that measurements are independent.

A critical value for B is 1. If B is greater than 1, then the population grows over time. If it is less than 1, then the population is headed toward extinction. So, for example, if our estimate B̂ = 1.03, then we may or may not be confident that the actual value of B is greater than 1 depending on the way the estimator B̂ distributes its values.

The central limit theorem can directly determine appropriate normal distributions to approximate F̄, the sample mean of the number of female fledglings per successful nest, p̂, the sample proportion of surviving nests, and N̄, the sample mean number of nests built per female per year. If our estimate B̂ of the fecundity is the product of these three numbers, then what is a good approximation for the distribution of this statistic?

Propagation of error analysis suggests that we find a linear approximation to B. Because the measurements for F, p, and N are independent, then σ_B², the variance of B̂ 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 analysis by noting that, by the central limit theorem, the random quantity given by the linear approximation can be approximated by a normal distribution. Returning to the question of estimating fecundity, the delta method tells us that B̂ can be approximated by a normal distribution. The variance can be determined from σ_F², the variance in the fecundity measurement and σ_N², the variance in the number of nests built per adult female per year. By combining this information, we derive in the class an expression that highlights in the contribution to the variance in the estimate of fecundity originating from each of the three measurements: This formula now plays an essential role in settling on a data collecting protocol. With a goal to bring the standard deviation σ_B down as much as possible given available resources, the choices of n₁, n₂, and n₃ are under control of the biologists. In addition, if independence of observations between the estimation of F, p, and N is an issue, we can extend the propagation of error and with it the delta method by including the covariances between the pairs of the sets of observations.

This application of the calculus greatly extends the applicability of the central limit theorem in the practical application of statistics. This idea is explored by the students who apply the delta method to describe the estimator for the focal length f of a convex lens based on repeated measurement of the distance, s₁, to an object and the distance, s₂, to its image using the thin lens formula, (See Figure 5.)

The appearances of the delta method do not end here. We shall use this technique to determine the variance for a method of moments estimator. Later, we can use these ideas to construct test statistics for hypothesis tests beyond those generally encountered in more elementary courses.

The study of statistical hypotheses begins with a shot of jargon that will take a student some time to absorb. Understanding the terminology is important because it codifies the relationship between hypothesis testing and scientific advancement. Under a simple hypothesis, we write the test as: H₀ is called the null hypothesis. H₁ is called the alternative hypothesis. The possible actions are:

Reject the hypothesis. Rejecting the hypothesis when it is true is called a type I error or a false positive. Its probability α is called the size of the test or the significance level.

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 L₁ as low as possible. The natural start is to pick values of x so that L₁ (x) = 0. Then, the points you collect begin to add up without your opponent gaining anything.

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

Understanding the next choice is crucial. Candidates are After some discussion, the class will come to the conclusion that having a high ratio is more valuable than having a high difference. The choice 6 against 1 is better than 9 against 2 because choosing 6 against 1 twice will put us in a better place than the single choice of 9 against 2. Now we can pick the next few candidates, keeping track of the size and the power of the test with the choice of critical region being the values of x not yet chosen (see Table 2).

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

Let L₀ denote the likelihood function for the random variable X corresponding to H₀ and L₁ denote the likelihood function for the random variable X corresponding to H₁. If there exists a critical region C of size α and a nonnegative constant k such that then C is the most powerful critical region of size α.

Using R, we can complete the table for L₀ total and L₁ total.

> o<-order(L1/L0)

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

Now the students are prepared to understand the reasoning behind the use of critical values for the t statistic, the χ² statistic, or the F statistic as related to the critical value k in extensions of the ideas of likelihood ratios and can extend the likelihood ratio tests to more novel situations.

These examples have been presented here in such a way as to emphasize transitions from the use of algebra to the use of calculus in the learning of concepts in statistics. However, students do not notice such a shift in the sense that they do not see calculus as a special tool. Limits need to be evaluated, rates change, areas under curves need to be determined, functions need to be maximized, and their concavity needs to be assessed. Moreover, calculus is more than a powerful computational tool: it provides a way of thinking that enhances the students' view of what is possible. I asked a student how much is calculus used in the course. The answer she gave, “So much we don't think about it,“ is a goal for this course.

For many years, quantitative subjects as diverse as physics and economics have developed distinct pedagogies for an algebra-based and for a calculus-based course. For example, Newton's second law applied to projectile motion or to the motion of springs gives, for the calculus student, simple differential equations from which the basic algebraic or trigonometric relationships for the variables position, velocity, acceleration, and time can be derived. For the algebra student, these relationships have to be taken on faith. In a similar way, students can receive a more comprehensive and elegant understanding of the fundamental principles of statistical science if they are equipped with a working knowledge of calculus.

Human populations and the languages that they speak change over time. The movement of people and the innovations they make in their languages are difficult to observe and quantify over short periods and impossible to witness over long periods. Consequently, researchers have been forced to undertake indirect approaches to infer associations between human languages and human genes. Many well-known studies focus their questions on the movement of people on continental scales. These studies led Diamond and Bellwood (2003) to suggest that many of the correlations that we presently see in languages and genes result from the movement of prehistoric farmers from the places in which these agricultural techniques first arose.

Using the Indonesian island of Sumba as a test case, Lansing et al. (2007) look to see the degree in which these correlations can be seen at much smaller scales. Both genetic and archeological evidence place the first migration of anatomically modern humans in Southeast Asia and Oceana between 40,000 and 45,000 years ago. Further archeological evidence places the transition on Sumba from the original hunter-gatherer technology to the neolithic technology between 3500 and 4000 years ago. At that time, a small number of farmers speaking an Austronesian language likely came into contact with resident foragers speaking presumably a Papuan language.

To investigate the paternal histories of the Sumbanese, Lansing et al. (2007) obtained genetic samples from 352 men inhabiting eight villages. Their genetic information is derived from the Y chromosome. Based on the Y Chromosome Consortium worldwide genealogical tree of paternal ancestry, haplogroup O appears to be associated with the expansion of Austronesian societies from southeast Asia to Indonesia and Oceana.

The linguistic data consist of 29 200-word Swadesh lists from sites well distributed throughout the island. Swadesh lists are built from words ascribed to meanings that are basic to everyday life. Using the methodology of comparative linguistics, some words from different language lists can be traced to a common ancestral word. These techniques have been used previously to construct a Proto-Austronesian (PAn) language. In addition, a phylogenetic tree built from these 29 word lists branches to form five major language subgroups and leads us to the conclusion that the present day Sumbanese all speak a language derived from a single common ancestral Proto-Sumbanese. From this we can count the number of words on the Swadesh word lists that are derived from the PAn language. This information is summarized in Figure 7 (Lansing et al., 2007).

One of the most widely used relations in chemistry is the Arrhenius relation, Here k is a reaction rate, k_B is Boltzmann's constant, T is the absolute temperature, and ΔG^‡ is the free energy (see, Nelson and Cox [2005], pp. 489–506.) Svante Arrhenius could not have imagined the development of micromanipulation techniques that allow measurement and control of piconewton size forces and nanometer size displacements with time resolution as short as a millisecond (see Figure 9). Such modern techniques have made it possible to probe, under mechanical load, the kinetics of biomolecular processes at the single-molecule level. The goal of this research is to examine the Arrhenius relationship at this level.

Taking displacement to be our reaction coordinate, the free energy ΔG^‡ may be split into the sum of the unloaded free energy, and the work done against or assisted by the external force, F, where δ is the displacement to the transition state. The sign of δ depends on whether the applied force helps or hinders the reaction. Setting k₀ to be the zero-force rate, we have for the form of the force-dependent rates in the system.

At the level of a single interaction, then for a simple reaction, describing an irreversible detachment of two molecules with no substeps, the physics leads us to two assumptions:

The dwell time is based on thermal fluctuations and thus possesses the memorylessness property. In other words, the dwell times are random and follow an exponential distribution (see Ross [2009]). Thus, the density of dwell times takes the form for some rate λ > 0. The mean of this random variable is 1/λ.

Previously published methods used seat of the pants bootstrapping techniques. The graduate student who brought this problem to my attention performed some numerical experimentation, and it appears that the estimators do not even converge as the number of observations increases. However, students can apply the concepts in the course to design most powerful tests and with it asymptotically narrowest possible confidence intervals.