This chapter reviews differential calculus, which gave a tremendous impetus to the development of both pure and applied mathematics. It covers optimization and polynomial approximation of functions, together with a powerful numerical method for finding roots. It also shows how calculus can provide all the ‘pre-calculus’ results about roots and turning-points for the basic functions, including an analysis of more complicated differentiable functions. The chapter emphasizes the crucial step of interpreting the function or graph in order to extract significant information from which practical conclusions can be drawn. It highlights the use of the function f (x) and its first and second derivatives f '(x) and f ''(x) to identify significant features of the graph of a general smooth function f (x).
Applications of differentiation
Applying our simulation approach beyond null hypothesis testing: parameter estimation, bayesian, and model-selection contexts
This chapter talks about the performance of a simulation analysis in the contexts of the precision of a parameter estimate, Bayesian statistics, and model selection. It describes null hypothesis testing as an absolute dominant form of statistical inquiry that remains important to a lot of researchers in many different areas of the life sciences. It also reviews toolkits of statistical approaches which are in common use alongside null hypothesis testing. The chapter reviews the simulation approach to power in the context of null hypothesis testing and other contexts that estimate the size of a parameter or select from a number of alternative models. It uses examples to explore simulation analysis in the context of precision of parameter estimation.
Arithmetic and algebra
This chapter gives an abstract approach that builds up the rules from basic concepts of real numbers and arithmetic operations. It presents mathematics in a bioscience context and includes case studies that show how a range of different mathematical techniques are needed in the development of a biological topic. It also introduces geometric growth, which is used to model colonies of certain populations in situations where the birth rate exceeds the mortality rate, there are sufficient supplies of food, and there is an absence of predators or diseases. The chapter describes a mathematical model that expresses the rates of change of the densities of the three types of cell: healthy, stage one cancer, and angiogenic cells. It examines the evaluation of numerical expressions and algebraic expressions involving fractions, exponents, and roots.
Biomeasurement provides an introduction to the use of statistics in the biosciences, emphasizing why statistical tools are essential tools for bioscientists. It begins by placing data analysis in the context of the wider scientific method and introducing key statistical concepts. It discusses inferring and estimating before showing how to choose the right test and graph and introducing null hypothesis significance testing. The following chapters focus on a variety of test types, including tests of difference and tests of relationship, such as regression and correlation. The book then introduces the generalized linear model, including logistic and loglinear models.
Choosing the right test and graph
This chapter details the process of choosing the right test and graph for analysing bioscience data. There are four main reasons for producing graphs: to understand the data; to explore the data for potential patterns; to assess whether the data conform to the assumptions of a particular analysis; and to help communicate the results. The rules for choosing the right graph are less hard and fast than those for choosing the right test. When using graphs for communication, the bottom line is that the graph should communicate the pattern in the data fairly and clearly. The chapter considers frequency distributions, pie charts, boxplots, errorplots, scatterplots, and line graphs. The chapter also explains when to choose null hypothesis significance testing (NHST) and which test to use, differentiating between tests of frequencies, tests of relationship, and tests of difference. Finally, it demonstrates how to use SPSS to produce four useful graph types.
This chapter describes communication in the context of biomedical science practice ranging from the everyday interactions between colleagues regarding clinical matters to the dissemination of the results of research. It outlines how to communicate effectively with various colleagues, such as presenting the findings of studies internally or externally as abstracts to learned societies, writing formal documents, and preparing data for publication in a journal. It also describes the key aspects of the layout of a PowerPoint presentation, the major sections of a written communication, the significant elements of an abstract, and the leading features of a research paper. The chapter promotes the correct passage of information, which is crucial for the safe and efficient working of a routine laboratory and is a key part of good laboratory practice. It emphasizes how failing to communicate in a clinical setting endangers life or can lead to poor experimental or process results.
This chapter brings together a range of different techniques which provide measures of association and agreement between theoretical models and experimental data, and also between different experimental measurements of the same quantities. It begins by developing the statistics of parametric correlation and introducing methods of nonparametric correlation. The chapter then looks at statistics used for testing association, and discusses Fisher's exact test and the ability to test for progression in factors. It also considers the strength of the association between factors, and reviews a range of possible methods of measurement. Finally, the chapter assesses the concept of agreement in various contexts, including the 'goodness of fit' of analytical models, and agreements between variables and within contingency tables.
Martin B Reed
Core Maths for the Biosciences consists of two parts. Part 1 looks atconsiders arithmetic, algebra, and functions. Here, chapters cover precision and accuracy, data tables, graphs, molarity and dilutions, variables, functions, equations, and linear functions. They also look at quadratic and polynomial functions, fitting curves, periodic functions, and exponential and logarithmic functions. Part 2 looks atfocusses on calculus and differential equations. It Chapters examines instantaneous rate-of-change, the rules of differentiation, applications of differentiation, techniques of integration, and the definite integral.
Andrew D. Blann
Data Handling starts off with an analysis of information in the biomedical sciences. It then considers handling quantities which encompasses mass, volume, and concentration. It moves on to obtaining and verifying data. Next, it looks at presenting data in graphic form. Another chapter considers quality, audit, and good laboratory practice. The next three chapters are about research, setting the scene, the analysis of modest data sets, and large data sets. Finally, the text ends with an examination of communication methods.
Data tables, graphs, interpolation
This chapter talks about finding patterns as the essence of mathematics, which can be done if the data involves measurement of two or more physical variables. It illustrates how to plot graphs of data-points on paper and in MS Excel, and it demonstrates the basic patterns to look for, such as direct and inverse proportion, and linear relations. It also elaborates how to construct a data table and a data plot, which begins by tabulating the data and then plotting the data values as points on a graph. The chapter highlights a typical experiment, wherein the experimenter allows the independent variable to increase or decrease and measures the value of the dependent variable. It clarifies how data-point plots show the relationship between the variables through a graph.
Dealing with multiple hypotheses
This chapter explains how to evaluate the powers for study designs with multiple hypotheses, emphasizing that no single design will offer exactly the same power for all the hypotheses. It points out that ranking hypotheses in terms of their importance can help with the process of finding a design that offers sufficient power for all the hypotheses of prime importance. It also discusses how to evaluate whether the power available for testing ancillary hypotheses is sufficient to continue to include them in the study after one has settled on a design with sufficient power for the most important hypotheses. The chapter explores how to select a single experiment offering good power to address all hypotheses and involve further simulations after carrying out power analyses for each of the hypotheses in isolation. It examines power analyses that flag up situations where bonus investigations are of little value. This is because an experiment optimized to address questions of primary interest means there is low power to address bonus hypotheses.
The definite integral
This chapter provides a geometric and physical interpretation of the integral, which leads to the concept of the definite integral as a numerical quantity instead of a function. It highlights practical applications of the definite integral, including a powerful method for evaluating definite integrals numerically. It also demonstrates how to calculate a definite integral by working out the indefinite integral F(x)| = (x) dx and omitting the constant of integration. The chapter points out that areas below the x-axis are counted as negative, noting the importance of checking that the function does not cross the x-axis anywhere within the interval. It talks about a concept of the definite integral that is independent of the idea of differentiation, which can be used to define the definite integral mathematically as a process that involves taking the limit called Riemann integration.
Describing a single sample
This chapter discusses how to conduct analyses to describe a single sample of data. We can do this by producing summary numbers, names, tables, or pictures (otherwise known as graphs, charts, or figures). The chapter begins by explaining when and how to calculate descriptive statistics for the central tendency (mean, median, and mode) and variability (range, interquartile range, standard deviation, and variance) of a sample. Descriptive statistics are numbers or names used to summarize the information in a sample. The chapter then considers the attributes, types, and importance of frequency distribution tables and graphs. It also introduces pie charts, boxplots, and error bars before demonstrating how to use Statistical Package for the Social Sciences (SPSS) to produce descriptive statistics and graphs of a single sample.
Differential equations I
This chapter deals with the differential equation, which is an equation involving a derivative and its solution requires finding y as a function of x. It points out that finding solutions for differential equations require basic algebra, the theory of functions, and differential and integral calculus. It also classifies the different types of differential equation, including methods for solving first-order differential equations. The chapter covers solution techniques applied to mathematical models of biological processes and numerical techniques implemented in MS Excel, which produce the solution as a set of data-points that can be graphed. It mentions the main classification of differential equations in terms of their order and the concept of boundary conditions.
Differential equations II
This chapter introduces numerical methods that can be used on problems that are too hard to solve by standard algebra, such as finding roots of complicated equations. It demonstrates numerical methods for solving a first-order ordinary differential equations (ODE) with an initial condition. It also looks at problems involving two or more simultaneous ODEs, particularly the equations relating the growth of populations of different species in a predator–prey situation. The chapter explains how a numerical method produces a set of data-points that lie on or close to the true solution curve. It analyses the accuracy of Euler's method, which can be improved by reducing the steplength or taking shorter steps between data-points.
Exponential and logarithmic functions
This chapter deals with exponential and logarithmic functions, which constitute arguably the most important area of mathematics for bioscientists. This is because many nonlinear natural processes involve some form of exponential growth or exponential decay. The chapter talks about the natural process of exponential growth and relates it to the geometric growth model. It also cites an example wherein a cell culture of 100 cells doubles in size each hour, which can be modelled by the function N(t) = 100.2t . The chapter introduces allometry, which is the study of power-law relationships between physical characteristics of an organism; these usually take the mass of the organism as the basic characteristic.
Extension: dynamical systems
This chapter focuses on dynamical systems, wherein the simplest nonlinear mathematical models could give rise to incredibly complex behaviour. It mentions scientists studying biological and physical processes of nature that were in the forefront of the revolution in mathematics for discovering dynamical systems. It also introduces some basic concepts of complexity theory: equilibria and stability, bifurcations, and chaos. The chapter begins with a brief snapshot of the birth of chaos theory, particularly theories of pattern formation and ecological implications. It cites the truncation error, which is another source of error in numerical methods that happens by approximating the true function by a Taylor series that is truncated after the first two or three terms.
Extensions to other designs
This chapter covers how to simulate experimental studies where the predictor variable is continuous and there is a straight-line relationship between the response and the predictor. Moreover, it explores simulations that include other kinds of relationships, such as a quadratic one. It also demonstrates how to deal with response variables which are not continuous, as well as with variation that does not follow a normal distribution. The chapter looks at another common type of biological study in which both the response and the predictor variable are continuous. It talks about the possibility of having a different number of replicates at each level and explores situations where the relationship is either linear or non-linear.
Fitting curves; rational and inverse functions
This chapter looks at two concepts that can provide functions: the reciprocal of a function and the inverse of a function. It discusses a class of functions called rational functions, which have important applications in the biosciences. It also explores how the functions produced can be used with real data, such as how to use data-points obtained from experiment or observation and how to determine the values of the parameters in the function. The chapter demonstrates how to find the values of the parameters m and c from two data-points, including how to put a best-fit straight line through to more than two data-points. It reviews a technique that can fit general curves to data and reveals how the equation of the best-fit straight line is calculated in MS Excel.
This chapter focuses on frequency data. It develops techniques which analyse the frequencies and probabilities of observed events. The chapter begins by presenting analyses relevant to single samples of categorical data, e.g. data descriptions and 'goodness of fit' test. It then develops the statistics of the contingency table and the use of cross-tabulation to generate the table from individual observations. Finally, the chapter looks at the analysis of binary systems using logistic regression and the description of probabilities using receiver operating characteristics (ROC) plots. The name 'receiver operating characteristics' derives from its early use to describe the properties of signal detectors in recording either a true or false signal.