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1 Chapter 1Figure 1.1 Using statistics for predictions. Age‐ and sex‐specific prevalenc...Figure 1.2 Examples of commonly used ordinal scales.Figure 1.3 Difference between an ordinal and an interval scale.Figure 1.4 Comparison of the mean and the median in an asymmetrical distribu...Figure 1.5 Classical view of the purpose of sampling.Figure 1.6 Relationship between representativeness and sample size in the cl...Figure 1.7 Modern view of the purpose of sampling. The purpose of sampling i...Figure 1.8 Inference with binary attributes.Figure 1.9 Inference with interval attributes I.Figure 1.10 Inference with interval attributes II.Figure 1.11 Measures of dispersion derived from measures of location.Figure 1.12 The n divisor of the sum of squares.Figure 1.13 The n − 1 divisor of the sum of squares.Figure 1.14 Relationship between a proportion and its variance.Figure 1.15 Two random variables with uniform distribution.Figure 1.16 Properties of means and variances.Figure 1.17 Table of mean and variance properties.Figure 1.18 Tabulation of nominal data.Figure 1.19 Typical presentation of several nominal attributes in a single t...Figure 1.20 Tabulation of ordinal and interval data.Figure 1.21 Table with summary statistics describing the information collect...Figure 1.22 Illustration of the phenomenon of sampling variation. Above are ...Figure 1.23 Frequency distributions of some biological variables.Figure 1.24 The origin of the normal distribution.Figure 1.25 Frequency distribution of sums of identical variables with unifo...Figure 1.26 Properties of the normal distribution.Figure 1.27 Relationship between the area under the normal curve and the sta...Figure 1.28 The total obtained from the throw of six dice may be seen as the...Figure 1.29 Distribution of sample means of different sample sizes.Figure 1.30 Comparison of the computation of sample and population statistic...Figure 1.31 Illustration of the phenomenon of sampling variation. Above, pie...Figure 1.32 Probability distribution of a proportion: the binomial distribut...Figure 1.33 The convergence of the binomial to the normal distribution.

2 Chapter 2Figure 2.1 The path to causality. Types of research studies.Figure 2.2 Classification of descriptive studies and corresponding populatio...Figure 2.3 Schema of a prevalence study.Figure 2.4 Steps in the inference of the value of the population mean from t...Figure 2.5 Statistical table of the normal distribution I.Figure 2.6 Statistical table of the normal distribution II.Figure 2.7 Relationship between the sample size and the spread of the sample...Figure 2.8 The error in the determination of confidence limits from standard...Figure 2.9 Several Student’s t distributions with different degrees of freed...Figure 2.10 Number of estimated standard errors on each side of the mean tha...Figure 2.11 Steps in the construction of 95% confidence intervals using Stud...Figure 2.12 Example of utilization of a statistical table of Student’s t dis...Figure 2.13 Steps in the construction of the 95% confidence interval of the ...Figure 2.14 Example of a statistical table of the binomial distribution.Figure 2.15 Relationship between sample size requirements and desired error ...Figure 2.16 Types of events.Figure 2.17 Organization of an incidence study.Figure 2.18 Schema of a cohort study.Figure 2.19 Closed and open cohort. Horizontal lines represent subjects obse...Figure 2.20 Illustration of the person‐time method.Figure 2.21 Illustration of the person‐time method applied to an event that ...Figure 2.22 The direct standardization method.Figure 2.23 The indirect standardization method.Figure 2.24 Time‐to‐event data.Figure 2.25 The actuarial method.Figure 2.26 Illustration of the actuarial method showing on the left the stu...Figure 2.27 Actuarial curve.Figure 2.28 Illustration of the Kaplan–Meier method showing on the left the ...Figure 2.29 Kaplan–Meier curve.Figure 2.30 Kaplan–Meier curve with 95% confidence bands.Figure 2.31 Representation of simple and systematic sampling. Each square re...Figure 2.32 Comparison of the binomial and the hypergeometric distributions....Figure 2.33 Decreasing variance by splitting the population into strata.Figure 2.34 The different types of stratified sampling.Figure 2.35 Sample size requirements for simple and stratified sampling.Figure 2.36 Multistage and cluster sampling.Figure 2.37 Illustration of two‐stage sampling combined with sampling with p...

3 Chapter 3Figure 3.1 Data from a prevalence study of COPD in the general population.Figure 3.2 Probability distributions of proportions, odds, and logits.Figure 3.3 Attributable fraction in the population and among the exposed.Figure 3.4 Classification of observational analytical studies.Figure 3.5 Schema of an uncontrolled cross‐sectional analytical study. Black...Figure 3.6 Schema of an uncontrolled analytical cohort study. Black subjects...Figure 3.7 Models of the relationship between exposure and outcome. In (a), ...Figure 3.8 Schema of a case–control study. Black subjects are the exposed.Figure 3.9 Schema of a cohort study. Black subjects are the cases.Figure 3.10 Steps in the construction of 95% confidence intervals for the di...Figure 3.11 Steps in the construction of 95% confidence intervals for the di...Figure 3.12 Steps in the construction of 95% confidence intervals for the di...

4 Chapter 4Figure 4.1 Rationale of the z‐test for large samples.Figure 4.2 Finding the exact p‐value using a statistical table of the normal...Figure 4.3 The rationale of Student’s t‐test in the case of small samples.Figure 4.4 Using the table of the t distribution to find exact p‐values.Figure 4.5 Observed and expected frequencies under H₀.Figure 4.6 The chi‐square distribution under the null hypothesis for a 2 × 2...Figure 4.7 The chi‐square distribution with 1 to 10 degrees of freedom. The ...Figure 4.8 Statistical table of the chi‐square distribution.Figure 4.9 Observations in three samples of distinct populations (left panel...Figure 4.10 Procedure of ANOVA.Figure 4.11 Procedure of ANOVA with unequal sample sizes.Figure 4.12 Left ventricular ejection fraction in three groups classified ac...Figure 4.13 The F distribution with 2 and 90 degrees of freedom. The dark ar...Figure 4.14 Section of a statistical table of the F distribution.Figure 4.15 ANOVA table of the data of Figure 4.12.

5 Chapter 5Figure 5.1 Difference between two‐sided and one‐sided tests.Figure 5.2 Distribution of differences between sample means under a one‐side...Figure 5.3 The alpha and beta errors of a statistical test.Figure 5.4 Distribution of the differences between sample means under the nu...Figure 5.5 Effect of increasing sample size ratios on the statistical power ...Figure 5.6 Comparison of the distribution of the differences between means u...Figure 5.7 Commonly used transformations to the normal distribution.Figure 5.8 Rationale of a non‐parametric test for comparing two independent ...

6 Chapter 6Figure 6.1 Representation of the relationship between two interval variables...Figure 6.2 Rationale of the least squares method I.Figure 6.3 Rationale of the least squares method II.Figure 6.4 Rationale of the least‐squares method III.Figure 6.5 Sampling variation of the intercepts and slopes of regression lin...Figure 6.6 The slope m of a line connecting to an observation (x, y) is ....Figure 6.7 Steps in the calculation of the standard error of the regression ...Figure 6.8 Scatterplots with regression lines illustrating different situati...Figure 6.9 Illustration of the relationship between the total variance of Y ...Figure 6.10 Under H₀ the variance of Y accounts for the difference (y − y^*...Figure 6.11 Regression analysis of FVC on height.Figure 6.12 Multiple regression analysis of FVC on sex adjusted for height t...Figure 6.13 Multiple regression analysis of FVC on two predictors.Figure 6.14 Residual plot of the regression of FVC on height and age.Figure 6.15 Patterns in residual plots suggestive of non‐normality of the re...Figure 6.16 Influence of an outlier (open circle) on the regression estimate...Figure 6.17 Results of the regression of FVC on three dummy variables encodi...Figure 6.18 Results of the regression of PEF on sex and height.Figure 6.19 Scatterplot of the data of the regression of PEF on sex and heig...Figure 6.20 Results of the regression of PEF on sex and height with a term f...Figure 6.21 Exemplification of an interaction effect on the relationship of ...Figure 6.22 Comparison of models obtained by linear regression (top) and pol...

7 Chapter 7Figure 7.1 Illustration of the problems of fitting a least‐squares line when...Figure 7.2 The least‐squares line fitted to the data of Figure 7.1 but with ...Figure 7.3 Transformation of the dependent variable back into proportions af...Figure 7.4 Likelihood function. The plotted values p are binomial probabilit...Figure 7.5 Illustration of the maximum likelihood estimation of regression c...Figure 7.6 Logistic regression line obtained with the method of maximum like...Figure 7.7 Results of logistic regression analysis of sex on height.Figure 7.8 Classification table of observed to predicted.Figure 7.9 The ROC curve.Figure 7.10 Development of a risk stratification system.Figure 7.11 ROC curves on the testing set and validation set.Figure 7.12 Illustration of the Hosmer–Lemeshow goodness‐of‐fit test.

8 Chapter 8Figure 8.1 Rationale and procedure of the paired t‐test.Figure 8.2 Most commonly used non‐parametric statistical tests.Figure 8.3 Overall regression line fit over clustered observations (left) an...Figure 8.4 The logrank test is used to compare two or more Kaplan–Meier curv...Figure 8.5 Illustration of the logrank test for the comparison of Kaplan–Mei...Figure 8.6 Illustration of the adjusted logrank test.Figure 8.7 Hazard rate in women diagnosed with breast cancer.Figure 8.8 Illustration of the proportionality assumption.Figure 8.9 Visual assessment of the proportionality assumption with log–log ...Figure 8.10 Example of the result of a Cox regression.

9 Chapter 9Figure 9.1 Dataset and correlation matrix of the factor analysis example.Figure 9.2 Data points representing the value of the questionnaire items def...Figure 9.3 Factor analysis. The first factor is positioned through the cente...Figure 9.4 Factor analysis. The second factor is positioned at a right angle...Figure 9.5 Factor analysis. Eigenvalues for the five factors. The eigenvalue...Figure 9.6 Scree plot.Figure 9.7 Factor loadings of the item questionnaires on the two retained fa...Figure 9.8 Orthogonal rotation of the factors shown in Figure 9.4.Figure 9.9 Oblique rotation of the factors shown in Figure 9.4.Figure 9.10 Factor loadings after orthogonal (left) and oblique rotation (ri...Figure 9.11 Covariance matrix. The diagonal elements are shaded.Figure 9.12 Relationship of Cronbach’s alpha with the number of k uncorrelat...Figure 9.13 Item analysis.Figure 9.14 Three situations with equal correlation coefficients but differe...Figure 9.15 Illustration of the results of a test–retest study with ANOVA ta...Figure 9.16 Cohen’s kappa.Figure 9.17 Weighted kappa.Figure 9.18 Bland–Altman diagram.

10 Chapter 10Figure 10.1 Conditions for the presumption of causality.Figure 10.2 Control groups of typical experimental designs and clinical tria...Figure 10.3 Pseudo‐replicates and replicates: sacrifice replication.Figure 10.4 Simple pseudo‐replication.Figure 10.5 Randomization scheme in a completely randomized design.Figure 10.6 Effect of two different treatments A and B on the same interval ...Figure 10.7 Alternative experimental design that requires only half the numb...Figure 10.8 Procedure of a two‐way ANOVA.Figure 10.9 Two‐way ANOVA table.Figure 10.10 Comparison of results in the absence of interaction between two...Figure 10.11 Procedure of ANOVA for a full factorial design.Figure 10.12 Full factorial design for the data of an experiment with two tr...Figure 10.13 ANOVA table of the data in Figure 10.12.Figure 10.14 Multiple regression analysis of the data in Figure 10.12. These...Figure 10.15 Interaction between two treatments: the difference between leve...Figure 10.16 ANOVA model II.Figure 10.17 ANOVA table for model I and model II.Figure 10.18 Full factorial ANOVA.Figure 10.19 Components of variance of a fixed effects factorial ANOVA model...Figure 10.20 Components of variance of a random effects factorial ANOVA (mod...Figure 10.21 Components of variance of a mixed effects factorial ANOVA (mode...Figure 10.22 Steps to define the expected mean squares in a mixed model ANOV...

11 Chapter 11Figure 11.1 Schematic representation of the process of allocation of experim...Figure 11.2 Procedure of a factorial ANOVA without replication for the analy...Figure 11.3 Example of an experiment with a randomized complete block design...Figure 11.4 ANOVA table from an experiment with a randomized complete block ...Figure 11.5 Experiment with a generalized randomized block design, that is, ...Figure 11.6 ANOVA table of the data in Figure 11.5.Figure 11.7 Treatment × block interaction. The differences between treatment...Figure 11.8 Examples of balanced incomplete block designs. To the left, four...Figure 11.9 Layout of an experiment based on a factorial design with randomi...Figure 11.10 Expected mean squares (left) and ANOVA table of the model and d...Figure 11.11 Latin square. On the left, the table with which the planning of...Figure 11.12 Matrix representation of the square in Figure 11.11, on the lef...Figure 11.13 Data of the experiment with a Latin square design and ANOVA tab...Figure 11.14 Overlapping two 3 × 3 Latin squares to form a Greco‐Latin squar...Figure 11.15 Matrix representation of the square in Figure 11.14.

12 Chapter 12Figure 12.1 Value of the alpha error as a function of the number of comparis...Figure 12.2 Data for illustration of Bonferroni correction.Figure 12.3 Data for illustration of the calculation of confidence intervals...Figure 12.4 Holm–Bonferroni procedure.Figure 12.5 Table of the studentized range distribution.Figure 12.6 Studentized range distribution for k = 5 observations. The black...Figure 12.7 Procedure of Tukey’s test.Figure 12.8 Data for the calculation of Tukey’s test.Figure 12.9 Illustration of the procedure for the Student–Newman–Keuls test....Figure 12.10 Data for the exemplification of the Student–Newman–Keuls test....Figure 12.11 Statistical table of Dunnett’s test for α = 0.05.Figure 12.12 Comparison between several post hoc tests of the minimum differ...Figure 12.13 Examples of the use of contrasts.Figure 12.14 Data for exemplifying the test of a linear contrast.Figure 12.15 ANOVA from the test of a linear contrast.Figure 12.16 Examples of orthogonal contrasts and ANOVA table of the simulta...Figure 12.17 Coefficients of orthogonal contrast for three, four, and five g...Figure 12.18 ANOVA table of the test of non‐orthogonal contrasts.

13 Chapter 13Figure 13.1 Layout of a 3 × 3 × 2 factorial design.Figure 13.2 ANOVA table of the data in Figure 13.1.Figure 13.3 Interaction ARB × CCB. The differences in the means of the depen...Figure 13.4 Design matrices for 2² and 2³ factorial designs.Figure 13.5 Initial analysis matrix for 2² and 2³ factorial designs.Figure 13.6 Analysis matrix for the 2³ factorial design of Figure 13.5 after...Figure 13.7 Illustration of the steps in the creation of the design matrix f...Figure 13.8 Final design matrix with center points (to the left) and the lay...Figure 13.9 Results of the experiment of the example in Figure 13.8.Figure 13.10 ANOVA table of the experiment illustrated in Figure 13.9.Figure 13.11 Multiple regression of the experiment illustrated in Figure 13....Figure 13.12 Procedure for creating the design matrix for a 2³ factorial exp...Figure 13.13 Design matrix of a full 2³ factorial design and an alternative ...Figure 13.14 Procedure for creating a 2³⁻¹ factorial design.

14 Chapter 14Figure 14.1 Scheme of a restricted randomization: for each level of factor A...Figure 14.2 Scheme of the allocation process of the experimental units to th...Figure 14.3 Layout of a split–plot design.Figure 14.4 ANOVA table of a split–plot design.Figure 14.5 Nested design.Figure 14.6 Scheme of the process of allocation of the experimental units to...Figure 14.7 Results of an experiment with a nested design, with cages nested...Figure 14.8 Determination of expected mean squares in a mixed model nested A...Figure 14.9 ANOVA table for the data in Figure 14.7.Figure 14.10 Results of an experiment with a mixed model nested design.Figure 14.11 Nested ANOVA table of the data in Figure 14.10.Figure 14.12 Results of an experiment with a model II nested design.Figure 14.13 Determination of the expected mean squares in pure model II nes...Figure 14.14 Model II nested ANOVA table of the data in Figure 14.12.

15 Chapter 15Figure 15.1 Verification of the sphericity assumption: the variances of the ...Figure 15.2 Covariance matrix (left) and centered covariance matrix (right) ...Figure 15.3 Two‐way ANOVA table of the data in Figure 15.2.Figure 15.4 Layout of an experiment with two factors and repeated measuremen...Figure 15.5 Table of the two‐way ANOVA of the data in Figure 15.4.Figure 15.6 Steps in the definition of the expected mean squares and error t...

16 Chapter 16Figure 16.1 Diagram of a controlled clinical trial.Figure 16.2 Uses of placebo for maintaining double‐blinding. The placebo is ...Figure 16.3 Randomization methods used in controlled clinical trials.Figure 16.4 Concealed randomization using opaque envelopes.Figure 16.5 The one‐group post‐test‐only design.Figure 16.6 The one‐group pre‐test/post‐test design.Figure 16.7 The one‐group pre‐test/post‐test with double pre‐test design.Figure 16.8 The one‐group pre‐test/post‐test with a non‐equivalent dependent...Figure 16.9 The removed‐treatment design.Figure 16.10 The repeated‐treatment design.Figure 16.11 Illustration of the phenomenon of regression toward the mean. T...Figure 16.12 The non‐equivalent groups with post‐test‐only design.Figure 16.13 The non‐equivalent groups with pre‐test and post‐test design.Figure 16.14 The non‐equivalent groups with double pre‐test and post‐test de...Figure 16.15 The non‐equivalent groups using pre‐test and post‐test with swi...Figure 16.16 The parallel group design.Figure 16.17 The crossover design.Figure 16.18 The 2 × 2 factorial design.Figure 16.19 Randomized withdrawal design. In the first period, all trial su...Figure 16.20 Delayed start design. In the first period, trial subjects are r...Figure 16.21 Illustration of the results of a delayed start design with (a) ...Figure 16.22 Designs of cluster randomized trials. Each horizontal bar in th...Figure 16.23 Interpretation of the results of equivalence (left) and non‐inf...Figure 16.24 Testing for non‐inferiority.Figure 16.25 Opportunities for optimization of a clinical trial design: (a) ...Figure 16.26 Nominal p‐values for different methods of defining stopping bou...Figure 16.27 Example of an alpha spending function illustrating two interim ...Figure 16.28 Types of alpha spending functions: Pocock type (gray line) and ...Figure 16.29 Illustration of a sequence of interim analyses in a clinical tr...

17 Chapter 17Figure 17.1 Several populations that can be identified in a clinical trial....Figure 17.2 Study populations eligible for analysis.Figure 17.3 Serial and parallel gatekeeping strategies.

18 Chapter 18Figure 18.1 Data to illustrate the calculations of the heterogeneity test in...Figure 18.2 Data to illustrate the calculations of the inverse variance meth...Figure 18.3 Rationale of the fixed and random effects models in meta‐analysi...Figure 18.4 Data for the illustration of the DerSimonian–Laird method.Figure 18.5 L’Abbé plot: solid line, line of equality; dashed line, meta‐ana...Figure 18.6 Galbraith plot for the detection of heterogeneity.Figure 18.7 Funnel plot for the identification of publication bias: solid li...Figure 18.8 Begg’s test: (a) absence of publication bias; (b) presence of pu...Figure 18.9 Egger’s test: (a) absence of publication bias; (b) presence of p...Figure 18.10 Forest plots. The same data analyzed with the fixed effects mod...