Introduction to Statistics at DTU

1 Introduction to Statisticsat DTUPer B. Brockhoff, Jan K. M ller, Elisabeth W. AndersenPeder Bacher, Lasse E. Christiansen2018 Fall(with minor updates 2022 May) xDensity xDensityn= 1X N(0,1) xDensity xDensity xDensity xDensityn= 5X U(0,1) xDensity xDensity xDensity xn= 30 DensityX Exp(1) xDensity xDensity X N( , 2n)Chapter 0 Contents1 Introduction , descriptive Statistics , R and data is Statistics - a primer .. at DTU Compute .. - why, what, how? .. Statistics .. of centrality .. of variability .. of relation: correlation and covariance .. to R and RStudio .. and scripts .. and vectors .. Statistics .. of R in the course and at the exam.

2 , graphics - data visualisation .. distributions and the histogram .. distributions .. box plot and the modified box plot .. Scatter plot .. plots and Pie charts .. plots in R? ..392 Probability and variable .. random variables .. to simulation .. and variance .. distributions .. distribution .. distribution .. distribution .. random variables .. and Variance .. distributions .. distribution .. distribution .. distribution .. distribution .. of random variables .. for the mean and variance .. and correlation .. of random variables .. Functions of normal random variables .. The 2-distribution.

3 Thet-distribution .. TheF-distribution .. Exercises .. 1123 Statistics for one and two from one-sample quantitative data .. of the sample mean .. the precision of the sample mean - the confi-dence interval .. language of Statistics and the process of learning fromdata .. we cannot assume a normal distribution: the Cen-tral Limit Theorem .. sampling interpretation of confidence intervals . interval for the variance .. testing, evidence, significance and thep-value and how to check them .. towards normality .. from two-sample quantitative data .. two independent means - confidence two independent means - hypothesis test.

4 Paired design and analysis .. of assumptions with normality investigations . a study: wanted precision and power .. Size for wanted precision .. size and statistical power .. size in two-sample setup .. 1934 Simulation Based and Simulation .. as a general computational tool .. of error .. parametric bootstrap .. confidence interval for .. confidence interval for any feature assumingany distribution .. confidence intervals assuming any distribu-tions .. non-parametric bootstrap .. confidence interval for .. confidence interval for any feature .. confidence intervals .. a further perspective.

5 Bootstrapping with theboot-package .. 2405 Simple Linear regression and least squares .. estimates and estimators .. are central .. of estimators .. and testing of parameters .. and prediction intervals for the line .. formulation of simple linear regression .. on the sample correlation coefficient .. and regression .. validation .. 2826 Multiple Linear estimation .. and prediction intervals for the line .. regression .. analysis .. regression in R .. formulation .. and prediction intervals for the line .. 3127 Inference for data .. of single proportions .. hypotheses.

6 Size determination .. proportions in two populations .. several proportions .. of Contingency Tables .. several groups .. between the two categorical variables .. 3438 Comparing means of multiple groups - .. ANOVA .. structure and model .. of variability, the ANOVA table .. hoc comparisons .. control .. complete worked through example: plastic types forlamps .. ANOVA .. structure and model .. of variability and the ANOVA table .. hoc comparisons .. control .. complete worked through example: Car tires .. 383 Glossaries393 Acronyms398A Collection of formulas and R Introduction , descriptive Statistics , R and data visualization.

7 Probability and Simulation .. Distributions .. Statistics for one and two samples .. Simulation based Statistics .. Simple linear regression .. Multiple linear regression .. Inference for proportions .. Comparing means of multiple groups - ANOVA .. 414 The plot on the front page is an illustration of the Central Limit Theorem (CLT). Toput it shortly, it states that when sampling a population: as the sample size increases,then the mean of the sample converges to a normal distribution no matter the distri-bution of the population. The thumb rule is that the normal distribution can be usedfor the sample mean when the sample sizenis above 30 observations (nis the numberobservations in the sample).

8 The plot is created by simulating 100000 sample means X= ni=1Xi(whereXiis an observation from a distribution) and plotting their his-togram with the CLT distribution on top (the red linie). The upper is for the normal, themid is for the uniform and the lower is for the exponential distribution. We can thus seethat asnincrease, then the distribution of the simulated sample means xapproachesthe distribution stated by the CLT (it is the normal distribution X N( , 2n), where is the mean and is the standard deviation of the population), see more in 11 Chapter 1 Introduction , descriptive Statistics , Rand data visualizationThis is the first chapter in the eight-chapter DTU Introduction to Statistics consists of eight chapters:1.

9 Introduction , descriptive Statistics ,Rand data visualization2. Probability and simulation3. Statistical analysis of one and two sample data4. Statistics by simulation5. Simple linear regression6. Multiple linear regression7. Analysis of categorical data8. Analysis of variance (analysis of multi-group data)In this first chapter the idea of Statistics is introduced together with some of thebasic summary Statistics and data visualization methods. The software usedthroughout the book for working with Statistics , probability and data analysis isthe open source environmentR. An Introduction toRis included in this WHAT IS Statistics - A What is Statistics - a primerTo catch your attention we will start out trying to give an impression of theimportance of Statistics in modern science and the well respectedNew England Journal of medicinea millennium editorial onthe development of medical research in a thousand years was written:EDITORIAL: Looking Back on the Millennium in Medicine,N Engl J Med, 342:42-49, January 6, 2000, came up with a list of 11 points summarizing the most important devel-opments for the health of mankind in a millennium.

10 Elucidation of human anatomy and physiology Discovery of cells and their substructures Elucidation of the chemistry of life Application of Statistics to medicine Development of anaesthesia Discovery of the relation of microbes to disease Elucidation of inheritance and genetics Knowledge of the immune system Development of body imaging Discovery of antimicrobial agents Development of molecular pharmacotherapyThe reason for showing the list here is pretty obvious: one of the points isAp-plication of Statistics to Medicine! Considering the other points on the list, andwhat the state of medical knowledge was around 1000 years ago, it is obviouslya very impressive list of developments.