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Descriptive Statistics and Exploratory Data Analysis - Bivariate • Quantitative (continuous) variables • Scatterplots (two variables; use color or symbol to add 3rd variable) • Starplots • Correlation coefficient • Qualitative (categorical) variables • Contingency (two-way) tables • Joint, marginal, conditional distribution • Simpson’s paradox (confounding) • Interaction Fall 2002 Biostat 511 50 Scatterplot A scatterplot offers a convenient way of visualizing the relationship between pairs of quantitative variables. Many interesting features can be seen in a scatterplot including the overall pattern (i.e. linear, nonlinear, periodic), strength and direction of the relationship, and outliers (values which are far from the bulk of the data). Fall 2002 Biostat 511 51 Scatterplot showing nonlinear relationship O O O Scatterplot showing daily rainfall amount (mm) at nearby stations in SW Australia. Note outliers (O). Are they data errors … or interesting science?! Fall 2002 Biostat 511 52 Presentation matters! Fall 2002 Biostat 511 53 - Important information can be seen in two dimensions that isn’t obvious in one dimension Fall 2002 Biostat 511 54 Use symbols or colors to add a third variable Fall 2002 Biostat 511 55 Plots for Multivariate data Star plots are used to display multivariate data • Each ray corresponds to a variable • Rays scaled from smallest to largest value in dataset Fall 2002 Biostat 511 56 Correlation How can we summarize the “strength of association” between two variables in a scatterplot? Fall 2002 Biostat 511 57 Correlation When two variables are measured on a scale in which order is meaningful, you can calculate a correlation coefficient that measures the strength of the association between the two variables. There are two common correlation measures: 1. Pearson Correlation Coefficient: Based on the actual data values. Measure of linear association. Natural when each variable has a normal distribution. 2. Spearman Rank Correlation: Based on ranks of each variable (ranks assigned separately). Useful measure of the monotone association, which may not be linear. Fall 2002 Biostat 511 58 Pearson’s Correlation Coefficient The correlation between two variables X and Y is: Properties: • No distinction between x and y. • The correlation is constrained: -1 £ R £ +1 • | R | = 1 means “perfect linear relationship” • The correlation is a scale free measure (correlation doesn’t change if there is a linear change in units). • Pearson’s correlation only measures strength of linear relationship. • Pearson’s correlation is sensitive to outliers. Fall 2002 Biostat 511 59 Perfect positive correlation (R = 1) Perfect negative correlation (R = -1) Uncorrelated (R = 0) but dependent Fall 2002 Biostat 511 60 Fall 2002 Biostat 511 61 Pearson’s Correlation Coefficient Correlation = .8776 Suppose we restrict the range of X … Correlation = .5111 • relationship between LSAT and GPA among law school students • relationship between height and basketball ability among NBA players Fall 2002 Biostat 511 62 Spearman Rank Correlation • A nonparametric analogue to Pearson’s correlation coefficient is Spearman’s rank correlation coefficient. Use Spearman’s correlation when the assumption of normality of X and Y is not met. • A measure of monotonic association (not necessarily linear) • Based on the ranked data • Rank each sample separately (1 … N) • Compute Pearson’s correlation on the ranks • -1 < Rs < 1 Fall 2002 Biostat 511 63 Two-way (Contingency) Tables Now we turn our attention to relationships between pairs of qualitative (categorical, discrete) measures. Types of Categorical Data: •Nominal •Ordinal Often we wish to assess whether two factors are related. To do so we construct an R x C table that cross-classifies the observations according to the two factors. Such a table is called a two-way or contingency table. Fall 2002 Biostat 511 64 Two-way tables Example. Education versus willingness to participate in a study of a vaccine to prevent HIV infection if the study was to start tomorrow. Counts, percents and row and column totals are given. The table displays the joint distribution of education and willingness to participate. Fall 2002 Biostat 511 65 Two-way tables The marginal distributions of a two-way table are simply the distributions of each measure summed over the other. E.g. Willingness to participate Fall 2002 Biostat 511 66 Two-way tables A conditional distribution is the distribution of one measure conditional on (given the) value of the other measure. E.g. Willingness to participate among those with a college education. Fall 2002 Biostat 511 67 Two-way tables What proportion of individuals … • will definitely participate? • have less than college education? • will probably or definitely participate given less than college education? • who will probably or definitely participate have have less than college education? • have a graduate/prof degree and will definitely not participate? Fall 2002 Biostat 511 68 Three-way tables There are two phenomena that can confuse our interpretation of two-way tables. In each case a third measure is involved. Simpson’s Paradox - Also known as confounding in the epidemiology literature. MM refer to this as the “lurking variable” problem. Aggregating over a third (lurking) variable results in incorrect interpretation of the association between the two primary variables of interest. Interaction - Also known as effect modification in the epidemiology literature. The degree of association between the two primary variables depends on a third variable. Fall 2002 Biostat 511 69 Simpson’s Paradox (aka Confounding) “Condom Use increases the risk of STD” BUT ... Explanation: Individuals with more partners are more likely to use condoms. But individuals with more partners are also more likely to get STD. Fall 2002 Biostat 511 70 Interaction (aka Effect Modification) Fall 2002 Biostat 511 71 Summary • Quantitative (continuous) variables Scatterplots - display relationship between two quantitative measures. Use colors or symbols to add a third (categorical) dimension. Starplots - display multivariate data. Correlation coefficient - summarizes the strength of the linear (Pearson’s) or monotonic (Spearman’s) relationship between two quantitative measures. • Qualitative (categorical) variables Contingency table – shows the joint distribution of the two variables, the marginal distributions of each variable and the conditional distribution of one variable for a fixed level of the other variable. Simpson’s paradox and interactions can occur if a third variable influences the association between the two variables of interest. Fall 2002 Biostat 511 72 Guidelines for Tables and Graphs • Tables • Good for showing exact values, small amounts of data • Guidelines • Graphs • Good for showing qualitative trends, large amounts of data • Guidelines for graphical integrity Fall 2002 Biostat 511 73 Tables and Graphs • Compact presentation of data • Visual appeal; readers feel that they are “seeing the data” • Tables are better for showing exact numerical values, small amounts of data and/or multiple localized comparisons • Graphs are better for highlighting qualitative aspects of the data and displaying large amounts of data. Fall 2002 Biostat 511 74 Guidelines for Tables (Ehrenberg, 1977) 1. Give marginal averages to provide a visual focus. 2. Order rows/columns by marginal averages or some other measure of size. 3. Put groups to be compared in rows (i.e. scanning down columns for comparisons) 4. Round to 2 effective digits 5. Use layout to facilitate comparisons 6. Give brief verbal summaries to lead reader to patterns and exception. 7. Clearly label rows and columns, give units, source (if appropriate), title. Fall 2002 Biostat 511 75 Unemployment in Great Britain(source: Facts in Focus, CSO, 1974). Note use of marginal averages and rounding. Table has been reordered so the reader can scan down the column for a time trend. Fall 2002 Biostat 511 76 Fall 2002 Biostat 511 77 Statistical Graphics “Modern data graphics can do much more than simply substitute for small statistical tables. At their best, graphics are instruments for reasoning about quantitative information. Often the most effective way to describe, explore, and summarize a set of numbers - even a very large set - is to look at pictures of those numbers.” Edward R. Tufte The Visual Display of Quantitative Information Graphics Press, 1983 Fall 2002 Biostat 511 78 Graphical Integrity 1. The representation of numbers, as physically measured on the surface of the graphic, should be directly proportional to the numerical quantities represented (e.g. purchasing power). 2. Clear, detailed and thorough labeling should be used to defeat graphical distortion and ambiguity. Write out explanations of the data on the graphic itself. Label important events in the data. (e.g. Minard’s graphic) 3. Focus on the data, not the design and maximize the data:ink ratio (counter e.g. USA Today) 4. The number of information-carrying (variable) dimensions depicted should not exceed the number of dimensions in the data (e.g. OPEC Oil) 5. Do not quote data out of context (e.g. traffic deaths) Fall 2002 Biostat 511 79 Fall 2002 Biostat 511 80 A less distorted view … Fall 2002 Biostat 511 81 Fall 2002 Biostat 511 82 Data density - Compare ... Fall 2002 Biostat 511 83 Fall 2002 Biostat 511 84 Fall 2002 Biostat 511 85 Fall 2002 Biostat 511 86 Summary • Tables • Good for showing exact values, small amounts of data • Guidelines • Graphs • Good for showing qualitative trends, large amounts of data • Guidelines for graphical integrity Fall 2002 Biostat 511 87 Designing Studies • Design issues • Types of studies a. Experimental studies - Control, randomization, replication b. Observational • Controls • Blinding • Hawthorne effect • Longitudinal/cross-sectional • Dropout • Population vs Sample • Bias • Variability Fall 2002 Biostat 511 88 Experimental Design “Obtaining valid results from a test program calls for commitment to sound statistical design. In fact, proper experimental design is more important than sophisticated statistical analysis. Results of a well- planned experiment are often evident from simple graphical analyses. However, the world’s best statistical analysis cannot rescue a poorly planned experiment.” Gerald Hahn, Encyclopedia of Statistical Science, page 359, entry for Design of Experiments Fall 2002 Biostat 511 89 Types of Studies Most scientific studies can be classified into one of two broad categories: 1) Experimental Studies The investigator deliberately sets one or more factors to a specific level. 2) Observational Studies The investigator collects data from an existing situation and does not (intentionally) interfere with the running of the system. Fall 2002 Biostat 511 90 Experimental Studies • Sources of (major) variability are controlled by the researcher • Randomization is often used to ensure that uncontrolled factors do not bias results • The experiment is replicated on many subjects (units) to reduce the effect of chance variation • Easier to make the case for causation Examples • effect of pesticide exposure on hatching of eggs • comparison of two treatments for preventing perinatal transmission of HIV Fall 2002 Biostat 511 91 Example: control of variability by matching Hypothesis: Lotions A and B equally effective at softening skin Fall 2002 Biostat 511 92 Design 1: Ignore pairing, randomly assign half of the hands to each lotion. What is the distribution of the sample mean difference in softness, if the “true” difference is 3? Design 2: Randomly assign lotion to one hand within each pair. What is the distribution of the sample mean difference in softness, if the true difference is 3? Fall 2002 Biostat 511 93 Observational Studies • Sources of variability (in the outcome) are not controlled by the researcher • Adjustment for imbalances between groups, if possible, occurs at the analysis phase • Randomization usually not an option; samples are assumed to be “representative” • Can identify association, but usually difficult to infer causation Examples • natural history of HIV infection • study of partners of individuals with gonorrhea • condom use and STD prevention • association between chess playing and reading skill in elementary school children Fall 2002 Biostat 511 94 Other Study Design Issues •Selection of controls •Blinding •Hawthorne effect •Longitudinal vs Cross-sectional •Dropouts Fall 2002 Biostat 511 95 Longitudinal vs Cross-sectional Studies • Longitudinal studies are more expensive and involve additional analytical complications. • Longitudinal studies allow one to study changes over time in individuals and populations (similar to idea of pairing or matching) Fall 2002 Biostat 511 96 Reading Ability Age Age Age Hypothetical data on the relationship between reading ability and age. Fall 2002 Biostat 511 97 Populations vs Samples So far we haven’t thought very hard about where our data come from. However, in almost all cases there is an implicit assumption that the conclusions we draw from our data analysis apply to some larger group than just the individuals we measured. Population Sample •set of all “units” •a subset of “units” •real or hypothetical •estimates/statistics •parameters e.g. population - all US households with a TV(~95 million) sample - Nielsen sample (~5000) The objective of statistics is to make valid inferences about the population from the sample. Fall 2002 Biostat 511 98 Fall 2002 sam ple of siz samp en le of size n sample of size n n of size sample Biostat 511 iz en Population of X’s ofs (true proportion = p) le samp 99 In making such inferences, there are two ways we can go wrong … Bias • Do I expect that, on average, the estimate from my sample will equal the parameter of the population of interest? If so, the estimate is unbiased. e.g. Ann Landers survey Pap smear study • In general, statistical methods do not correct for bias (Sampling) Variability •If I repeat an experiment (draw a new sample), I don’t expect to get exactly the same results. The sample estimates are variable. •The aim of experimental design and statistical analysis is to quantify/control effects of variability. Fall 2002 Biostat 511 100 Fall 2002 Biostat 511 101 Types of samples in medical studies - a hierarchy 1) Probability samples (e.g. simple random sample, stratified samples, multistage samples) 2) Representative samples (no obvious bias, but …) 3) Convienence samples (biases likely …) 4) Anecdotal, Case reports Fall 2002 Biostat 511 102 Problems in Design/Data Collection Example: 33% reduction in blood pressure after treatment with medication in a sample of 60 hypertensive men. Problem: Example: Daytime telephone interview of voting preferences Problem: Example: Higher proportion of “abnormal” values on tests performed in 1990 than a comparable sample taken in 1980. Problem: Fall 2002 Biostat 511 103 Summary 1. Statistics plays a role from study conception to study reporting. 2. Statistics is concerned with making valid inferences about populations from samples that are subject to various sources of variability. 3. Different studies require different statistical approaches. You must understand the study design and sampling procedures before you can hope to interpret the data!! Fall 2002 Biostat 511 104

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