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Introduction to Biostatistics for Clinical and Translational Researchers KUMC Departments of Biostatistics & Internal Medicine University of Kansas Cancer Center FRONTIERS: The Heartland Institute of Clinical and Translational Research

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Course Information Jo A. Wick, PhD Office Location: 5028 Robinson Lectures are recorded and posted at under ‘Educational Opportunities’

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Course Objectives Understand the role of statistics in the scientific process Understand features, strengths and limitations of descriptive, observational and experimental studies Distinguish between association and causation Understand roles of chance, bias and confounding in the evaluation of research

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Course Calendar June 29: Descriptive Statistics and Core Concepts July 6: Hypothesis Testing July 13: Linear Regression & Survival Analysis July 20: Clinical Trial & Experimental Design

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“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.” Albert Einstein ( )

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Basic Concepts

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Statistics is a collection of procedures and principles for gathering data and analyzing information to help people make decisions when faced with uncertainty. In research, we observe something about the real world. Then we must infer details about the phenomenon that produced what we observed. A fundamental problem is that, very often, more than one phenomenon can give rise to the observations at hand!

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Example: Infertility Suppose you are concerned about the difficulties some couples have in conceiving a child. It is thought that women exposed to a particular toxin in their workplace have greater difficulty becoming pregnant compared to women who are not exposed to the toxin. You conduct a study of such women, recording the time it takes to conceive.

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Example: Infertility Suppose you are concerned about the difficulties some couples have in conceiving a child. Of course, there is natural variability in time-to- pregnancy attributable to many causes aside from the toxin. Nevertheless, suppose you finally determine that those females with the greatest exposure to the toxin had the most difficulty getting pregnant.

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Example: Infertility Suppose you are concerned about the difficulties some couples have in conceiving a child. But what if there is a variable you did not consider that could be the cause? No study can consider every possibility.

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Example: Infertility It turns out that women who smoke while they are pregnant reduce the chance their daughters will be able to conceive because the toxins involved in smoking effect the eggs in the female fetus. If you didn’t record whether or not the females had mothers who smoked when they were pregnant, you may draw the wrong conclusion about the industrial toxin.

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The Role of Statistics The conclusions we draw—the inferences we make—always come with some amount of uncertainty. We must quantify that uncertainty in order to know how “good” our conclusions are. This is the role that statistics plays in the scientific process.

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The Role of Statistics Scientists use statistical inference to help model the uncertainty inherent in their investigations.

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How to Talk to a Statistician? “It’s all Greek to me...”

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Why Do I Need a Statistician? Planning a study Proposal writing Data analysis and interpretation Presentation and manuscript development

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When Should I Seek a Statistician’s Help? Literature interpretation Defining the research questions Deciding on data collection instruments Determining appropriate study size

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What Does the Statistician Need to Know? General idea of the research What has been done before Rationale for the study Budget constraints

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Vocabulary Hypotheses: a statement of the research question that sets forth the appropriate statistical evaluation Null hypothesis “H 0 ”: statement of no differences or association between variables Alternative hypothesis “H 1 ”: statement of differences or association between variables

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“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.” Albert Einstein ( )

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Disproving the Null If someone claims that all swans are white, confirmatory evidence (in the form of lots of white swans) cannot prove the assertion to be true. Contradictory evidence (in the form of a single black swan) makes it clear the claim is invalid.

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The Scientific Method ObservationHypothesisExperimentResults Evidence supports H Evidence inconsistent with H Revise H

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Hypothesis Testing By hypothesizing that the mean BMI of a population is 26.3, I am saying that I expect the mean of a sample drawn from that population to be ‘close to’ 26.3:

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Hypothesis Testing What if, in collecting data to test my hypothesis, I observe a sample mean of 26? What conclusion might I draw?

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Hypothesis Testing What if, in collecting data to test my hypothesis, I observe a sample mean of 27.5? What conclusion might I draw?

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Hypothesis Testing What if, in collecting data to test my hypothesis, I observe a sample mean of 30? What conclusion might I draw?

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Hypothesis Testing If the observed sample mean seems odd or unlikely under the assumption that H 0 is true, then we reject H 0 in favor of H 1. We typically use the p-value as a measure of the strength of evidence against H 0.

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What is a P-value? Null distribution Observed sample mean p-value If H 1 states that the mean is greater than 26.3, the p-value is as shown. A p-value the probability of getting a sample mean as favorable or more favorable to H 1 than what was observed, assuming H 0 is true. The tail of the distribution it is in is determined by H 1.

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Vocabulary One-tailed hypothesis: outcome is expected in a single direction (e.g., administration of experimental drug will result in a decrease in systolic BP) Two-tailed hypothesis: the direction of the effect is unknown (e.g., experimental therapy will result in a different response rate than that of current standard of care)

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Vocabulary Type I Error (α): a true H 0 is incorrectly rejected “An innocent man is proven GUILTY in a court of law” Commonly accepted rate is α = 0.05 Type II Error (β): failing to reject a false H 0 “A guilty man is proven NOT GUILTY in a court of law” Commonly accepted rate is β = 0.2 Power (1 – β): correctly rejecting a false H 0 “Justice has been served” Commonly accepted rate is 1 – β = 0.8

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Decisions

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Hypothesis Testing We will cover these concepts more fully on July 6 when we discuss Hypothesis Testing

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Statistical Power Primary factors that influence the power of your study: Effect size: as the magnitude of the difference you wish to find increases, the power of your study will increase Variability of the outcome measure: as the variability of your outcome decreases, the power of your study will increase Sample size: as the size of your sample increases, the power of your study will increase

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Statistical Power Secondary factors that influence the power of your study: Dropouts Nuisance variation Confounding variables Multiple hypotheses Post-hoc hypotheses

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Types of Studies Purpose of research 1) To explore 2) To describe or classify 3) To establish relationships 4) To establish causality Strategies for accomplishing these purposes: 1) Naturalistic observation 2) Case study 3) Survey 4) Quasi-experiment 5) Experiment Ambiguity Control

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Design of Experiments We will cover these concepts more fully on July 20 when we discuss Design of Experiments

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Descriptive Statistics

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Field of Statistics Statistics Descriptive Statistics Methods for processing, summarizing, presenting and describing data Experimental Design Techniques for planning and conducting experiments Inferential Statistics Evaluation of the information generated by an experiment or through observation

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Field of Statistics StatisticsDescriptiveGraphicalNumericalInferentialEstimation Hypothesis Testing Experimental Design

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Field of Statistics Descriptive statistics Summarizing and describing the data Uses numerical and graphical summaries to characterize sample data Inferential statistics Uses sample data to make conclusions about a broader range of individuals—a population—than just those who are observed (a sample)

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Field of Statistics Experimental Design Formulation of hypotheses Determination of experimental conditions, measurements, and any extraneous conditions to be controlled Specification of the number of subjects required and the population from which they will be sampled Specification of the procedure for assigning subjects to experimental conditions Determination of the statistical analysis that will be performed

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Descriptive Statistics Descriptive statistics is one branch of the field of Statistics in which we use numerical and graphical summaries to describe a data set or distribution of observations. StatisticsDescriptiveGraphsStatisticsInferential Hypothesis Testing Interval Estimates

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Types of Data All data contains information. It is important to recognize that the hierarchy implied in the level of measurement of a variable has an impact on (1) how we describe the variable data and (2) what statistical methods we use to analyze it.

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Levels of Measurement Nominal: difference Ordinal: difference, order Interval: difference, order, equivalence of intervals Ratio: difference, order, equivalence of intervals, absolute zero discrete qualitative continuous quantitative

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Types of Data NOMINAL ORDINAL INTERVAL RATIO Information increases

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Ratio Data Ratio measurements provide the most information about an outcome. Different values imply difference in outcomes. 6 is different from 7. Order is implied. 6 is smaller than 7.

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Ratio Data Intervals are equivalent. The difference between 6 and 7 is the same as the difference between 101 and 102. Zero indicates a lack of what is being measured. If item A weighs 0 ounces, it weighs nothing.

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Ratio Data Ratio measurements provide the most information about an outcome. Can make statements like: “Person A (t = 10 minutes) took twice as long to complete a task as Person B (t = 5 minutes).” This is the only type of measurement where statements of this nature can be made. Examples: age, birth weight, follow-up time, time to complete a task, dose

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Interval Data Interval measurements are one step down on the “information” scale from ratio measurements. Difference and order are implied and intervals are equivalent. BUT, zero no longer implies an absence of the outcome. What is the interpretation of 0 C? 0 K? The Celsius and Fahrenheit scales of temperature are interval measurements, Kelvin is a ratio measurement.

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Interval Data Interval measurements are one step down on the “information” scale from ratio measurements. You can tell what is better, and by how much, but ratios don’t make sense due to the lack of a ‘starting point’ on the scale. 60 F is greater than 30 F, but not twice as hot since 0 F doesn’t represent an absence of heat. Examples: temperature, dates

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Ordinal Data Ordinal measurements are one step down on the “information” scale from interval measurements. Difference and order are implied. BUT, intervals are no longer equivalent. For instance, the differences in performance between the 1st and 2nd ranked teams in basketball isn’t necessary equivalent to the differences between the 2nd and 3rd ranked teams. The ranking only implies that 1st is better than 2nd, 2nd is better than 3rd, and so on... but it doesn’t try to quantify the ‘betterness’ itself.

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Ordinal Data Ordinal measurements are one step down on the “information” scale from interval measurements. Examples: Highest level of education achieved, tumor grading, survey questions (e.g., likert-scale quality of life)

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Nominal Data Nominal measurements collect the least amount of information about the outcome. Only difference is implied. Observations are classified into mutually exclusive categories. Examples: Gender, ID numbers, pass/fail response

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Levels of Measurement It is important to recognize that the hierarchy implied in the level of measurement of a variable has an impact on (1) how we describe the variable data and (2) what statistical methods we use to analyze it. The levels are in increasing order of mathematical structure—meaning that more mathematical operations and relations are defined—and the higher levels are required in order to define some statistics.

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Levels of Measurement At the lower levels, assumptions tend to be less restrictive and the appropriate data analysis techniques tend to be less sensitive. In general, it is desirable to have a higher level of measurement. A summary of the appropriate statistical summaries and mathematical relations or operations is given in Table 1. We will discuss these topics in more detail later.

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Levels of Measurement LevelStatistical Summary Mathematical Relation/Operation NominalModeone-to-one transformations OrdinalMedianmonotonic transformations IntervalMean, Standard Deviationpositive linear transformations RatioGeometric Mean, Coefficient of Variationmultiplication by c 0 We must know where an outcome falls on the measurement scale--this not only determines how we describe the data (descriptive statistics) but how we analyze it (inferential statistics).

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Using Graphs to Describe Data Nominal and ordinal measurements are discrete and qualitative, even if they are represented numerically. Rank: 1, 2, 3 Gender: male = 1, female = 0 We typically use frequencies, percentages, and proportions to describe how the data is distributed among the levels of a qualitative variable. Bar and pie charts are even more useful.

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Example: Myopia A survey of n = 479 children found that those who had slept with a nightlight or in a fully lit room before the age of 2 had a higher incidence of nearsightedness later in childhood. No MyopiaMyopia High Myopia Total Darkness155 (90%)15 (9%)2 (1%)172 (100%) Nightlight153 (66%)72 (31%)7 (3%)232 (100%) Full Light34 (45%)26 (48%)5 (7%)75 (100%) Total342 (71%)123 (26%)14 (3%)479 (100%)

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Example: Myopia High Some None

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Example: Myopia As the amount of sleep time light increases, the incidence of myopia increases. This study does not prove that sleeping with the light causes myopia in more children. There may be some confounding factor that isn’t measured or considered possibly genetics. Children whose parents have myopia are more likely to suffer from it themselves. It’s also possible that those parents are more likely to provide light while their children are sleeping.

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Example: Nausea How many subjects experienced drug-related nausea?

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Example: Nausea With unequal sample sizes across doses, it is more meaningful to use percent rather than frequency.

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Bar & Pie Charts

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Using Graphs to Describe Data Interval and Ratio variables are continuous and quantitative and can be graphically and numerically represented with more sophisticated mathematical techniques. Height Survival Time We typically use means, standard deviations, medians, and ranges to describe how the variables tend to behave. Histograms and boxplots are even more useful.

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Example: Time-to-death Suppose that we record the variable x = time-to- death of n = 100 patients in a study.

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Example: Time-to-death We can quickly observe several characteristics of the data from the histogram: For most subjects, death occurred between 0 and 5 months For a few subjects, death occurred past 15 months From this picture, we may wish to identify the distinguishing characteristics of the individuals with unusually long times.

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Example: Weight Suppose we record the weight in pounds of n = 100 subjects in a study.

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Example: Tooth Growth Boxplots represent the same information, but are more useful for comparing characteristics between several data sets. Right: distributions of tooth growth for two supplements and three dose levels

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Using Numbers to Describe Data Nominal and ordinal measurements are discrete and qualitative, even if they are represented numerically. Rank: 1, 2, 3 Gender: male = 1, female = 0 Interval and Ratio variables are continuous and quantitative and can be graphically and numerically represented with more sophisticated mathematical techniques. Height Survival Time

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Using Numbers to Describe Data Nominal and ordinal measurements are qualitative, even if they are represented numerically. We typically describe qualitative data using frequencies and percentages in tables. Measures of central tendency and variability don’t make as much sense with categorical data, though the mode can be reported.

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Describing Data Interval and ratio measurements are quantitative. When dealing with a quantitative measurements, we typically describe three aspects of its distribution. Central tendency: a single value around which data tends to fall. Variability: a value that represents how scattered the data is around that central value--large values are indicative of high scatter. We also want to describe the shape of the distribution of the sample data values.

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Central Tendency location Mean: arithmetic average of data Median: approximate middle of data Mode:most frequently occurring value

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Central Tendency Mode, Mo The most frequently occurring value in the data set. May not exist or may not be uniquely defined. It is the only measure of central tendency that can be used with nominal variables, but it is also meaningful for quantitative variables that are inherently discrete (e.g., performance of a task). Its sampling stability is very low (i.e., it varies greatly from sample to sample).

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Central Tendency: Mode Mo

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Central Tendency: Mode Mo

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Central Tendency Median, M The middle value (Q 2, the 50 th percentile) of the variable. It is appropriate for ordinal measures and for skewed interval or ratio measures because it isn’t affected by extreme values. It’s unaffected (robust to outliers) because it takes into account only the relative ordering and number of observations, not the magnitude of the observations themselves. It has low sampling stability.

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Example: Median Suppose we have a set of observations: The median for this set is M = 2. Now suppose we accidentally mismeasured the last observation: The median for this new set is still M = 2.

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Central Tendency: Median MoM

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Central Tendency Mean, The arithmetic average of the variable x. It is the preferred measure for interval or ratio variables with relatively symmetric observations. It has good sampling stability (e.g., it varies the least from sample to sample), implying that it is better suited for making inferences about population parameters. It is affected by extreme values because it takes into account the magnitude of every observation. It can be thought of as the center of gravity of the variable’s distribution.

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Example: Mean Suppose we have a set of observations: The median for this set is M = 2, the mean is Now suppose we accidentally mismeasured the last observation: The median for this new set is still M = 2, but the new mean is

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Central Tendency: Median MoM

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Variability spread Range: difference between min and max values Standard deviation: measures the spread of data about the mean, measured in the same units as the data

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Variability Measures of variability depict how similar observations of a variable tend to be. Variability of a nominal or ordinal variable is rarely summarized numerically. The more familiar measures of variability are mathematical, requiring measurement to be of the interval or ratio scale.

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Variability Range, R The distance from the minimum to the maximum observation. Easy to calculate. Influenced by extreme values (outliers) R = 10 1 = R = 100 1 = 99

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Variability Interquartile Range, IQR The distance from the 1 st quartile (25 th percentile) to the 3 rd quartile (75 th percentile), Q 3 Q 1. Unlike the range, IQR is not influenced by extreme values.

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Variability: IQR

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Variability Standard deviation, s Represents the average spread of the data around the mean. Expressed in the same units as the data. “Average deviation” from the mean.

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Variability Variance, s 2 The standard deviation squared. “Average squared deviation” from the mean.

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Shape shape

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Distribution Shapes

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Summary Basic Concepts Definition and role of statistics Vocabulary lesson Brief introduction to Hypothesis Testing (July 6, 13) Brief introduction to Design concepts (July 20) Descriptive Statistics Levels of Measurement Graphical summaries Numerical summaries Next time: Hypothesis Testing T-tests, ANOVA, Chi-square, etc.

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