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حيمبسم هللا الرحمن الرّ ّ www.biostat.ir 1 Biostatistics Academic Preview Descriptive Statistics www.biostat.ir 2 What Is Statistics? Statistics is the science of describing or making inferences about the world from a sample of data. Descriptive statistics are numerical estimates that organize and sum up or present the data. Inferential statistics is the process of inferring from a sample to the population. www.biostat.ir 3 Statistics has two major chapters: Descriptive Statistics Inferential statistics www.biostat.ir 4 Two types of Statistics Descriptive statistics Used to summarize, organize and simplify data What was the average height score? What was the highest and lowest score? What is the most common response to a question? Inferential statistics Techniques that allow us to study samples and then make generalizations about the populations from which they were selected Are 5th grade boys taller than 5th grade girls? Does a treatment suitable? www.biostat.ir 5 Population and Samples The Population under study is the set off all individuals of interest for the research. That part of the population for which we collect measurements is called sample. The number of individuals in a sample is denoted by n. www.biostat.ir 6 Variables www.biostat.ir 7 Definitions Variable: a characteristic that changes or varies over time and/or different subjects under consideration. Changing over time Blood pressure, height, weight Changing across a population gender, race www.biostat.ir 8 Types of variables Data Variables Quantitative Qualitative (numeric) (categorical) Discrete Continuous Nominal Ordinal www.biostat.ir 9 Types of variables : Definitions Quantitative variables (numeric): measure a numerical quantity of amount on each experimental unit Qualitative variables (categorical): measure a non numeric quality or characteristic on each experimental unity by classifying each subject into a category www.biostat.ir 10 Types of variables : Quantitative variables Discrete variables: can only take values from a list of possible values Number of brushing per day Continuous variables: can assume the infinitely many values corresponding to the points on a line interval weight, height www.biostat.ir 11 Types of variables : Categorical variables Nominal: unordered categories Race Gender Ordinal: ordered categories likert scales( disagree, neutral, agree ) Income categories www.biostat.ir 12 Types of Variables A discrete variable has gaps between its values. For example, number of brushing per day is a discrete variable. A continuous variable has no gaps between its values. All values or fractions of values have meaning. Age is an example of continuous variable. www.biostat.ir 13 Levels of Measurement Reflects type of information measured and helps determine what descriptive statistics and which statistical test can be used. www.biostat.ir 14 Four Levels of Measurement Nominal lowest level, categories, no rank Ordinal second lowest, ranked categories Interval next to highest, ranked categories with known units between rankings Ratio highest level, ranked categories with known intervals and an absolute zero www.biostat.ir 15 Scales of Measurement Temperature Interval Men/Women Nominal Good/Better/Best Ordinal Weight Ratio Republicans/Democrats/ Independents Nominal Volume Ratio IQ Interval Not at all/A little/A lot Ordinal www.biostat.ir 16 Descriptive Statistics Qualitative Quantitative Frequency Measures of Central Tendency Relative frequency Measures of spread Percentage Five number system Tables Tables Histograms Pie Charts Box plots Bar Graphs Bar charts Line charts www.biostat.ir 17 Descriptive Measures Central Tendency measures. They are computed in order to give a “center” around which the measurements in the data are distributed. Relative Standing measures. They describe the relative position of a specific measurement in the data. Variation or Variability measures. They describe “data spread” or how far away the measurements are from the center. www.biostat.ir 18 Measures of Central Tendency Mean: Sum of all measurements in the data divided by the number of measurements. Median: A number such that at most half of the measurements are below it and at most half of the measurements are above it. Mode: The most frequent measurement in the data. www.biostat.ir 19 n xi i 1 Summary Statistics: Measures of central tendency (location) Mean: The mean of a data set is the sum of the observations divided by the number of observation Population mean: 1 n Sample mean: xi 1 n x xi n i 1 n i 1 Median: The median of a data set is the “middle value” For an odd number of observations, the median is the observation exactly in the middle of the ordered list For an even number of observation, the median is the mean of the two middle observation is the ordered list Mode: The mode is the single most frequently occurring data value www.biostat.ir 20 Skewness The skewness of a distribution is measured by comparing the relative positions of the mean, median and mode. Distribution is symmetrical Mean = Median = Mode Distribution skewed right Median lies between mode and mean, and mode is less than mean Distribution skewed left Median lies between mode and mean, and mode is greater than mean www.biostat.ir 21 Relative positions of the mean and median for (a) right-skewed, (b) symmetric, and (c) left-skewed distributions Note: The mean assumes that the data is normally distributed. If this is not the case it is better to report the median as the measure of location. www.biostat.ir 22 Frequency Distributions and Histograms Histograms for symmetric and skewed distributions. www.biostat.ir 23 Normal curves same mean but different standard deviation www.biostat.ir 24 Further Notes When the Mean is greater than the Median the data distribution is skewed to the Right. When the Median is greater than the Mean the data distribution is skewed to the Left. When Mean and Median are very close to each other the data distribution is approximately symmetric. www.biostat.ir 25 Summary statistics Measures of spread (scale) Variance: The average of the squared deviations of each sample value from the sample mean, except that instead of dividing the sum of the squared deviations by the sample size N, the sum is divided by N-1. 1 n s 2 xi x 2 n 1 i 1 Standard deviation: The square root of the sample variance s 1 n x x 2 n 1 i 1 i Range: the difference between the maximum and minimum values in the sample. www.biostat.ir 26 Summary statistics: measures of spread (scale) We can describe the spread of a distribution by using percentiles. The pth percentile of a distribution is the value such that p percent of the observations fall at or below it. Median=50th percentile Quartiles divide data into four equal parts. First quartile—Q1 25% of observations are below Q1 and 75% above Q1 Second quartile—Q2 50% of observations are below Q2 and 50% above Q2 Third quartile—Q3 75% of observations are below Q3 and 25% above Q3 www.biostat.ir 27 Quartiles Q1 Q2 Q3 25% 25% 25% 25% www.biostat.ir 28 Five number system Maximum Minimum Median=50th percentile Lower quartile Q1=25th percentile Upper quartile Q3=75th percentile www.biostat.ir 29 Graphical display of numerical variables (histogram) Class Interval Frequency 20 20-under 30 6 30-under 40 18 Frequency 40-under 50 11 10 50-under 60 11 60-under 70 3 70-under 80 1 0 0 10 20 30 40 50 60 70 80 Years www.biostat.ir 30 Frequency Distributions and Histograms A histogram of the compressive strength data with 17 bins. www.biostat.ir 31 Frequency Distributions and Histograms A histogram of the compressive strength data with nine bins. www.biostat.ir 32 Frequency Distributions and Histograms Histogram of compressive strength data. www.biostat.ir 33 Graphical display of numerical variables (box plot) Median Minimum Q1 Q2 Q3 Maximum www.biostat.ir 34 Graphical display of numerical variables (box plot) S<0 S=0 S>0 Negatively Symmetric Positively Skewed (Not Skewed) Skewed www.biostat.ir 35 Univariate statistics (categorical variables) Summary measures Count=frequency Percent=frequency/total sample The distribution of a categorical variable lists the categories and gives either a count or a percent of individuals who fall in each category www.biostat.ir 36 Displaying categorical variables Rank Cause of Frequency Death (%) 1 Heart 710,760 Disease (43%) 2 Cancer 553,091 (33%) 3 Stroke 167,661 heart cancer stroke CLRD accident (11%) 4 CLRD 122,009 60 ( 7%) 40 5 Accidents 97,900 20 ( 6%) Total All five 1,651,421 0 heart cancer stroke CLRD accident causes www.biostat.ir 37 Response and explanatory variables Response variable: the variable which we intend to model. we intend to explain through statistical modeling Explanatory variable: the variable or variables which may be used to model the response variable values may be related to the response variable www.biostat.ir 38 Bivariate relationships An extension of univariate descriptive statistics Used to detect evidence of association in the sample Two variables are said to be associated if the distribution of one variable differs across groups or values defined by the other variable www.biostat.ir 39 Bivariate Relationships Two quantitative variables Scatter plot Side by side stem and leaf plots Two qualitative variables Tables Bar charts One quantitative and one qualitative variable Side by side box plots Bar chart www.biostat.ir 40 Two quantitative variables Correlation A relationship between two variables. Explanatory Response (Independent)Variable (Dependent)Variable x y Hours of Training Number of Accidents Shoe Size Height Cigarettes smoked per day Lung Capacity Height IQ What type of relationship exists between the two variables and is the correlation significant? www.biostat.ir 41 Scatter Plots and Types of Correlation x = hours of training Accidents y = number of accidents 60 50 40 30 20 10 0 0 2 4 6 8 10 12 14 16 18 20 Hours of Training Negative Correlation as x increases, y decreases www.biostat.ir 42 Scatter Plots and Types of Correlation x = SAT score GPA 4.00 y = GPA 3.75 3.50 3.25 3.00 2.75 2.50 2.25 2.00 1.75 1.50 300 350 400 450 500 550 600 650 700 750 800 Math SAT Positive Correlation as x increases y increases www.biostat.ir 43 Scatter Plots and Types of Correlation x = height y = IQ IQ 160 150 140 130 120 110 100 90 80 60 64 68 72 76 80 Height No linear correlation www.biostat.ir 44 Correlation Coefficient A measure of the strength and direction of a linear relationship between two variables nxy xy r nx 2 x ny 2 (y ) 2 2 The range of r is from -1 to 1. -1 0 1 If r is close to -1 If r is close to If r is close to 1 there is a strong 0 there is no there is a strong negative linear positive correlation correlation correlation www.biostat.ir 45 Positive and negative correlation 1 If two variables x and y are positively correlated this means that: large values of x are associated with large values of y, and small values of x are associated with small values of y 2 If two variables x and y are negatively correlated this means that: large values of x are associated with small values of y, and small values of x are associated with large values of y www.biostat.ir 46 Positive correlation www.biostat.ir 47 Negative correlation www.biostat.ir 48 Two qualitative variables (Contingency Tables) Categorical data is usually displayed using a contingency table, which shows the frequency of each combination of categories observed in the data value The rows correspond to the categories of the explanatory variable The columns correspond the categories of the response variable www.biostat.ir 49 Example Aspirin and Heart Attacks Explanatory variable=drug received placebo Aspirin Response variable=heart attach status yes no www.biostat.ir 50 Contingency table: heart attack example Heart Attack No Heart Total Attack Aspirin 104 10,933 11,037 placebo 189 10,845 11,034 Total 293 21,778 22,071 www.biostat.ir 51 Two qualitative variables Marijuana Use in College: x=parental use, y=student use Both Neither One 60 50 Never 17 141 68 226 40 Occasional 11 54 44 109 30 20 Regular 19 40 51 110 10 0 Total 47 235 163 445 Both Neither One Never Occasional Regular www.biostat.ir 52 One quantitative, One qualitative Box plot of age by low birth weight Mean age by low birth weight 50 24 23.66 40 23.5 23 a g 30 22.31 22.5 e 20 22 21.5 yes no 10 0 1 low birth weight l bw low birth weight www.biostat.ir 53 Trivariate Relationships An extension of bivariate descriptive statistics We focus on description that helps us decide about the role variables might play in the ultimate statistical analyses Identify variables that can increase the precision of the data analysis used to answer associations between two other variables www.biostat.ir 54 Confounding and effect modification A factor, Z, is said to confound a relationship between a risk factor, X, and an outcome, Y, if it is not an effect modifier and the unadjusted strength of the relationship between X and Y differs from the common strength of the relationship between X and Y for each level of Z. A factor, Z, is said to be an effect modifier of a relationship between a risk factor, X, and an outcome measure, Y, if the strength of the relationship between the risk factor, X, and the outcome, Y, varies among the levels of Z. www.biostat.ir 55 Example: confounding In our low birth weight data suppose we wish to investigate the association between race and low birth weight. Our ability to detect this association might be affected by: Smoking status being associated with low birth weight Smoking status being associated with race www.biostat.ir 56 Multiple Models Allows one to calculated the association between and response and outcome of interest, after controlling for potential confounders. Allows for one to assess the association between an outcome and multiple response variables of interest. www.biostat.ir 57 Time Sequence Plots • A time series or time sequence is a data set in which the observations are recorded in the order in which they occur. • A time series plot is a graph in which the vertical axis denotes the observed value of the variable (say x) and the horizontal axis denotes the time (which could be minutes, days, years, etc.). • When measurements are plotted as a time series, we often see •trends, •cycles, or •other broad features of the data www.biostat.ir 58 Time Sequence Plots Company sales by year (a) and by quarter (b). www.biostat.ir 59 Tests comparing difference between 2 or more groups Test Dependent Independent variable variable Paired Interval/ratio pre Nominal and post tests (dependent t-test) Unpaired Interval/ratio Nominal (2 grps) (independent t-test) ANOVA F-test Interval/ratio Nominal (>2 grps) Chi-Square Nominal Nominal (Dichotomous) (Nonparametric) www.biostat.ir 60 Tests demonstrating association between two groups Test Dependent var. Independent var. Spearman rho Ordinal Ordinal Mann-Whitney U Ordinal Nominal Non-parametric Pearson’s r Interval/ratio Interval/ratio www.biostat.ir 61 Tests demonstrating association between two groups, controlling for third variable Test Dependent Independent Logistic Nominal Nominal regression Linear regression Interval/ratio Interval/ratio Pearson partial r Interval/ratio Interval/ratio Kendall’s partial r Ordinal Ordinal www.biostat.ir 62