بسم الله الرّحمن الرّحيم

بسم الله الرّحمن الرّحيم

Biostatistics Academic Preview (2006): Session 2
08/28/06 Biostatistics Academic Preview Descriptive Statistics

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.

Statistics has two major chapters:
Descriptive Statistics Inferential statistics

Two types of Statistics
Biostatistics Academic Preview (2006): Session 2 08/28/06 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? We will cover methods of descriptive data in the beginning of the semester and in the middle and the end of the semester we will be working with inferential stats. Honestly, as a researcher, I use both kinds daily.

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.

Variables

08/28/06 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

08/28/06 Types of variables

Types of variables : Definitions
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

Types of variables : Quantitative variables
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

Types of variables : Categorical variables
Biostatistics Academic Preview (2006): Session 2 08/28/06 Types of variables : Categorical variables Nominal: unordered categories Race Gender Ordinal: ordered categories likert scales( disagree, neutral, agree ) Income categories

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.

Levels of Measurement Reflects type of information measured and helps determine what descriptive statistics and which statistical test can be used.

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

08/28/06 Scales of Measurement Temperature Men/Women Good/Better/Best Weight Republicans/Democrats/ Independents Volume IQ Not at all/A little/A lot Interval Nominal Ordinal Ratio - IQ: can’t have an absence of IQ. Also, we can’t say that a person with an IQ of

08/28/06

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.

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.

Summary Statistics: Measures of central tendency (location)
Biostatistics Academic Preview (2006): Session 2 08/28/06 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: Sample mean: 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

Skewness 08/28/06 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

08/28/06 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.

Frequency Distributions and Histograms
Histograms for symmetric and skewed distributions.

Normal curves same mean but different standard deviation
Biostatistics Academic Preview (2006): Session 2 08/28/06 Normal curves same mean but different standard deviation

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.

Summary statistics Measures of spread (scale)
Biostatistics Academic Preview (2006): Session 2 08/28/06 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. Standard deviation: The square root of the sample variance Range: the difference between the maximum and minimum values in the sample.

Summary statistics: measures of spread (scale)
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

08/28/06 Quartiles 25% Q3 Q2 Q1 18

08/28/06 Five number system Maximum Minimum Median=50th percentile Lower quartile Q1=25th percentile Upper quartile Q3=75th percentile

Graphical display of numerical variables (histogram)
Biostatistics Academic Preview (2006): Session 2 08/28/06 Graphical display of numerical variables (histogram) Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 50-under 60 11 60-under 70 3 70-under 80 1 18

A histogram of the compressive strength data with 17 bins.

A histogram of the compressive strength data with nine bins.

Histogram of compressive strength data.

Graphical display of numerical variables (box plot)
Biostatistics Academic Preview (2006): Session 2 08/28/06 Graphical display of numerical variables (box plot) Q1 Q3 Q2 Minimum Maximum Median 53

Graphical display of numerical variables (box plot)
Biostatistics Academic Preview (2006): Session 2 08/28/06 Graphical display of numerical variables (box plot) Negatively Skewed Positively Symmetric (Not Skewed) S < 0 S = 0 S > 0 54

Univariate statistics (categorical variables)
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

08/28/06 Displaying categorical variables Rank Cause of Death Frequency (%) 1 Heart Disease 710,760 (43%) 2 Cancer 553,091 (33%) 3 Stroke 167,661 (11%) 4 CLRD 122,009 ( 7%) 5 Accidents 97,900 ( 6%) Total All five causes 1,651,421

Response and explanatory variables
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

Bivariate relationships
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

Bivariate Relationships
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

Two quantitative variables Correlation
Biostatistics Academic Preview (2006): Session 2 08/28/06 Two quantitative variables Correlation A relationship between two variables. Explanatory (Independent)Variable Response (Dependent)Variable x y Hours of Training Number of Accidents Shoe Size Height Cigarettes smoked per day Lung Capacity In this chapter we will be concerned with linear correlation. (How the points fit to a straight line) In more advanced courses you may study other types of correlation. Height IQ What type of relationship exists between the two variables and is the correlation significant?

Scatter Plots and Types of Correlation
Biostatistics Academic Preview (2006): Session 2 08/28/06 Scatter Plots and Types of Correlation x = hours of training y = number of accidents Accidents Start with a scatter plot. It can give a picture of the relationship between the two variables. Negative Correlation as x increases, y decreases

Biostatistics Academic Preview (2006): Session 2 08/28/06 Scatter Plots and Types of Correlation x = SAT score y = GPA GPA Positive Correlation as x increases y increases

Biostatistics Academic Preview (2006): Session 2 08/28/06 Scatter Plots and Types of Correlation x = height y = IQ IQ There is no particular pattern here. No linear correlation

Correlation Coefficient
Biostatistics Academic Preview (2006): Session 2 08/28/06 Correlation Coefficient A measure of the strength and direction of a linear relationship between two variables The range of r is from -1 to 1. -1 1 If r is close to -1 there is a strong negative correlation If r is close to 1 there is a strong positive correlation Give several examples r = -0.97, r = 0.02 and ask for the strength of the correlation. For values like 0.63 a hypothesis test is necessary to determine whether it is strong or not. If r is close to 0 there is no linear correlation

Positive and negative correlation
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

08/28/06 Positive correlation

08/28/06 Negative correlation

Two qualitative variables (Contingency Tables)
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

08/28/06 Example Aspirin and Heart Attacks Explanatory variable=drug received placebo Aspirin Response variable=heart attach status yes no

Contingency table: heart attack example
Biostatistics Academic Preview (2006): Session 2 08/28/06 Contingency table: heart attack example Heart Attack No Heart Attack Total Aspirin 104 10,933 11,037 placebo 189 10,845 11,034 293 21,778 22,071

Two qualitative variables
Biostatistics Academic Preview (2006): Session 2 08/28/06 Two qualitative variables Marijuana Use in College: x=parental use, y=student use Both Neither One Never 17 141 68 226 Occasional 11 54 44 109 Regular 19 40 51 110 Total 47 235 163 445

08/28/06 One quantitative, One qualitative Box plot of age by low birth weight Mean age by low birth weight low birth weight

Trivariate Relationships
Biostatistics Academic Preview (2006): Session 2 08/28/06 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

Confounding and effect modification
Biostatistics Academic Preview (2006): Session 2 08/28/06 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.

08/28/06 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

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.

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

Company sales by year (a) and by quarter (b).
Time Sequence Plots Company sales by year (a) and by quarter (b).

Tests comparing difference between 2 or more groups
Dependent variable Independent variable Paired (dependent t-test) Interval/ratio pre and post tests Nominal Unpaired (independent t-test) Interval/ratio Nominal (2 grps) ANOVA F-test Nominal (>2 grps) Chi-Square (Nonparametric) Nominal (Dichotomous)

Tests demonstrating association between two groups
Dependent var. Independent var. Spearman rho Ordinal Mann-Whitney U Non-parametric Nominal Pearson’s r Interval/ratio

Tests demonstrating association between two groups, controlling for third variable
Dependent Independent Logistic regression Nominal Linear regression Interval/ratio Pearson partial r Kendall’s partial r Ordinal

بسم الله الرّحمن الرّحيم

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