This document provides an introduction to probability distributions and biostatistics. It discusses three fundamental probability distributions used in statistics: the binomial distribution, Poisson distribution, and normal distribution. For the binomial distribution, it provides examples of how to calculate probabilities using the binomial probability formula. It also gives an example showing how the binomial distribution can be used to analyze results from a clinical trial on a new kidney cancer therapy.
This document provides information about the Poisson distribution. It begins with background on the mathematician Simeon Denis Poisson who developed the distribution in 1837. The Poisson distribution models the number of random events occurring in a fixed interval of time or space. The key properties are that the probability of an event is constant and events are independent. Several examples of real-world applications are given such as disease occurrences, mutations, and telephone calls. The document then provides the Poisson probability mass function equation and explains how to calculate probabilities for specific values. It also discusses Poisson processes and fitting observed data to a Poisson distribution. Finally, it compares the Poisson distribution to the binomial distribution and outlines their key differences.
The document provides information about binomial probability distributions including:
- Binomial experiments have a fixed number (n) of independent trials with two possible outcomes and a constant probability (p) of success.
- The binomial probability distribution gives the probability of getting exactly x successes in n trials. It is calculated using the binomial coefficient and p and q=1-p.
- The mean, variance and standard deviation of a binomial distribution are np, npq, and √npq respectively.
- Examples demonstrate calculating probabilities of outcomes for binomial experiments and determining if results are significantly low or high using the range rule of μ ± 2σ.
PG STAT 531 Lecture 5 Probability DistributionAashish Patel
This document provides an overview of probability distributions including binomial, Poisson, and normal distributions. It discusses key concepts such as:
- Binomial distributions describe experiments with two possible outcomes and fixed number of trials.
- Poisson distributions model rare events with sample sizes so large one outcome is much more common.
- Normal distributions produce bell-shaped curves defined by the mean and standard deviation. They are widely used in statistics.
1. The Poisson distribution models the number of discrete events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event.
2. It was first introduced by Siméon Denis Poisson in 1837 to study the number of wrongful convictions in a country.
3. The Poisson distribution can be used when the probability of an event is small but the number of trials is large, such as births in a hospital, particle emissions, or telephone calls to a switchboard.
The document discusses the binomial and Poisson distributions. The binomial distribution describes the probability of success in a fixed number of yes/no trials, while the Poisson distribution models the occurrence of rare events. Examples of applications include defective items in manufacturing and accidents. Key characteristics of both distributions including their formulas are provided. The document concludes by noting the Poisson distribution is useful for modeling rare events with small probabilities of success.
1. The document discusses the Poisson probability distribution, which models random processes with discrete outcomes.
2. A Poisson experiment has properties including a known average number of successes (μ) that is proportional to the region size, with extremely small regions having virtually zero probability of success.
3. Examples of Poisson applications include the number of car accidents per month or network failures per day.
This presentation discusses binomial probability distributions through the following key points:
- It defines basic terminology related to random experiments, events, and variables. The binomial distribution specifically describes discrete data from Bernoulli processes.
- It outlines the notation and assumptions for binomial distributions, including that there are two possible outcomes for each trial (success/failure), a fixed number of trials, and constant probabilities of success/failure.
- It presents three methods for calculating binomial probabilities: the binomial probability formula, table method, and using technology like Excel.
- It discusses measures of central tendency and dispersion for binomial distributions and how the shape of the distribution depends on the number of trials and probability of success.
- Real-world
This document provides information about the Poisson distribution. It begins with background on the mathematician Simeon Denis Poisson who developed the distribution in 1837. The Poisson distribution models the number of random events occurring in a fixed interval of time or space. The key properties are that the probability of an event is constant and events are independent. Several examples of real-world applications are given such as disease occurrences, mutations, and telephone calls. The document then provides the Poisson probability mass function equation and explains how to calculate probabilities for specific values. It also discusses Poisson processes and fitting observed data to a Poisson distribution. Finally, it compares the Poisson distribution to the binomial distribution and outlines their key differences.
The document provides information about binomial probability distributions including:
- Binomial experiments have a fixed number (n) of independent trials with two possible outcomes and a constant probability (p) of success.
- The binomial probability distribution gives the probability of getting exactly x successes in n trials. It is calculated using the binomial coefficient and p and q=1-p.
- The mean, variance and standard deviation of a binomial distribution are np, npq, and √npq respectively.
- Examples demonstrate calculating probabilities of outcomes for binomial experiments and determining if results are significantly low or high using the range rule of μ ± 2σ.
PG STAT 531 Lecture 5 Probability DistributionAashish Patel
This document provides an overview of probability distributions including binomial, Poisson, and normal distributions. It discusses key concepts such as:
- Binomial distributions describe experiments with two possible outcomes and fixed number of trials.
- Poisson distributions model rare events with sample sizes so large one outcome is much more common.
- Normal distributions produce bell-shaped curves defined by the mean and standard deviation. They are widely used in statistics.
1. The Poisson distribution models the number of discrete events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event.
2. It was first introduced by Siméon Denis Poisson in 1837 to study the number of wrongful convictions in a country.
3. The Poisson distribution can be used when the probability of an event is small but the number of trials is large, such as births in a hospital, particle emissions, or telephone calls to a switchboard.
The document discusses the binomial and Poisson distributions. The binomial distribution describes the probability of success in a fixed number of yes/no trials, while the Poisson distribution models the occurrence of rare events. Examples of applications include defective items in manufacturing and accidents. Key characteristics of both distributions including their formulas are provided. The document concludes by noting the Poisson distribution is useful for modeling rare events with small probabilities of success.
1. The document discusses the Poisson probability distribution, which models random processes with discrete outcomes.
2. A Poisson experiment has properties including a known average number of successes (μ) that is proportional to the region size, with extremely small regions having virtually zero probability of success.
3. Examples of Poisson applications include the number of car accidents per month or network failures per day.
This presentation discusses binomial probability distributions through the following key points:
- It defines basic terminology related to random experiments, events, and variables. The binomial distribution specifically describes discrete data from Bernoulli processes.
- It outlines the notation and assumptions for binomial distributions, including that there are two possible outcomes for each trial (success/failure), a fixed number of trials, and constant probabilities of success/failure.
- It presents three methods for calculating binomial probabilities: the binomial probability formula, table method, and using technology like Excel.
- It discusses measures of central tendency and dispersion for binomial distributions and how the shape of the distribution depends on the number of trials and probability of success.
- Real-world
This document provides an overview of probability distributions and related concepts. It defines key probability distributions like the binomial, beta, multinomial, and Dirichlet distributions. It also describes probability distribution equations like the cumulative distribution function and probability density function. Additionally, it outlines descriptive parameters for distributions like mean, variance, skewness and kurtosis. Finally, it briefly discusses probability theorems such as the law of large numbers, central limit theorem, and Bayes' theorem.
This document defines and provides examples of binomial probability distributions. It explains that a binomial probability distribution is a table that lists the possible numbers of successes from n trials paired with each outcome's probability. It also outlines how to calculate the mean, variance, and standard deviation of a binomial distribution. Finally, it demonstrates how to use the distribution table to calculate various probabilities, such as the probability of exactly, at least, more than, or within a range of successes.
Hypergeometric probability distributionNadeem Uddin
The document discusses hypergeometric probability distribution. It provides examples of hypergeometric experiments involving selecting items from a population without replacement, where the probability of success changes with each trial. The key points are:
- A hypergeometric experiment has a fixed population with a specified number of successes, samples items without replacement, and the probability of success changes on each trial.
- The hypergeometric distribution gives the probability of getting x successes in n draws from a population of N items with K successes.
- Examples demonstrate calculating hypergeometric probabilities and approximating it as a binomial when the population is large compared to the sample size.
The document discusses the Poisson distribution, which models rare events. It describes how the Poisson distribution can be used when the number of events is large but the probability of each individual event is small. The key conditions for applying the Poisson distribution are that events occur independently and the rate of occurrence is constant. The mean and variance of the Poisson distribution are equal to the parameter μ, which represents the average number of events. Examples of phenomena that follow a Poisson distribution include traffic accidents, website visits, and product demand. A formula for calculating Poisson probabilities is provided.
Discrete and continuous probability distributions ppt @ bec domsBabasab Patil
The document discusses various probability distributions including discrete and continuous distributions. It covers the binomial, hypergeometric, Poisson, and normal distributions. It provides the characteristics and formulas for each distribution and examples of how to calculate probabilities using the distributions.
Probability And Probability Distributions Sahil Nagpal
This document provides an overview of key concepts in probability and probability distributions. It defines important terms like probability, sample space, events, mutually exclusive events, independent events, and conditional probability. It also covers rules of probability like addition rules, complement rules, and Bayes' theorem. Finally, it introduces discrete and continuous random variables and discusses properties of discrete probability distributions like expected value and standard deviation.
Binomial and Poission Probablity distributionPrateek Singla
The document discusses binomial and Poisson distributions. Binomial distribution describes random events with two possible outcomes, like success/failure. Poisson distribution models rare, independent events occurring randomly over an interval of time/space. An example calculates the probability of defective thermometers using binomial distribution. It also fits a Poisson distribution to automobile accident data from a 50-day period.
This document discusses several probability distributions:
1. The gamma distribution, which has a probability density function involving the gamma function. It includes the exponential distribution as a special case.
2. The Poisson distribution, which models the number of discrete events occurring in a fixed interval of time or space. The waiting time between arrivals in a Poisson process has an exponential distribution.
3. The chi-squared distribution, which is a special case of the gamma distribution where the shape parameter equals degrees of freedom. It is used to model sums of squared random variables.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
The document provides information about normal probability distributions and how to solve problems using normal distributions. It defines the normal distribution and standard normal distribution. It gives the equation for a normal distribution and how to standardize a normal variable. Examples are provided on finding probabilities and areas under the normal curve. The document also discusses using normal approximations to the binomial and Poisson distributions and provides continuity correction rules for such approximations.
1. Sampling error occurs because sample means are not equal to the population mean and differ from each other.
2. The distribution of sample means follows a normal distribution if drawn from a normal population, and approximates a normal distribution if drawn from a non-normal population as the sample size increases.
3. A confidence interval for the population mean or probability can be constructed given the sample size, mean or probability, and standard deviation. The confidence level indicates the probability the true population parameter falls within the interval.
1. A geometric experiment involves independent trials with two possible outcomes (success/failure), where the probability of success (p) is constant across trials.
2. The geometric random variable (X) represents the number of trials until the first success. It has a geometric distribution where P(X=x) = p(1-p)^(x-1).
3. Examples are provided to illustrate calculating the probability of success on a given trial (x), as well as the mean and standard deviation, for geometric distributions with different probabilities of success (p).
The PPT covered the distinguish between discrete and continuous distribution. Detailed explanation of the types of discrete distributions such as binomial distribution, Poisson distribution & Hyper-geometric distribution.
1) The document discusses probability proportional to size (PPS) sampling techniques, including PPS sampling with and without replacement.
2) Cumulative total and Lahiri's methods are described for PPS sampling with replacement, while general selection procedure, Sen-Midzuno method, and Narain's scheme are covered for PPS sampling without replacement.
3) Estimators for population totals, means, variances, and inclusion probabilities are defined for PPS samples. Both ordered and unordered estimators are discussed.
The document provides information about the binomial distribution, including its definition, assumptions, and properties. The binomial distribution expresses the probability of success/failure outcomes from Bernoulli trials. It assumes a fixed number of independent trials, a constant probability of success on each trial, and only two possible outcomes per trial (success or failure). The mean and variance of the binomial distribution are provided. Examples are given to demonstrate how to calculate binomial probabilities.
The document discusses the Poisson distribution, which describes the probability of rare events. It has one parameter, the mean (m), and is used when the number of trials is large but the probability of an individual success is small. Examples of Poisson distributions given include defects per box of screws and printing mistakes per page. The key characteristics are outlined, such as being discrete and positively skewed. The document provides an example problem calculating probabilities based on the average number of accidents per month at an intersection. It also demonstrates how to fit data to a Poisson distribution by calculating expected frequencies based on the observed mean number of events.
Bernoullis Random Variables And Binomial Distributionmathscontent
Bernoulli and binomial random variables are used to model success/failure experiments. A Bernoulli variable represents a single trial with outcomes success (1) and failure (0). A binomial variable counts the number of successes in n independent Bernoulli trials. The probability of x successes in n trials is given by the binomial distribution. Its mean and variance can be derived. The moment generating function of the binomial distribution helps compute moments like variance.
Chi square test- a test of association, Pearson's chi square test of independence, Goodness of fit test, chi square test of homogeneity, advantages and disadvantages of chi square test.
This document discusses probability distributions and provides examples of calculating probabilities using binomial distributions. It begins by defining a probability distribution as a table, graph or formula used to specify the possible values and probabilities of a discrete random variable. It then gives examples of probability distributions for number of assistance programs used by families and calculates related probabilities. The document introduces binomial distribution and provides two examples of calculating probabilities of outcomes for binomial processes, such as number of full term births out of total births. It describes key concepts like Bernoulli trials, processes and use of combinations and factorials to calculate probabilities for larger sample sizes.
This document provides an overview of probability distributions and related concepts. It defines key probability distributions like the binomial, beta, multinomial, and Dirichlet distributions. It also describes probability distribution equations like the cumulative distribution function and probability density function. Additionally, it outlines descriptive parameters for distributions like mean, variance, skewness and kurtosis. Finally, it briefly discusses probability theorems such as the law of large numbers, central limit theorem, and Bayes' theorem.
This document defines and provides examples of binomial probability distributions. It explains that a binomial probability distribution is a table that lists the possible numbers of successes from n trials paired with each outcome's probability. It also outlines how to calculate the mean, variance, and standard deviation of a binomial distribution. Finally, it demonstrates how to use the distribution table to calculate various probabilities, such as the probability of exactly, at least, more than, or within a range of successes.
Hypergeometric probability distributionNadeem Uddin
The document discusses hypergeometric probability distribution. It provides examples of hypergeometric experiments involving selecting items from a population without replacement, where the probability of success changes with each trial. The key points are:
- A hypergeometric experiment has a fixed population with a specified number of successes, samples items without replacement, and the probability of success changes on each trial.
- The hypergeometric distribution gives the probability of getting x successes in n draws from a population of N items with K successes.
- Examples demonstrate calculating hypergeometric probabilities and approximating it as a binomial when the population is large compared to the sample size.
The document discusses the Poisson distribution, which models rare events. It describes how the Poisson distribution can be used when the number of events is large but the probability of each individual event is small. The key conditions for applying the Poisson distribution are that events occur independently and the rate of occurrence is constant. The mean and variance of the Poisson distribution are equal to the parameter μ, which represents the average number of events. Examples of phenomena that follow a Poisson distribution include traffic accidents, website visits, and product demand. A formula for calculating Poisson probabilities is provided.
Discrete and continuous probability distributions ppt @ bec domsBabasab Patil
The document discusses various probability distributions including discrete and continuous distributions. It covers the binomial, hypergeometric, Poisson, and normal distributions. It provides the characteristics and formulas for each distribution and examples of how to calculate probabilities using the distributions.
Probability And Probability Distributions Sahil Nagpal
This document provides an overview of key concepts in probability and probability distributions. It defines important terms like probability, sample space, events, mutually exclusive events, independent events, and conditional probability. It also covers rules of probability like addition rules, complement rules, and Bayes' theorem. Finally, it introduces discrete and continuous random variables and discusses properties of discrete probability distributions like expected value and standard deviation.
Binomial and Poission Probablity distributionPrateek Singla
The document discusses binomial and Poisson distributions. Binomial distribution describes random events with two possible outcomes, like success/failure. Poisson distribution models rare, independent events occurring randomly over an interval of time/space. An example calculates the probability of defective thermometers using binomial distribution. It also fits a Poisson distribution to automobile accident data from a 50-day period.
This document discusses several probability distributions:
1. The gamma distribution, which has a probability density function involving the gamma function. It includes the exponential distribution as a special case.
2. The Poisson distribution, which models the number of discrete events occurring in a fixed interval of time or space. The waiting time between arrivals in a Poisson process has an exponential distribution.
3. The chi-squared distribution, which is a special case of the gamma distribution where the shape parameter equals degrees of freedom. It is used to model sums of squared random variables.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
The document provides information about normal probability distributions and how to solve problems using normal distributions. It defines the normal distribution and standard normal distribution. It gives the equation for a normal distribution and how to standardize a normal variable. Examples are provided on finding probabilities and areas under the normal curve. The document also discusses using normal approximations to the binomial and Poisson distributions and provides continuity correction rules for such approximations.
1. Sampling error occurs because sample means are not equal to the population mean and differ from each other.
2. The distribution of sample means follows a normal distribution if drawn from a normal population, and approximates a normal distribution if drawn from a non-normal population as the sample size increases.
3. A confidence interval for the population mean or probability can be constructed given the sample size, mean or probability, and standard deviation. The confidence level indicates the probability the true population parameter falls within the interval.
1. A geometric experiment involves independent trials with two possible outcomes (success/failure), where the probability of success (p) is constant across trials.
2. The geometric random variable (X) represents the number of trials until the first success. It has a geometric distribution where P(X=x) = p(1-p)^(x-1).
3. Examples are provided to illustrate calculating the probability of success on a given trial (x), as well as the mean and standard deviation, for geometric distributions with different probabilities of success (p).
The PPT covered the distinguish between discrete and continuous distribution. Detailed explanation of the types of discrete distributions such as binomial distribution, Poisson distribution & Hyper-geometric distribution.
1) The document discusses probability proportional to size (PPS) sampling techniques, including PPS sampling with and without replacement.
2) Cumulative total and Lahiri's methods are described for PPS sampling with replacement, while general selection procedure, Sen-Midzuno method, and Narain's scheme are covered for PPS sampling without replacement.
3) Estimators for population totals, means, variances, and inclusion probabilities are defined for PPS samples. Both ordered and unordered estimators are discussed.
The document provides information about the binomial distribution, including its definition, assumptions, and properties. The binomial distribution expresses the probability of success/failure outcomes from Bernoulli trials. It assumes a fixed number of independent trials, a constant probability of success on each trial, and only two possible outcomes per trial (success or failure). The mean and variance of the binomial distribution are provided. Examples are given to demonstrate how to calculate binomial probabilities.
The document discusses the Poisson distribution, which describes the probability of rare events. It has one parameter, the mean (m), and is used when the number of trials is large but the probability of an individual success is small. Examples of Poisson distributions given include defects per box of screws and printing mistakes per page. The key characteristics are outlined, such as being discrete and positively skewed. The document provides an example problem calculating probabilities based on the average number of accidents per month at an intersection. It also demonstrates how to fit data to a Poisson distribution by calculating expected frequencies based on the observed mean number of events.
Bernoullis Random Variables And Binomial Distributionmathscontent
Bernoulli and binomial random variables are used to model success/failure experiments. A Bernoulli variable represents a single trial with outcomes success (1) and failure (0). A binomial variable counts the number of successes in n independent Bernoulli trials. The probability of x successes in n trials is given by the binomial distribution. Its mean and variance can be derived. The moment generating function of the binomial distribution helps compute moments like variance.
Chi square test- a test of association, Pearson's chi square test of independence, Goodness of fit test, chi square test of homogeneity, advantages and disadvantages of chi square test.
This document discusses probability distributions and provides examples of calculating probabilities using binomial distributions. It begins by defining a probability distribution as a table, graph or formula used to specify the possible values and probabilities of a discrete random variable. It then gives examples of probability distributions for number of assistance programs used by families and calculates related probabilities. The document introduces binomial distribution and provides two examples of calculating probabilities of outcomes for binomial processes, such as number of full term births out of total births. It describes key concepts like Bernoulli trials, processes and use of combinations and factorials to calculate probabilities for larger sample sizes.
Sample size in clinical research 2021 aprilINAAMUL HAQ
This document discusses sample size calculations in clinical research. It begins by introducing common questions around determining appropriate sample sizes. It then provides background on key terms like hypotheses, effect sizes, and type I and type II errors. Examples are given for calculating sample sizes for common study designs like case-control studies, cohort studies, and randomized controlled trials. Formulas, online calculators, and published tables are presented as methods for performing sample size calculations. Worked examples are shown for several clinical scenarios.
Barbara Osimani, Problems with Evidence of Pharmaceutical Harm. King's Colleg...Barbara Osimani
This document discusses evidence hierarchies and randomized controlled trials (RCTs) for assessing causal effects of pharmaceuticals. It covers:
1. Philosophical debates on evidence hierarchies and whether RCTs should be considered the highest form of evidence. Alternative approaches consider all relevant evidence.
2. RCTs aim to balance groups and isolate causal effects, but their results may not be generalizable due to unknown confounders and differences between study and target populations. In particular, RCTs may not adequately assess unintended effects of pharmaceuticals.
3. Observational studies and mechanisms-based reasoning have distinct roles to play in causal assessment, especially for unintended effects. Evidence hierarchies should consider these differences rather than have a one
This document discusses various probability distributions including the binomial, Poisson, and normal distributions. It provides definitions and formulas for calculating probabilities for each distribution. For the binomial distribution, it covers the binomial probability formula and using the binomial table. For the Poisson distribution, it discusses the Poisson probability formula and Poisson table. It also addresses calculating the mean, variance, and standard deviation for the binomial and Poisson distributions. Finally, it introduces the normal distribution as the most important continuous probability distribution.
4 1 probability and discrete probability distributionsLama K Banna
This document discusses probabilities and probability distributions. It begins by defining an experiment and sample space. A random variable is defined as a numerical value determined by the outcome of an experiment. Random variables can be discrete or continuous. Probability distributions show all possible outcomes of an experiment and their probabilities. The binomial distribution is discussed as modeling discrete experiments with binary outcomes and fixed probabilities. Key properties of the binomial include the mean, variance, and use of the binomial probability formula and tables to calculate probabilities of various outcomes.
The document discusses key concepts in probability and epidemiology, including:
1) Definitions of probability, sensitivity, specificity, positive predictive value, and negative predictive value in diagnostic testing.
2) Methods for calculating probabilities of events using formulas like multiplication rule and addition rule.
3) Examples of how to determine the probability of outcomes from throwing dice, coins, and other chance experiments.
4) Explanations of dependent vs independent events and the use of Bayes' theorem.
5) Illustrations of how test characteristics impact positive and negative predictive values.
This paper studies an identification problem that arises when clinicians seek to personalize patient care by predicting health outcomes conditional on observed patient covariates. Let y be an outcome of interest and let (x = k, w = j) be observed patient covariates. Suppose a clinician wants to choose a care option that maximizes a patient's expected utility conditional on the observed covariates. To accomplish this, the clinician needs to know the conditional probability distribution P(y|x = k, w = j). It is common to have a trustworthy evidence-based risk assessment that predicts y conditional on a subset of the observed covariates, say x, but not conditional on (x, w). Then the clinician knows P(y|x = k) but not P(y|x = k, w = j). Research on the ecological inference problem studies partial identification of P(y∣x, w) given knowledge of P(y|x) and P(w|x). Combining this knowledge with structural assumptions yields tighter conclusions. A psychological literature comparing actuarial predictions and clinical judgments has concluded that clinicians should not attempt to subjectively predict patient outcomes conditional on covariates that are not utilized in evidence-based risk assessments. I argue that formalizing clinical judgment through analysis of the identification problem can improve risk assessments and care decisions.
1) The document discusses evaluating medical literature to answer a clinical question about whether duct tape is an effective treatment for warts in children.
2) A randomized controlled trial studied 61 patients comparing duct tape to cryotherapy treatment for common warts. It found that duct tape was significantly more effective, with an absolute risk reduction of 25%.
3) Key points to evaluate in studies include similarity of patients, interventions and outcomes measured, study design, results, and statistics reported like absolute risk reduction and number needed to treat.
This document discusses chi square distribution and its use in analyzing frequency data. Chi square tests can be used to test goodness of fit, independence, and homogeneity. It provides examples of chi square tests for goodness of fit to determine if sample data fits a theoretical distribution, and tests of independence to determine if two classification criteria are independent. The document also outlines the steps for conducting chi square tests, including calculating test statistics, determining degrees of freedom, and comparing results to critical values to reject or fail to reject the null hypothesis.
This document provides an overview of cross-sectional studies. It defines cross-sectional studies as studies that measure prevalence by observing exposures and outcomes in a population at a single point in time. It discusses key aspects of cross-sectional study design such as sampling, data collection methods, analysis of prevalence data, and potential biases like selection bias.
This document provides an overview of cross-sectional studies, including what they are, their uses, methodology, advantages, and disadvantages. A cross-sectional study involves observing a population at a single point in time to determine prevalence of disease. It is a quick and inexpensive way to describe characteristics of a population and identify associations between variables. However, it cannot determine causation due to its observational nature.
The document discusses binomial distributions, which model outcomes that can be classified as successes or failures, with a constant probability of success for each trial. A binomial distribution is defined by the number of trials (n) and the probability of success (p) for each trial. The mean is np and the standard deviation is npq. Examples are given of calculating binomial probabilities, such as the probability of a certain number of patients recovering from a disease out of a sample size.
There are three main probability distributions: binomial, Poisson, and normal. The binomial distribution calculates the probability of a certain number of successes in a fixed number of trials when the probability of success is constant. The Poisson distribution calculates the probability of a number of random events occurring in an interval. It applies when occurrences are independent and the average number of occurrences in an interval is known. The normal distribution is the most important continuous probability distribution and describes variables that can take any value within a range.
This document summarizes key probability distributions: binomial, Poisson, and normal. The binomial distribution describes the number of successes in fixed number of trials where the probability of success is constant. The Poisson distribution approximates the binomial when the number of trials is large and the probability of success is small. The normal distribution describes many continuous random variables and is symmetric with two parameters: mean and standard deviation. The document also discusses when binomial and Poisson distributions can be approximated as normal distributions.
This document discusses evidence-based medicine (EBM) and key concepts in evaluating medical evidence. It defines EBM as the conscientious use of current best evidence in patient care. Randomized controlled trials are considered the gold standard for evaluating new therapies or tests. However, observational studies can also provide valuable evidence when RCTs are not possible or ethical. Systematic reviews provide a critical summary of all relevant randomized trials on a topic to determine the state of evidence and guide clinical practice and policy.
Parmetric and non parametric statistical test in clinical trailsVinod Pagidipalli
The document discusses parametric and non-parametric statistical tests used in clinical trials. Parametric tests like the z-test, t-test, ANOVA, and correlation tests are used when data follows a normal distribution. Non-parametric tests like the chi-square test, Fisher's exact test, and binomial test are used when data cannot be assumed to be normally distributed. Several statistical tests are described, including how to apply them in clinical trials to compare treatment groups, analyze associations between variables, and test hypotheses about population proportions.
Introduction to stats important for third proff and hivhly recommended for no...udayaditya6446
This document provides an overview of epidemiology and study designs used in public health research. It discusses key epidemiology concepts like incidence, prevalence, relative risk, and odds ratio. Different study designs are described including cross-sectional studies, case-control studies, cohort studies, and randomized controlled trials. Sampling methods like simple random sampling, stratified random sampling, and cluster sampling are also covered. The document uses examples and case studies to illustrate important epidemiological concepts and how different study designs can be applied to investigate health outcomes.
Choosing appropriate statistical test RSS6 2104RSS6
This document discusses choosing appropriate statistical tests based on study design and data type. It covers descriptive studies that measure prevalence and incidence, as well as analytic studies like randomized controlled trials, cohort studies, and case-control studies. For data type, it discusses approaches for continuous and categorical variables, including t-tests, ANOVA, chi-square tests, and regression. It also discusses measures of disease frequency, effect, and impact like risk difference, risk ratio, and odds ratio.
Categorical data analysis refers to methods for analyzing discrete or categorical response variables. Common distributions for categorical data include the Bernoulli, binomial, Poisson, and multinomial distributions. Chi-square tests can be used to test goodness of fit, independence, and homogeneity for categorical data. The chi-square test statistic compares observed and expected frequencies in one or more categories. A larger chi-square value provides more evidence to reject the null hypothesis of a good fit or independence between variables.
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
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This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
1. Biostatistics
Introduction to BIOSTATISTICS
Lecturer:
Jalal Karimi, MSc, PhD of Epidemiology
Reference:
Introduction to Biostatistics and Research Methods, Fifth Edition
By Sunder Rao
Department of Community Medicine
Third session
Jalal Karimi, Epidemiologist, PhD,
Community Medicine Department 1
2. Probability distribution
For making inferences from samples, we found that we have to think in
terms of the part played by chance.
This done by considering the sampling distribution and calculating the
probability.
Three such families witch are fundamental in the theory of statistics are:
Binomial distribution
Poisson distribution
Normal distribution
Jalal Karimi, Epidemiologist, PhD,
Community Medicine Department 2
3. Binomial distribution
Very often we are interested in knowing what proportion of individual in a
population possess a particular character.
For example:
The proportion persons of a locality who are sick at a particular point of
time.
An estimate of this proportion is calculated on the basis of a suitably
drown sample from this population and the corresponding sampling
distribution
In this type of problem the sampling distribution is given by a theoretical
frequency distribution known Binomial distribution.
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Community Medicine Department 3
4. Example:
In a morbidity survey in a village, it is found that the proportion of sick persons is 40%.
A sample of 4 person can be any one of the five types having no sick person in the
sample or having 1,2,3, or,4 sick person
Assuming random sampling, there are sixteen ways in witch we will get such sample as
shown in the diagram
4
5. Binomial distribution, generally
XnX
n
X
pp
)1(
1-p = probability
of failure
p =
probability of
success
X = #
successes
out of n
trials
n = number of trials
Note the general pattern emerging if you have only two possible
outcomes (call them 1/0 or yes/no or success/failure) in n independent
trials, then the probability of exactly X “successes”=
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Community Medicine Department 5
6. The Binomial Distribution
Overview
However, if order is not important, then
where is the number of ways to obtain x successes
in n trials, and i! = i (i – 1) (i – 2) … 2 1
n!
x!(n – x)!
px qn – xP(x) =
n!
x!(n – x)!
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Community Medicine Department 6
7. **All probability distributions are characterized by
an expected value and a variance:
If X follows a binomial distribution with
parameters n and p: X ~ Bin (n, p)
Then:
x= E(X) = np
x
2 =Var (X) = np(1-p)
x =SD (X)= )1( pnp
Note: the variance will
always lie between
0*N-.25 *N
p(1-p) reaches maximum at
p=.5
P(1-p)=.25
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Community Medicine Department 7
8. A binomial random variable X is defined to the number
of “successes” in n independent trials where the
P(“success”) = p is constant.
Notation: X ~ BIN(n,p)
In the definition above notice the following conditions
need to be satisfied for a binomial experiment:
1. There is a fixed number of n trials carried out.
2. The outcome of a given trial is either a “success”
or “failure”.
3. The probability of success (p) remains constant
from trial to trial.
4. The trials are independent, the outcome of a trial is
not affected by the outcome of any other trial.
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Community Medicine Department 8
9. Binomial Distribution
If X ~ BIN(n, p), then
where
.,...,1,0)1(
)!(!
!
)1()( nxpp
xnx
n
pp
x
n
xXP xnxxnx
psuccessP
nx
nnnn
)"("
trials.insuccesses""
obtaintowaysofnumberthex"choosen"
x
n
11!and10!also,1...)2()1(!
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Community Medicine Department 9
10. Binomial Distribution
If X ~ BIN(n, p), then
E.g. when n = 3 and p = .50 there are 8 possible equally
likely outcomes (e.g. flipping a coin)
SSS SSF SFS FSS SFF FSF FFS FFF
X=3 X=2 X=2 X=2 X=1 X=1 X=1 X=0
P(X=3)=1/8, P(X=2)=3/8, P(X=1)=3/8, P(X=0)=1/8
Now let’s use binomial probability formula instead…
.,...,1,0)1(
)!(!
!
)1()( nxpp
xnx
n
pp
x
n
xXP xnxxnx
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Community Medicine Department 10
11. Binomial Distribution
If X ~ BIN(n, p), then
E.g. when n = 3, p = .50 find P(X = 2)
.,...,1,0)1(
)!(!
!
)1()( nxpp
xnx
n
pp
x
n
xXP xnxxnx
8
3or375.)5)(.5(.3)5(.5.
2
3
)2(
ways3
1)12(
123
!1!2
!3
)!23(!2
!3
2
3
12232
XP
SSF
SFS
FSS
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Community Medicine Department 11
12. Example: Treatment of Kidney
Cancer
Suppose we have n = 40 patients who will be
receiving an experimental therapy which is
believed to be better than current treatments
which historically have had a 5-year survival rate
of 20%, i.e. the probability of 5-year survival is
p = .20.
Thus the number of patients out of 40 in our
study surviving at least 5 years has a binomial
distribution, i.e. X ~ BIN(40,.20).
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Community Medicine Department 12
13. Results and “The Question”
Suppose that using the new treatment we find
that 16 out of the 40 patients survive at least 5
years past diagnosis.
Q: Does this result suggest that the new therapy
has a better 5-year survival rate than the current,
i.e. is the probability that a patient survives at
least 5 years greater than .20 or a 20% chance
when treated using the new therapy?
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Community Medicine Department 13
14. What do we consider in answering
the question of interest?
We essentially ask ourselves the following:
If we assume that new therapy is no better than
the current what is the probability we would see
these results by chance variation alone?
More specifically what is the probability of
seeing 16 or more successes out of 40 if the
success rate of the new therapy is .20 or 20% as
well?
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Community Medicine Department 14
15. Connection to Binomial
This is a binomial experiment situation…
There are n = 40 patients and we are counting the
number of patients that survive 5 or more years. The
individual patient outcomes are independent and IF
WE ASSUME the new method is NOT better then the
probability of success is p = .20 or 20% for all patients.
So X = # of “successes” in the clinical trial is binomial
with n = 40 and p = .20,
i.e. X ~ BIN(40,.20)
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Community Medicine Department 15
16. Example: Treatment of Kidney Cancer
X ~ BIN(40,.20), find the probability that exactly 16
patients survive at least 5 years.
This requires some calculator gymnastics and some
scratchwork!
Also, keep in mind we need to find the probability of
having 16 or more patients surviving at least 5 yrs.
001945.80.20.
16
40
)16( 2416
XP
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Community Medicine Department 16
17. Example: Treatment of Kidney
Cancer
So we actually need to find:
P(X > 16) = P(X = 16) + P(X = 17) + … + P(X = 40)
+
…
+
= .002936
001945.80.20.
16
40
)16( 2416
XP
000686.80.20.
17
40
)17( 2317
XP
080.20.
40
40
)40( 040
XP
The chance that we would see
16 or more patients out of 40
surviving at least 5 years if the
new method has the same
chance of success as the current
methods (20%) is VERY
SMALL, .0029!!!!
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Community Medicine Department 17
18. Conclusion
Because it is high unlikely (p = .0029) that we would
see this many successes in a group 40 patients if the
new method had the same probability of success as the
current method we have to make a choice, either …
A) we have obtained a very rare result by dumb luck.
OR
B) our assumption about the success rate of the new
method is wrong and in actuality the new method has a
better than 20% 5-year survival rate making the
observed result more plausible.
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Community Medicine Department 18
19. The Poisson Distribution
When there is a large number of trials, but a small
probability of success, binomial calculation becomes
impractical
Example: Number of spells of diarrhea observed in a
group of infants over a predetermined period can be
counted but not the number of spells that did not
occur.
The probability of observing one spell, two spells,
etc., in a given sample in such cases, can theoretically
be found out by the use of Poisson distribution
P(x) =
e -µµx
x!
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Community Medicine Department 19
20. Assuming these are independent random events, the number
of people killed in a given year therefore has a Poisson
distribution:
Answer:
Poisson distribution
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Community Medicine Department 20
22. The Normal Distribution
Properties of the Normal Distribution
Shapes of Normal Distributions
Standard (Z) Scores
The Standard Normal Distribution
Transforming Z Scores into Proportions
Transforming Proportions into Z Scores
Finding the Percentile Rank of a Raw Score
Finding the Raw Score for a Percentile
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Community Medicine Department 22
23. Normal Distribution – A bell-shaped and
symmetrical theoretical distribution, with the
mean, the median, and the mode all coinciding at its
peak and with frequencies gradually decreasing at
both ends of the curve.
Normal Distributions
• The normal distribution is a theoretical ideal
distribution. Real-life empirical distributions never
match this model perfectly. However, many things
in life do approximate the normal distribution, and
are said to be “normally distributed.”
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Community Medicine Department 23
24. Scores “Normally Distributed?”
Is this distribution normal?
There are two things to initially examine: (1) look at
the shape illustrated by the bar chart, and (2)
calculate the mean, median, and mode.
Table 10.1 Final Grades in Social Statistics of 1,200 Students (1983-1993)
Midpoint
Score Frequency Bar Chart Freq.
Cum. Freq.
(below) %
Cum %
(below)
40 * 4 4 0/33 0/33
50 ******* 78 82 6/5 6/83
60 *************** 275 357 22/92 29/75
70 *********************** 483 840 40/25 70
80 *************** 274 1114 22/83 92/83
90 ******* 81 1195 6/75 99/58
100 * 5 1200 0/42 100
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Community Medicine Department 24
25. Scores Normally Distributed!
The Mean = 70.07
The Median = 70
The Mode = 70
Since all three are essentially equal, and this is
reflected in the bar graph, we can assume that these
data are normally distributed.
Also, since the median is approximately equal to
the mean, we know that the distribution is
symmetrical.
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Community Medicine Department 25
26. The Shape of a Normal Distribution:
The Normal Curve
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Community Medicine Department 26
27. The Shape of a Normal Distribution
Notice the shape of the normal curve in this graph. Some normal
distributions are tall and thin, while others are short and wide. All
normal distributions, though, are wider in the middle and
symmetrical.
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Community Medicine Department 27
28. Notice that the standard deviation changes the relative width of the
distribution; the larger the standard deviation, the wider the curve.
Different Shapes of the Normal Distribution
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Community Medicine Department 28
29. Areas Under the Normal Curve by
Measuring Standard Deviations
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30. Standard (Z) Scores
A standard score (also called Z score) is
the number of standard deviations that a
given raw score is above or below the
mean.
yS
YY
Z
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Community Medicine Department 30
31. The Standard Normal Table
A table showing the area (as a proportion,
which can be translated into a percentage) under
the standard normal curve corresponding to
any Z score or its fraction
Area up to
a given score
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Community Medicine Department 31
32. The Standard Normal Table
A table showing the area (as a proportion,
which can be translated into a percentage) under
the standard normal curve corresponding to
any Z score or its fraction
Area beyond
a given score
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Community Medicine Department 32
33. Finding the Area Between the Mean
and a Positive Z Score
Using the data presented in Table 10.1, find the
percentage of students whose scores range from the
mean (70.07) to 85.
(1) Convert 85 to a Z score:
Z = (85-70.07)/10.27 = 1.45
(2) Look up the Z score (1.45) in next slide
finding the proportion (.4265)
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Community Medicine Department 33
35. Finding the Area Between the
Mean and a Positive Z Score
(3) Convert the proportion (.4265) to a percentage (42.65%); this
is the percentage of students scoring between the mean and 85 in
the course.
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Community Medicine Department 35
36. Finding the Area Between the
Mean and a Negative Z Score
Using the data presented in Table 10.1, find
the percentage of students scoring between
65 and the mean (70.07)
(1) Convert 65 to a Z score:
Z = (65-70.07)/10.27 =
•(2) Since the curve is symmetrical and
negative area does not exist, use .49 to find
the area in the standard normal table:
-.49
.1879
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Community Medicine Department 36
38. Finding the Area Between the
Mean and a Negative Z Score
(3) Convert the proportion (.1879) to a percentage (18.79%); this is the
percentage of students scoring between 65 and the mean (70.07)
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Community Medicine Department 38
39. Finding the Area Between 2 Z Scores
on the Same Side of the Mean
Using the same data presented in Table 10.1, find the
percentage of students scoring between 74 and 84.
(1) Find the Z scores for 74 and 84:
Z = .38 and Z = 1.36
(2) Look up the corresponding areas for those Z scores:
.1480 and .4131
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Community Medicine Department 39
40. Finding the Area Between 2 Z Scores on the Same
Side of the Mean
(3) To find the highlighted area above, subtract the smaller area
from the larger area (.4131-.1480 = ).2651
Now, we have the percentage of students scoring
between 74 and 84.
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Community Medicine Department 40
41. Finding the Area Between 2 Z Scores on Opposite
Sides of the Mean
Using the same data, find the percentage of students
scoring between 62 and 72.
(1) Find the Z scores for 62 and 72:
Z = (72-70.07)/10.27 = .19
-.79
.3605
Z = (62-70.07)/10.27 =
(2) Look up the areas between these Z scores and
the mean, like in the previous 2 examples:
Z = .19 is .0753 and Z = -.79 is .2852
(3) Add the two areas together: .0753 + .2852 =
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Community Medicine Department 41
42. Finding the Area Between 2 Z Scores
on Opposite Sides of the Mean
(4) Convert the proportion (.3605) to a percentage (36.05%); this
is the percentage of students scoring between 62 and 72.
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Community Medicine Department 42
43. Finding Area Above a Positive Z Score or Below a
Negative Z Score
Find the percentage of students who did (a) very well,
scoring above 85, and (b) those students who did
poorly, scoring below 50.
(a) Convert 85 to a Z score, then look up the value in
Column C of the Standard Normal Table:
Z = (85-70.07)/10.27 = 1.45
(b) Convert 50 to a Z score, then look up the value
(look for a positive Z score!) in Column C:
Z = (50-70.07)/10.27 = -1.95
7.35%
2.56%
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Community Medicine Department 43
44. Finding Area Above a Positive Z
Score or Below a Negative Z Score
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Community Medicine Department 44
45. Finding a Z Score Bounding an Area Above It
Find the raw score that bounds the top 10 percent of
the distribution (Table 10.1)
(1) 10% = a proportion of .10
(2) Using the Standard Normal Table, look in Column
C for .1000, then take the value in Column A; this is
the Z score (1.28)
(3) Finally convert the Z score to a raw score:
Y=70.07 + 1.28 (10.27) = 83.22
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Community Medicine Department 45
46. Finding a Z Score Bounding an Area Above It
(4) 83.22 is the raw score that bounds the upper 10% of the
distribution. The Z score associated with 83.22 in this
distribution is 1.28
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Community Medicine Department 46
47. Finding a Z Score Bounding an Area
Below It
Find the raw score that bounds the lowest 5 percent of
the distribution (Table 10.1)
(1) 5% = a proportion of .05
(2) Using the Standard Normal Table, look in Column
C for .05, then take the value in Column A; this is the
Z score (-1.65); negative, since it is on the left side of
the distribution
(3) Finally convert the Z score to a raw score:
Y=70.07 + -1.65 (10.27) = 53.12
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Community Medicine Department 47
48. Finding a Z Score Bounding an Area Below It
(4) 53.12 is the raw score that bounds the lower 5% of the
distribution. The Z score associated with 53.12 in this
distribution is -1.65
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Community Medicine Department 48