Probability Biostatics and Research Methodology
- 1. Probability Ms. Nigar K.Mujawar Assistant Professor, Shri.Balasaheb Mane Shikshan Prasarak Mandal Ambap Womens College of Pharmacy, Peth-Vadgaon, Kolhapur, MH, INDIA.
- 2. Probability Definition: In research, probability refers to the likelihood of an event OR outcome occurring within the context of a study or experiment. It's a fundamental concept in statistical analysis that helps researchers quantify uncertain y and make inferences about populations based on sample data. The key roles of probability Quantifies the likelihood of events Provides a basis for inference Guides experimental design Ensures representative sampling Helps quantify uncertainty Essential in statistical analysis Enables prediction and decision-making Widely applicable across disciplines
- 4. Examples: 1.Probability of Adverse Events: Pharmaceutical companies often calculate the probability of adverse events associated with their drugs. This helps in assessing the safety profile of the drug and informing healthcare professionals and patients about potential risks. For example, the probability of a severe allergic reaction to a vaccine is an important consideration in vaccine development and administration. 2.Probability of Drug Interactions: Pharmaceuticals can interact with each other in the body, affecting their efficacy or safety. Understanding the probability of drug interactions is crucial in medication management and can influence prescribing practices to minimize adverse effects. For example, the probability of a drug interaction between a statin and a medication metabolized by the cytochrome P450 enzyme system may influence dosing recommendations.
- 5. Binomial distribution The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials (Independent Bernoulli trials refer to a series of experiments or events where each trial has only two possible outcomes - success or failure - and the outcome of each trial is independent of the outcomes of the previous trials. These trials are named after Swiss mathematician Jacob Bernoulli). where each trial has only two possible outcomes: success or failure.
- 6. Properties of Binomial Distribution Discrete Probability Distribution: The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials. 1. Two Possible Outcomes: Each trial in a binomial experiment can result in one of two possible outcomes: success (usually denoted as "1") or failure (usually denoted as "0"). 2. Fixed Number of Trials: The number of trials in a binomial experiment, denoted as "n", is fixed in advance. 3. Independent Trials: Each trial is independent of the others, meaning the outcome of one trial does not affect the outcome of another. 4. Constant Probability of Success: The probability of success on each trial, denoted as "p", remains constant from trial to trial. 5. Approximation with Normal Distribution: When the number of trials (n) is large and both np and n(1-p) are greater than or equal to 5, the binomial distribution can be approximated by a normal distribution.
- 7. Pharmaceutical Examples of Binomial Distribution 1. Drug Efficacy Trials: Assessing new drug efficacy compared to placebos or standard treatments by modeling positive responses among trial participants. 2. Vaccine Efficacy: Modeling the proportion of vaccinated individuals developing immunity against diseases. 3. Drug Response Rates: Modeling different response rates based on genetic variations among patients receiving the same medication. 4. Clinical Trials for Adverse Events: Modeling occurrences of adverse events during drug trials. 5. Quality Control in Drug Manufacturing: Modeling defective units in drug batches during quality control processes. 6. Drug Resistance in Microorganisms: Modeling proportions of microbial populations developing resistance to antibiotics. 7. Patient Compliance in Clinical Trials: Modeling the probability of patients adhering to prescribed treatment regimens in trials.
- 8. Normal Distribution The normal distribution, also known as the Gaussian distribution or bell curve, is a fundamental probability distribution that is symmetric and bell-shaped. Properties of Normal Distribution: 1. Continuous Probability Distribution: The normal distribution is a continuous probability distribution. 2. Bell-shaped Curve: It is characterized by a symmetric, bell-shaped curve when graphed. 3. Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution. 4. Standard Deviation: The spread of data in a normal distribution is described by the standard deviation. About 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations.
- 9. 5. Empirical Rule: Also known as the 68-95-99.7 rule, it states the percentage of data that falls within one, two, and three standard deviations from the mean in a normal distribution. 6. Symmetry: The normal distribution is symmetric around its mean. 7. Inflection Points: The points of inflection(Modification), where the curvature of the curve changes, are located at one standard deviation from the mean. 8. Central Limit Theorem: It states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. Pharmaceutical Examples of Normal Distribution 1. Drug Dosage Optimization: Normal distribution aids in determining optimal drug dosage by analyzing drug concentrations in patients' bloodstreams. 2. Biological Measurements: Pharmacokinetic and pharmacodynamic studies utilize normal distribution to interpret parameters like drug clearance rates and receptor binding affinity.
- 10. 3. Clinical Trial Outcomes: Data collected in clinical trials, such as changes in disease symptoms, often follow a normal distribution, enabling hypothesis testing and treatment effect estimation. 4. Population Pharmacokinetics: Normal distribution assists in modeling pharmacokinetic variability across a population, aiding in drug development and interpretation. 5. Quality Control in Manufacturing: Normal distribution is used in pharmaceutical manufacturing for quality control of critical attributes like tablet weight and drug content uniformity. 6. Biostatistical Analysis: Epidemiological(health-related events within populations, focusing on their distribution, causes, and control measures)studies rely(depend on) on normal distribution for analyzing disease prevalence(proportion of a particular condition), incidence rates, and risk factors, enabling robust statistical analysis.
- 11. Poisson distribution The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, when these events occur with a known average rate and independently of the time since the last event. Properties of Poisson distribution 1.Discrete Probability Distribution: The Poisson distribution is a discrete(Separate) probability distribution used to model the number of events occurring within a fixed interval of time or space. 2.Single Parameter: The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average rate of event occurrences in the given interval.
- 12. 3.Independent Events: The occurrence of events in a Poisson process is assumed to be independent of one another. 4.Approximation of Binomial Distribution: The Poisson distribution can be used as an approximation to the binomial distribution under certain conditions, such as when the number of trials is large and the probability of success is small. 5.Used for Rare Events: The Poisson distribution is often used to model rare events, such as the number of arrivals at a service point in a given time period or the number of radioactive decay events in a sample. 6.Applications: It is widely used in various fields, including queuing theory (branch of applied mathematics that studies the behavior of queues or waiting lines), reliability analysis, epidemiology, and telecommunications, where the occurrence of events can be modeled as a random process.
- 13. Pharmaceutical Examples of Poisson distribution 1. Drug Dispensing Errors: Modeling the number of dispensing errors in a pharmacy over a fixed time period. 2. Medication Adverse Events: Analyzing the number of adverse events reported for a specific drug within a certain timeframe. 3. Hospital Admissions: Modeling the number of patient admissions to a hospital's emergency department within a day or week. 4. Drug Manufacturing Defects: Predicting the number of defective drug units produced in a pharmaceutical manufacturing process. 5. Patient Arrivals at Clinics: Analyzing the number of patient arrivals at a clinic or healthcare facility during specific hours of the day. 6. Drugstore Sales Transactions: Modeling the number of sales transactions occurring at a drugstore during peak hours.
- 14. • Population: In statistics, a population refers to the entire group of individuals OR items that we are interested in studying, and from which we collect data. It's the entire pool from which a statistical sample is drawn. • Sample: A sample is a subset of the population that is selected for study. It's important that the sample is representative of the population from which it is drawn to ensure the validity of statistical inferences. • Large Sample: In statistics, a large sample refers to a sample size that is sufficiently large enough to provide reliable estimates of population parameters. Generally, as the sample size increases, the sample statistics (such as mean or proportion) tend to converge to the population parameters.
- 15. • Small Sample: Conversely, a small sample refers to a relatively small sample size. Small samples may be subject to greater variability and may not provide accurate estimates of population parameters without appropriate statistical techniques. • Null Hypothesis (H0): The null hypothesis is a statement that there is no significant difference or effect. It represents the default assumption or the status quo(existing condition or situation or current situation). In hypothesis testing, we test whether there is enough evidence to reject the null hypothesis in favor of an alternative hypothesis. • Alternative Hypothesis (H1 or Ha): The alternative hypothesis is a statement that contradicts(Opposite/Deny) the null hypothesis. It represents the researcher's claim or the effect they are trying to detect. In hypothesis testing, rejecting the null hypothesis provides evidence in support of the alternative hypothesis.
- 16. Sr.No . Null Hypothesis(H0) Alternative Hypothesis(H1 or Ha) 1 Is formulated in a negative form. Is formulated in an affirmative(Positive) form. 2 Assumes that there is no relationship between the variables. Assumes a specific relationship between variables. 3 Implies that the outcome will not be observed if the condition is met. Implies that the outcome will be observed if one condition is met. 4 The result of null hypothesis indicates no changes in opinions or actions The results are observed as a result of some real causes 5 The results are observed as a result of chance The results of an alternative hypothesis causes changes. 6 If the null hypothesis is accepted the results of the study become insignificant. If an alternative hypothesis is accepted the results of the study become significant.
- 17. 7 Denoted by symbol H0 Denoted by H1 or Ha 8 If the P Value is greater than the level of significance, the null hypothesis is accepted. If the P value is smaller than the level of significance an alternative hypothesis is accepted. 9 Predicts there NO relationship between the 2 variable. Predicts there a relationship between the 2 variables. 10 Statement about the value of a population parameter. Statement about the value of a population parameter that must be true if the null hypothesis is false. 11 Always stated as an Equality Stated in on of three forms > < ≠
- 19. • Sampling: Sampling is the process of selecting a subset (sample) from a larger group (population) in order to study that subset and make inferences(conclusions) about the larger group. It's essential in statistics because it allows researchers to gather information about a population without having to study every individual or item within that population. • Essence of Sampling: Sampling is crucial in statistics because it allows researchers to draw conclusions about populations without having to study every individual or item within those populations. By selecting representative samples and using statistical techniques, researchers can make valid inferences about populations, saving time and resources. Sampling methods must be carefully chosen to ensure that samples accurately represent the populations of interest and that statistical analyses are valid.
- 20. A. Random Sampling 1. Simple Random Sampling 2. Systematic Random Sampling 3. Stratified Random Sampling 4. Cluster Random Sampling B. Non-Random Sampling 1. Convenience Sampling 2. Purposive Sampling 3. Quota Sampling 4. Snowball Sampling C. Mixed Methods Sampling 1. Sequential Sampling 2. Concurrent Sampling 3. Triangulation Sampling D. Probability Sampling 1. Random Sampling Techniques 2. Systematic Sampling 3. Stratified Sampling
- 21. ERROR I: Random Sampling 1. Simple Random Sampling 2. Systematic Random Sampling 3. Stratified Random Sampling 4. Cluster Random Sampling ERROR II: Non-Random Sampling 1. Convenience Sampling 2. Purposive Sampling 3. Snowball Sampling 4. Quota Sampling
- 22. ERROR I: Random Sampling Random sampling involves selecting a sample from a population in such a way that each member of the population has an equal chance of being chosen. This method helps in reducing bias and ensures that the sample is representative of the population. Types: 1. Simple Random Sampling: Each member of the population is equally likely to be selected, and each sample of a given size has an equal chance of selection. 2. Systematic Random Sampling: Selecting every nth member of the population after a random start. 3. Stratified Random Sampling: Dividing the population into distinct subgroups (or strata) and then randomly selecting samples from each subgroup. 4. Cluster(a grouping or collection of similar items, entities, or phenomena) Random Sampling: Dividing the population into clusters and then randomly selecting entire clusters to sample.
- 23. ERROR II: Non-Random Sampling Non-random sampling methods do not involve random selection of participants from the population. Instead, participants are chosen based on specific criteria or convenience, which may introduce bias (systematic error or deviation from the true value in a measurement) into the sample. Types: 1. Convenience Sampling: Selecting individuals who are conveniently available or easy to reach. 2. Purposive Sampling: Selecting participants based on specific characteristics or criteria relevant to the research. 3. Snowball Sampling: Using existing participants to recruit new participants, is often used when the population of interest is difficult to access. 4. Quota Sampling: Dividing the population into subgroups and then setting quotas(predetermined targets or proportions set for specific categories or groups within a larger population) for each subgroup to ensure representation, but the actual selection within each subgroup is non-random.
- 24. Mixed Methods Sampling: • Definition: Combines qualitative and quantitative sampling strategies to gather diverse data. • Purpose: Offers a comprehensive understanding of pharmaceutical issues by merging different data types. • Example: Surveying patients about drug efficacy (quantitative) and conducting interviews to explore their experiences (qualitative). 1. Sequential Sampling: • Definition: Involves collecting data from one group before moving to another. • Purpose: Tracks changes over time or assesses different populations sequentially. • Example: Administering a drug to a small group initially, then expanding to larger populations to monitor effects. 2. Concurrent Sampling: • Definition: Simultaneously collecting data from multiple groups or sources. • Purpose: Allows for immediate comparison between groups or conditions. • Example: Testing two formulations of a drug concurrently to determine efficacy and side effects.
- 25. 3.Triangulation Sampling: • Definition: Utilizes multiple methods or sources to validate findings. • Purpose: Enhances the credibility and depth of research by corroborating evidence(support for a statement). • Example: Gathering data on drug effectiveness from clinical trials, patient records, and expert opinions for a comprehensive assessment.
- 26. Probability Sampling: • Definition: Probability sampling involves selecting a sample from a population in a way that each member of the population has a known and non-zero chance of being selected. • Purpose: It ensures that the sample is representative of the population, allowing for generalization of findings. • Example: Selecting patients from a hospital registry using random sampling techniques to study the prevalence(Proportion) of a certain disease in the population. 1.Random Sampling Techniques: • Definition: Random sampling techniques involve selecting a sample from a population where each member has an equal chance of being selected. • Purpose: It ensures that the sample is unbiased and representative of the population. • Example: Using simple random sampling to select vials from a batch of pharmaceutical products for quality control testing.
- 27. 2.Systematic Sampling: • Definition: Systematic sampling involves selecting every nth member from a population after randomly selecting the first member. • Purpose: It provides a simple and efficient way to select a representative sample when a list of the population is available. • Example: Selecting every 10th patient from a hospital admission list to study patient satisfaction levels. 3.Stratified Sampling: • Definition: Stratified sampling involves dividing the population into subgroups (strata) based on certain characteristics and then randomly selecting samples from each subgroup. • Purpose: It ensures that each subgroup of the population is represented proportionally in the sample, allowing for comparisons between subgroups. • Example: Stratifying patients by age groups and then randomly selecting samples from each age group to study the efficacy of a drug across different age categories. These sampling techniques are commonly used in pharmaceutical research to ensure that study samples are representative of the target population, thereby enhancing the validity and generalizability of research findings.
- 28. Standard Error of the Mean (SEM) The standard error of the mean (SEM) is a measure of how much the sample mean is likely to vary from the true population mean. In other words, it quantifies the precision(accurate, or exact) of the sample mean estimate. Definition: The standard error of the mean is the standard deviation of the sampling distribution of the sample means. In simpler terms, it tells us how much the sample means from multiple samples are expected to differ from each other. Calculation: Mathematically, the standard error of the mean is calculated by dividing the standard deviation of the population (σ) by the square root of the sample size (n). If the population standard deviation is unknown, the sample standard deviation (s) can be used as an estimate.
- 29. • Interpretation: A smaller standard error indicates that the sample mean is likely to be close to the true population mean, while a larger standard error suggests that the sample mean may be less reliable and could deviate further from the population mean. • `Use: 1. Researchers often report the standard error along with the sample mean to provide a measure of the precision of the estimate. 2. It helps in determining the confidence interval around the sample mean and assessing the reliability of the findings. •Example: Suppose the SEM for the group receiving the new drug is smaller than the SEM for the placebo group (refers to a group of participants in a scientific study or clinical trial who receive a treatment that appears identical to the active treatment being tested but does not contain the active ingredient or therapeutic component). This would indicate that the mean blood pressure reduction observed in the drug group is more reliable and precise compared to the placebo group, suggesting that the new drug may indeed be effective in lowering blood pressure. In summary, SEM helps pharmaceutical researchers assess the precision of their findings in clinical trials, providing valuable information about the reliability of the observed effects of a drug or treatment.