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

3. Properties of Probability

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.