DESIGN OF EXPERIMENTS (DOE)
DOE is invented by Sir Ronald Fisher in 1920βs and 1930βs.
The following designs of experiments will be usually followed:
Completely randomised design(CRD)
Randomised complete block design(RCBD)
Latin square design(LSD)
Factorial design or experiment
Confounding
Split and strip plot design
FACTORIAL DESIGN
When a several factors are investigated simultaneously in a single experiment such experiments are known as factorial experiments. Though it is not an experimental design, indeed any of the designs may be used for factorial experiments.
For example, the yield of a product depends on the particular type of synthetic substance used and also on the type of chemical used.
ADVANTAGES OF FACTORIAL DESIGN.
Factorial experiments are advantageous to study the combined effect of two or more factors simultaneously and analyze their interrelationships. Such factorial experiments are economic in nature and provide a lot of relevant information about the phenomenon under study. It also increases the efficiency of the experiment.
It is an advantageous because a wide range of factor combination are used. This will give us an idea to predict about what will happen when two or more factors are used in combination.
DISADVANTAGES
It is disadvantageous because the execution of the experiment and the statistical analysis becomes more complex when several treatments combinations or factors are involved simultaneously.
It is also disadvantageous in cases where may not be interested in certain treatment combinations but we are forced to include them in the experiment. This will lead to wastage of time and also the experimental material.
2(square) FACTORIAL EXPERIMENT
A special set of factorial experiment consist of experiments in which all factors have 2 levels such experiments are referred to generally as 2n factorials.
If there are four factors each at two levels the experiment is known as 2x2x2x2 or 24 factorial experiment. On the other hand if there are 2 factors each with 3 levels the experiment is known as 3x3 or 32 factorial experiment. In general if there are n factors each with p levels then it is known as pn factorial experiment.
The calculation of the sum of squares is as follows:
Correction factor (CF) = (πΊπ)2/π
GT = grand total
n = total no of observations
Total sum of squares = ββγπ₯2βπΆπΉγ
Replication sum of squares (RSS) = ((π 1)2+(π 2)2+β¦+(π π)2)/π - CF
Or
1/π ββπ 2βπΆπΉ
2(Cube) FACTORIAL DESIGN
In this type of design, one independent variable has 2 levels, and the other independent variable has 3 levels.
Estimating the effect:
In a factorial design the main effect of an independent variable is its overall effect averaged across all other independent variable.
Effect of a factor A is the average of the runs where A is at the high level minus the average of the runs
you can know about the central composite design, historical design, optimisation techniques and also about the TYPES OF CENTRAL COMPOSITE DESIGN, BOX-BEHNKEN DESIGN, DATA COLLECTION, CRITICISM OF DATA, PRESENTATION OF FACTS, PURPOSE, OPTIMISATION PROCESS, DIFFERENT TYPES PRESENT IN IT AND THEIR CLASSIFICATION AND EXPLANATION.
you can know about the central composite design, historical design, optimisation techniques and also about the TYPES OF CENTRAL COMPOSITE DESIGN, BOX-BEHNKEN DESIGN, DATA COLLECTION, CRITICISM OF DATA, PRESENTATION OF FACTS, PURPOSE, OPTIMISATION PROCESS, DIFFERENT TYPES PRESENT IN IT AND THEIR CLASSIFICATION AND EXPLANATION.
Todayβs overwhelming number of techniques applicable to data analysis makes it extremely difficult to define the most beneficial approach while considering all the significant variables.
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.
Sir Ronald Fisher pioneered the development of ANOVA for analyzing results of agricultural experiments.1 Today, ANOVA is included in almost every statistical package, which makes it accessible to investigators in all experimental sciences. It is easy to input a data set and run a simple ANOVA, but it is challenging to choose the appropriate ANOVA for different experimental designs, to examine whether data adhere to the modeling assumptions, and to interpret the results correctly. The purpose of this report, together with the next 2 articles in the Statistical Primer for Cardiovascular Research series, is to enhance understanding of ANVOA and to promote its successful use in experimental cardiovascular research. My colleagues and I attempt to accomplish those goals through examples and explanation, while keeping within reason the burden of notation, technical jargon, and mathematical equations.
Experimental design is a way to carefully plan experiments in advance so that results are both objective and valid. Ideally, an experimental design should:
β’ Describe how participants are allocated to experimental groups. A common method is completely randomized design, where participants are assigned to groups at random. A second method is randomized block design, where participants are divided into homogeneous blocks (for example, age groups) before being randomly assigned to groups.
β’ Minimize or eliminate confounding variables, which can offer alternative explanations for the experimental results.
β’ Allows making inferences about the relationship between independent variables and dependent variables.
β’ Reduce variability, to make it easier to find differences in treatment outcomes.
Types of Experimental Design
1. Between Subjects Design.
2. Completely Randomized Design.
3. Factorial Design.
4. Matched-Pairs Design.
5. Observational Study
β’ Longitudinal Research
β’ Cross Sectional Research
6. Pretest-Posttest Design.
7. Quasi-Experimental Design.
8. Randomized Block Design.
9. Randomized Controlled Trial
10. Within subjects Design.
Introduction & Basics of DoE
Terminologies
Key steps in DOE
Softwares used for DOE
Factorial Designs ( Full and Fractional)
Mixture Designs
Response Surface Methodology
Central Composite Design
Box -Behnken Design
Conclusion
References
Non Parametric Test
1. Wilcoxon Signed Rank Test: (WSRT)
In this test the difference in positive and negative value is taken into consideration without assigning any weightage to the magnitude of the differences as a result, the sign test is often used in practice.
The Wilcoxon Sign Rank test can be used to overcome this limitation.
2. Wilcoxon Rank Sum test: (WRST)
This is also called as Mann- Whitney U test.
WRST is used to compare two independent sample while WSRT compare two related or two dependent samples.
This test is applicable if the data are at least ordinal {i.e. the observation can be ordered}
3. MANN-WHITNEY U-TEST
It is a non-parametric method used to determine whether two independent samples have been drawn from populations with same distribution. This test is also known as U-Test.
This test enables us to test the null hypothesis that both population medians are equal(or that the two samples are drawn from a single population).
4. KRUSKAL WALLIS TEST
This test is employed when more then 2 population are involved where as Man Whitney test is used when there are 2 populations. The use of this test will enable us to determine weather independent samples have been drawn from the sample population (or) different populations have the same distribution.
5. FRIEDMAN TEST
It is a non-parametric test applied to a data i.e. at least ranked and it is in the form of a 2 way ANOVA design. This test which may be applied to ranked or Interval or Ratio type of data is used when more than 2 treatment, group are included in the experiment.
Through this ppt you could learn what is Wilcoxon Signed Ranked Test. This will teach you the condition and criteria where it can be run and the way to use the test.
Graphs(Biostatistics and Research Methodology) B.pharmacy(8th sem.)Pranjal Saxena
Β
This slides contains the description about the Graphs(Histograms, Pie-Chart, Cubic Graph, Response surface Plot, Counter surface plot ) mainly Histograms with advantages, disadvantages and examples, Pie-chart with advantages, disadvantages and examples, Cubic Graph with examples, Response surface plot and Counter plot with examples and uses.
CROSSOVER STUDY DESIGN, DESIGN OF PHARMACOKINETIC STUDIES, FACTORS INFLUENCING BIOAVAILABILITY STUDIES, STUDY DESIGN, PARALLEL DESIGN, CROSS-OVER STUDIES, LATIN SQUARE DESIN, TWO-PERIOD CROSSOVER STUDY DESIGN, BALANCED INCOMPLETE BLOCK DESIGN (BIBD), REPLICATE CROSSOVER STUDY DESIGN , DIFFERENCE BETWEEN PARALLEL AND CROSSOVER STUDY DESIGN.
Application of Excel and SPSS software for statistical analysis- Biostatistic...Himanshu Sharma
Β
This slide contains B.Pharm Biostatistics and Research methodology 8th Sem. Unit-3 L2 topic- "Statistical Analysis using Software"
It contains topics:
1. MS Excel
2. SPSS
3. MiniTab
#StatisticalAnalysisusingMSExcel
#StatisticalAnalysisusingMiniTab
#StatisticalAnalysisusingSPSS
Todayβs overwhelming number of techniques applicable to data analysis makes it extremely difficult to define the most beneficial approach while considering all the significant variables.
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.
Sir Ronald Fisher pioneered the development of ANOVA for analyzing results of agricultural experiments.1 Today, ANOVA is included in almost every statistical package, which makes it accessible to investigators in all experimental sciences. It is easy to input a data set and run a simple ANOVA, but it is challenging to choose the appropriate ANOVA for different experimental designs, to examine whether data adhere to the modeling assumptions, and to interpret the results correctly. The purpose of this report, together with the next 2 articles in the Statistical Primer for Cardiovascular Research series, is to enhance understanding of ANVOA and to promote its successful use in experimental cardiovascular research. My colleagues and I attempt to accomplish those goals through examples and explanation, while keeping within reason the burden of notation, technical jargon, and mathematical equations.
Experimental design is a way to carefully plan experiments in advance so that results are both objective and valid. Ideally, an experimental design should:
β’ Describe how participants are allocated to experimental groups. A common method is completely randomized design, where participants are assigned to groups at random. A second method is randomized block design, where participants are divided into homogeneous blocks (for example, age groups) before being randomly assigned to groups.
β’ Minimize or eliminate confounding variables, which can offer alternative explanations for the experimental results.
β’ Allows making inferences about the relationship between independent variables and dependent variables.
β’ Reduce variability, to make it easier to find differences in treatment outcomes.
Types of Experimental Design
1. Between Subjects Design.
2. Completely Randomized Design.
3. Factorial Design.
4. Matched-Pairs Design.
5. Observational Study
β’ Longitudinal Research
β’ Cross Sectional Research
6. Pretest-Posttest Design.
7. Quasi-Experimental Design.
8. Randomized Block Design.
9. Randomized Controlled Trial
10. Within subjects Design.
Introduction & Basics of DoE
Terminologies
Key steps in DOE
Softwares used for DOE
Factorial Designs ( Full and Fractional)
Mixture Designs
Response Surface Methodology
Central Composite Design
Box -Behnken Design
Conclusion
References
Non Parametric Test
1. Wilcoxon Signed Rank Test: (WSRT)
In this test the difference in positive and negative value is taken into consideration without assigning any weightage to the magnitude of the differences as a result, the sign test is often used in practice.
The Wilcoxon Sign Rank test can be used to overcome this limitation.
2. Wilcoxon Rank Sum test: (WRST)
This is also called as Mann- Whitney U test.
WRST is used to compare two independent sample while WSRT compare two related or two dependent samples.
This test is applicable if the data are at least ordinal {i.e. the observation can be ordered}
3. MANN-WHITNEY U-TEST
It is a non-parametric method used to determine whether two independent samples have been drawn from populations with same distribution. This test is also known as U-Test.
This test enables us to test the null hypothesis that both population medians are equal(or that the two samples are drawn from a single population).
4. KRUSKAL WALLIS TEST
This test is employed when more then 2 population are involved where as Man Whitney test is used when there are 2 populations. The use of this test will enable us to determine weather independent samples have been drawn from the sample population (or) different populations have the same distribution.
5. FRIEDMAN TEST
It is a non-parametric test applied to a data i.e. at least ranked and it is in the form of a 2 way ANOVA design. This test which may be applied to ranked or Interval or Ratio type of data is used when more than 2 treatment, group are included in the experiment.
Through this ppt you could learn what is Wilcoxon Signed Ranked Test. This will teach you the condition and criteria where it can be run and the way to use the test.
Graphs(Biostatistics and Research Methodology) B.pharmacy(8th sem.)Pranjal Saxena
Β
This slides contains the description about the Graphs(Histograms, Pie-Chart, Cubic Graph, Response surface Plot, Counter surface plot ) mainly Histograms with advantages, disadvantages and examples, Pie-chart with advantages, disadvantages and examples, Cubic Graph with examples, Response surface plot and Counter plot with examples and uses.
CROSSOVER STUDY DESIGN, DESIGN OF PHARMACOKINETIC STUDIES, FACTORS INFLUENCING BIOAVAILABILITY STUDIES, STUDY DESIGN, PARALLEL DESIGN, CROSS-OVER STUDIES, LATIN SQUARE DESIN, TWO-PERIOD CROSSOVER STUDY DESIGN, BALANCED INCOMPLETE BLOCK DESIGN (BIBD), REPLICATE CROSSOVER STUDY DESIGN , DIFFERENCE BETWEEN PARALLEL AND CROSSOVER STUDY DESIGN.
Application of Excel and SPSS software for statistical analysis- Biostatistic...Himanshu Sharma
Β
This slide contains B.Pharm Biostatistics and Research methodology 8th Sem. Unit-3 L2 topic- "Statistical Analysis using Software"
It contains topics:
1. MS Excel
2. SPSS
3. MiniTab
#StatisticalAnalysisusingMSExcel
#StatisticalAnalysisusingMiniTab
#StatisticalAnalysisusingSPSS
In this ppt the viewer will able to know about designing of experiments. How experimental design helps to improve the quality & purity of the products. In this example, our experimental design is a planned experiment that is used to determine how reactor temperature and residence time affect purity so we can find the optimum operating conditions. Experimental design is needed to rectify the error in materials, methods & machines.
Portion explained:
1. Introduction to the problem
2. EXPERIMENTAL DESIGN TERMINOLOGY
3. EXPERIMENTAL DESIGN DATA
4. EFFECTS AND MAIN EFFECTS
5. INTERACTIONS BETWEEN FACTORS
6. ARE THE EFFECTS, MAIN EFFECTS AND INTERACTIONS SIGNIFICANT?
Population in statistics means the whole of the information which comes under the preview of statistical investigation.
In other words, an aggregate of objects animate or in animate under study is the population.
It is also known as βUniverseβ.
Microsoft ExcelΒ is a spreadsheet program used to record and analyse numerical and statistical data. Microsoft Excel provides multiple features to perform various operations like calculations, pivot tables, graph tools, macro programming, etc.
An Excel spreadsheet can be understood as a collection of columns and rows that form a table. Alphabetical letters are usually assigned to columns, and numbers are usually assigned to rows. The point where a column and a row meet is called a cell.
SPSS (Statistical Package for the Social Sciences) is a versatile and responsive program designed to undertake a range of statistical procedures.Β SPSS software is widely used in a range of disciplines and is available from all computer pools within the University of South Australia.
DOE is an essential tool to ensure products and processes satisfy Quality by Design requirements imposed by regulatory agencies.Β Using aβ―QbD approach to develop your testing process can help you reduce waste, meet compliance criteria and get to market faster.Β Β
DOE helps you create a reliable QbD process for assessing formula robustness, determining critical quality attributes and predicting shelf life by using a few months of historical data.
Minitab is a statistics package developed at the Pennsylvania State University by researchers Barbara F. Ryan, Thomas A. Ryan, Jr., and Brian L. Joiner in conjunction with Triola Statistics Company in 1972.
It began as a light version of OMNITAB 80, a statistical analysis program by NIST, which was conceived by Joseph Hilsenrath in years 1962-1964 as OMNITAB program for IBM 7090. The documentation for OMNITAB 80 was last published 1986, and there has been no significant development since then.
R is a language and environment for statistical computing and graphics."
"R provides a wide variety of statistical (linear and nonlinear modelling, classical statistical tests, time-series analysis, classification, clustering) and graphical techniques, and is highly extensible."
"One of R's strengths is the ease with which well-designed publication-quality plots can be produced, including mathematical symbols and formulae where needed.β
SAMPLE SIZE DETERMINATION
Sample size determination is the essential step of research methodology. It is an act of choosing the number of observers or replicates to include in a statistical sample.
Sample size determinationΒ is the act of choosing the number of observations orΒ replicatesΒ to include in aΒ statistical sample. The sample size is an important feature of any empirical study in which the goal is to makeΒ inferencesΒ about aΒ populationΒ from a sample.
Precision
A measure of how close an estimate is to the true value of a population parameter. Or it can be thought of as the amount of fluctuation from the population parameter that we can expect by chance alone in sample estimates.
Degree of Precision
This is presented in the form of a confidence interval (Range of values within which confidence lies).
RESEARCH REPORT
A research report is considered a major component of any research study as the research remains incomplete till the report has been presented or written. No matter how good a research study, and how meticulously the research study has been conducted, the findings of the research are of little value unless they are effectively documented and communicated to others.
TYPES OF RESEARCH REPORT
The research report is classified based on 2 things; Nature of research and Target audience.
COHORT STUDIES
A research study that compares a particular outcome in groups of individuals who are alike in many ways but differ by a certain characteristic is called as Cohort study.
Cohort studies are a type of research design that follow groups of people over time. Researchers use data from cohort studies to understand human health and the environmental and social factors that influence it.
CLINICAL TRIALS
A clinical trial, also known as a clinical research study, is a protocol to evaluate the effects and efficacy of experimental medical treatments or behavioral interventions on health outcomes. This type of study gathers data from volunteer human subjects and is typically funded by a medical institution, university orΒ nonprofitΒ group, or by pharmaceutical companies and government agencies.
Clinical trial vs. clinical study
A clinical study is research conducted with the intent of gaining medical knowledge. Observational and interventional are the two main types of clinical studies. A clinical trial is an interventional study.
NEED FOR RESEARCH
Research is a systemic process of collecting and analyzing information to increase the understanding of the phenomenon under study.
It strengthens pharmacist-provided services, builds the evidence base for developing and commissioning new services, improves patient care and contributes to health service knowledge.
Phase I studies: Are done on healthy volunteers who agree to take the study drug to help the doctors determine how safe the drug is and if there are any side effects. Usually a small number of subjects (20-100) participate in Phase I studies. Approximately 70% of new drugs will pass this phase.
Phase II studies: Measure the effect of the new drug in patients with the disease or disorder to be treated. The main purpose is to determine safety and effectiveness of the new drug. Usually several hundred patients participate. These studies are usually βDouble-blinded, randomized and controlledβ.
Phase III studies: also use patients with the disorder to be treated by the new drug. These studies are done to gain a more thorough understanding of the effectiveness, benefits and side effects of the study drug.Β
NEED FOR DESIGN OF EXPERIMENTS
Design of experiments (DOE) is defined as a branch of applied statistics that deals with planning, conducting, analyzing, and interpreting controlled tests to evaluate the factors that control the value of a parameter or group of parameters.
DOE is a powerfulΒ data collection and analysis tool that can be used in a variety of experimental situations.
1. PRE-EXPERIMENTAL DESIGN
In pre-experimental research design, either a group or various dependent groups are observed for the effect of the application of an independent variable which is presumed to cause change.
It is the simplest form of experimental research design and is treated with no control group
2. TRUE EXPERIMENTAL DESIGN
The true experimental research design relies on statistical analysis to approve or disprove a hypothesis. It is the most accurate type of experimental design and may be carried out with or without a pretest on at least 2 randomly assigned dependent subjects.
The true experimental research design must contain a control group, a variable that can be manipulated by the researcher, and the distribution must be random.
3. QUASI EXPERIMENTAL DESIGN
The word "quasi" means partial, half, or pseudo. Therefore, the quasi-experimental research bearing a resemblance to the true experimental research, but not the same.Β Β In quasi-experiments, the participants are not randomly assigned, and as such, they are used in settings where randomization is difficult or impossible.
This is very common in educational research, where administrators are unwilling to allow the random selection of students for experimental samples.
PLAGIARISM
The word Plagiarism is derived from the Latin word Plagiarius, which means abducting, kidnapping, seducing, or plundering.
Correlation- If two variables are so inter-related in such a manner that change in one variable brings about change in the other variable, then this type of relation of variable is known as correlation.
Types of Correlation.
1.Based on the direction of change of variables
a. Positive
correlation
b. Negative
correlation
2. Based upon the number of variables studied
a. Simple
correlation
b. Partial correlation
c. Multiple correlation
3. Based upon the constancy of the ratio of change between the variables
a. Linear correlation
b. Non-linear correlation
METHODS OF STUDYING CORRELATION
1) GRAPHIC
METHODS
A) SCATTER DIAGRAM
B) CORRELATION
GRAPH
2). ALGEBRIC METHOD
A) KARL PEARSON COEFFICIENT OF CORRELATION
B) RANK CORRELATION METHOD
C) CONCURRENT DEVIATION METHOD
Uses of Correlation.
Merits of Correlation.
Demerits of Correlation.
Dispersion- It is a statistical term that describes the size of the distribution of values expected for a particular variable and can be measured by several different statistics, such as Range, Variance and standard deviation.
Method of Dispersion-A measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of their distribution.
Methods of Dispersion.
1.Relative Dispersion
a. Coefficient of Mean Deviation
b. Coefficient of Quartile Deviation
c. Coefficient of Range
d. Coefficient of Variation
2. Absolute Dispersion
a. Range
b. Quartile range
c. Standard deviation
d. Mean Deviation
Range- It is the difference between smallest & largest values in the dataset. Also the relative measure of range is known as Coefficient of Range.
Advantages and disadvantages of Range.
Calculation of Range by different Methods.
b. Quartile Range- The interquartile range of a group of observations is the interval between the values of upper quartile and the lower quartile for that group.
Advantages and Disadvantages of Quartile Range.
Calculation of Quartile Range by different Methods.
c. Standard Deviation- It measures the absolute dispersion (or) variability of a distribution. A small standard deviation means a high degree of uniformity of the observations as well as homogeneity in the series.
Advantages and Disadvantages of Quartile Range.
Calculation of Standard Deviation using.
i) Direct Method
ii) Short-cut Method
iii) Step Deviation Method.
Mean- Mean is an essential concept in mathematics and statistics. The mean is the average or the most common value in a collection of numbers
Types of Mean
A. Arithmetic Mean
a. Simple Arithmetic Mean
b. Weighted Arithmetic Mean
B. Geometric Mean
C. Harmonic Mean
1.Calculation of Simple Arithmetic Mean
a) Direct Method
b) Shortcut Method
c) Step Deviation Method
2. Calculation of Weighted Arithmetic Mean
a) Direct Method
b) Shortcut Method
Merits and Demerits of Different types of Mean.
Introduction to Mode.
Calculation of modes by different methods.
Merits and Demerits of Mode.
Mode is the value which occurs the maximum number of times in a series of observations and has the highest frequency.
Calculation of Mode
1. Calculation of mode in a series of individual observations (Ungrouped data)
2. Calculation of mode in a discrete series (Grouped data)
3. Calculation of mode in a continuous series (Grouped data)
4. Calculation of mode in a unequal class intervals (Grouped data)
Median
Middle value in a distribution is known as Median.
Calculation of median.
1. Calculation of median in a series of individual observations or Calculation of median for ungrouped data
2. Calculation of median for grouped data
a) Calculation of median in a discrete series.
b) Calculation of median in a continuous series.
c) Calculation of median in unequal class intervals.
d) Calculation of median in open-end classes.
Merits and Demerits of Median.
Frequency distribution, types of frequency distribution.
Ungrouped frequency distribution
Grouped frequency distribution
Cumulative frequency distribution
Relative frequency distribution
Relative cumulative frequency distribution
Graphical representation of frequency distribution
I. Representation of Grouped data
1.Line graphs
2.Bar diagrams
a) Simple bar diagram
b)Multiple/Grouped bar diagram
c)Sub-divided bar diagram.
d) % bar diagram
3. Pie charts
4.Pictogram
II. Graphical representation of ungrouped data
1, Histogram
2.Frequency polygon
3.Cumulative change diagram
4. Proportional change diagram
5. Ratio diagram
Introduction to biostatistics and its application in various sectors.
Introduction to variables and variation.
Different types of variables and their introduction.
Use of biostatistics in various fields.
I am Mrs. G. Sreelatha, Assistant Professor, CMR College Of Pharmacy, Hyderabad.
I will be uploading notes on Biostatistics And Research Methodology (BRM) of B.Pharmacy, 4th year II sem based on PCI syllabus - JNTUH.
Topic included in this PPT are Origin and History of Statistics.
Hope it will be useful for your studies and will clear your all the doubts.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
Β
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as βdistorted thinkingβ.
How to Create Map Views in the Odoo 17 ERPCeline George
Β
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
Palestine last event orientationfvgnh .pptxRaedMohamed3
Β
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Β
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
Β
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdf
Β
factorial design.pptx
1. BIOSTATISTICS AND RESEARCH METHODOLOGY
Unit-5: design and analysis of experiments (Factorial design)
PRESENTED BY
Gokara Madhuri
B. Pharmacy IV Year
UNDER THE GUIDANCE OF
Gangu Sreelatha M.Pharm., (Ph.D)
Assistant Professor
CMR College of Pharmacy, Hyderabad.
email: sreelatha1801@gmail.com
2. β’ DOE is invented by Sir Ronald Fisher in 1920βs and 1930βs.
β’ The brief history of design of experiments is summarized in the following 4 eras:
1. The agricultural origin,1918 to 1940 ( R. A .Fisher Frank Yatis ).
β’ Profound impact on agricultural sciences.
β’ Development of factorial designs and ANOVA.
2. The first industrial era 1951 to late 1970βs ( Box and Wilson).
β’ Applications in the chemical and process industries.
3. The second industrial era late 1970βs to 1990 (W. Edward Beming).
β’ Quality improvement initiatives in many companies.
β’ Total quality management(TQM) and continuous quality improvement(CQI) were important ideas
and became management goals.
4. The modern era beginning from 1990, when economic competitiveness and globalization is driving
all sectors of economy to be more competitive.
DESIGN OF EXPERIMENTS (DOE)
3. β’ The following designs of experiments will be usually followed:
1. Completely randomised design(CRD)
2. Randomised complete block design(RCBD)
3. Latin square design(LSD)
4. Factorial design or experiment
5. Confounding
6. Split and strip plot design
οΆFACTORIAL DESIGN:
β’ When a several factors are investigated simultaneously in a single experiment such experiments are
known as factorial experiments. Though it is not an experimental design, indeed any of the designs
may be used for factorial experiments.
οFor example, the yield of a product depends on the particular type of synthetic substance used and
also on the type of chemical used.
β’ If there are βpβ different varieties or types then we shall say that there are p levels of the factor variety
or type.
β’ Similarly, the second factor chemical may have q levels i.e there may be q different chemicals or q
different concentrations of the same chemical.
4. β’ These factorial design will now be called as pxq experiment.
β’ In an example the two factors may be 2 different chemicals i.e acetonitrile and methanol and at p and
q different concentrations respectively. This will also give a pxq experiment. We shall consider only
the simplest cases, of βnβ factors such as two or three levels are known as 2n ,3n experiments were βnβ
is any positive integer β₯ 2 .
β’ If two factors are considered simultaneously the experiment is called two factor factorial experiment.
β’ If these factors are studied simultaneously as done in the case of latin square were we block two
factors and studied the effect of treatment only, the experiment is called as three factor factorial
experiment.
β’ The total no of treatments in a factorial experiment is the product of the levels in each factor, in two
square factorial example, the no. of treatments is 2x2=4 and in the 23 factorial the no. of treatments
is 2x2x2=8. The no. of treatments increases rapidly with an increase in the no. of factors or an
increase in the levels in each factor.
β’ For a factorial experiment involving 5 varieties or types the total no. of treatments would be
5x4x3=60.
5. ADVANTAGES OF FACTORIAL DESIGN
οThere are many advantages of the factorial experiments:
1. Factorial experiments are advantageous to study the combined effect of two or more factors
simultaneously and analyze their interrelationships. Such factorial experiments are economic in
nature and provide a lot of relevant information about the phenomenon under study. It also
increases the efficiency of the experiment.
2. It is an advantageous because a wide range of factor combination are used. This will give us an idea
to predict about what will happen when two or more factors are used in combination.
3. The factorial approach will result in considerable saving of the time and the experimental materials.
It is because the time required for the combined experiment is less than that required for the
separate experiments.
4. In single factor experiments, the results may not be satisfactory because of the changes in
environmental conditions. However, in factorial experiments such type of difficulties will not arise
even after several factors are investigated simultaneously.
5. Information may obtained from factorial experiments is more complete than that obtained from a
series of single factor experiments, in the sense that factorial experiments permit the evaluation of
interaction effects.
6. DISADVANTAGES
οThe disadvantages of the factorial experiments are:
β’ It is disadvantageous because the execution of the experiment and the statistical analysis becomes more
complex when several treatments combinations or factors are involved simultaneously.
β’ It is also disadvantageous in cases where may not be interested in certain treatment combinations but
we are forced to include them in the experiment. This will lead to wastage of time and also the
experimental material.
β’ In factorial experiments, the number of treatment combinations will increase if the factors are
increased. This will also lead to the increase in block size, which in turn will increase the
heterogenicity in the experimental material. Because of this it will lead to the increased experimental
error and will decrease the precision in the experiment. Appropriate block size must be maintained.
7. 22 FACTORIAL EXPERIMENT
β’ A special set of factorial experiment consist of experiments in which all factors have 2 levels such
experiments are referred to generally as 2n factorials.
β’ If there are four factors each at two levels the experiment is known as 2x2x2x2 or 24 factorial
experiment. On the other hand if there are 2 factors each with 3 levels the experiment is known as 3x3
or 32 factorial experiment. In general if there are n factors each with p levels then it is known as pn
factorial experiment.
β’ The calculation of the sum of squares is as follows:
Correction factor (CF) =
πΊπ 2
π
GT = grand total
n = total no of observations
β’ Total sum of squares = π₯2 β πΆπΉ
β’ Replication sum of squares (RSS) =
π 1 2
+ π 2 2
+β―+ π π 2
π
- CF
Or
1
π
π 2 β πΆπΉ
8. β’ Treatment sum of squares (TSS) =
π1 2
+ π2 2
+β―+ ππ 2
π
β CF
Or
1
π
π2 β πΆπΉ
β’ Residual sum of square = error sum of squares = total sum of square- replication sum of square - TSS
οΆANOVA TABLE
οExample: An experiment was conducted on 4 rabbits in a randomised block design with a four
replications. Analyse the data and conclude the results.
Source of variation Degree of freedom Sum of squares Mean sum of
squares
Factor
Replications r-1 RSS RSS/r-1 RMS/EMS
Treatments t-1 TSS TSS/t-1 TMS/EMS
Error (r-1) (t-1) ESS ESS/(r-1) (t-1) -
Total rt-1 - - -
9. οSolution: Null hypothesis ( H0 ) the data do not differ with respect to blocks and treatments.
BLOCK TREATMENT COMBINATIONS
1 l(23) k(25) p(22) kp(38)
2 p(40) l(26) k(36) kp(38)
3 l(29) k(20) kp(30) p(20)
4 kp(34) k(31) p(24) l(28)
Treatment
combination
1 2 3 4 Treatment
totals
l 23 26 29 28 106
k 25 36 20 31 112
p 22 40 20 24 106
kp 38 38 30 34 140
Block total 108 140 99 17 464
11. =
54616
4
- 13456
= 13654 -13456
= 198
β’ Residual sum of square = ESS = TSS βRSS -TSS
= 660 -232.5 -198
= 224.5
Source of variation Degree of Freedom Sum of squares Mean sum of squares Factor(f)
Replication 4-1
=3
RSS = 232.5 232.5/3
= 77.5
RMS/EMS =
77.5
25.5
= 3.03
Treatment 4-1
=3
TSS = 198 = 198/3
= 66
TMS/EMS =
66
25.5
= 2.58
Error (r-1) (t-1)
=3X3
=9
229.5 = 229.5/9
= 25.5
Total rt-1
=4X4-1
=16-1
=15
12. Conclusion: The tabulated f value at 5% significance for three degree of freedom is 3.86. The
calculated f value is less than the tabulated value. Therefore, the values are not significant. The data
with respect to blocks and treatments do not differ significantly. So, the null hypothesis is accepted.
13. 23 FACTORIAL DESIGN
β’ In this type of design, one independent variable has 2 levels, and the other independent variable has 3
levels as shown in the figure 1
β’ Example: let us imagine that the researcher wants to find the effectiveness of a diet pill A over a
placebo, a dietary regimen B versus usual diet and an exercise regimen C versus usual exercise, on a
weight loss lasting at least a year in adult men.
β’ In this situation a factorial design can be more efficient and informative because, it allows the
researchers to study not only all the treatments A B C (factors) but allow the precise interactions AB,
BC, AC and also ABC
14. β’ We have 3 factors of only 2 levels. The levels may be low or high denoted by β-β and β+β respectively.
β’ The treatment combination is given as
οNote (Yates notation)
β’ The high level of any factor is denoted by the corresponding letter (for example in the above table 2nd
row A is having high level of factor(+) and is denoted by the corresponding letters (a) whereas B and
C are having low levels so the treatment will be denoted by the absence of the corresponding letter.
A B C TREATMENTS
- - - (1)
+ - - a
- + - b
+ + - ab
- - + c
+ - + ac
- + + bc
+ + + abc
15. β’ The above table can also be represented in a graphical notation
β’ Design table for treatment
β’ 1 A B AB C AC BC ABC TREATM
ENTS
+ - - + - + + - 1
+ + - - - - + + a
+ - + - - + - + b
+ + + + - - - - ab
+ - - + + - - + c
+ + - - + + - - ac
+ - + - + - + - bc
+ + + + + + + + abc
16. β’ In the above table 1st row, A and B are having negative signs(β-β) so AB becomes(β+β) (- x - = +) and
in the 2nd row A is having β+β sign and B is having β-β sign so AB becomes (β-β) (+ x - = -).
β’ In 3rd row A is having (β-β) sign and B is having (β+β)sign so AB becomes(β-β) ( - x + = -) and in the 4th
row A and B are having (β+β) signs so AB becomes(β+β) (+ x + = +)
β’ In ABC it can be written as AB and C and also as BC and A.
Estimating the effect:
β’ In a factorial design the main effect of an independent variable is its overall effect averaged across all
other independent variable.
β’ Effect of a factor A is the average of the runs where A is at the high level minus the average of the
runs where A is at the low level.
β’ Symbolically effect of factor A =
β’ The equation is rearranged and given as
β’ Main effect of A =
1
4π
π + ππ + ππ + πππ β 1 β π β π β ππ
β’ n = denotes replicate
β’ 4 is taken because there are 4 positive values (a, ab, ac, abc)
17. Effect of A
β’ When we put these values in graphical rotation.
This indicates that in an experiment if we get higher values then they will be placed/fall in the
right side of the graph which indicates all positive values and vice versa. (This is for factor A)
+ -
a (1)
ab b
ac c
abc bc
18. β’ For factor C it is shown in round balls. (High and low values)
β’ For factor B it is shown in dotted lines (high and low).
Estimating simple effect:
Simple effect of A at low level of B and C=
π
π
β
(1)
π
Simple effect of A at high level of B and low level of C =
ππ
π
β
π
π
Simple effect of A at low level of B and high level of C=
ππ
π
-
π
π
Simple effect of A at high level of B and C=
πππ
π
β
ππ
π
Estimating the main effect:
Main effect of A can be written in short as
β’ Main effect of A =
πβ1 (πβ1)(π+1)
4π
a-1 means it should be kept in mind that 1 is given in brackets i.e., [a-(1)].
β’ Main effect of B =
1
4π
[b+ab+bc+abc-(1)-a-c-ac]
19. In short ,
β’ Main effect of B =
(π+1)(πβ1)(π+1)
4π
β’ Main effect of C=
1
4π
π + ππ + ππ + πππ β 1 β π β π β ππ
β’ Main effect of c =
(π+1)(π+1)(πβ1)
4π
Estimating the interaction effect:
β’ Interaction effect of AB =
1
4π
[(1)+ab+c+abc-a-b-ac-bc]
In short,
β’ Interaction of AB =
(πβ1)(πβ1)(π+1)
4π
β’ Interaction effect of AC =
1
4π
1 + π + ππ + πππ β π β ππ β π β ππ
In short,
β’ Interaction of AC=
(π+1)(π+1)(πβ1)
4π
β’ Interaction effect of BC =
1
4π
[(1)+a+bc+abc-b-ab-c-ac]
In short,
β’ Interaction of BC=
(π+1)(πβ1)(πβ1)
4π
20. β’ Interaction effect of ABC =
1
4π
πππ + π + π + π β ππ β ππ β ππ β 1
In short,
β’ Interaction effect of AB=
πβ1 (πβ1)(πβ1)
4π
n= no of replicate
Statistical testing using ANOVA:
Source of variation Sum of squares Df Mean of square f
A SSA 1 MSA =
πππ΄
1
fA=
πππ΄
πππΈ
B SSB 1 MSB=
πππ΅
1
fB=
πππ΅
πππΈ
AB SSAB 1 MSAB=
πππ΄π΅
1
fAB=
πππ΄π΅
πππΈ
C SSC 1 MSC=
πππΆ
1
fc=
πππΆ
πππΈ
AC SSAC 1 MSAC=
πππ΄πΆ
1
fAC=
πππ΄πΆ
πππΈ
BC SSBC 1 MSBC=
πππ΅πΆ
1
fBC=
πππ΅πΆ
πππΈ
ABC SSABC 1 MSABC=
πππ΄π΅πΆ
1
fABC=
πππ΄π΅πΆ
πππΈ
error SSE 8(n-1) MSE=
πππΈ
8(πβ1)
Total SST 8n-1
21. β’ Calculation of sum of squares:
β’ In the 23 design with n replicates, the sum of squares for any effect is given by
SS=
πΆπππ‘πππ π‘ 2
8π
2x2x2=8 [So 8n is taken]
(23)
β’ Note: The number calculated in the effect of any factors which is inside the bracket is called the
contrast. Also we can take the numerator from the shortcut formula.
For Example:
β’ Main effect of A =
1
4π
π + ππ + ππ + πππ β 1 β π β ππ
β’ The terms which are in bracket are taken as contrast.
In short,
β’ Main effect of A=
πβ1 (π+1)(π+1)
4π
Numerator is contrast.
22. Example:
β’ A 23 factorial design was used to develop a model for preparing a nanoparticle of optimised size. For
this purpose two concentrations A,B and a powder based drug C were used.
β’ The experiment is run in 2 replicates and the result is tabulated as follows. The response variable is
the particle size.
Solution:
β’ Here nanoparticle optimised size is taken as yield.
β’ A, B taken as liquid.
β’ C taken as solid.
23. Run A B C Replicate
1
Replicate
2
Total Rotation
1 - - - 550 604 1154 (1)
2 + - - 669 650 1319 a
3 - + - 633 601 1234 b
4 + + - 642 635 1277 ab
5 - - + 1044.5 1044.5 2089 c
6 + - + 749 868 1617 ac
7 - + + 1069 1069 2138 bc
8 - - - 794.5 794.5 1589 abc
25. β’ Estimating the main effect
a = 1319, b = 1234, c = 2089, ab = 1277, ac = 1617, bc = 2138, abc = 1589, (1) = 1154
Main effect of A =
1
4π
π + ππ + ππ + πππ β 1 β π β π β ππ
=
1
8
(1319 + 1277 + 1617 + 1589 β 1154 β 1234 β 2084 β 2138)
=
1
8
(-813) = -101.625
Similarly for B
=
1
4π
π + ππ + ππ + πππ β 1 β π β π β ππ
=
1
8
(1234 + 1277 + 2138 + 1589 β 1154 β 1319 β 2089 β 1617)
=
1
8
(59)
= 7.375
26. β’ For factor C
=
1
4π
(c +ac + bc + abc- (1) β a-b-ab)
=
1
8
( 2089+1617+2138+ 1589- 1154-1319-1234-1277) =
1
8
( 2449) = 306.125
β’ Interaction effect of AB
=
1
4π
1 + as + c + abc β a β b β ac β bc
=
1
8
1154 + 1277 + 2089 + 1589 + 1319 β 1234 β 1617 β 2138 = -24.875
β’ Interaction effect of AC
=
1
4π
1 + π + ππ + πππ β π β ππ β π β ππ
=
1
8
1154 + 1234 + 1617 + 1589 β 1319 β 1277 β 2089 β 2138
=
1
8
[-1229]
= -153.625
27. β’ Interaction effect of BC
=
1
4π
[ (1) + a + bc + abc β b β ab β c β ac]
=
1
8
[ 1154 + 1319 + 2138 + 1589 β 1234 -1277 β 2089 β 1617]
=
1
8
[ -17]
= -2.125
β’ Interaction effect of ABC
=
1
4π
[abc + a + b+ c βab β ac- bc -1]
=
1
8
[ 1589 + 1319 + 1234+ 2089 β 1277 β 1617-2138 β 1154] =
1
8
[45] = 5.625.
οFrom the calculations
β’ The largest effects is for
β’ Factor C = 306.125
β’ Factor A = -101.625
β’ AC = -153.625
28. οcalculate total sum of squares
β’ SSA =
β813 2
16
=
ππππ‘πππ π‘ 2
8π
= 41310.5625 since here 2 replicates are there 8 x 2 = 16
β’ SSB =
59 2
16
= 217.5625
β’ SSC =
2449 2
16
= 374850.0625
β’ SSAB =
β199 2
16
= 2475.0625
β’ SSAC =
β1229 2
16
= 94402.5625
β’ SSBC =
β17 2
16
= 18.0625
β’ SSABC =
45 2
16
= 126.5625
29. From the table we find that factor A and factor C are highly significant. And the interaction AC is highly
significant .
So there is a strong interaction between factor A and factor C.
Note: Factorial design can be done using regression also.
Source of
variation
Sum of squares DF Mean square F P value
A 41310.5625 1 41310.5625 18.34 0.0027
B 217.5625 1 217.5625 0.10 0.7639
C 374850.0625 1 374850.0625 166.41 0.0001
AB 2475.0625 1 2475.0625 1.10 0.5252
AC 94402.5625 1 94402.5625 41.91 0.0002
BC 18.0625 1 18.0625 0.01 0.9308
ABC 126.5625 1 126.5625 0.06 0.8186
Error 18020.5000 8 2252.5625
Total 531420.9375 15