- The document discusses multiple correlation and the multiple correlation coefficient.
- Multiple correlation is the study of the combined influence of two or more variables on a single variable.
- The multiple correlation coefficient is defined as the simple correlation coefficient between a variable and its estimate based on two or more other variables.
- A formula is derived for the multiple correlation coefficient that incorporates the total correlation coefficients between each variable pair.
This document discusses factorial design for pharmaceutical experiments. It defines factorial design as an experiment where two or more factors are each studied at different levels or values. The document then describes different types of factorial designs, including full factorial designs with two or three levels, and fractional factorial designs used when there are more than five factors. It also explains how factors and levels are coded numerically for the experiments.
- Response surface methodology (RSM) is a statistical technique used to optimize processes and develop new products. It was developed in the 1950s to improve chemical processes.
- RSM uses experimental designs and mathematical/statistical techniques to model and analyze the relationship between inputs and outputs or responses. The goal is to optimize the response by selecting the best setting of each input variable.
- Common RSM methods include steepest ascent/descent, central composite design, and Box-Behnken design. They are used to estimate coefficients in a polynomial regression model and determine optimal settings for the inputs.
Factorial design ,full factorial design, fractional factorial designSayed Shakil Ahmed
This document discusses factorial designs and their application in formulation. It defines factorial experiments as those involving two or more factors, each with different levels. Full factorial designs involve every combination of all factors and levels, while fractional factorial designs examine multiple factors efficiently with fewer runs. Applications mentioned include formulation and processing, clinical chemistry, and studying the effects of factors like disintegrant and lubricant concentration on tablet dissolution. IVIVC and BCS classification are also discussed in relation to predicting oral absorption of immediate release formulations.
This document discusses factorial design, which is an experimental design technique used for optimization. It involves studying the effects of two or more factors simultaneously. There are two main types: full factorial design, which tests all possible combinations of factors and levels, and fractional factorial design, which reduces the number of runs when there are many factors. Factorial designs allow evaluation of both main effects and interaction effects. They are useful for formulation development and method optimization in chromatography. Software is used to analyze the results of factorial experiments.
Correlation measures the strength and direction of the linear relationship between two variables. A correlation coefficient value close to 1 indicates a strong positive linear relationship, while a value close to -1 indicates a strong negative linear relationship. A value of 0 indicates no linear relationship between the variables. Some examples of correlation types include positive correlation between income and expenses, and negative correlation between price and demand. Rank correlation measures the correspondence between rankings of the same items. The coefficient of rank correlation is used to assess if rankings correspond statistically significantly. Applications of correlation include understanding dependency between variables to build linear models for prediction and analyzing relationships like the impact of population on poverty rates.
Unit-III Non Parametric tests: Wilcoxon Rank Sum Test, Mann-Whitney U test, Kruskal-Wallis
test, Friedman Test. BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory)
Unit ibp801 t l multiple correlation a24022022ashish7sattee
The multiple correlation coefficient denotes the correlation between one variable and multiple other variables. It is represented as R1.234...k, where 1 is the variable being correlated and 2, 3, 4, etc. are the other variables. As an example, R1.23 would represent correlating variable 1 with variables 2 and 3 simultaneously by creating a linear combination of 2 and 3. The document then discusses using multiple correlation to correlate academic achievement with a linear combination of anxiety and intelligence.
BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory), Unit-II, Regression: Curve fitting by the method of least squares, fitting the lines y= a + bx and x
= a + by, Multiple regression, standard error of regression– Pharmaceutical Examples, • Regression: how well a certain independent variable
predict dependent variable?
• Regression: a measure of the relation between
the mean value of one variable (e.g. output) and
corresponding values of other variables (e.g.
time and cost).
This document discusses factorial design for pharmaceutical experiments. It defines factorial design as an experiment where two or more factors are each studied at different levels or values. The document then describes different types of factorial designs, including full factorial designs with two or three levels, and fractional factorial designs used when there are more than five factors. It also explains how factors and levels are coded numerically for the experiments.
- Response surface methodology (RSM) is a statistical technique used to optimize processes and develop new products. It was developed in the 1950s to improve chemical processes.
- RSM uses experimental designs and mathematical/statistical techniques to model and analyze the relationship between inputs and outputs or responses. The goal is to optimize the response by selecting the best setting of each input variable.
- Common RSM methods include steepest ascent/descent, central composite design, and Box-Behnken design. They are used to estimate coefficients in a polynomial regression model and determine optimal settings for the inputs.
Factorial design ,full factorial design, fractional factorial designSayed Shakil Ahmed
This document discusses factorial designs and their application in formulation. It defines factorial experiments as those involving two or more factors, each with different levels. Full factorial designs involve every combination of all factors and levels, while fractional factorial designs examine multiple factors efficiently with fewer runs. Applications mentioned include formulation and processing, clinical chemistry, and studying the effects of factors like disintegrant and lubricant concentration on tablet dissolution. IVIVC and BCS classification are also discussed in relation to predicting oral absorption of immediate release formulations.
This document discusses factorial design, which is an experimental design technique used for optimization. It involves studying the effects of two or more factors simultaneously. There are two main types: full factorial design, which tests all possible combinations of factors and levels, and fractional factorial design, which reduces the number of runs when there are many factors. Factorial designs allow evaluation of both main effects and interaction effects. They are useful for formulation development and method optimization in chromatography. Software is used to analyze the results of factorial experiments.
Correlation measures the strength and direction of the linear relationship between two variables. A correlation coefficient value close to 1 indicates a strong positive linear relationship, while a value close to -1 indicates a strong negative linear relationship. A value of 0 indicates no linear relationship between the variables. Some examples of correlation types include positive correlation between income and expenses, and negative correlation between price and demand. Rank correlation measures the correspondence between rankings of the same items. The coefficient of rank correlation is used to assess if rankings correspond statistically significantly. Applications of correlation include understanding dependency between variables to build linear models for prediction and analyzing relationships like the impact of population on poverty rates.
Unit-III Non Parametric tests: Wilcoxon Rank Sum Test, Mann-Whitney U test, Kruskal-Wallis
test, Friedman Test. BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory)
Unit ibp801 t l multiple correlation a24022022ashish7sattee
The multiple correlation coefficient denotes the correlation between one variable and multiple other variables. It is represented as R1.234...k, where 1 is the variable being correlated and 2, 3, 4, etc. are the other variables. As an example, R1.23 would represent correlating variable 1 with variables 2 and 3 simultaneously by creating a linear combination of 2 and 3. The document then discusses using multiple correlation to correlate academic achievement with a linear combination of anxiety and intelligence.
BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory), Unit-II, Regression: Curve fitting by the method of least squares, fitting the lines y= a + bx and x
= a + by, Multiple regression, standard error of regression– Pharmaceutical Examples, • Regression: how well a certain independent variable
predict dependent variable?
• Regression: a measure of the relation between
the mean value of one variable (e.g. output) and
corresponding values of other variables (e.g.
time and cost).
The document describes a 23 factorial design used to optimize chromatographic conditions. Three factors (temperature, ethanol concentration, and mobile phase flow rate) were each tested at two levels in a 23 factorial design. Resolution was used as the response. Regression analysis was performed on the results to develop a polynomial equation relating the factors and their interactions to resolution. This allowed determination of optimum conditions for chromatographic separation.
Experimental design techniques involve controlling variables to measure their effects. There are three main types of designs: pre-experimental (no control group), quasi-experimental (no random assignment), and true experimental (control group, random assignment, manipulation). True experiments allow cause-and-effect conclusions through statistical analysis and include variations like post-test only, pre-test post-test, and Solomon four-group designs. Factorial designs test multiple hypotheses simultaneously by manipulating multiple independent variables. Randomized block and cross-over designs help control for differences between subjects. The goal is to draw valid conclusions about variable relationships through controlled experimentation.
The reduced model is better than the full model based on the following criteria:
1. The reduced model has a higher R-squared and adjusted R-squared value indicating it fits the data better.
2. The predicted R-squared of the reduced model is closer to the adjusted R-squared indicating it has better predictive power.
3. The PRESS value which indicates prediction accuracy is lower for the reduced model.
4. The lack of fit F-value is higher (better) for the reduced model indicating it fits the data as well as the full quadratic model without the extra terms.
Therefore, the reduced model is more statistically significant and has better predictive ability compared to the full quadratic model based on these
The Friedman test is a non-parametric alternative to the one-way ANOVA with repeated measures that is used to test for differences between groups when the dependent variable is ordinal. It can be used with continuous data that violates the assumptions of one-way ANOVA. The test assumes that samples are from a population and measured at the ordinal or higher level, and that the samples are not normally distributed. It involves ranking the data and calculating a test statistic M to determine if there are statistically significant differences between treatment columns. An example is provided comparing the rankings of three violins rated by 10 subjects.
The document provides an overview of biostatistics and research methodology topics. It defines key biostatistics concepts like population and parameter. It also introduces several non-parametric tests like the Wilcoxon Rank Sum test, Mann-Whitney U test, Kruskal-Wallis test, and Friedman test. Additionally, it discusses important aspects of research like the need for research, types of research, the research process, and challenges researchers may encounter.
This document provides an overview of different types of graphs that can be used to present statistical data, including histograms, pie charts, bar charts, line charts, cubic graphs, response surface plots, and contour plots. It discusses the purpose and construction of each graph type, advantages and disadvantages, and provides examples of how and when each type of graph might be used. The overall goal is to help students identify, construct, and properly label different graphs to effectively communicate statistical data.
This document describes a two-factor factorial design experiment involving two factors, each with two levels. It defines what a factorial design is, how it is more efficient than single-factor experiments, and allows investigating interactions between factors. It provides an example of a 2x2 factorial design measuring the effects of two materials and two temperatures on battery life. Analysis of variance is performed on the example to investigate main effects and interactions between the factors.
The Friedman test is a non-parametric statistical test used to detect differences in treatments across multiple test attempts. It compares three or more related/matched samples, such as the same measure repeated over time. The test ranks the data and calculates a statistic to determine if the null hypothesis that there are no differences between treatments can be rejected. For the given problem, the Friedman test was performed and calculated a chi-square value of 2.33, which is less than the critical value of 5.991 for 2 degrees of freedom. Therefore, the null hypothesis that there is no difference between the three weeks cannot be rejected at the 0.05 significance level.
The document discusses parametric and non-parametric tests. It provides examples of commonly used non-parametric tests including the Mann-Whitney U test, Kruskal-Wallis test, and Wilcoxon signed-rank test. For each test, it gives the steps to perform the test and interpret the results. Non-parametric tests make fewer assumptions than parametric tests and can be used when the data is ordinal or does not meet the assumptions of parametric tests. They provide a distribution-free alternative for analyzing data.
This document discusses correlation and different aspects of studying correlation. It defines correlation as the association or relationship between two variables that do not cause each other. It describes different types of correlation including positive, negative, linear, non-linear, simple, multiple and partial correlation. It also discusses various methods of studying correlation including graphic methods like scattered diagrams and correlation graphs, and algebraic methods like Karl Pearson's correlation coefficient and Spearman's rank correlation coefficient. The document explains concepts like coefficient of determination and hypothesis testing in correlation. It emphasizes that correlation indicates association but does not necessarily imply causation between variables.
this ppt gives you adequate information about Karl Pearsonscoefficient correlation and its calculation. its the widely used to calculate a relationship between two variables. The correlation shows a specific value of the degree of a linear relationship between the X and Y variables. it is also called as The Karl Pearson‘s product-moment correlation coefficient. the value of r is alwys lies between -1 to +1. + 0.1 shows Lower degree of +ve correlation, +0.8 shows Higher degree of +ve correlation.-0.1 shows Lower degree of -ve correlation. -0.8 shows Higher degree of -ve correlation.
This document provides information about the Kruskal-Wallis test, a non-parametric statistical method for comparing three or more groups of independent samples. It discusses the history of the test, its assumptions, how to calculate it, examples of its use, and references for further information. The Kruskal-Wallis test is used as an alternative to one-way ANOVA when assumptions of normality or equal variance are not met, to compare population medians among three or more groups. An example is provided to demonstrate how to perform the test on serum protein levels from three groups of patients.
This document discusses factorial design in pharmaceutical research. It defines key terms like factors, levels, and effects. Factorial design is used to study the effect of different factors and their interactions on a response. It presents examples of 2^2, 2^3, and 3^2 factorial designs and how to compute main effects and interactions. Data analysis methods like Yates' method and ANOVA are described. The design allows fitting of a polynomial equation to optimize a response based on factor levels. Advantages include efficiency in estimating effects and revealing interactions across factor levels.
This document discusses design of experiments (DoE) and its application in formulation development. It defines key terms like independent variables, dependent variables, levels, quality target product profile, critical process parameters, and critical quality attributes. It describes different types of DoE like factorial designs, response surface designs, central composite design, and Box-Behnken design. It provides an example of using a three-factor, three-level Box-Behnken design to investigate formulation variables affecting droplet size, drug release, and solubility of a fenofibrate SMEDDS formulation. Statistical software was used to fit the experimental results to a quadratic mathematical model.
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.
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 determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population 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 or nonprofit 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.
The document describes the Wilcoxon Rank-Sum Test, a non-parametric statistical hypothesis test used to assess whether one of two independent samples of observations tends to have larger values than the other when normality cannot be assumed. It provides details on running the test, including ranking the combined observations and computing the test statistic to determine if it is less than or equal to the critical value, rejecting the null hypothesis. An example applies the test to compare the nicotine content of two cigarette brands, finding no significant difference between their medians.
Unit-I, BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory)
Correlation: Definition, Karl Pearson’s coefficient of correlation, Multiple correlations -
Pharmaceuticals examples.
Correlation: is there a relationship between 2
variables.
A statistical criterion for reducing indeterminacy in linear causal modelingGianluca Bontempi
This document proposes a new statistical criterion called C to help distinguish between causal patterns in completely connected triplets when inferring causal relationships from observational data. The criterion is based on differences in values of the term S, which is derived from the covariance matrix, between different causal hypotheses. This criterion informs an algorithm called RC that incorporates both relevance and causal measures to iteratively select variables. Experiments on linear and nonlinear networks show RC has higher accuracy than other algorithms at inferring network structure. The criterion C and RC algorithm help address challenges of causal inference from complex data where dependencies are frequent.
Bayesian Inferences for Two Parameter Weibull DistributionIOSR Journals
This document discusses Bayesian inference methods for estimating the parameters of a two-parameter Weibull distribution. It begins by introducing the Weibull distribution and defining its probability density function. Maximum likelihood estimation is derived for the scale and shape parameters. Approximate Bayesian methods are then explored, including the Lindley and Laplace approximations, to obtain expressions for the marginal posterior densities since closed-form solutions are not available. The results indicate that the posterior variances for the scale parameter obtained with the Laplace method are smaller than those from the Lindley approximation or asymptotic variances of the maximum likelihood estimates.
The document describes a 23 factorial design used to optimize chromatographic conditions. Three factors (temperature, ethanol concentration, and mobile phase flow rate) were each tested at two levels in a 23 factorial design. Resolution was used as the response. Regression analysis was performed on the results to develop a polynomial equation relating the factors and their interactions to resolution. This allowed determination of optimum conditions for chromatographic separation.
Experimental design techniques involve controlling variables to measure their effects. There are three main types of designs: pre-experimental (no control group), quasi-experimental (no random assignment), and true experimental (control group, random assignment, manipulation). True experiments allow cause-and-effect conclusions through statistical analysis and include variations like post-test only, pre-test post-test, and Solomon four-group designs. Factorial designs test multiple hypotheses simultaneously by manipulating multiple independent variables. Randomized block and cross-over designs help control for differences between subjects. The goal is to draw valid conclusions about variable relationships through controlled experimentation.
The reduced model is better than the full model based on the following criteria:
1. The reduced model has a higher R-squared and adjusted R-squared value indicating it fits the data better.
2. The predicted R-squared of the reduced model is closer to the adjusted R-squared indicating it has better predictive power.
3. The PRESS value which indicates prediction accuracy is lower for the reduced model.
4. The lack of fit F-value is higher (better) for the reduced model indicating it fits the data as well as the full quadratic model without the extra terms.
Therefore, the reduced model is more statistically significant and has better predictive ability compared to the full quadratic model based on these
The Friedman test is a non-parametric alternative to the one-way ANOVA with repeated measures that is used to test for differences between groups when the dependent variable is ordinal. It can be used with continuous data that violates the assumptions of one-way ANOVA. The test assumes that samples are from a population and measured at the ordinal or higher level, and that the samples are not normally distributed. It involves ranking the data and calculating a test statistic M to determine if there are statistically significant differences between treatment columns. An example is provided comparing the rankings of three violins rated by 10 subjects.
The document provides an overview of biostatistics and research methodology topics. It defines key biostatistics concepts like population and parameter. It also introduces several non-parametric tests like the Wilcoxon Rank Sum test, Mann-Whitney U test, Kruskal-Wallis test, and Friedman test. Additionally, it discusses important aspects of research like the need for research, types of research, the research process, and challenges researchers may encounter.
This document provides an overview of different types of graphs that can be used to present statistical data, including histograms, pie charts, bar charts, line charts, cubic graphs, response surface plots, and contour plots. It discusses the purpose and construction of each graph type, advantages and disadvantages, and provides examples of how and when each type of graph might be used. The overall goal is to help students identify, construct, and properly label different graphs to effectively communicate statistical data.
This document describes a two-factor factorial design experiment involving two factors, each with two levels. It defines what a factorial design is, how it is more efficient than single-factor experiments, and allows investigating interactions between factors. It provides an example of a 2x2 factorial design measuring the effects of two materials and two temperatures on battery life. Analysis of variance is performed on the example to investigate main effects and interactions between the factors.
The Friedman test is a non-parametric statistical test used to detect differences in treatments across multiple test attempts. It compares three or more related/matched samples, such as the same measure repeated over time. The test ranks the data and calculates a statistic to determine if the null hypothesis that there are no differences between treatments can be rejected. For the given problem, the Friedman test was performed and calculated a chi-square value of 2.33, which is less than the critical value of 5.991 for 2 degrees of freedom. Therefore, the null hypothesis that there is no difference between the three weeks cannot be rejected at the 0.05 significance level.
The document discusses parametric and non-parametric tests. It provides examples of commonly used non-parametric tests including the Mann-Whitney U test, Kruskal-Wallis test, and Wilcoxon signed-rank test. For each test, it gives the steps to perform the test and interpret the results. Non-parametric tests make fewer assumptions than parametric tests and can be used when the data is ordinal or does not meet the assumptions of parametric tests. They provide a distribution-free alternative for analyzing data.
This document discusses correlation and different aspects of studying correlation. It defines correlation as the association or relationship between two variables that do not cause each other. It describes different types of correlation including positive, negative, linear, non-linear, simple, multiple and partial correlation. It also discusses various methods of studying correlation including graphic methods like scattered diagrams and correlation graphs, and algebraic methods like Karl Pearson's correlation coefficient and Spearman's rank correlation coefficient. The document explains concepts like coefficient of determination and hypothesis testing in correlation. It emphasizes that correlation indicates association but does not necessarily imply causation between variables.
this ppt gives you adequate information about Karl Pearsonscoefficient correlation and its calculation. its the widely used to calculate a relationship between two variables. The correlation shows a specific value of the degree of a linear relationship between the X and Y variables. it is also called as The Karl Pearson‘s product-moment correlation coefficient. the value of r is alwys lies between -1 to +1. + 0.1 shows Lower degree of +ve correlation, +0.8 shows Higher degree of +ve correlation.-0.1 shows Lower degree of -ve correlation. -0.8 shows Higher degree of -ve correlation.
This document provides information about the Kruskal-Wallis test, a non-parametric statistical method for comparing three or more groups of independent samples. It discusses the history of the test, its assumptions, how to calculate it, examples of its use, and references for further information. The Kruskal-Wallis test is used as an alternative to one-way ANOVA when assumptions of normality or equal variance are not met, to compare population medians among three or more groups. An example is provided to demonstrate how to perform the test on serum protein levels from three groups of patients.
This document discusses factorial design in pharmaceutical research. It defines key terms like factors, levels, and effects. Factorial design is used to study the effect of different factors and their interactions on a response. It presents examples of 2^2, 2^3, and 3^2 factorial designs and how to compute main effects and interactions. Data analysis methods like Yates' method and ANOVA are described. The design allows fitting of a polynomial equation to optimize a response based on factor levels. Advantages include efficiency in estimating effects and revealing interactions across factor levels.
This document discusses design of experiments (DoE) and its application in formulation development. It defines key terms like independent variables, dependent variables, levels, quality target product profile, critical process parameters, and critical quality attributes. It describes different types of DoE like factorial designs, response surface designs, central composite design, and Box-Behnken design. It provides an example of using a three-factor, three-level Box-Behnken design to investigate formulation variables affecting droplet size, drug release, and solubility of a fenofibrate SMEDDS formulation. Statistical software was used to fit the experimental results to a quadratic mathematical model.
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.
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 determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population 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 or nonprofit 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.
The document describes the Wilcoxon Rank-Sum Test, a non-parametric statistical hypothesis test used to assess whether one of two independent samples of observations tends to have larger values than the other when normality cannot be assumed. It provides details on running the test, including ranking the combined observations and computing the test statistic to determine if it is less than or equal to the critical value, rejecting the null hypothesis. An example applies the test to compare the nicotine content of two cigarette brands, finding no significant difference between their medians.
Unit-I, BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory)
Correlation: Definition, Karl Pearson’s coefficient of correlation, Multiple correlations -
Pharmaceuticals examples.
Correlation: is there a relationship between 2
variables.
A statistical criterion for reducing indeterminacy in linear causal modelingGianluca Bontempi
This document proposes a new statistical criterion called C to help distinguish between causal patterns in completely connected triplets when inferring causal relationships from observational data. The criterion is based on differences in values of the term S, which is derived from the covariance matrix, between different causal hypotheses. This criterion informs an algorithm called RC that incorporates both relevance and causal measures to iteratively select variables. Experiments on linear and nonlinear networks show RC has higher accuracy than other algorithms at inferring network structure. The criterion C and RC algorithm help address challenges of causal inference from complex data where dependencies are frequent.
Bayesian Inferences for Two Parameter Weibull DistributionIOSR Journals
This document discusses Bayesian inference methods for estimating the parameters of a two-parameter Weibull distribution. It begins by introducing the Weibull distribution and defining its probability density function. Maximum likelihood estimation is derived for the scale and shape parameters. Approximate Bayesian methods are then explored, including the Lindley and Laplace approximations, to obtain expressions for the marginal posterior densities since closed-form solutions are not available. The results indicate that the posterior variances for the scale parameter obtained with the Laplace method are smaller than those from the Lindley approximation or asymptotic variances of the maximum likelihood estimates.
1. Regression analysis is a statistical technique used to model relationships between variables and make predictions. It can be used to describe relationships, estimate coefficients, make predictions, and control systems.
2. Linear regression models describe straight-line relationships between variables, while non-linear models describe curved relationships. The goodness of fit of a model can be evaluated using the coefficient of determination.
3. The least squares method is used to fit regression lines by minimizing the sum of the squared vertical distances between observed and estimated y-values for a regression of y on x, or minimizing the sum of squared horizontal distances for a regression of x on y.
The document discusses various statistical concepts related to correlation and regression. It defines the coefficient of correlation as a measure of the strength and direction of the relationship between variables ranging from +1 to -1. A value close to 0 indicates no relationship, while values close to +1 or -1 indicate a strong positive or negative linear relationship, respectively. It also discusses the covariance and correlation of random variables, Pearson correlation coefficient, Spearman rank correlation, partial correlation coefficient, and multiple correlation coefficients. Finally, it provides a definition of regression as a technique to determine the mathematical relationship between two variables using a regression line equation.
This document discusses linear regression analysis. It defines simple and multiple linear regression, and explains that regression examines the relationship between independent and dependent variables. The document provides the equations for linear regression analysis, and discusses calculating the slope, intercept, standard error of the estimate, and coefficient of determination. It explains that regression analysis is widely used for prediction and forecasting in areas like advertising and product sales.
To get a copy of the slides for free Email me at: japhethmuthama@gmail.com
You can also support my PhD studies by donating a 1 dollar to my PayPal.
PayPal ID is japhethmuthama@gmail.com
To get a copy of the slides for free Email me at: japhethmuthama@gmail.com
You can also support my PhD studies by donating a 1 dollar to my PayPal.
PayPal ID is japhethmuthama@gmail.com
This document discusses methods for estimating the parameters of a multivariate normal distribution when certain constraints are placed on the parameters. Specifically, it considers:
1) Estimating the mean sub-vector when the constraint Rμ1 = r is placed, where R and r are known. It derives the maximum likelihood estimator for this case when the covariance matrix is known and unknown.
2) Estimating a submatrix of the covariance matrix Σ when the constraint Σ11 = kΣ110 is placed, where Σ110 is a known submatrix and k is an unknown parameter. It derives the maximum likelihood estimator for k.
3) Provides the likelihood function when a general constraint of Σ = k
Chapter summary and solutions to end-of-chapter exercises for "Data Visualization: Principles and Practice" book by Alexandru C. Telea
Chapter provides an overview of a number of methods for visualizing tensor data. It explains principal component analysis as a technique used to process a tensor matrix and extract from it information that can directly be used in its visualization. It forms a fundamental part of many tensor data processing and visualization algorithms. Section 7.4 shows how the results of the principal component analysis can be visualized using the simple color-mapping techniques. Next parts of the chapter explain how same data can be visualized using tensor glyphs, and streamline-like visualization techniques.
In contrast to Slicer, which is a more general framework for analyzing and visualizing 3D slice-based data volumes, the Diffusion Toolkit focuses on DT-MRI datasets, and thus offers more extensive and easier to use options for fiber tracking.
The document discusses correlation and regression, explaining that correlation describes the strength of a linear relationship between two variables, while regression tells us how to draw the straight line described by the correlation. It provides examples of using correlation coefficients to determine the strength and direction of relationships between independent and dependent variables, and discusses calculating correlation coefficients and using regression analysis to predict variable relationships and outcomes.
This chapter discusses elastic and viscoplastic self-consistent theories for predicting the effective mechanical behavior of polycrystalline materials from their microstructure. It reviews how crystal elasticity and plasticity theories can be used in mean-field homogenization methods like the self-consistent approximation. Examples are given of applying the elastic self-consistent method to predict effective properties of composites and textured polycrystals. The viscoplastic self-consistent method is also described for modeling the rate-dependent behavior of polycrystals based on linearizing the nonlinear viscoplastic behavior at the grain scale.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
This document discusses correlation and regression analysis. It begins by outlining the chapter's objectives and providing an introduction to investigating relationships between variables using statistical analysis. The document then presents examples of collecting data to study potential relationships between variables like stone dimensions, human heights and weights, and sprint and long jump performances. It introduces various statistical measures for quantifying relationships in data, including covariance, Pearson's product moment correlation coefficient, and Spearman's rank correlation coefficient. Examples are provided to demonstrate calculating and interpreting these statistics. Limitations of correlation analysis are also noted.
Extreme bound analysis based on correlation coefficient for optimal regressio...Loc Nguyen
Regression analysis is an important tool in statistical analysis, in which there is a demand of discovering essential independent variables among many other ones, especially in case that there is a huge number of random variables. Extreme bound analysis is a powerful approach to extract such important variables called robust regressors. In this research, a so-called Regressive Expectation Maximization with RObust regressors (REMRO) algorithm is proposed as an alternative method beside other probabilistic methods for analyzing robust variables. By the different ideology from other probabilistic methods, REMRO searches for robust regressors forming optimal regression model and sorts them according to descending ordering given their fitness values determined by two proposed concepts of local correlation and global correlation. Local correlation represents sufficient explanatories to possible regressive models and global correlation reflects independence level and stand-alone capacity of regressors. Moreover, REMRO can resist incomplete data because it applies Regressive Expectation Maximization (REM) algorithm into filling missing values by estimated values based on ideology of expectation maximization (EM) algorithm. From experimental results, REMRO is more accurate for modeling numeric regressors than traditional probabilistic methods like Sala-I-Martin method but REMRO cannot be applied in case of nonnumeric regression model yet in this research.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Lecture notes on Johansen cointegrationMoses sichei
This document discusses the Johansen cointegration procedure and error correction models. It provides an example where there are 3 variables (short-term interest rate, 3-year interest rate, and 10-year interest rate) that are cointegrated with 2 cointegrating relationships. The error correction form of the vector autoregression is shown, with the 2 cointegrating vectors entering each equation. Restrictions can be tested on the coefficients of the cointegrating vectors (beta) using likelihood ratio tests. This allows testing of economic theory restrictions on the long-run relationships between the variables.
Numerical solution of oscillatory reaction–diffusion system of $\lambda-omega$ type was done by using two finite difference schemes . The first one is the explicit scheme and the second one is the implicit Crank– Nicholson scheme with averaging of f(u,v) and without averaging of f(u,v). The comparison showed that Crank–Nicholson scheme is better than explicit scheme and Crank–Nicholson scheme with averaging of f(u,v) is more accurate but it needs more time and double storage and double time steps than Crank–Nicholson scheme without averaging of f(u,v).
This presentation covered the following topics:
1. Definition of Correlation and Regression
2. Meaning of Correlation and Regression
3. Types of Correlation and Regression
4. Karl Pearson's methods of correlation
5. Bivariate Grouped data method
6. Spearman's Rank correlation Method
7. Scattered diagram method
8. Interpretation of correlation coefficient
9. Lines of Regression
10. regression Equations
11. Difference between correlation and regression
12. Related examples
Similar to Unit 1 BP801T t h multiple correlation examples (20)
Unti 1 bp801 t ga correlation worksheet24022022ashish7sattee
The document contains instructions for calculating Pearson's correlation coefficient from sample data and calculating multiple correlation coefficients from additional sample data. It provides data sets with marks in pharmacognosy and statistics, as well as data for variables X1, X2, and X3. The goal is to calculate the correlation coefficients and interpret the values.
Unit i bp801 t k spearman coefficient of correlation24022022ashish7sattee
The Spearman's rank correlation coefficient is a nonparametric measure of the monotonic relationship between two ranked variables. It assesses how well the relationship between two variables can be described using a monotonic function. Unlike Pearson's correlation, the Spearman's correlation does not assume a linear relationship between variables and can be used when assumptions of interval/ratio scaling and bivariate normality are not met. The Spearman's correlation measures statistical dependence between two variables using a monotonic function to represent their relationship.
Unit i bp801 t j introduction to correlation 24022022ashish7sattee
This document provides an introduction to correlation. It defines correlation as quantifying the relationship between two quantitative variables. A positive correlation means high values of one variable are associated with high values of the other, while a negative correlation means high values of one variable are associated with low values of the other. The document uses a data set measuring socioeconomic status and bicycle helmet use across neighborhoods to illustrate these concepts. It shows the data follows a negative correlation, with higher socioeconomic status associated with lower helmet use. The correlation coefficient is introduced as a measure of the strength and direction of correlation.
Unit 1 bp801 t g correlation analysis24022022ashish7sattee
Correlation is a statistical measure that indicates the extent to which two or more variables fluctuate together. A positive correlation indicates that the variables increase or decrease in parallel, while a negative correlation indicates that one variable increases as the other decreases. A correlation of -0.97 represents a strong negative correlation, while 0.10 represents a weak positive correlation. Correlations above 0.4 are generally considered relatively strong. Even if there is no linear relationship between two variables, it does not necessarily mean that no relationship exists at all.
Unit 1 bp801 t f standard deviation 24022022ashish7sattee
Standard deviation is a measure of how dispersed the values in a data set are from the mean. It is calculated as the square root of the variance, which is the average of the squared distances from the mean. A lower standard deviation indicates values are closer to the mean, while a higher standard deviation means values are more spread out from the mean. Standard deviation can be used to calculate both grouped and ungrouped data, as well as discrete and continuous variables.
The document discusses the statistical concept of range. Range is defined as the difference between the highest and lowest values in a data set. To calculate range, values must first be arranged from lowest to highest. Range is then found using the formula: Range = Highest Value - Lowest Value. While range indicates variability, it can be misleading when outliers are present, as a single extreme value will impact the calculated range. Two examples are provided to demonstrate calculating range.
The mode is the value that occurs most frequently in a data set. It represents the observation with the highest frequency. For an ungrouped data set, the mode is the value with the maximum number of occurrences. A data set can have no mode, one mode (unimodal), or multiple modes (bimodal, trimodal, or multimodal). When data is grouped, the mode is calculated using the frequency and limits of the modal class along with the preceding and succeeding classes. There is also an empirical relationship between the mean, median, and mode of a data set.
Unit 1 bp801 t c median with solved examplesashish7sattee
The document defines and provides examples to illustrate the median, which represents the middle-most value of a data set. The median is calculated by arranging the data points in ascending order and selecting the middle value for an odd number of points, or averaging the two middle values for an even number of points. The median is useful for summarizing large data sets with a single value and is less influenced by outliers than the mean. Examples are provided to demonstrate calculating the median for both discrete and grouped data.
Unit 1 bp801 t b frequency distributionashish7sattee
The document discusses frequency distribution, which organizes collected data into a table showing the frequency of items or values. It provides an example of students' quiz scores organized into a frequency distribution table showing the number of students who scored each mark. The document also discusses various graphical representations of frequency distributions, including bar graphs, histograms, pie charts, and frequency polygons, which provide visual displays of the frequency of data values.
Unit 1 BP801T a introduction mean median mode ashish7sattee
This document provides information and examples for calculating various statistical measures - mean, median, mode, range - from data sets. It explains that the mean is the average found by adding all values and dividing by the number of values, the median is the middle number of a data set arranged in order, and the mode is the most frequently occurring value. Examples are given to demonstrate calculating each measure from sets of test score data. Frequency and cumulative frequency are also defined.
Standard deviation is a measure of how dispersed data values are from the mean. It is calculated as the square root of the variance, which is the average of the squared differences from the mean. A lower standard deviation means values are closer to the mean, while a higher standard deviation means values are more spread out from the mean. The document provides steps to calculate standard deviation for both grouped and ungrouped data, as well as formulas for finding the standard deviation of a sample or entire population.
MRM301T Research Methodology and Biostatistics: Euthanasia An Indian perspec...ashish7sattee
In our society, the palliative care and quality of life issues in patients with terminal illnesses like advanced cancer and AIDS have become an important concern for clinicians.
Parallel to this concern has arisen another controversial issue-euthanasia or “mercy –killing” of terminally ill patients.
MRM301T Research Methodology and Biostatistics: Confidentiality 1 22102021ashish7sattee
Ethicists rely heavily on case studies for research and teaching, but using patient information without consent raises confidentiality issues. Obtaining consent is difficult as patients may be incompetent, sensitive details are often essential to cases, and harms include violation of privacy. While public interest in medical ethics exists, there is no consensus on what constitutes public interest to justify publication without consent. Balancing privacy protections with contributions to ethical discussions remains challenging.
This document discusses different types of experimental designs, including informal and formal designs. Informal designs use less sophisticated analysis based on magnitude differences, while formal designs offer more control and use statistical analysis. Important informal designs are before-and-after without control, after-only with control, and before-and-after with control. Formal designs discussed include completely randomized, randomized block, Latin square, and factorial designs. Random replication design provides control for extraneous variables and randomizes differences among experiment conductors.
This document provides an overview of statistical tools used in research. It begins with an introduction to statistics and discusses descriptive and inferential statistics. Descriptive statistics summarize data through measures like the mean, median and mode, while inferential statistics make inferences about a population based on a sample. Both parametric and non-parametric statistical tests are covered. Common parametric tests include the t-test and ANOVA, which assume a normal distribution, while non-parametric tests like the chi-squared test are used when distributions are unknown. The document also reviews variables, types of data, statistical software options and includes examples and quizzes.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
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.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
How to Build a Module in Odoo 17 Using the Scaffold Method
Unit 1 BP801T t h multiple correlation examples
1. 37
Multiple Correlation
UNIT 11 MULTIPLE CORRELATION
Structure
11.1 Introduction
Objectives
11.2 Coefficient of Multiple Correlation
11.3 Properties of Multiple Correlation Coefficient
11.4 Summary
11.5 Solutions / Answers
11.1 INTRODUCTION
In Unit 9, you have studied the concept of regression and linear regression.
Regression coefficient was also discussed with its properties. You learned
how to determine the relationship between two variables in regression and
how to predict value of one variable from the given value of the other
variable. Plane of regression for trivariate, properties of residuals and
variance of the residuals were discussed in Unit 10 of this block, which are
basis for multiple and partial correlation coefficients. In Block 2, you have
studied the coefficient of correlation that provides the degree of linear
relationship between the two variables.
If we have more than two variables which are interrelated in someway and
our interest is to know the relationship between one variable and set of
others. This leads us to multiple correlation study.
In this unit, you will study the multiple correlation and multiple correlation
coefficient with its properties .To understand the concept of multiple
correlation you must be well versed with correlation coefficient. Before
starting this unit, you go through the correlation coefficient given in Unit 6
of the Block 2. You should also clear the basics given in Unit 10 of this
block to understand the mathematical formulation of multiple correlation
coefficients.
Section 11.2 discusses the concept of multiple correlation and multiple
correlation coefficient. It gives the derivation of the multiple correlation
coefficient formula. Properties of multiple correlation coefficients are
described in Section 11.3
Objectives
After reading this unit, you would be able to
describe the concept of multiple correlation;
define multiple correlation coefficient;
derive the multiple correlation coefficient formula; and
explain the properties of multiple correlation coefficient.
2. Regression and Multiple
Correlation
38
11.2 COEFFICIENT OF MULTIPLE
CORRELATION
If information on two variables like height and weight, income and
expenditure, demand and supply, etc. are available and we want to study the
linear relationship between two variables, correlation coefficient serves our
purpose which provides the strength or degree of linear relationship with
direction whether it is positive or negative. But in biological, physical and
social sciences, often data are available on more than two variables and value
of one variable seems to be influenced by two or more variables. For
example, crimes in a city may be influenced by illiteracy, increased
population and unemployment in the city, etc. The production of a crop may
depend upon amount of rainfall, quality of seeds, quantity of fertilizers used
and method of irrigation, etc. Similarly, performance of students in
university exam may depend upon his/her IQ, mother’s qualification, father’s
qualification, parents income, number of hours of studies, etc. Whenever we
are interested in studying the joint effect of two or more variables on a single
variable, multiple correlation gives the solution of our problem.
In fact, multiple correlation is the study of combined influence of two or
more variables on a single variable.
Suppose, 1
X , 2
X and 3
X are three variables having observations on N
individuals or units. Then multiple correlation coefficient of 1
X on 2
X and
3
X is the simple correlation coefficient between 1
X and the joint effect of
2
X and 3
X . It can also be defined as the correlation between 1
X and its
estimate based on 2
X and 3
X .
Multiple correlation coefficient is the simple correlation coefficient between
a variable and its estimate.
Let us define a regression equation of 1
X on 2
X and 3
X as
3
2
.
13
2
3
.
12
1 X
b
X
b
a
X
Let us consider three variables 3
2
1 x
and
x
,
x measured from their respective
means. The regression equation of 1
x depends upon 3
2 x
and
x is given by
3
2
.
13
2
3
.
12
1 x
b
x
b
x
… (1)
3
3
3
2
2
2
1
1
1 x
X
X
and
x
X
X
,
x
X
X
Where
0
x
x
x 3
2
1
Right hand side of equation (1) can be considered as expected or estimated
value of 1
x based on 2
x and 3
x which may be expressed as
3
2
.
13
2
3
.
12
23
.
1 x
b
x
b
x
… (2)
Residual 23
.
1
e (see definition of residual in Unit 5 of Block 2 of MST 002) is
written as
23
.
1
e = 3
2
.
13
2
3
.
12
1 x
b
x
b
x
= 23
.
1
1 x
x
3. 39
Multiple Correlation
23
.
1
1
23
.
1 x
x
e
23
.
1
1
23
.
1 e
x
x
… (3)
The multiple correlation coefficient can be defined as the simple correlation
coefficient between 1
x and its estimate 23
.
1
e . It is usually denoted by 23
.
1
R
and defined as
)
x
(
V
)
x
(
V
)
x
,
x
(
Cov
R
23
.
1
1
23
.
1
1
23
.
1 … (4)
Now,
23
.
1
23
.
1
1
1
23
.
1
1 x
x
x
x
N
1
)
x
,
x
(
Cov
(By the definition of covariance)
Since, 2
1 x
,
x and 3
x are measured from their respective means, so
0
x
x
x 3
2
1
0
x
x
x 3
2
1
and consequently
0
x
b
x
b
x 3
2
.
13
2
3
.
12
23
.
1
(From equation (2))
Thus,
)
x
,
x
(
Cov 23
.
1
1 23
.
1
1x
x
N
1
)
e
x
(
x
N
1
23
.
1
1
1
(From equation (3))
23
.
1
1
2
1 e
x
N
1
x
N
1
(By third property of residuals)
2
23
.
1
2
1 e
N
1
x
N
1
2
23
.
1
2
1
(From equation (29) of Unit10)
Now
2
23
.
1
23
.
1
23
.
1 )
x
x
(
N
1
)
x
(
V
2
23
.
1 )
x
(
N
1
(Since 23
.
1
x = 0)
= 2
23
.
1
1 )
e
x
(
N
1
(From equation (3))
)
e
x
2
e
x
(
N
1
23
.
1
1
2
23
.
1
2
1
4. Regression and Multiple
Correlation
40
23
.
1
1
2
23
.
1
2
1 e
x
N
1
2
e
N
1
x
N
1
2
23
.
1
2
23
.
1
2
1 e
N
1
2
e
N
1
x
N
1
(By third property of residuals)
2
23
.
1
2
1 e
N
1
x
N
1
)
x
(
V 23
.
1
2
23
.
1
2
1
(From equation (29) of Unit 10)
Substituting the value of )
x
,
x
(
Cov 23
.
1
1 and )
x
(
V 23
.
1 in equation (4),
we have
)
(
R
2
23
.
1
2
1
2
1
2
23
.
1
2
1
23
.
1
)
(
)
(
R 2
23
.
1
2
1
2
1
2
2
23
.
1
2
1
2
23
.
1
2
1
2
23
.
1
2
1
2
23
.
1
2
1
2
23
.
1
σ
σ
1
σ
σ
σ
R
here, 2
23
.
1
is the variance of residual, which is
)
r
r
r
2
r
r
r
1
(
r
1
13
23
12
2
13
2
12
2
23
2
23
2
1
2
23
.
1
(From equation (30) of unit 10)
Then,
2
23
.
1
R
)
r
1
(
)
r
r
r
2
r
r
r
1
(
1 2
23
2
1
23
13
12
2
23
2
13
2
12
2
1
2
23
.
1
R 2
23
23
13
12
2
23
2
13
2
12
r
1
r
r
r
2
r
r
r
1
1
2
23
.
1
R 2
23
23
13
12
2
23
2
13
2
12
2
23
r
1
r
r
r
2
r
r
r
1
r
1
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
23
.
1
R = 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
… (5)
which is required formula for multiple correlation coefficient.
where, 12
r is the total correlation coefficient between variable 1
X and 2
X ,
23
r is the total correlation coefficient between variable 2
X and 3
X ,
5. 41
Multiple Correlation
13
r is the total correlation coefficient between variable 1
X and 3
X .
Now let us solve a problem on multiple correlation coefficients.
Example 1: From the following data, obtain 23
.
1
R and 13
.
2
R
1
X 65 72 54 68 55 59 78 58 57 51
2
X 56 58 48 61 50 51 55 48 52 42
3
X 9 11 8 13 10 8 11 10 11 7
Solution: To obtain multiple correlation coefficients 23
.
1
R and ,
R 13
.
2 we
use following formulae
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
and
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
We need 23
13
12 r
and
r
,
r which are obtained from the following table:
S. No. X1 X2 X3 (X1)2
(X2)2
(X3)2
X1X2 X1X3 X2X3
1 65 56 9 4225 3136 81 3640 585 504
2 72 58 11 5184 3364 121 4176 792 638
3 54 48 8 2916 2304 64 2592 432 384
4 68 61 13 4624 3721 169 4148 884 793
5 55 50 10 3025 2500 100 2750 550 500
6 59 51 8 3481 2601 64 3009 472 408
7 78 55 11 6084 3025 121 4290 858 605
8 58 48 10 3364 2304 100 2784 580 480
9 57 52 11 3249 2704 121 2964 627 572
10 51 42 7 2601 1764 49 2142 357 294
Total 617 521 98 38753 27423 990 32495 6137 5178
Now we get the total correlation coefficient 23
13
12 r
and
r
,
r
2
2
2
2
2
1
2
1
2
1
2
1
12
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
521
(
)
521
(
)
27423
10
(
)
617
(
)
617
(
)
38753
10
(
)
521
(
)
617
(
)
32495
10
(
r12
6. Regression and Multiple
Correlation
42
80
.
0
01
.
4368
3493
2789
6841
3493
r12
2
3
2
3
2
1
2
1
3
1
3
1
13
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
98
98
(
)
990
10
(
)
617
617
(
)
38753
10
(
)
98
(
)
617
(
)
6137
10
(
r13
64
.
0
00
.
1423
904
296
6841
904
r13
and
2
3
2
3
2
2
2
2
3
2
3
2
23
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
98
98
(
)
990
10
(
)
521
521
(
)
27423
10
(
)
98
(
)
521
(
)
5178
10
(
r23
79
.
0
59
.
908
722
296
2789
722
r23
Now, we calculate 23
.
1
R
We have, 80
.
0
r12 , 64
.
0
r13 and 79
.
0
r23 , then
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
2
2
2
79
.
0
1
79
.
0
64
.
0
80
.
0
2
64
.
0
80
.
0
63
.
0
38
.
0
24
.
0
R
62
.
0
1
81
.
0
41
.
0
64
.
0
2
23
.
1
Then
79
.
0
R 23
.
1 .
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
2
2
2
64
.
0
1
79
.
0
64
.
0
80
.
0
2
79
.
0
80
.
0
88
.
0
51
.
0
45
.
0
49
.
0
1
81
.
0
62
.
0
64
.
0
7. 43
Multiple Correlation
Thus,
13
.
2
R = 0.94
Example 2: From the following data, obtain 23
.
1
R , 13
.
2
R and 12
.
3
R
1
X 2 5 7 11
2
X 3 6 10 12
3
X 1 3 6 10
Solution: To obtain multiple correlation coefficients 23
.
1
R 13
.
2
R and R3.12,
we use following formulae
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
,
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
and
2
12
.
3
R 2
12
23
13
12
2
23
2
13
r
1
r
r
r
2
r
r
We need 23
13
12 r
and
r
,
r which are obtained from the following table:
S. No. X1 X2 X3 (X1)2
(X2)2
(X3)2
X1X2 X1X3 X2X3
1 2 3 1 4 9 1 6 2 3
2 5 6 3 25 36 9 30 15 18
3 7 10 6 49 100 36 70 42 60
4 11 12 10 121 144 100 132 110 120
Total 25 31 20 199 289 146 238 169 201
Now we get the total correlation coefficient 23
13
12 r
and
r
,
r
2
2
2
2
2
1
2
1
2
1
2
1
12
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
31
(
)
31
(
)
289
4
(
)
25
(
)
25
(
)
199
4
(
)
31
(
)
25
(
)
238
4
(
r12
97
.
0
.
0
61
.
182
177
195
171
177
r12
2
3
2
3
2
1
2
1
3
1
3
1
13
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
8. Regression and Multiple
Correlation
44
)
20
20
(
)
146
4
(
)
25
25
(
)
199
4
(
)
20
(
)
25
(
)
169
4
(
r13
99
.
0
38
.
177
176
184
171
176
r13
and
2
3
2
3
2
2
2
2
3
2
3
2
23
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
20
20
(
)
146
4
(
)
31
31
(
)
289
4
)
20
(
)
31
(
)
201
4
(
r23
97
.
0
42
.
189
184
184
195
184
r23
Now, we calculate 23
.
1
R
We have, 97
.
0
r12 , 99
.
0
r13 and 97
.
0
r23 , then
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
2
2
2
97
.
0
1
97
.
0
99
.
0
97
.
0
2
99
.
0
97
.
0
98
.
0
059
.
0
058
.
0
Then
99
.
0
R 23
.
1 .
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
2
2
2
99
.
0
1
97
.
0
99
.
0
97
.
0
2
97
.
0
97
.
0
95
.
0
20
.
0
19
.
0
Thus,
13
.
2
R = 0.97
2
12
.
3
R 2
12
23
13
12
2
23
2
13
r
1
r
r
r
2
r
r
2
2
2
97
.
0
1
97
.
0
99
.
0
97
.
0
2
97
.
0
99
.
0
9. 45
Multiple Correlation
981
.
0
591
.
0
58
.
0
Thus,
12
.
3
R = 0.99
Example 3: The following data is given:
1
X 60 68 50 66 60 55 72 60 62 51
2
X 42 56 45 64 50 55 57 48 56 42
3
X 74 71 78 80 72 62 70 70 76 65
Obtain 23
.
1
R , 13
.
2
R and 12
.
3
R
Solution: To obtain multiple correlation coefficients 23
.
1
R , 13
.
2
R and 12
.
3
R
we use following formulae:
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
,
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
and
2
12
.
3
R 2
12
23
13
12
2
23
2
13
r
1
r
r
r
2
r
r
We need 23
13
12 r
and
r
,
r which are obtained from the following table:
S.
No.
X1 X2 X3 d1=
X1 -60
d2 =
X2 -50
d3=
X3 -70
(d1)2
(d2)2
(d3)2
d1d2 d1d3 d2d3
1 60 42 74 0 -8 4 0 64 16 0 0 -32
2 68 56 71 8 6 1 64 36 1 48 8 6
3 50 45 78 -10 -5 8 100 25 64 50 -80 -40
4 66 64 80 6 14 10 36 196 100 84 60 140
5 60 50 72 0 0 2 0 0 4 0 0 0
6 55 55 62 -5 5 -8 25 25 64 -25 40 -40
7 72 57 70 12 7 0 144 49 0 84 0 0
8 60 48 70 0 -2 0 0 4 0 0 0 0
9 62 56 76 2 6 6 4 36 36 12 12 36
10 51 42 65 -9 -8 -5 81 64 25 72 45 40
Total 4 15 18 454 499 310 325 85 110
10. Regression and Multiple
Correlation
46
Here, we can also use shortcut method to calculate r12, r13 & r23,
Let d1 = X1− 60
d2 = X2− 50
d3 = X1− 70
Now we get the total correlation coefficient 23
13
12 r
and
r
,
r
2
2
2
2
2
1
2
1
2
1
2
1
12
)
d
(
)
d
(
N
)
d
(
)
d
(
N
)
d
(
)
d
(
)
d
d
(
N
r
)
15
(
)
15
(
)
499
10
(
)
4
(
)
4
(
)
454
10
(
)
15
(
)
4
(
)
325
10
(
r12
69
.
0
94
.
4642
3190
4765
4524
3190
r12
2
3
2
3
2
1
2
1
3
1
3
1
13
)
d
(
)
d
(
N
)
d
(
)
d
(
N
)
d
(
)
d
(
)
d
d
(
N
r
)
18
18
(
)
310
10
(
)
4
4
(
)
454
10
(
)
18
(
)
4
(
)
85
10
(
r13
22
.
0
81
.
3543
778
2776
4524
778
r13
and
2
3
2
3
2
2
2
2
3
2
3
2
23
)
d
(
)
d
(
N
)
d
(
)
d
(
N
)
d
(
)
d
(
)
d
d
(
N
r
)
18
18
(
)
310
10
(
)
15
15
(
)
499
10
(
)
18
(
)
15
(
)
110
10
(
r23
23
.
0
98
.
3636
830
2776
4765
830
r23
Now, we calculate 23
.
1
R
We have, 69
.
0
r12 , 22
.
0
r13 and 23
.
0
r23 , then
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
2
2
2
23
.
0
1
23
.
0
22
.
0
69
.
0
2
22
.
0
69
.
0
4801
.
0
9471
.
0
4547
.
0
Then
69
.
0
R 23
.
1
11. 47
Multiple Correlation
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
2
2
2
22
.
0
1
23
.
0
22
.
0
69
.
0
2
23
.
0
69
.
0
4825
.
0
9516
.
0
4592
.
0
Thus,
13
.
2
R = 0.69
2
12
.
3
R 2
12
23
13
12
2
23
2
13
r
1
r
r
r
2
r
r
2
2
2
69
.
0
1
23
.
0
22
.
0
69
.
0
2
23
.
0
22
.
0
0601
.
0
5239
.
0
0315
.
0
Thus,
12
.
3
R = 0.25
Now let us solve some exercises.
E1) In bivariate distribution, 54
.
0
r
r
,
6
.
0
r 31
23
12
, then calculate
23
.
1
R .
E2) If 54
.
0
r
and
74
.
0
r
,
70
.
0
r 23
13
12
, calculate multiple correlation
coefficient 13
.
2
R .
E3) Calculate multiple correlation coefficients 23
.
1
R and 13
.
2
R from the
following information: 42
.
0
r
and
57
.
0
r
,
82
.
0
r 13
23
12
.
E4) From the following data,
1
X 22 15 27 28 30 42 40
2
X 12 15 17 15 42 15 28
3
X 13 16 12 18 22 20 12
Obtain 23
.
1
R , 13
.
2
R and 12
.
3
R
E5) The following data is given:
1
X 50 54 50 56 50 55 52 50 52 51
2
X 42 46 45 44 40 45 43 42 41 42
3
X 72 71 73 70 72 72 70 71 75 71
By using the short-cut method obtain 23
.
1
R , 13
.
2
R and 12
.
3
R
12. Regression and Multiple
Correlation
48
11.3 PROPERTIES OF MULTIPLE
CORRELATION COEFFICIENT
The following are some of the properties of multiple correlation coefficients:
1. Multiple correlation coefficient is the degree of association between
observed value of the dependent variable and its estimate obtained by
multiple regression,
2. Multiple Correlation coefficient lies between 0 and 1,
3. If multiple correlation coefficient is 1, then association is perfect and
multiple regression equation may said to be perfect prediction formula,
4. If multiple correlation coefficient is 0, dependent variable is uncorrelated
with other independent variables. From this, it can be concluded that
multiple regression equation fails to predict the value of dependent
variable when values of independent variables are known,
5. Multiple correlation coefficient is always greater or equal than any total
correlation coefficient. If 23
.
1
R is the multiple correlation coefficient
than 23
13
12
23
.
1 r
or
r
or
r
R , and
6. Multiple correlation coefficient obtained by method of least squares
would always be greater than the multiple correlation coefficient
obtained by any other method.
11.4 SUMMARY
In this unit, we have discussed:
1. The multiple correlation, which is the study of joint effect of a group of
two or more variables on a single variable which is not included in that
group,
2. The estimate obtained by regression equation of that variable on other
variables,
3. Limit of multiple correlation coefficient, which lies between 0 and +1,
4. The numerical problems of multiple correlation coefficient, and
5. The properties of multiple correlation coefficient.
11.5 SOLUTIONS / ANSWERS
E1) We have,
54
.
0
r
r
,
6
.
0
r 31
23
12
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
13. 49
Multiple Correlation
42
.
0
71
.
0
30
.
0
71
.
0
35
.
0
29
.
0
36
.
0
Then
65
.
0
R 23
.
1
E2) We have
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
49
.
0
45
.
0
22
.
0
55
.
0
1
56
.
0
29
.
0
49
.
0
Thus
13
.
2
R = 0.70.
E3) We have
42
.
0
r
57
.
0
r
,
82
.
0
r 13
23
12
.
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
68
.
0
68
.
0
46
.
0
68
.
0
39
.
0
18
.
0
67
.
0
Then
23
.
1
R = 0.82
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
73
.
0
82
.
0
60
.
0
82
.
0
39
.
0
32
.
0
67
.
0
Thus,
13
.
2
R = 0.85.
E4) To obtain multiple correlation coefficients 23
.
1
R , 13
.
2
R and 12
.
3
R
we use following formulae:
14. Regression and Multiple
Correlation
50
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
,
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
and
2
12
.
3
R 2
12
23
13
12
2
23
2
13
r
1
r
r
r
2
r
r
We need 23
13
12 r
and
r
,
r which are obtained from the following
table:
S. No. X1 X2 X3 (X1)2
(X2)2
(X3)2
X1X2 X1X3 X2X3
1 22 12 13 484 144 169 264 286 156
2 15 15 16 225 225 256 225 240 240
3 27 17 12 729 289 144 459 324 204
4 28 15 18 784 225 324 420 504 270
5 30 42 22 900 1764 484 1260 660 924
6 42 15 20 1764 225 400 630 840 300
7 40 28 12 1600 784 144 1120 480 336
Total 204 144 113 6486 3656 1921 4378 3334 2430
Now, we get the total correlation coefficient 23
13
12 r
and
r
,
r
2
2
2
2
2
1
2
1
2
1
2
1
12
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
144
(
)
144
(
)
3656
7
(
)
204
(
)
204
(
)
6486
7
(
)
144
(
)
204
(
)
4378
7
(
r12
4856
3786
1270
r12
30
.
0
75
.
4287
1270
2
3
2
3
2
1
2
1
3
1
3
1
13
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
113
113
(
)
1921
7
(
)
204
204
(
)
6486
7
(
)
113
(
)
204
(
)
3334
7
(
r13
15. 51
Multiple Correlation
678
3786
286
r13
18
.
0
16
.
1602
286
and
2
3
2
3
2
2
2
2
3
2
3
2
23
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
113
113
(
)
1921
7
(
)
144
144
(
)
3656
7
)
113
(
)
144
(
)
2430
7
(
r23
678
4856
738
r23
41
.
0
49
.
1814
738
Now, we calculate 23
.
1
R
We have, 30
.
0
r12 , 18
.
0
r13 and 41
.
0
r23 , then
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
2
2
2
)
41
.
0
(
1
41
.
0
18
.
0
30
.
2
18
.
0
30
.
0
9380
.
0
8319
.
0
0781
.
0
Then
30
.
0
R 23
.
1 .
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
2
2
2
18
.
0
1
41
.
0
18
.
0
30
.
2
41
.
0
30
.
0
221
.
0
9676
.
0
2138
.
0
Thus,
13
.
2
R = 0.47
2
12
.
3
R 2
12
23
13
12
2
23
2
13
r
1
r
r
r
2
r
r
16. Regression and Multiple
Correlation
52
2
2
2
30
.
0
1
41
.
0
18
.
0
30
.
0
2
41
.
0
18
.
0
1717
.
0
9100
.
0
1562
.
0
Thus,
12
.
3
R = 0.41
E5) To obtain multiple correlation coefficients 23
.
1
R , 13
.
2
R and 12
.
3
R
we use following formulae
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
,
2
12
.
3
R 2
12
23
13
12
2
23
2
13
r
1
r
r
r
2
r
r
and
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r
We need 23
13
12 r
and
r
,
r which are obtained from the following
table:
S.
No.
X1 X2 X3 d1=
X1 -50
d2 =
X2 -40
d3=
X3 -70
(d1)2
(d2)2
(d3)2
d1d2 d1d3 d2d3
1 50 42 72 0 2 2 0 4 4 0 0 4
2 54 46 71 4 6 1 16 36 1 24 4 6
3 50 45 73 0 5 3 0 25 9 0 0 15
4 56 44 70 6 4 0 36 16 0 24 0 0
5 50 40 72 0 0 2 0 0 4 0 0 0
6 55 45 72 5 5 2 25 25 4 25 10 10
7 52 43 70 2 3 0 4 9 0 6 0 0
8 50 42 71 0 2 1 0 4 1 0 0 2
9 52 41 75 2 1 5 4 1 25 2 10 5
10 51 42 71 1 2 1 1 4 1 2 1 2
Total 20 30 17 86 124 49 83 25 44
Now, we get the total correlation coefficient 23
13
12 r
and
r
,
r
2
2
2
2
2
1
2
1
2
1
2
1
12
)
d
(
)
d
(
N
)
d
(
)
d
(
N
)
d
(
)
d
(
)
d
d
(
N
r
)
30
(
)
30
(
)
124
10
(
)
20
(
)
20
(
)
86
10
(
)
30
(
)
20
(
)
83
10
(
r12
17. 53
Multiple Correlation
340
460
230
r12
58
.
0
47
.
395
230
2
3
2
3
2
1
2
1
3
1
3
1
13
)
d
(
)
d
(
N
)
d
(
)
d
(
N
)
d
(
)
d
(
)
d
d
(
N
r
)
17
17
(
)
49
10
(
)
20
20
(
)
86
10
(
)
17
(
)
20
(
)
25
10
(
r13
201
460
90
r13
30
.
0
07
.
304
90
and
2
3
2
3
2
2
2
2
3
2
3
2
23
)
X
(
)
X
(
N
)
X
(
)
X
(
N
)
X
(
)
X
(
)
X
X
(
N
r
)
17
17
(
)
49
10
(
)
20
20
(
)
124
10
(
)
17
(
)
30
(
)
44
10
(
r23
201
340
70
r23
27
.
0
42
.
261
70
Now, we calculate 23
.
1
R
We have, 58
.
0
r12 , 30
.
0
r13
and 27
.
0
r23
, then
2
23
.
1
R 2
23
23
13
12
2
13
2
12
r
1
r
r
r
2
r
r
2
2
2
27
.
0
1
27
.
0
30
.
0
58
.
0
2
30
.
0
58
.
0
36
.
0
9271
.
0
3324
.
0
Then
60
.
0
R 23
.
1 .
2
13
.
2
R 2
13
23
13
12
2
23
2
12
r
1
r
r
r
2
r
r