Logistic regression is used to model dichotomous outcome variables based on one or more predictor variables. It models the probability of an outcome as a function of the predictors using a logistic link function. The model estimates regression coefficients that represent the log odds ratios of the predictors. Odds ratios above 1 indicate increased odds of the outcome with higher predictor values, while odds ratios below 1 indicate decreased odds. Logistic regression allows estimation of effects of individual predictors while controlling for other predictors. Loglinear models generalize logistic regression to situations with only categorical variables by modeling associations between variable pairs.
This document provides guidance on performing and interpreting logistic regression analyses in SPSS. It discusses selecting appropriate statistical tests based on variable types and study objectives. It covers assumptions of logistic regression like linear relationships between predictors and the logit of the outcome. It also explains maximum likelihood estimation, interpreting coefficients, and evaluating model fit and accuracy. Guidelines are provided on reporting logistic regression results from SPSS outputs.
This document provides an overview of correlation and linear regression. It defines key terms like independent variable, dependent variable, correlation coefficient, and regression coefficients. It explains how to calculate the correlation coefficient and regression coefficients using the least squares method. Properties of the regression coefficients are also discussed. Examples are provided to demonstrate how to interpret correlation, draw scatter plots, calculate coefficients, and predict values using the linear regression equation.
This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.
This document provides an overview of different types of variables and methods for summarizing clinical data, including descriptive statistics. It discusses categorical variables like gender and ordinal variables like disease staging. For continuous variables it explains measures of central tendency like mean, median and mode, and measures of variation like range, standard deviation, and interquartile range. Graphs for summarizing univariate data are also covered, such as bar charts for categorical variables and histograms and box plots for continuous variables.
This document provides an overview of logistic regression. It begins by explaining that linear regression is not appropriate when the dependent variable is dichotomous. Logistic regression uses an S-shaped logistic function to model the probabilities of different outcomes. The logistic function transforms the non-linear probabilities into linear-looking data that can be modeled using linear regression. Examples are provided to demonstrate how logistic regression can be used to predict the probability of coronary heart disease based on age and to analyze the relationship between patient satisfaction and residence.
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.
Correlation _ Regression Analysis statistics.pptxkrunal soni
This document discusses correlation and related statistical concepts. Correlation measures the strength and direction of association between two quantitative variables. A correlation of 0 means no association, 1 means perfect positive association, and -1 means perfect negative association. Correlation is independent of measurement units and scaling of variables. Hypothesis testing is used to make inferences about the population correlation based on a sample correlation. The null hypothesis is that the population correlation is 0, and alternative hypotheses specify a non-zero correlation. The test statistic used is Student's t distribution. The null is rejected if the calculated t exceeds the critical value or if the p-value is less than the significance level.
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
This document provides guidance on performing and interpreting logistic regression analyses in SPSS. It discusses selecting appropriate statistical tests based on variable types and study objectives. It covers assumptions of logistic regression like linear relationships between predictors and the logit of the outcome. It also explains maximum likelihood estimation, interpreting coefficients, and evaluating model fit and accuracy. Guidelines are provided on reporting logistic regression results from SPSS outputs.
This document provides an overview of correlation and linear regression. It defines key terms like independent variable, dependent variable, correlation coefficient, and regression coefficients. It explains how to calculate the correlation coefficient and regression coefficients using the least squares method. Properties of the regression coefficients are also discussed. Examples are provided to demonstrate how to interpret correlation, draw scatter plots, calculate coefficients, and predict values using the linear regression equation.
This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.
This document provides an overview of different types of variables and methods for summarizing clinical data, including descriptive statistics. It discusses categorical variables like gender and ordinal variables like disease staging. For continuous variables it explains measures of central tendency like mean, median and mode, and measures of variation like range, standard deviation, and interquartile range. Graphs for summarizing univariate data are also covered, such as bar charts for categorical variables and histograms and box plots for continuous variables.
This document provides an overview of logistic regression. It begins by explaining that linear regression is not appropriate when the dependent variable is dichotomous. Logistic regression uses an S-shaped logistic function to model the probabilities of different outcomes. The logistic function transforms the non-linear probabilities into linear-looking data that can be modeled using linear regression. Examples are provided to demonstrate how logistic regression can be used to predict the probability of coronary heart disease based on age and to analyze the relationship between patient satisfaction and residence.
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.
Correlation _ Regression Analysis statistics.pptxkrunal soni
This document discusses correlation and related statistical concepts. Correlation measures the strength and direction of association between two quantitative variables. A correlation of 0 means no association, 1 means perfect positive association, and -1 means perfect negative association. Correlation is independent of measurement units and scaling of variables. Hypothesis testing is used to make inferences about the population correlation based on a sample correlation. The null hypothesis is that the population correlation is 0, and alternative hypotheses specify a non-zero correlation. The test statistic used is Student's t distribution. The null is rejected if the calculated t exceeds the critical value or if the p-value is less than the significance level.
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
Logistic regression vs. logistic classifier. History of the confusion and the...Adrian Olszewski
Despite the wrong (yet widespread) claim, that "logistic regression is not a regression", it's one of the key regression tool in experimental research, like the clinical trials. It is used also for advanced testing hypotheses.
The logistic regression is part of the GLM (Generalized Linear Model) regression framework. I expanded this topic here: https://medium.com/@r.clin.res/is-logistic-regression-a-regression-46dcce4945dd
This document provides an introduction to simple linear regression and correlation. It defines key terms like independent and dependent variables, and explains how to estimate regression coefficients using the least squares method. Graphs like scatter plots are used to visualize the linear relationship between two variables. The correlation coefficient measures the strength of the linear association. Regression seeks to predict a dependent variable from an independent variable, while correlation simply measures the degree of association between two variables.
This document discusses various statistical tests used to analyze dental research data, including parametric and non-parametric tests. It provides information on tests of significance such as the t-test, Z-test, analysis of variance (ANOVA), and non-parametric equivalents. Key points covered include the differences between parametric and non-parametric tests, assumptions and applications of the t-test, Z-test, ANOVA, and non-parametric alternatives like the Mann-Whitney U test and Kruskal-Wallis test. Examples are provided to illustrate how to perform and interpret common statistical analyses used in dental research.
This document discusses standard and hierarchical multiple regression. It provides examples using data on academic achievement (GPA) predicted from minutes spent studying, motivation, and anxiety. Standard multiple regression is used to assess how much variance in GPA is explained collectively by the three predictors. Specifically, it finds the predictors explain 65% of variance in GPA. It also describes interpreting individual predictor importance through coefficients like beta weights. Hierarchical regression is mentioned but not demonstrated.
This document discusses logistic regression, including:
- Logistic regression can be used when the dependent variable is binary and predicts the probability of an event occurring.
- The logistic regression equation calculates the log odds of an event occurring based on independent variables.
- Logistic regression is commonly used in medical research when variables are a mix of categorical and continuous.
This document provides an overview of regression models and analysis techniques. It introduces simple and multiple linear regression, as well as logistic regression. It discusses assessing regression models, cross-validation, model selection, and using regression models for prediction. Additionally, it covers the similarities and differences between linear and logistic regression, and assessing correlation without inferring causation. Scatter plots, correlation coefficients, and computing regression equations are also summarized.
- A sample is a small group selected from a population to represent that population. Sampling provides benefits like being less time-consuming, less expensive, and allowing results to be repeated.
- There are two main types of samples: probability and non-probability. Probability samples include simple random, systematic, stratified, and cluster samples. Sample size is determined based on factors like the type of study, expected results, costs, and available resources.
- Inferential statistics allow generalization from a sample to a population through hypothesis testing and significance tests. Tests include t-tests, F-tests, chi-squared tests, and correlation/regression to analyze relationships between variables. Significant results suggest differences are likely not due to chance
This document discusses statistical techniques for investigating relationships between variables, including correlation, regression, and t-tests. Correlation determines if a linear relationship exists between two variables, while simple linear regression describes the relationship. Scatterplots are used to visualize paired data and the correlation coefficient measures the strength of the linear relationship. Regression finds the line of best fit that minimizes the residuals between observed and predicted values.
univariate and bivariate analysis in spss Subodh Khanal
this slide will help to perform various tests in spss targeting univariate and bivariate analysis along with the way of entering and analyzing multiple responses.
1) Statistics are important in analytical chemistry to objectively analyze experimental data, communicate significance, and optimize experimental design.
2) Key statistical terms include mean, median, population, sample, standard deviation, and accuracy vs precision.
3) Spreadsheet software can be used to calculate statistical values like standard deviation and perform regressions for calibration curves.
This document discusses key statistical concepts used in analytical chemistry, including accuracy, precision, standard deviation, probability distributions, and significance testing. It explains how statistics are applied to evaluate experimental data quality and validate analytical methods. Spreadsheets and linear regression are also summarized as tools for statistical data analysis.
The document provides an overview of statistical hypothesis testing and various statistical tests used to analyze quantitative and qualitative data. It discusses types of data, key terms like null hypothesis and p-value. It then outlines the steps in hypothesis testing and describes different tests of significance including standard error of difference between proportions, chi-square test, student's t-test, paired t-test, and ANOVA. Examples are provided to demonstrate how to apply these statistical tests to determine if differences observed in sample data are statistically significant.
Chapter 9
Multivariable Methods
Objectives
• Define and provide examples of dependent and
independent variables in a study of a public
health problem
• Explain the principle of statistical adjustment
to a lay audience
• Organize data for regression analysis
Objectives
• Define and provide an example of confounding
• Define and provide an example of effect
modification
• Interpret coefficients in multiple linear and
multiple logistic regression analysis
Definitions
• Confounding – the distortion of the effect of a
risk factor on an outcome
• Effect Modification – a different relationship
between the risk factor and an outcome
depending on the level of another variable
Confounding
• A confounder is related to the risk factor and
also to the outcome
• Assessing confounding
– Formal tests of hypothesis
– Clinically meaningful associations
Example 9.1.
Confounding
We wish to assess the association between obesity and
incident cardiovascular disease.
Incident
CVD
No
CVD
Total
Obese 46 254 300
Not
Obese
60 640 700
Total 106 894 1000
1.78
0.086
0.153
60/700
46/300
RR
CVD
Example 9.1.
Confounding
Is age a confounder?
Age
< 50
CVD No
CVD
Total Age
50+
CVD No
CVD
Total
Obese 10 90 100 Obese 36 164 200
Not
Obese
35 465 500 Not
Obese
25 175 200
Total 45 555 600 Total 65 335 400
1.44
0.13
0.18
RR and 1.43
0.07
0.10
RR
50 Age|CVD50Age|CVD
Example 9.2.
Effect Modification
A clinical trial is run to assess the efficacy of a new drug
to increase HDL cholesterol.
N Mean Std Dev
New drug 50 40.16 4.46
Placebo 50 39.21 3.91
H0: m1m2 versus H1:m1≠m2
Z=-1.13 is not statistically significant
Example 9.2.
Effect Modification
Is there effect modification by gender?
Women N Mean Std Dev
New drug 40 38.88 3.97
Placebo 41 39.24 4.21
Men N Mean Std Dev
New drug 10 45.25 1.89
Placebo 9 39.06 2.22
Effect Modification
34
36
38
40
42
44
46
Women Men
M
e
a
n
H
D
L
Gender
Placebo
New Drug
Cochran-Mantel-Haenszel Method
• Technique to estimate association between risk
factor and outcome accounting for
confounding
• Data are organized into stratum and
associations are estimated in each stratum and
combined
Correlation and Simple Linear Regression
Analysis
• Two continuous variables
– Y= dependent, outcome variable
– X=independent, predictor variable
Relationship between age and SBP, number of
hours of exercise and percent body fat, caffeine
consumption and blood sugar level.
Correlation and Simple Linear Regression
• Correlation – nature and strength of linear
association between variables
• Regression – equation that best describes
relationship between variables
Scatter Diagram
0
5
10
15
20
25
0 5 10 15 20 25 30 35 40 45
X
Y
Correlation Coefficient
• Population correlation r
• Sample correlation r, -1 < r < +1
• Sign indicates nature of relationship (positive
or direct, negative o.
The document discusses bivariate and multivariate linear regression analysis, explaining how to estimate regression coefficients using software like SPSS and interpret their results. It covers topics such as estimating and interpreting intercept and slope coefficients, measuring predictive power using R-squared, and testing the significance of individual regression coefficients and the overall regression model through techniques like t-tests and F-tests.
The document discusses hypothesis testing to determine if districts with smaller class sizes have higher test scores. It summarizes the steps taken: 1) Estimation to calculate the difference in average test scores between districts with low vs high student-teacher ratios (STRs), 2) Hypothesis testing to determine if the difference is statistically significant by calculating a t-statistic and comparing it to a critical value, 3) Construction of a confidence interval for the difference between the means. The analysis found the difference in average test scores between low and high STR districts was statistically significant based on a t-statistic greater than the critical value.
The document provides information about performing chi-square tests and choosing appropriate statistical tests. It discusses key concepts like the null hypothesis, degrees of freedom, and expected versus observed values. Examples are provided to illustrate chi-square tests for goodness of fit and comparison of proportions. The document also compares parametric and non-parametric tests, providing examples of when each would be used.
This document discusses key statistical concepts including random variables, probability distributions, expected value, variance, and correlation. It defines discrete and continuous random variables and explains how probability distributions assign probabilities to the possible values of a random variable. It also defines important metrics like expected value and variance, and how they are calculated for discrete and continuous random variables. The document concludes by explaining correlation, how the correlation coefficient measures the strength and direction of linear association between two variables, and how it is calculated.
This document discusses multiple regression analysis techniques. It begins by stating the goals of developing a statistical model to predict dependent variables from independent variables and using multiple regression when more than one independent variable is useful for prediction. It then provides an introduction to simple and multiple regression. The rest of the document discusses key aspects of multiple regression analysis, including linear models, the method of least squares, standard error of estimate, coefficient of multiple determination, hypothesis testing, and selection of predictor variables.
The document discusses various measures of central tendency, dispersion, and shape used to describe data numerically. It defines terms like mean, median, mode, variance, standard deviation, coefficient of variation, range, interquartile range, skewness, and quartiles. It provides formulas and examples of how to calculate these measures from data sets. The document also discusses concepts like normal distribution, empirical rule, and how measures of central tendency and dispersion do not provide information about the shape or symmetry of a distribution.
Logistic regression vs. logistic classifier. History of the confusion and the...Adrian Olszewski
Despite the wrong (yet widespread) claim, that "logistic regression is not a regression", it's one of the key regression tool in experimental research, like the clinical trials. It is used also for advanced testing hypotheses.
The logistic regression is part of the GLM (Generalized Linear Model) regression framework. I expanded this topic here: https://medium.com/@r.clin.res/is-logistic-regression-a-regression-46dcce4945dd
This document provides an introduction to simple linear regression and correlation. It defines key terms like independent and dependent variables, and explains how to estimate regression coefficients using the least squares method. Graphs like scatter plots are used to visualize the linear relationship between two variables. The correlation coefficient measures the strength of the linear association. Regression seeks to predict a dependent variable from an independent variable, while correlation simply measures the degree of association between two variables.
This document discusses various statistical tests used to analyze dental research data, including parametric and non-parametric tests. It provides information on tests of significance such as the t-test, Z-test, analysis of variance (ANOVA), and non-parametric equivalents. Key points covered include the differences between parametric and non-parametric tests, assumptions and applications of the t-test, Z-test, ANOVA, and non-parametric alternatives like the Mann-Whitney U test and Kruskal-Wallis test. Examples are provided to illustrate how to perform and interpret common statistical analyses used in dental research.
This document discusses standard and hierarchical multiple regression. It provides examples using data on academic achievement (GPA) predicted from minutes spent studying, motivation, and anxiety. Standard multiple regression is used to assess how much variance in GPA is explained collectively by the three predictors. Specifically, it finds the predictors explain 65% of variance in GPA. It also describes interpreting individual predictor importance through coefficients like beta weights. Hierarchical regression is mentioned but not demonstrated.
This document discusses logistic regression, including:
- Logistic regression can be used when the dependent variable is binary and predicts the probability of an event occurring.
- The logistic regression equation calculates the log odds of an event occurring based on independent variables.
- Logistic regression is commonly used in medical research when variables are a mix of categorical and continuous.
This document provides an overview of regression models and analysis techniques. It introduces simple and multiple linear regression, as well as logistic regression. It discusses assessing regression models, cross-validation, model selection, and using regression models for prediction. Additionally, it covers the similarities and differences between linear and logistic regression, and assessing correlation without inferring causation. Scatter plots, correlation coefficients, and computing regression equations are also summarized.
- A sample is a small group selected from a population to represent that population. Sampling provides benefits like being less time-consuming, less expensive, and allowing results to be repeated.
- There are two main types of samples: probability and non-probability. Probability samples include simple random, systematic, stratified, and cluster samples. Sample size is determined based on factors like the type of study, expected results, costs, and available resources.
- Inferential statistics allow generalization from a sample to a population through hypothesis testing and significance tests. Tests include t-tests, F-tests, chi-squared tests, and correlation/regression to analyze relationships between variables. Significant results suggest differences are likely not due to chance
This document discusses statistical techniques for investigating relationships between variables, including correlation, regression, and t-tests. Correlation determines if a linear relationship exists between two variables, while simple linear regression describes the relationship. Scatterplots are used to visualize paired data and the correlation coefficient measures the strength of the linear relationship. Regression finds the line of best fit that minimizes the residuals between observed and predicted values.
univariate and bivariate analysis in spss Subodh Khanal
this slide will help to perform various tests in spss targeting univariate and bivariate analysis along with the way of entering and analyzing multiple responses.
1) Statistics are important in analytical chemistry to objectively analyze experimental data, communicate significance, and optimize experimental design.
2) Key statistical terms include mean, median, population, sample, standard deviation, and accuracy vs precision.
3) Spreadsheet software can be used to calculate statistical values like standard deviation and perform regressions for calibration curves.
This document discusses key statistical concepts used in analytical chemistry, including accuracy, precision, standard deviation, probability distributions, and significance testing. It explains how statistics are applied to evaluate experimental data quality and validate analytical methods. Spreadsheets and linear regression are also summarized as tools for statistical data analysis.
The document provides an overview of statistical hypothesis testing and various statistical tests used to analyze quantitative and qualitative data. It discusses types of data, key terms like null hypothesis and p-value. It then outlines the steps in hypothesis testing and describes different tests of significance including standard error of difference between proportions, chi-square test, student's t-test, paired t-test, and ANOVA. Examples are provided to demonstrate how to apply these statistical tests to determine if differences observed in sample data are statistically significant.
Chapter 9
Multivariable Methods
Objectives
• Define and provide examples of dependent and
independent variables in a study of a public
health problem
• Explain the principle of statistical adjustment
to a lay audience
• Organize data for regression analysis
Objectives
• Define and provide an example of confounding
• Define and provide an example of effect
modification
• Interpret coefficients in multiple linear and
multiple logistic regression analysis
Definitions
• Confounding – the distortion of the effect of a
risk factor on an outcome
• Effect Modification – a different relationship
between the risk factor and an outcome
depending on the level of another variable
Confounding
• A confounder is related to the risk factor and
also to the outcome
• Assessing confounding
– Formal tests of hypothesis
– Clinically meaningful associations
Example 9.1.
Confounding
We wish to assess the association between obesity and
incident cardiovascular disease.
Incident
CVD
No
CVD
Total
Obese 46 254 300
Not
Obese
60 640 700
Total 106 894 1000
1.78
0.086
0.153
60/700
46/300
RR
CVD
Example 9.1.
Confounding
Is age a confounder?
Age
< 50
CVD No
CVD
Total Age
50+
CVD No
CVD
Total
Obese 10 90 100 Obese 36 164 200
Not
Obese
35 465 500 Not
Obese
25 175 200
Total 45 555 600 Total 65 335 400
1.44
0.13
0.18
RR and 1.43
0.07
0.10
RR
50 Age|CVD50Age|CVD
Example 9.2.
Effect Modification
A clinical trial is run to assess the efficacy of a new drug
to increase HDL cholesterol.
N Mean Std Dev
New drug 50 40.16 4.46
Placebo 50 39.21 3.91
H0: m1m2 versus H1:m1≠m2
Z=-1.13 is not statistically significant
Example 9.2.
Effect Modification
Is there effect modification by gender?
Women N Mean Std Dev
New drug 40 38.88 3.97
Placebo 41 39.24 4.21
Men N Mean Std Dev
New drug 10 45.25 1.89
Placebo 9 39.06 2.22
Effect Modification
34
36
38
40
42
44
46
Women Men
M
e
a
n
H
D
L
Gender
Placebo
New Drug
Cochran-Mantel-Haenszel Method
• Technique to estimate association between risk
factor and outcome accounting for
confounding
• Data are organized into stratum and
associations are estimated in each stratum and
combined
Correlation and Simple Linear Regression
Analysis
• Two continuous variables
– Y= dependent, outcome variable
– X=independent, predictor variable
Relationship between age and SBP, number of
hours of exercise and percent body fat, caffeine
consumption and blood sugar level.
Correlation and Simple Linear Regression
• Correlation – nature and strength of linear
association between variables
• Regression – equation that best describes
relationship between variables
Scatter Diagram
0
5
10
15
20
25
0 5 10 15 20 25 30 35 40 45
X
Y
Correlation Coefficient
• Population correlation r
• Sample correlation r, -1 < r < +1
• Sign indicates nature of relationship (positive
or direct, negative o.
The document discusses bivariate and multivariate linear regression analysis, explaining how to estimate regression coefficients using software like SPSS and interpret their results. It covers topics such as estimating and interpreting intercept and slope coefficients, measuring predictive power using R-squared, and testing the significance of individual regression coefficients and the overall regression model through techniques like t-tests and F-tests.
The document discusses hypothesis testing to determine if districts with smaller class sizes have higher test scores. It summarizes the steps taken: 1) Estimation to calculate the difference in average test scores between districts with low vs high student-teacher ratios (STRs), 2) Hypothesis testing to determine if the difference is statistically significant by calculating a t-statistic and comparing it to a critical value, 3) Construction of a confidence interval for the difference between the means. The analysis found the difference in average test scores between low and high STR districts was statistically significant based on a t-statistic greater than the critical value.
The document provides information about performing chi-square tests and choosing appropriate statistical tests. It discusses key concepts like the null hypothesis, degrees of freedom, and expected versus observed values. Examples are provided to illustrate chi-square tests for goodness of fit and comparison of proportions. The document also compares parametric and non-parametric tests, providing examples of when each would be used.
This document discusses key statistical concepts including random variables, probability distributions, expected value, variance, and correlation. It defines discrete and continuous random variables and explains how probability distributions assign probabilities to the possible values of a random variable. It also defines important metrics like expected value and variance, and how they are calculated for discrete and continuous random variables. The document concludes by explaining correlation, how the correlation coefficient measures the strength and direction of linear association between two variables, and how it is calculated.
This document discusses multiple regression analysis techniques. It begins by stating the goals of developing a statistical model to predict dependent variables from independent variables and using multiple regression when more than one independent variable is useful for prediction. It then provides an introduction to simple and multiple regression. The rest of the document discusses key aspects of multiple regression analysis, including linear models, the method of least squares, standard error of estimate, coefficient of multiple determination, hypothesis testing, and selection of predictor variables.
The document discusses various measures of central tendency, dispersion, and shape used to describe data numerically. It defines terms like mean, median, mode, variance, standard deviation, coefficient of variation, range, interquartile range, skewness, and quartiles. It provides formulas and examples of how to calculate these measures from data sets. The document also discusses concepts like normal distribution, empirical rule, and how measures of central tendency and dispersion do not provide information about the shape or symmetry of a distribution.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"
chapter15c.ppt
1. Logistic Regression
• Logistic Regression - Dichotomous Response
variable and numeric and/or categorical
explanatory variable(s)
– Goal: Model the probability of a particular as a function
of the predictor variable(s)
– Problem: Probabilities are bounded between 0 and 1
• Distribution of Responses: Binomial
• Link Function:
1
log
)
(
g
2. Logistic Regression with 1 Predictor
• Response - Presence/Absence of characteristic
• Predictor - Numeric variable observed for each case
• Model - p(x) Probability of presence at predictor level x
x
x
e
e
x
p
1
)
(
• = 0 P(Presence) is the same at each level of x
• > 0 P(Presence) increases as x increases
• < 0 P(Presence) decreases as x increases
3. Logistic Regression with 1 Predictor
, are unknown parameters and must be
estimated using statistical software such as SPSS,
SAS, or STATA
· Primary interest in estimating and testing
hypotheses regarding
· Large-Sample test (Wald Test):
· H0: = 0 HA: 0
)
(
:
:
.
.
:
.
.
2
2
2
1
,
2
2
^
^
2
^
obs
obs
obs
X
P
val
P
X
R
R
X
S
T
5. Example - Rizatriptan for Migraine (SPSS)
t
5
7
9
1
0
0
0
5
6
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3
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. E
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ig
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x 165
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490
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000
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819
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037
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1
,
05
.
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P
X
RR
X
S
T
H
H
obs
obs
A
6. Odds Ratio
• Interpretation of Regression Coefficient ():
– In linear regression, the slope coefficient is the change
in the mean response as x increases by 1 unit
– In logistic regression, we can show that:
)
(
1
)
(
)
(
)
(
)
1
(
x
x
x
odds
e
x
odds
x
odds
p
p
• Thus e represents the change in the odds of the outcome
(multiplicatively) by increasing x by 1 unit
• If = 0, the odds and probability are the same at all x levels (e=1)
• If > 0 , the odds and probability increase as x increases (e>1)
• If < 0 , the odds and probability decrease as x increases (e<1)
7. 95% Confidence Interval for Odds Ratio
• Step 1: Construct a 95% CI for :
^
^
^
^
^
^
^
^
^
96
.
1
,
96
.
1
96
.
1
• Step 2: Raise e = 2.718 to the lower and upper bounds of the CI:
^
^
^
^
^
^
96
.
1
96
.
1
,
e
e
• If entire interval is above 1, conclude positive association
• If entire interval is below 1, conclude negative association
• If interval contains 1, cannot conclude there is an association
8. Example - Rizatriptan for Migraine
)
2375
.
0
,
0925
.
0
(
)
037
.
0
(
96
.
1
165
.
0
:
%
95
037
.
0
165
.
0 ^
^
^
CI
• 95% CI for :
• 95% CI for population odds ratio:
)
27
.
1
,
10
.
1
(
, 2375
.
0
0925
.
0
e
e
• Conclude positive association between dose and
probability of complete relief
9. Multiple Logistic Regression
• Extension to more than one predictor variable (either
numeric or dummy variables).
• With k predictors, the model is written:
k
k
k
k
x
x
x
x
e
e
p
1
1
1
1
1
• Adjusted Odds ratio for raising xi by 1 unit, holding
all other predictors constant:
i
e
ORi
• Many models have nominal/ordinal predictors, and
widely make use of dummy variables
10. Testing Regression Coefficients
• Testing the overall model:
)
(
.
.
))
log(
2
(
))
log(
2
(
.
.
0
all
Not
:
0
:
2
2
2
,
2
1
0
2
1
0
obs
k
obs
obs
i
A
k
X
P
P
X
R
R
L
L
X
S
T
H
H
• L0, L1 are values of the maximized likelihood function, computed by
statistical software packages. This logic can also be used to compare
full and reduced models based on subsets of predictors. Testing for
individual terms is done as in model with a single predictor.
11. Example - ED in Older Dutch Men
• Response: Presence/Absence of ED (n=1688)
• Predictors: (p=12)
– Age stratum (50-54*, 55-59, 60-64, 65-69, 70-78)
– Smoking status (Nonsmoker*, Smoker)
– BMI stratum (<25*, 25-30, >30)
– Lower urinary tract symptoms (None*, Mild,
Moderate, Severe)
– Under treatment for cardiac symptoms (No*, Yes)
– Under treatment for COPD (No*, Yes)
* Baseline group for dummy variables
12. Example - ED in Older Dutch Men
Predictor b sb Adjusted OR (95% CI)
Age 55-59 (vs 50-54) 0.83 0.42 2.3 (1.0 – 5.2)
Age 60-64 (vs 50-54) 1.53 0.40 4.6 (2.1 – 10.1)
Age 65-69 (vs 50-54) 2.19 0.40 8.9 (4.1 – 19.5)
Age 70-78 (vs 50-54) 2.66 0.41 14.3 (6.4 – 32.1)
Smoker (vs nonsmoker) 0.47 0.19 1.6 (1.1 – 2.3)
BMI 25-30 (vs <25) 0.41 0.21 1.5 (1.0 – 2.3)
BMI >30 (vs <25) 1.10 0.29 3.0 (1.7 – 5.4)
LUTS Mild (vs None) 0.59 0.41 1.8 (0.8 – 4.3)
LUTS Moderate (vs None) 1.22 0.45 3.4 (1.4 – 8.4)
LUTS Severe (vs None) 2.01 0.56 7.5 (2.5 – 22.5)
Cardiac symptoms (Yes vs No) 0.92 0.26 2.5 (1.5 – 4.3)
COPD (Yes vs No) 0.64 0.28 1.9 (1.1 – 3.6)
Interpretations: Risk of ED appears to be:
• Increasing with age, BMI, and LUTS strata
• Higher among smokers
• Higher among men being treated for cardiac or COPD
13. Loglinear Models with
Categorical Variables
• Logistic regression models when there is a clear
response variable (Y), and a set of predictor
variables (X1,...,Xk)
• In some situations, the variables are all responses,
and there are no clear dependent and independent
variables
• Loglinear models are to correlation analysis as
logistic regression is to ordinary linear regression
14. Loglinear Models
• Example: 3 variables (X,Y,Z) each with 2 levels
• Can be set up in a 2x2x2 contingency table
• Hierarchy of Models:
– All variables are conditionally independent
– Two of the pairs of variables are conditionally
independent
– One of the pairs are conditionally independent
– No pairs are conditionally independent, but each
association is constant across levels of third variable
(no interaction or homogeneous association)
– All pairs are associated, and associations differ
among levels of third variable
15. Loglinear Models
• To determine associations, must have a measure: the
odds ratio (OR)
• Odds Ratios take on the value 1 if there is no
association
• Loglinear models make use of regressions with
coefficients being exponents. Thus, tests of whether
odds ratios are 1, is equivalently to testing whether
regression coefficients are 0 (as in logistic regression)
• For a given partial table, OR=e, software packages
estimate and test whether =0
16. Example - Feminine Traits/Behavior
3 Variables, each at 2 levels (Table contains observed counts):
Feminine Personality Trait (Modern/Traditional)
Female Role Behavior (Modern/Traditional)
Class (Lower Classman/Upper Classman)
*
C
3
5
8
1
3
4
4
8
2
9
3
2
0
5
5
9
8
7
M
T
P
T
M
T
P
T
C
L
U
d
i t
E B
o t
17. Example - Feminine Traits/Behavior
• Expected cell counts under model that allows for association
among all pairs of variables, but no interaction (association
between personality and role is same for each class, etc).
Model:(PR,PC,RC)
– Evidence of personality/role association (see odds ratios)
Class=Lower Class=Lower Class=Upper Class=Upper
Role=M Role=T Role=M Role=T
Personality=M 34.1 23.9 17.9 14.1
Personality=T 19.9 54.1 11.1 33.9
88
.
3
)
1
.
11
(
1
.
14
)
9
.
33
(
9
.
17
:
Upper
Class
88
.
3
)
9
.
19
(
9
.
23
)
1
.
54
(
1
.
34
:
Lower
Class
OR
OR
Note that under the no
interaction model, the odds
ratios measuring the
personality/role association
is same for each class
19. Example - Feminine Traits/Behavior
• Intuitive Results:
– Controlling for class in school, there is an
association between personality trait and role
behavior (ORLower=ORUpper=3.88)
– Controlling for role behavior there is no association
between personality trait and class (ORModern=
ORTraditional=1.06)
– Controlling for personality trait, there is no
association between role behavior and class
(ORModern= ORTraditional=1.12)
20. SPSS Output
• Statistical software packages fit regression type models, where
the regression coefficients for each model term are the log of the
odds ratio for that term, so that the estimated odds ratio is e
raised to the power of the regression coefficient.
Note: e1.3554 = 3.88 e.0605 = 1.06 e.1166 = 1.12
Parameter Estimates
Asymptotic 95% CI
Parameter Estimate SE Z-value Lower Upper
Constant 3.5234 .1651 21.35 3.20 3.85
Class .4674 .2050 2.28 .07 .87
Personality -.8774 .2726 -3.22 -1.41 -.34
Role -1.1166 .2873 -3.89 -1.68 -.55
C*P .0605 .3064 .20 -.54 .66
C*R .1166 .3107 .38 -.49 .73
R*P 1.3554 .2987 4.54 .77 1.94
21. Interpreting Coefficients
• The regression coefficients for each variable
corresponds to the lowest level (in alphanumeric
ordering of symbols). Computer output will print a
“mapping” of coefficients to variable levels.
Cell (C,P,R) Class Prsnlty Role C*P C*R P*R Expected Count
L,M,M 0.4674 -0.8774 -1.1166 0.0605 0.1166 1.3554 34.1
L,M,T 0.4674 -0.8774 0 0.0605 0 0 23.9
L,T,M 0.4674 0 -1.1166 0 0.1166 0 19.9
L,T,T 0.4674 0 0 0 0 0 54.1
U,M,M 0 -0.8774 -1.1166 0 0 1.3554 17.9
U,M,T 0 -0.8774 0 0 0 0 14.1
U,T,M 0 0 -1.1166 0 0 0 11.1
U,T,T 0 0 0 0 0 0 33.9
To obtain the expected cell counts, add the constant (3.5234) to
each of the s for that row, and raise e to the power of that sum
22. Goodness of Fit Statistics
• For any logit or loglinear model, we will have
contingency tables of observed (fo) and
expected (fe) cell counts under the model being
fit.
• Two statistics are used to test whether a model
is appropriate: the Pearson chi-square statistic
and the likelihood ratio (aka Deviance) statistic
e
o
o
e
e
o
f
f
f
G
f
f
f
log
2
:
Ratio
-
Likelihood
)
(
:
square
-
Chi
Pearson
2
2
2
23. Goodness of Fit Tests
• Null hypothesis: The current model is
appropriate
• Alternative hypothesis: Model is more complex
• Degrees of Freedom: Number of sample logits-
Number of parameters in model
• Distribution of Goodness of Fit statistics under
the null hypothesis is chi-square with degrees of
freedom given above
• Statistical software packages will print these
statistics and P-values.
25. Example - Feminine Traits/Behavior
Goodness of fit statistics/tests for all possible models:
Model G2
2
df P-value (G2
) P-value (2
)
(C,P,R) 22.21 22.46 4 .0002 .0002
(C,PR) 0.7199 0.7232 3 .8685 .8677
(P,CR) 21.99 22.24 3 .00007 .00006
(R,CP) 22.04 22.34 3 .00006 .00006
(CR,CP) 22.93 22.13 2 .00002 .00002
(CP,PR) 0.6024 0.6106 2 .7399 .7369
(CR,PR) 0.5047 0.5085 2 .7770 .7755
(CP,CR,PR) 0.4644 0.4695 1 .4946 .4932
The simplest model for which we fail to reject the null
hypothesis that the model is adequate is: (C,PR): Personality
and Role are the only associated pair.
26. Adjusted Residuals
• Standardized differences between actual and
expected counts (fo-fe, divided by its
standard error).
• Large adjusted residuals (bigger than 3 in
absolute value, is a conservative rule of
thumb) are cells that show lack of fit of
current model
• Software packages will print these for logit
and loglinear models
27. Example - Feminine Traits/Behavior
• Adjusted residuals for (C,P,R) model of all
pairs being conditionally independent:
Adj.
Factor Value Resid. Resid.
PRSNALTY Modern
ROLEBHVR Modern
CLASS1 Lower Classman 10.43 3.04**
CLASS1 Upper Classman 5.83 1.99
ROLEBHVR Traditional
CLASS1 Lower Classman -9.27 -2.46
CLASS1 Upper Classman -6.99 -2.11
PRSNALTY Traditional
ROLEBHVR Modern
CLASS1 Lower Classman -8.85 -2.42
CLASS1 Upper Classman -7.41 -2.32
ROLEBHVR Traditional
CLASS1 Lower Classman 7.69 1.93
CLASS1 Upper Classman 8.57 2.41
28. Comparing Models with G2 Statistic
• Comparing a series of models that increase in
complexity.
• Take the difference in the deviance (G2) for the
models (less complex model minus more
complex model)
• Take the difference in degrees of freedom for the
models
• Under hypothesis that less complex (reduced)
model is adequate, difference follows chi-square
distribution
29. Example - Feminine Traits/Behavior
• Comparing a model where only Personality
and Role are associated (Reduced Model)
with the model where all pairs are associated
with no interaction (Full Model).
• Reduced Model (C,PR): G2=.7232, df=3
• Full Model (CP,CR,PR): G2=.4695, df=1
• Difference: .7232-.4695=.2537, df=3-1=2
• Critical value (=0.05): 5.99
• Conclude Reduced Model is adequate
30. Logit Models for Ordinal Responses
• Response variable is ordinal (categorical
with natural ordering)
• Predictor variable(s) can be numeric or
qualitative (dummy variables)
• Labeling the ordinal categories from 1
(lowest level) to c (highest), can obtain the
cumulative probabilities:
c
j
j
Y
P
Y
P
j
Y
P ,
,
1
)
(
)
1
(
)
(
31. Logistic Regression for Ordinal Response
• The odds of falling in category j or below:
1
)
(
1
,
,
1
)
(
)
(
c
Y
P
c
j
j
Y
P
j
Y
P
• Logit (log odds) of cumulative probabilities are modeled
as linear functions of predictor variable(s):
1
,
,
1
)
(
)
(
log
)
(
logit
c
j
X
j
Y
P
j
Y
P
j
Y
P j
This is called the proportional odds model, and assumes the
effect of X is the same for each cumulative probability
33. SPSS Output
• Note that SPSS fits the model in the
following form:
1
,
,
1
)
(
)
(
log
)
(
logit
c
j
X
j
Y
P
j
Y
P
j
Y
P j
r E
7
2
3
1
0
7
8
5
4
0
1
0
0
1
1
3
0
1
3
3
0
0 a
.
.
0
.
.
.
[ A
[ A
T
[ R
[ R
L
m
E
a
d f
S i g
r B
r B
d e
L
T
a
Note that the race variable is not significant (or even close).
34. Fitted Equation
• The fitted equation for each group/category:
165
.
0
0
165
.
0
)
Nonwhite
|
2
(
)
Nonwhite
|
2
(
logit
:
te
Mod/Nonwhi
or
Neg
166
.
0
)
001
.
0
(
165
.
0
)
White
|
2
(
)
White
|
2
(
logit
:
Mod/White
or
Neg
027
.
1
)
0
(
027
.
1
)
Nonwhite
|
1
(
)
Nonwhite
|
1
(
logit
:
onwhite
Negative/N
026
.
1
)
001
.
0
(
027
.
1
)
White
|
1
(
)
White
|
1
(
logit
:
hite
Negative/W
Y
P
Y
P
Y
P
Y
P
Y
P
Y
P
Y
P
Y
P
For each group, the fitted probability of falling in that set of categories
is eL/(1+eL) where L is the logit value (0.264,0.264,0.541,0.541)
35. Inference for Regression Coefficients
• If = 0, the response (Y) is independent of X
• Z-test can be conducted to test this (estimate
divided by its standard error)
• Most software will conduct the Wald test, with the
statistic being the z-statistic squared, which has a
chi-squared distribution with 1 degree of freedom
under the null hypothesis
• Odds ratio of increasing X by 1 unit and its
confidence interval are obtained by raising e to the
power of the regression coefficient and its upper
and lower bounds
36. Example - Urban Renewal Attitudes
• Z-statistic for testing for race differences:
Z=0.001/0.133 = 0.0075 (recall model estimates -)
• Wald statistic: .000 (P-value=.993)
• Estimated odds ratio: e.001 = 1.001
• 95% Confidence Interval: (e-.260,e.263)=(0.771,1.301)
• Interval contains 1, odds of being in a given category or
below is same for whites as nonwhites
r E
7
2
3
1
0
7
8
5
4
0
1
0
0
1
1
3
0
1
3
3
0
0 a
.
.
0
.
.
.
[ A
[ A
T
[ R
[ R
L
m
E
a
d f
S i g
r B
r B
d e
L
T
a
37. Ordinal Predictors
• Creating dummy variables for ordinal
categories treats them as if nominal
• To make an ordinal variable, create a new
variable X that models the levels of the
ordinal variable
• Setting depends on assignment of levels
(simplest form is to let X=1,...,c for the
categories which treats levels as being
equally spaced)