This document presents a new statistical approach for analyzing longitudinal changes in amyloid PET scans from clinical trials. It proposes using a linear regression model (the "Δ-model") relating changes in target and reference region SUVs (ΔT and ΔR), rather than the standard SUV ratio (SUVr), to more powerfully detect treatment effects. The Δ-model performed better than ΔSUVr at detecting progression in a Phase 2 amyloid therapy trial (BLAZE) but not an Alzheimer's cohort (ADNI), likely due to different data characteristics between the studies. Simulations show the Δ-model has higher power than ΔSUVr to detect treatment effects using parameters from BLAZE, but similar power using ADNI parameters.
The document discusses various measures used to describe the dispersion or variability in a data set. It defines dispersion as the extent to which values in a distribution differ from the average. Several measures of dispersion are described, including range, interquartile range, mean deviation, and standard deviation. The document also discusses measures of relative standing like percentiles and quartiles, and how they can locate the position of observations within a data set. The learning objectives are to understand how to describe variability, compare distributions, describe relative standing, and understand the shape of distributions using these measures.
The document provides objectives and instructions for calculating standard deviation, variance, and student's t-test. It defines standard deviation as the positive square root of the arithmetic mean of the squared deviations from the mean. Standard deviation is considered the most reliable measure of variability. Variance is defined as the square of the standard deviation. Student's t-test is used to compare means of two samples and determine if they are statistically different. The document provides examples of calculating standard deviation, variance, and performing matched pairs and independent samples t-tests on sets of data.
The chi-square test is used to determine if an observed frequency distribution differs from an expected theoretical distribution. It can test goodness of fit, independence of attributes, and homogeneity. The test involves calculating chi-square by taking the sum of the squares of the differences between observed and expected frequencies divided by expected frequencies. For the test to be valid, certain conditions must be met regarding sample size, expected frequencies, independence, and randomness. The test has some limitations such as not measuring strength of association and being unreliable with small expected frequencies.
This document discusses various measures of dispersion used in statistics to quantify how spread out or varied a set of data values are. It defines dispersion as the state of being dispersed or spread out, and explains that measures of dispersion help interpret the variability in data by showing how squeezed or scattered the values are. The document then describes several common measures of absolute and relative dispersion, including range, quartile deviation, mean deviation, standard deviation, and coefficient of variation. For each measure, it provides a definition and formula to calculate it from a raw data set.
This document provides an introduction to the statistical concept of kurtosis. It defines kurtosis as a measure of the peakedness of a distribution that indicates how concentrated data is around the mean. There are three main types of kurtosis: leptokurtic distributions have higher peaks; platykurtic have lower peaks; and mesokurtic have normal peaks. Methods for calculating kurtosis include percentile measures and measures based on statistical moments. An example calculation demonstrates a leptokurtic distribution with a kurtosis value greater than 3. SPSS syntax for computing kurtosis from data is also presented.
This document provides an overview of key concepts in inferential statistics, including distributions, the normal distribution, the central limit theorem, estimators and estimates, confidence intervals, the Student's t-distribution, and formulas for calculating confidence intervals. It defines key terms and concepts, provides examples to illustrate statistical distributions and properties, and outlines the general formulas used to construct confidence intervals for different sampling situations.
The document discusses various measures used to describe the dispersion or variability in a data set. It defines dispersion as the extent to which values in a distribution differ from the average. Several measures of dispersion are described, including range, interquartile range, mean deviation, and standard deviation. The document also discusses measures of relative standing like percentiles and quartiles, and how they can locate the position of observations within a data set. The learning objectives are to understand how to describe variability, compare distributions, describe relative standing, and understand the shape of distributions using these measures.
The document provides objectives and instructions for calculating standard deviation, variance, and student's t-test. It defines standard deviation as the positive square root of the arithmetic mean of the squared deviations from the mean. Standard deviation is considered the most reliable measure of variability. Variance is defined as the square of the standard deviation. Student's t-test is used to compare means of two samples and determine if they are statistically different. The document provides examples of calculating standard deviation, variance, and performing matched pairs and independent samples t-tests on sets of data.
The chi-square test is used to determine if an observed frequency distribution differs from an expected theoretical distribution. It can test goodness of fit, independence of attributes, and homogeneity. The test involves calculating chi-square by taking the sum of the squares of the differences between observed and expected frequencies divided by expected frequencies. For the test to be valid, certain conditions must be met regarding sample size, expected frequencies, independence, and randomness. The test has some limitations such as not measuring strength of association and being unreliable with small expected frequencies.
This document discusses various measures of dispersion used in statistics to quantify how spread out or varied a set of data values are. It defines dispersion as the state of being dispersed or spread out, and explains that measures of dispersion help interpret the variability in data by showing how squeezed or scattered the values are. The document then describes several common measures of absolute and relative dispersion, including range, quartile deviation, mean deviation, standard deviation, and coefficient of variation. For each measure, it provides a definition and formula to calculate it from a raw data set.
This document provides an introduction to the statistical concept of kurtosis. It defines kurtosis as a measure of the peakedness of a distribution that indicates how concentrated data is around the mean. There are three main types of kurtosis: leptokurtic distributions have higher peaks; platykurtic have lower peaks; and mesokurtic have normal peaks. Methods for calculating kurtosis include percentile measures and measures based on statistical moments. An example calculation demonstrates a leptokurtic distribution with a kurtosis value greater than 3. SPSS syntax for computing kurtosis from data is also presented.
This document provides an overview of key concepts in inferential statistics, including distributions, the normal distribution, the central limit theorem, estimators and estimates, confidence intervals, the Student's t-distribution, and formulas for calculating confidence intervals. It defines key terms and concepts, provides examples to illustrate statistical distributions and properties, and outlines the general formulas used to construct confidence intervals for different sampling situations.
The document discusses various measures of central tendency and dispersion used in statistical analysis. It defines measures of central tendency like arithmetic mean, median and mode, and provides their formulas and properties. It also discusses measures of dispersion such as range, mean deviation, standard deviation, variance and their characteristics. The document provides examples and steps to calculate various averages and measures of dispersion for a given data set.
Analysis of Variance (ANOVA) is a generalized statistical
technique used to analyze sample variances to obtain information on comparing multiple
population means.
The document discusses basic statistical concepts for analyzing environmental data. It defines key terms like frequency distribution, measures of central tendency (mean, median, mode), standard deviation, and normal distribution. It also discusses the precision and accuracy of experimental data. Precision refers to the reproducibility of results and can be expressed through terms like average deviation, range, and standard deviation. Accuracy considers both determinate errors from issues like improper calibration and indeterminate random errors from small uncertainties that cumulatively can impact results.
The document discusses basic statistical concepts used to analyze environmental data. It provides an example of a frequency distribution based on 44 replicate analyses of water hardness. The results are classified into ranges and the number in each range is calculated. The mean, median, mode, and standard deviation are defined as measures of central tendency. Standard deviation measures how spread out the data is from the mean. Most large data sets conform to a normal distribution curve. The document also discusses precision, accuracy, and how to calculate the propagation of errors when taking sums, differences, products and quotients of data that each have an associated standard deviation.
PG STAT 531 Lecture 3 Graphical and Diagrammatic Representation of DataAashish Patel
The document discusses various methods of graphically and diagrammatically representing statistical data, including:
1) Bar diagrams, pie charts, and line graphs that use bars, circles, or lines to show relationships between data points;
2) Histograms that use rectangles to show frequency distributions; and
3) Frequency polygons and curves that smooth data points to reveal trends, and ogives that show cumulative frequencies. Graphical representations make trends and relationships easier for experts and non-experts to understand versus numerical representations alone.
This document discusses concepts related to sampling and sampling distributions. It begins with definitions of key terms like population, sample, parameter, and statistic. It then explains different sampling methods, focusing on simple random sampling. Different measures of central tendency and variability are outlined like mean, median, mode, range, variance, and standard deviation. The central limit theorem is introduced, which states that the sampling distribution of the mean will approximate a normal distribution for large sample sizes regardless of the population distribution. Examples are provided to illustrate these concepts.
This document provides an overview of key concepts in statistics for engineers and scientists. It discusses parameters and statistics, which are characteristics of populations and samples respectively. It then covers various measures of central tendency (mean, median, mode) and how to calculate them. It also discusses measures of variability such as range, variance, standard deviation, and coefficient of variation. Various distribution shapes are presented. Examples are provided to demonstrate calculating statistics like the mean, median, variance and coefficient of variation. The document aims to describe fundamental statistical concepts and calculations.
This document discusses various measures of dispersion in statistics including range, mean deviation, variance, and standard deviation. It provides definitions and formulas for calculating each measure along with examples using both ungrouped and grouped frequency distribution data. Box-and-whisker plots are also introduced as a graphical method to display the five number summary of a data set including minimum, quartiles, and maximum values.
The document discusses goodness-of-fit tests for categorical data. It introduces notation for categorical variables with multiple categories and hypotheses for goodness-of-fit tests. Expected counts are calculated based on hypothesized proportions. The chi-square statistic is used to calculate test statistics and P-values are found using the chi-square distribution. Examples demonstrate applying goodness-of-fit tests to determine if variable categories occur with equal frequency.
This document provides an overview of one-way analysis of variance (ANOVA). It defines ANOVA, explains its assumptions and steps, and provides an example to illustrate its use. Specifically:
1) ANOVA is used to compare the means of three or more groups and determine if they differ significantly. It partitions variance into between-groups and within-groups components.
2) The key assumptions are normality, homogeneity of variance, and independence of observations.
3) Steps include establishing a significance level, calculating an F-statistic to compare between-group and within-group variance, and determining whether results are statistically significant.
This document provides an introduction to statistics. It defines key statistical concepts such as descriptive statistics, inferential statistics, populations, samples, variables, and different types of data. It also discusses methods for organizing and summarizing data, including frequency distributions, histograms, frequency polygons, ogives, time series graphs and pie charts. The goal of statistics is to collect, organize, analyze and draw conclusions from data.
This document discusses measures of central tendency and dispersion. It begins by defining measures of central tendency as statistical measures that describe the position of a distribution. The most commonly used measures of central tendency for a univariate context are the mean, median, and mode. The document then discusses the arithmetic mean in detail, including how to calculate the mean for individual, discrete, and continuous data series using direct and shortcut methods. It also covers the geometric mean and how to calculate it using logarithms for individual, discrete, and continuous data series. Various examples and practice problems are provided.
ANOVA analysis was conducted to compare the effectiveness of 4 teaching methods on student grades. The analysis found a significant difference between the methods (F=79.61678, p<0.01), with Method 4 being most effective. A second ANOVA compared acceptability of luncheon meat from 3 sources using 20 panelists, finding significant differences between sources (F=99.59873, p<0.01) and panelists (F=5.605096, p<0.01).
Descriptive statistics offer nurse researchers valuable options for analysing and pre-senting large and complex sets of data, suggests Christine Hallett
This document outlines key concepts in descriptive statistics including measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and measures of shape (skewness, kurtosis). It defines these terms and concepts, provides examples of how to compute them, and explains how to interpret and compare distributions based on these measures. The learning objectives are to understand and be able to calculate various descriptive statistics and use them to analyze and describe data distributions.
PG STAT 531 Lecture 2 Descriptive statisticsAashish Patel
This document provides an overview of descriptive statistics. It discusses that descriptive statistics are used to describe basic features of data through simple summaries, without drawing inferences. The document outlines various measures of central tendency like mean, median and mode. It also discusses measures of dispersion such as range, variance and standard deviation that describe how spread out the data is. The key purpose of descriptive statistics is to present quantitative data in a simplified and manageable form.
Small sample theory deals with statistical inference when sample sizes are small (n ≤ 30). It involves t and F distributions which are defined in terms of degrees of freedom. The t-distribution was developed by William Gosset and is used when sample sizes are small. It has a bell shape but is more spread out than the normal distribution. The F-distribution is used to test if two variances are equal and is defined as the ratio of two chi-square variables. Both distributions depend on degrees of freedom.
1) The document presents information on different types of t-tests including the single sample t-test, independent sample t-test, and dependent/paired sample t-test. Equations and examples are provided for each.
2) The single sample t-test compares the mean of a sample to a hypothesized population mean. The independent t-test compares the means of two independent samples. The dependent t-test compares the means of two related samples, such as pre-and post-test scores.
3) A z-test is also discussed and compared to t-tests. The z-test is used when the population standard deviation is known and sample sizes are large, while t-tests are used
This document provides an overview of analysis of variance (ANOVA). It discusses two-way ANOVA and the design of experiments (DOE) including completely randomized design (CRD) and randomized block design (RBD). CRD is the simplest design where treatments are randomly allocated without blocking. RBD uses blocking to reduce experimental error by making comparisons only between treatments within the same block. The document provides formulas and examples for calculating ANOVA tables for one-way and two-way ANOVA to test for differences between sample means.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.2: Measures of Variation
The document provides an overview of the Goods and Services Tax (GST) system that is proposed to be implemented in India. It discusses what GST is, the need for GST to replace existing tax structures, the justification for GST at central and state levels, the proposed dual GST model, key features of GST including coverage, tax rates, registration requirements, invoices, and periodic tax payments. It also addresses taxes that may be subsumed under GST, treatment of exports and imports, inter-state transactions, and emerging issues related to implementation.
I have learned several TOEFL test-taking skills in class this month, such as how to answer questions. I've also gained the ability to overcome obstacles in my life. Additionally, I've picked up skimming and scanning techniques to spend less time on exercises.
The document discusses various measures of central tendency and dispersion used in statistical analysis. It defines measures of central tendency like arithmetic mean, median and mode, and provides their formulas and properties. It also discusses measures of dispersion such as range, mean deviation, standard deviation, variance and their characteristics. The document provides examples and steps to calculate various averages and measures of dispersion for a given data set.
Analysis of Variance (ANOVA) is a generalized statistical
technique used to analyze sample variances to obtain information on comparing multiple
population means.
The document discusses basic statistical concepts for analyzing environmental data. It defines key terms like frequency distribution, measures of central tendency (mean, median, mode), standard deviation, and normal distribution. It also discusses the precision and accuracy of experimental data. Precision refers to the reproducibility of results and can be expressed through terms like average deviation, range, and standard deviation. Accuracy considers both determinate errors from issues like improper calibration and indeterminate random errors from small uncertainties that cumulatively can impact results.
The document discusses basic statistical concepts used to analyze environmental data. It provides an example of a frequency distribution based on 44 replicate analyses of water hardness. The results are classified into ranges and the number in each range is calculated. The mean, median, mode, and standard deviation are defined as measures of central tendency. Standard deviation measures how spread out the data is from the mean. Most large data sets conform to a normal distribution curve. The document also discusses precision, accuracy, and how to calculate the propagation of errors when taking sums, differences, products and quotients of data that each have an associated standard deviation.
PG STAT 531 Lecture 3 Graphical and Diagrammatic Representation of DataAashish Patel
The document discusses various methods of graphically and diagrammatically representing statistical data, including:
1) Bar diagrams, pie charts, and line graphs that use bars, circles, or lines to show relationships between data points;
2) Histograms that use rectangles to show frequency distributions; and
3) Frequency polygons and curves that smooth data points to reveal trends, and ogives that show cumulative frequencies. Graphical representations make trends and relationships easier for experts and non-experts to understand versus numerical representations alone.
This document discusses concepts related to sampling and sampling distributions. It begins with definitions of key terms like population, sample, parameter, and statistic. It then explains different sampling methods, focusing on simple random sampling. Different measures of central tendency and variability are outlined like mean, median, mode, range, variance, and standard deviation. The central limit theorem is introduced, which states that the sampling distribution of the mean will approximate a normal distribution for large sample sizes regardless of the population distribution. Examples are provided to illustrate these concepts.
This document provides an overview of key concepts in statistics for engineers and scientists. It discusses parameters and statistics, which are characteristics of populations and samples respectively. It then covers various measures of central tendency (mean, median, mode) and how to calculate them. It also discusses measures of variability such as range, variance, standard deviation, and coefficient of variation. Various distribution shapes are presented. Examples are provided to demonstrate calculating statistics like the mean, median, variance and coefficient of variation. The document aims to describe fundamental statistical concepts and calculations.
This document discusses various measures of dispersion in statistics including range, mean deviation, variance, and standard deviation. It provides definitions and formulas for calculating each measure along with examples using both ungrouped and grouped frequency distribution data. Box-and-whisker plots are also introduced as a graphical method to display the five number summary of a data set including minimum, quartiles, and maximum values.
The document discusses goodness-of-fit tests for categorical data. It introduces notation for categorical variables with multiple categories and hypotheses for goodness-of-fit tests. Expected counts are calculated based on hypothesized proportions. The chi-square statistic is used to calculate test statistics and P-values are found using the chi-square distribution. Examples demonstrate applying goodness-of-fit tests to determine if variable categories occur with equal frequency.
This document provides an overview of one-way analysis of variance (ANOVA). It defines ANOVA, explains its assumptions and steps, and provides an example to illustrate its use. Specifically:
1) ANOVA is used to compare the means of three or more groups and determine if they differ significantly. It partitions variance into between-groups and within-groups components.
2) The key assumptions are normality, homogeneity of variance, and independence of observations.
3) Steps include establishing a significance level, calculating an F-statistic to compare between-group and within-group variance, and determining whether results are statistically significant.
This document provides an introduction to statistics. It defines key statistical concepts such as descriptive statistics, inferential statistics, populations, samples, variables, and different types of data. It also discusses methods for organizing and summarizing data, including frequency distributions, histograms, frequency polygons, ogives, time series graphs and pie charts. The goal of statistics is to collect, organize, analyze and draw conclusions from data.
This document discusses measures of central tendency and dispersion. It begins by defining measures of central tendency as statistical measures that describe the position of a distribution. The most commonly used measures of central tendency for a univariate context are the mean, median, and mode. The document then discusses the arithmetic mean in detail, including how to calculate the mean for individual, discrete, and continuous data series using direct and shortcut methods. It also covers the geometric mean and how to calculate it using logarithms for individual, discrete, and continuous data series. Various examples and practice problems are provided.
ANOVA analysis was conducted to compare the effectiveness of 4 teaching methods on student grades. The analysis found a significant difference between the methods (F=79.61678, p<0.01), with Method 4 being most effective. A second ANOVA compared acceptability of luncheon meat from 3 sources using 20 panelists, finding significant differences between sources (F=99.59873, p<0.01) and panelists (F=5.605096, p<0.01).
Descriptive statistics offer nurse researchers valuable options for analysing and pre-senting large and complex sets of data, suggests Christine Hallett
This document outlines key concepts in descriptive statistics including measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and measures of shape (skewness, kurtosis). It defines these terms and concepts, provides examples of how to compute them, and explains how to interpret and compare distributions based on these measures. The learning objectives are to understand and be able to calculate various descriptive statistics and use them to analyze and describe data distributions.
PG STAT 531 Lecture 2 Descriptive statisticsAashish Patel
This document provides an overview of descriptive statistics. It discusses that descriptive statistics are used to describe basic features of data through simple summaries, without drawing inferences. The document outlines various measures of central tendency like mean, median and mode. It also discusses measures of dispersion such as range, variance and standard deviation that describe how spread out the data is. The key purpose of descriptive statistics is to present quantitative data in a simplified and manageable form.
Small sample theory deals with statistical inference when sample sizes are small (n ≤ 30). It involves t and F distributions which are defined in terms of degrees of freedom. The t-distribution was developed by William Gosset and is used when sample sizes are small. It has a bell shape but is more spread out than the normal distribution. The F-distribution is used to test if two variances are equal and is defined as the ratio of two chi-square variables. Both distributions depend on degrees of freedom.
1) The document presents information on different types of t-tests including the single sample t-test, independent sample t-test, and dependent/paired sample t-test. Equations and examples are provided for each.
2) The single sample t-test compares the mean of a sample to a hypothesized population mean. The independent t-test compares the means of two independent samples. The dependent t-test compares the means of two related samples, such as pre-and post-test scores.
3) A z-test is also discussed and compared to t-tests. The z-test is used when the population standard deviation is known and sample sizes are large, while t-tests are used
This document provides an overview of analysis of variance (ANOVA). It discusses two-way ANOVA and the design of experiments (DOE) including completely randomized design (CRD) and randomized block design (RBD). CRD is the simplest design where treatments are randomly allocated without blocking. RBD uses blocking to reduce experimental error by making comparisons only between treatments within the same block. The document provides formulas and examples for calculating ANOVA tables for one-way and two-way ANOVA to test for differences between sample means.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.2: Measures of Variation
The document provides an overview of the Goods and Services Tax (GST) system that is proposed to be implemented in India. It discusses what GST is, the need for GST to replace existing tax structures, the justification for GST at central and state levels, the proposed dual GST model, key features of GST including coverage, tax rates, registration requirements, invoices, and periodic tax payments. It also addresses taxes that may be subsumed under GST, treatment of exports and imports, inter-state transactions, and emerging issues related to implementation.
I have learned several TOEFL test-taking skills in class this month, such as how to answer questions. I've also gained the ability to overcome obstacles in my life. Additionally, I've picked up skimming and scanning techniques to spend less time on exercises.
El documento proporciona instrucciones para el uso seguro de una pulidora angular, advirtiendo que instalar incorrectamente el disco puede causar lesiones por partículas sueltas. Recomienda seleccionar cuidadosamente el disco apropiado para el material, asegurar el trabajo antes de iniciar, y mantener un ángulo agudo y control al pulir para una operación eficiente.
El Puerto de la Libertad es uno de los destinos turísticos más representativos del país, con playas y un clima perfectos para los turistas. Fue habilitado en 1824 y hoy en día el turismo se concentra en el Malecón, un complejo turístico renovado recientemente. El puerto recibe un 70% de turismo nacional y un 30% extranjero, y ofrece restaurantes de mariscos, artesanías locales y vendedores en el muelle.
Este documento proporciona una introducción a los dispositivos lógicos programables (PLDs), incluidos los FPGA. Explica que los PLDs son circuitos integrados reconfigurables cuyas conexiones pueden ser programadas por el usuario para construir circuitos digitales. Luego compara diferentes tipos de PLDs como SPLDs, CPLDs y FPGAs, señalando que los FPGAs son los más avanzados con matrices de bloques lógicos programables. Finalmente, enumera algunas aplicaciones comunes de los FPGAs como lóg
El documento define varios tipos de virus informáticos y fraudes, incluyendo malware, spyware, phishing, adware, gusanos, y backdoors. También explica conceptos como antivirus, cortafuegos, crackers, y denegación de servicio. Finalmente, proporciona definiciones breves de términos técnicos comunes relacionados con la seguridad informática.
La WebQuest es una propuesta metodológica de investigación guiada que utiliza recursos de Internet. Se centra en tareas con múltiples resultados posibles que requieren pensamiento crítico. Consiste en una introducción, tarea, procesos, recursos, evaluación y conclusión.
Presentación Estadística y sus Términos Básicos Oliver Ramirez
Desarrollo de la Presentación:
•Definición, Tipos y Ejemplo de Variable.
◦Definición y Ejemplo de Población y Muestra.
•Definición y Ejemplo de Parámetros Estadísticos.
• Definición, Tipos y Ejemplo de Escalas de Medición.
• Definición y Ejemplo de Sumatoria Razón, Proporción, Tasa y Frecuencia.
Este documento describe un procedimiento para determinar la mezcla de carbonatos y bicarbonatos en el agua mineral mediante titulación acidimétrica. Se mide una muestra de agua mineral y se titula primero con HCl hasta el punto final del indicador fenolftaleína, anotando el volumen gastado V1, que corresponde a la mitad del carbonato. Luego se agrega indicador anaranjado de metilo y se continúa la titulación hasta el punto final, anotando el volumen V2, que valora el bicarbonato inicial más el formado
The most successful green projects are those that share the successful design strategies implemented, and are embraced by the community as an instrument for education and a source of inspiration for other building professionals.
This is the case for Gordon Estates; the most highly rated and certified green subdivision in the U.S., and recently the site of the G Street education series ‘Green Home 101’. Kudos to the teams at the City of Phoenix and Mandalay Communities for completing this first of its kind project and sharing a successful roadmap for others to follow.
About G Street education: G Street was contracted by the Municipality and the Builder to create and present CEU approved curriculum that shared the project’s sustainable design strategies, and to increase local and national awareness. When sustainable design is done right it includes education, allowing the project to reach its climax.
University teaching as epistemic fluency: Frames, conceptual blending and ex...Lina Markauskaite
University teaching as epistemic fluency: Frames, conceptual blending and experiential resources in teacher pedagogical and ICT choices, by Lina Markauskaite
Presented during sabbatical at University of Berkeley, Glasgow Caledonian Academy and Sheffield University during 2011 April-May
La práctica describe la preparación y estandarización de soluciones de tiosulfato de sodio y yodo utilizadas en métodos yodométricos de óxido-reducción. Se pesan cristales de tiosulfato de sodio pentahidratado para preparar una solución 0.1N, y se estandariza mediante la titulación con permanganato de potasio de una solución de yodo liberado por reacción. Las conclusiones discuten la aplicación de los métodos redox y la influencia del potencial en la reacción de valoración.
This presentation was delivered @ Open Source India Conference 2016 Bangalore on Oct 21st 2016. The idea was to introduce the concept very light way - not for advanced users :-)
Este documento describe tres especies de animales en peligro de extinción en Perú: 1) La Pava Aliblanca, un ave endémica de los bosques secos de la costa norte de Perú con menos de 250 especímenes restantes en estado salvaje. 2) El Mono Choro de cola amarilla, un mono endémico de los Andes peruanos con menos de 250 especímenes también. 3) La Rana del Lago Titicaca, una rana acuática única que no necesita pulmones ni branquias y realiza los intercambios
Este documento describe la anatomía de la columna vertebral humana. Se compone principalmente de vértebras y discos intervertebrales que protegen la médula espinal y soportan el peso del cuerpo. La columna vertebral se divide en cinco regiones (cervical, torácica, lumbar, sacra y coccis) que varían en tamaño, forma y detalles de las vértebras. Cada región permite diferentes tipos y rangos de movimiento.
- 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 provides an overview of linear regression analysis. It discusses (1) why regression is used, including for description, adjustment for covariates, identifying predictors, and prediction; (2) the basics of linear regression in predicting an interval outcome variable based on predictor variables; and (3) how to conduct univariate linear regression in SPSS, including interpreting results and ensuring assumptions are met. Key assumptions include no outliers, independent data points, normally distributed residuals with constant variance.
2.0.statistical methods and determination of sample sizesalummkata1
statistical methods and determination of sample size
These guidelines focus on the validation of the bioanalytical methods generating quantitative concentration data used for pharmacokinetic and toxicokinetic parameter determinations.
The t-test is used to test hypotheses about population means when the population variance is unknown. It is closely related to the z-test but uses the t distribution instead of the normal. There are three main types of t-tests: single sample, independent samples, and dependent samples. The t-test compares the sample mean to the population mean and takes into account factors like sample size and variability. Larger sample sizes and stronger associations between variables increase the power of the t-test to detect significant differences or relationships.
Statistics is the science of dealing with numbers and data. It involves collecting, summarizing, presenting, and analyzing data. There are four main steps: data collection, summarization by removing unwanted data and classifying/tabulating, presentation with diagrams/graphs/tables, and analysis using measures like average, dispersion, and correlation. Descriptive statistics summarize and describe data, while inferential statistics allow generalizing from samples to populations. Common descriptive statistics include measures of central tendency (mean, median, mode), variability (range, variance, standard deviation), and distribution properties. Inferential statistics techniques like hypothesis testing and ANOVA are used to make inferences about populations based on samples.
This document provides an introduction to biostatistics. It defines biostatistics as the branch of statistics dealing with biological data. It discusses different types of data, methods of data presentation including tables, charts and graphs. It also covers measures of central tendency and dispersion, sampling methods, tests of significance including chi-square test and t-test, and correlation and regression. The overall purpose is to introduce basic statistical concepts and methods used for analyzing health and medical data.
The document discusses correlation, regression, and hypothesis testing involving two variables. It defines correlation and the correlation coefficient r, which measures the strength of a linear relationship between two variables. Regression analyzes the relationship between variables to determine if it is positive/negative and linear/nonlinear. Hypothesis tests using r evaluate whether a linear correlation exists between two variables in a population. Confidence intervals and predictions can be made from significant relationships.
This document provides information about various statistical analysis techniques used in biology, including definitions of median, mode, mean, range, and standard deviation. It discusses how to calculate standard deviation using a graphing calculator. It also covers comparing data sets, significant vs. non-significant differences, using t-tests to evaluate differences between populations, and types of correlations between variables.
This document provides information about statistical tests that can be used to make inferences when comparing two samples or populations. Specifically, it discusses:
- Tests for comparing two proportions, means, variances or standard deviations from independent and dependent samples using z-tests, t-tests and F-tests.
- The assumptions and procedures for each test, including how to determine critical values and calculate test statistics.
- Examples of how to perform hypothesis tests and construct confidence intervals for various statistical comparisons between two samples or populations using a TI calculator.
1. Statistical analysis involves collecting, organizing, analyzing data, and drawing inferences about populations based on samples. It includes both descriptive and inferential statistics.
2. The document defines key terms used in statistical analysis like population, sample, statistical analysis, and discusses various statistical measures like mean, median, mode, interquartile range, and standard deviation.
3. The purposes of statistical analysis are outlined as measuring relationships, making predictions, testing hypotheses, and summarizing results. Both parametric and non-parametric statistical analyses are discussed.
Treatment comparisons in clinical trials with Covariates analysis of diastoli...Dr.Govind Nidigattu
This analysis compared the effects of a new drug (Treatment A) and placebo (Treatment B) on lowering diastolic blood pressure (DBP) in clinical trials. DBP was measured at baseline and monthly for 4 months. Treatment A decreased DBP by an average of 15 mmHG, more than the 5 mmHG decrease for Treatment B. Statistical tests found this 10.4 mmHG difference to be highly statistically significant. While some covariates like age, sex, and their interactions were significant predictors of DBP, the best fitting regression model showed Treatment and Age as the only statistically significant factors in lowering DBP.
This document provides an introduction to the t-statistic, which is used to test hypotheses about population means when the population standard deviation is unknown. It describes how the t-statistic is calculated using the sample standard deviation rather than the unknown population standard deviation. It also explains that the t-distribution, which the t-statistic is compared to, depends on the degrees of freedom and becomes closer to a normal distribution as the degrees of freedom increase. The document outlines the four-step process for a hypothesis test using the t-statistic and describes how effect size can be estimated.
The document discusses basic statistical concepts used to analyze environmental data. It provides an example of a frequency distribution based on 44 replicate analyses of water hardness. The data are classified into ranges and the number of values in each range are used to calculate the frequency. Central tendencies like the mean, median, and mode are defined. Standard deviation is described as a measure of how data points are clustered around the mean. The concept of normal distribution is introduced. Precision is defined as the reproducibility of results and accuracy as the closeness to the accepted value. Methods to calculate and express precision both absolutely and relatively are presented. The propagation of errors when results involve sums, differences, products and quotients is demonstrated through examples.
Chi-square is a non-parametric test used to compare observed data with expected data. It can test goodness of fit, independence of attributes, and homogeneity. The document provides an introduction to chi-square terms and calculations including contingency tables, expected and observed frequencies, degrees of freedom, and test steps. Examples demonstrate applying chi-square to test the effectiveness of chloroquine and inoculation. Both examples find the null hypothesis of no effect can be rejected, indicating the treatments were effective.
Biostatistics is the science of collecting, summarizing, analyzing, and interpreting data in the fields of medicine, biology, and public health. It involves both descriptive and inferential statistics. Descriptive statistics summarize data through measures of central tendency like mean, median, and mode, and measures of dispersion like range and standard deviation. Inferential statistics allow generalization from samples to populations through techniques like hypothesis testing, confidence intervals, and estimation. Sample size determination and random sampling help ensure validity and minimize errors in statistical analyses.
The document defines various statistical measures and types of statistical analysis. It discusses descriptive statistical measures like mean, median, mode, and interquartile range. It also covers inferential statistical tests like the t-test, z-test, ANOVA, chi-square test, Wilcoxon signed rank test, Mann-Whitney U test, and Kruskal-Wallis test. It explains their purposes, assumptions, formulas, and examples of their applications in statistical analysis.
Isotonic Regression is a statistical technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible. Isotonic Regression is limited to predicting numeric output so the dependent variable must be numeric in nature…
MSC III_Research Methodology and Statistics_Inferrential ststistics.pdfSuchita Rawat
This document discusses various statistical measures of dispersion and relationships. It defines dispersion as describing how spread out a set of data is, and lists common measures including range, variance, standard deviation, and interquartile range. It also discusses relative measures that allow comparison between datasets, and measures of relationships like covariance and correlation that indicate the strength and direction of relationships between variables. Finally, it provides formulas and explanations of common statistical tests like t-tests, chi-square tests, ANOVA, and simple and multiple linear regression analyses.
- The sample mean is the best estimate of the population mean and can be used to construct confidence intervals to estimate the true population mean.
- There are two situations when estimating a population mean: when the population standard deviation (σ) is known, and when σ is unknown.
- When σ is known, a z-test is used. When σ is unknown, a t-test is used since the sample standard deviation is used to estimate the population standard deviation.
This document provides an overview of key concepts in statistics for quantitative analysis, including:
- Statistics are mathematical tools used to describe and make judgments about data. The type of statistics discussed assumes data has a normal (bell-shaped) distribution.
- The normal distribution is characterized by a mean (μ) and standard deviation (σ or s). Standard deviation quantifies the spread of data around the mean.
- Common statistical tests covered include confidence intervals, comparing a measured value to a known value using a t-test, and comparing means of two data sets using an F-test and t-test.
- The F-test determines if the standard deviations of two data sets are significantly different before using
1. BACKGROUND
• Several recent clinical trials for amyloid-targeted therapies have used
florbetapir-PET to measure fibrillar amyloid burden.
• e de facto metric used in these trials: longitudinal change in SUVr, the
ratio of SUV values in a cortical target region to that of a disease-free
reference region.
• Recent reports indicate that such measurements improve when using a
reference region consisting of subcortical white matter, rather than a
region entirely in the hindbrain.
• Regardless, SUVr suffers from an inherent statistical problem: the
asymmetric property of ratios when the denominator contains
uncharacterized noise.
• Also, the numerator contains additive components: the true signal of
interest (binding to fibrillar amyloid) + nonspecific binding, and
normalizing to a reference region assumed to have similar binding
properties as the cortex is therefore a poor approximation to measuring
the signal of interest.
• We propose an empirically-motivated and intuitive linear data model
relating target- and reference signals with greater statistical power (than
SUVr) for detecting treatment effects on the target signal.
DATA
1. BLAZE: Phase 2 trial of crenezumab; mild-to-moderate AD
(MMSE>17); N=30 placebo, N=61 treatment; florbetapir-PET at baseline
and 69 weeks; all randomized subjects were assessed as amyloid-positive
by visual read; SUV measurements using PMOD AAL template with gray
matter masks from baseline T1 MRI.
2. ADNI: AD group (N=40); florbetapir-PET at baseline and 2 years; SUV
measurements using FreeSurfer method performed by the UC Berkeley
core lab, available on the LONI web site.
VARIABLE DEFINITIONS & NOTATION
• Ti(t) and Ri(t) are mean SUV’s of target and reference regions,
respectively, for patient i at visit t
• ΔTi = Ti (t2) − Ti (t1) is difference between values of T at visits t1 (baseline)
and t2 (follow-up); similar definitions for ΔR and ΔSUVr.
• α and β denote the intercept and slope parameters, respectively; ε is a
zero-mean residual of the linear regression, with standard deviation σε; Z
is the within-patient effect, with standard deviation σz
• SUVr(t) = T(t) ∕R(t)
Detecting treatment effects in clinical trials with florbetapir-PET: An alternative statistical approach to SUVr
Funan Shi1,2, Thomas Bengtsson1, David Clayton1, Peter Bickel2
1Genentech Inc., 2University of California, Berkeley
HAIC 2015 P14
FIG 5: The power to detect a simulated treatment effect reducing
progression from t1 to t2 by 50% as a function of σε (colors) and
σZ (x-axis). σε=.045, σZ=.1 in BLAZE (left) and σε=.065 , σZ=.25 in
ADNI (right).
METHODS
• ree dominant data-features for (all) combinations of target and
reference regions were observed in both BLAZE and ADNI:
1. Plots of Ri(t) vs. Ti(t) show strong linear relationships (Fig 2).
2. Plots of ΔRi vs. ΔTi show strong linear relationships (Fig 3).
3. Residuals from the regressions of T(t1) on R(t1) and T(t2) on R(t2)
are highly correlated (Fig 4), implying a strong longitudinal within-
patient effect.
DISCUSSION
• Under the data structure present in BLAZE and ADNI, the linear
regression based Δ-model provides a more statistically powerful
alternative to gauge amyloid accumulation in multi-center trials.
• e Δ-model has a clear advantage for detecting progression and
treatment effect over ΔSUVr in data with parameters motivated by
BLAZE, but this advantage is not present for parameters suggested
by ADNI.
• From theoretical calculations (not shown here) and simulations, we
observe 3 parameters that dictate the relative performances of the
two methods: σz, σε, and CVR; in particular: CVR(BLAZE) = .3,
while CVR(ADNI) = .1
• e Δ-model provides a more flexible framework; e.g., 1) to
incorporate predictors such as age, gender, cognitive scores, and 2)
to simultaneously evaluate treatment and progression at multiple
time points.
CONCLUSION
• Describing longitudinal changes in target SUV through a linear
regression framework allows for statistical inference with power
equal to or greater than that detectable through corresponding
changes on SUVr.
A NEW APPROACH TO ASSESSING
LONGITUDINAL CHANGES ON PET
• e data features of Figs 1 – 3 led us to an alternative approach to
describing longitudinal changes in the specific binding component of
T using simple linear regression techniques.
• We note that the empirical relationship between Ti(t) and Ri(t) is
easily expressed by the following linear model:
Ti(t) = α(t) + βRi(t) + Zi + εi(t)
- α(t) represents a specific binding component of the target signal
Ti(t) which remains unexplained by the reference signal Ri(t)
- β is a proportionality constant relating Ri(t) and Ti(t)
- zi is a longitudinally persistent patient level effect
- εi(t) is a random zero-mean error term.
• e above suggests that longitudinal changes in the target signal can
be assessed by testing if the intercept has changed over time: i.e. with
Δα = α(t2) - α(t1), we test Ho: Δα = 0, vs. Ha: Δα ≠ 0.
• Removing the statistically deleterious effects of Zi, the above
hypothesis is most efficiently modeled by regressing changes in the
target on changes in the reference region (cf. Fig 3), i.e., by fitting
ΔTi = Δα + βΔRi + √2εi
- We term this approach the Δ-model.
• Δα represents the expected group-level change in target binding when
there is no change in the reference uptake (i.e. when ΔR=0).
- is parameter is the proposed alternative to group-level mean
differences in ΔSUVr = T(t2) ∕R(t2) −T(t1) ∕R(t1)
FIG 2: Ri(t) vs.Ti(t). SUVs from SWM plotted vs. Frontal Cortex
(left BLAZE placebo cohort; right-ADNI); all data at baseline.
FIG 4: The patient level longitudinal effect (Zi; cf. equation 2).
Empirical residuals at t1, t2 from the linear regression of Frontal
Cortex SUVs on SWM SUVs (left-BLAZE placebo; right ADNI).
FIG 3: ΔRi vs. ΔTi . Linear change in SUV in SWM versus
change in Frontal Cortex (left-BLAZE placebo; right ADNI)
non-specific
binding
specific
binding
bloodflow
non-specific
binding
bloodflow
FIG 1: Signal decomposition in target and reference regions
based on two-compartment model
Target region ≈ R(t) + T(t)
Reference region ≈ R(t)
TABLE 2: Detecting Progression in ADNI (baseline to week 104).
• e preceding observations agree with intuitive reasoning based on
compartmental models of tracer binding (Fig 1) in which T and R are
both proportional to non-specific binding. us, targets and reference
region SUVs should be linearly proportional.
RESULTS
• Across various target regions, using p-values, we compared the Δ-
model with ΔSUVr for the BLAZE (Table 1) and ADNI (Table 2)
data. Subcortical White Matter was used in all analyses.
• As seen, in BLAZE, assuming progression is present, compared to
ΔSUVr, the Δ-model is be more sensitive to detecting an increase in
the target signal from baseline. However, for the ADNI cohort, this
observation is not recapitulated.
DETECTING TREATMENT EFFECTS
• Using simulations we compare the power of the Δ-model and
ΔSUVr to detect treatment effects.
• e simulated data was generated as follows: pairs Ri(t1), Ri(t2) are
bootstrapped from BLAZE/ADNI, and, with parameters set to
empirically motivated values suggested by BLAZE/ADNI, target
SUV data is generated at times t1 and t2 using the models
Ti(t1) = α(t1) + βRi(t1) + Zi + εi(t1)
Ti(t2) = α(t2) − δ(TX) + βRi(t2) + Zi + εi(t2)
- α(t1) = .02 and α(t2) = .05 (representing progression)
- δ(TX) = .015 for patients in the treatment arm; 0 for controls
- β = .8
- Zi ~ N(0, σZ) and εi(t) ~N(0, σε)
- 2:1 randomization with Ntx= 100 and Nct= 50.
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−0.5 0.0 0.5 1.0
−0.4−0.20.00.20.40.60.8
BLAZE
Change in Tar vs Change in Ref bwn entry and followup
Tar: Frontal, Ref: SWM
ΔR = R2 − R1
ΔT=T2−T1
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−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
−0.20.00.20.4
ADNI
Change in Tar vs Change in Ref btw entry and followup
Tar: Frontal, Ref: SWM
ΔR = R2 − R1
ΔT=T2−T1
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1.0 1.5 2.0
0.60.81.01.21.41.61.8
BLAZE
Tar vs Ref @ entry scan
Tar: Frontal, Ref: SWM
R1
T1
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1.6 1.8 2.0 2.2 2.4 2.6
1.01.21.41.61.82.0
ADNI
Tar vs Ref @ entry scan
Tar: Frontal, Ref: SWM
R1
T1
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−0.2 −0.1 0.0 0.1
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Tar~Ref Residuals @ entry vs Residuals @ followup
Tar: Frontal, Ref: SWM
Residual @ entry scan
Residual@followup
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−0.6 −0.4 −0.2 0.0 0.2
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Tar~Ref Residuals @ entry vs Residuals @ follwup
Tar: Frontal, Ref: SWM
Residual @ entry scan
Residual@followup
R(t) T(t)
R(t)
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.20.40.60.81.0
Power Curves for Detecting TX Effect by Parametric Boostrapping BLAZE Data
50%Treatment Effect
σZ
Power
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σε=0.01
σε=0.03
σε=0.045
Δ−model
SUVr
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power @ BLAZE parameters using Δ−model
power @ BLAZE
parameters using SUVr
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.20.40.60.81.0
Power Curves for Detecting TX Effect by Parametric Boostrapping ADNI Data
50%Treatment Effect
σZ
Power
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σε=0.01
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SUVr
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power @ ADNI parameters using Δ−model
power @ ADNI parameters using SUVr
Genentech Research and Early Development |
Tar ROIs ↵.pval SUV r.pval
frontal 0.09 0.03
cingulate 0.19 0.11
parietal 0.08 0.01
temporal 0.69 0.94
Tar ROIs ↵.pval SUV r.pval
frontal 0.0006 0.0091
post cingulum 0.0530 0.0245
parietal 0.1950 0.1871
lateral tmpr 0.0009 0.0019
medial tmpr 0.0030 0.0104
orbitofrontal 0.2913 0.6180
occipital 0.0002 0.0000
ant cingulum 0.0518 0.0871
rectus 0.2796 0.6058
caudate 0.1304 0.1369
putamen 0.0001 0.0002
thalamus 0.3211 0.1317
Detecting(Progression(by(the(Two(Methods
1
ADNI((bl(to(w104)BLAZE(PLACEBO((bl(to(w47)
Designated(BLAZE(Targets
TABLE 1: Detecting Progression in BLAZE (baseline to week 47).
Tar ROIs ↵.pval SUV r.pval
frontal 0.09 0.03
cingulate 0.19 0.11
parietal 0.08 0.01
temporal 0.69 0.94
Tar ROIs ↵.pval SUV r.pval
frontal 0.0006 0.0091
post cingulum 0.0530 0.0245
parietal 0.1950 0.1871
lateral tmpr 0.0009 0.0019
medial tmpr 0.0030 0.0104
orbitofrontal 0.2913 0.6180
occipital 0.0002 0.0000
ant cingulum 0.0518 0.0871
Detecting(Progression(by(the(Two(Methods
ADNI((bl(to(w104)BLAZE(PLACEBO((bl(to(w47)
Designated(BLAZE(Targets