Analysis of Variance
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This document discusses Analysis of Variance (ANOVA), a statistical technique used to analyze the differences between group means and their associated procedures. It can separate observed variance into components, allowing for additional tests. ANOVA requires assumptions of normality, equal variances, and random sampling. It calculates sums of squares, mean squares, and uses the F-value and p-value to determine if group means are significantly different. Checks of the model adequacy, like R-squared values, should also be considered before interpreting the ANOVA results.
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1. You must post at least three times each week.
2. Your initial post is due Tuesday of each week and the following two post are due before Sunday.
3. All post must be on separate days of the week.
4. Post must be at least 150 words and cite all of your references even it its the book.
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Describe how the lives of students with disabilities from culturally and/or linguistically diverse backgrounds have changed since the advent of IDEA. What do you feel are some things that can or should be implemented to better assist with students that have disabilities? Tell me about these ideas and how would you integrate them?
ANOVA
ANOVA
• Analysis of Variance
• Statistical method to analyzes variances to determine if the means from more than
two populations are the same
• compare the between-sample-variation to the within-sample-variation
• If the between-sample-variation is sufficiently large compared to the within-sample-
variation it is likely that the population means are statistically different
• Compares means (group differences) among levels of factors. No
assumptions are made regarding how the factors are related
• Residual related assumptions are the same as with simple regression
• Explanatory variables can be qualitative or quantitative but are categorized
for group investigations. These variables are often referred to as factors
with levels (category levels)
ANOVA Assumptions
• Assume populations , from which the response values for the groups
are drawn, are normally distributed
• Assumes populations have equal variances
• Can compare the ratio of smallest and largest sample standard deviations.
Between .05 and 2 are typically not considered evidence of a violation
assumption
• Assumes the response data are independent
• For large sample sizes, or for factor level sample sizes that are equal,
the ANOVA test is robust to assumption violations of normality and
unequal variances
ANOVA and Variance
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• A factor is fixed if its levels are chosen before the ANOVA investigation
begins
• Difference in groups are only investigated for the specific pre-selected factors
and levels
• A factor is random if its levels are choosen randomly from the
population before the ANOVA investigation begins
Randomization
• Assigning subjects to treatment groups or treatments to subjects
randomly reduces the chance of bias selecting results
ANOVA hypotheses statements
One-way ANOVA
One-Way ANOVA
Hypotheses statements
Test statistic
=
𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
𝑊𝑖𝑡ℎ𝑖𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Under the null hypothesis both the between and within group variances estimate the
variance of the random error so the ratio is assumed to be close to 1.
Null Hypothesis
Alternate Hypothesis
One-Way ANOVA
One-Way ANOVA
One-Way ANOVA Excel Output
Treatme
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Discussion Forum Instructions:
1. You must post at least three times each week.
2. Your initial post is due Tuesday of each week and the following two post are due before Sunday.
3. All post must be on separate days of the week.
4. Post must be at least 150 words and cite all of your references even it its the book.
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Describe how the lives of students with disabilities from culturally and/or linguistically diverse backgrounds have changed since the advent of IDEA. What do you feel are some things that can or should be implemented to better assist with students that have disabilities? Tell me about these ideas and how would you integrate them?
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variation it is likely that the population means are statistically different
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assumptions are made regarding how the factors are related
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for group investigations. These variables are often referred to as factors
with levels (category levels)
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• Assume populations , from which the response values for the groups
are drawn, are normally distributed
• Assumes populations have equal variances
• Can compare the ratio of smallest and largest sample standard deviations.
Between .05 and 2 are typically not considered evidence of a violation
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• Assumes the response data are independent
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the ANOVA test is robust to assumption violations of normality and
unequal variances
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• A factor is fixed if its levels are chosen before the ANOVA investigation
begins
• Difference in groups are only investigated for the specific pre-selected factors
and levels
• A factor is random if its levels are choosen randomly from the
population before the ANOVA investigation begins
Randomization
• Assigning subjects to treatment groups or treatments to subjects
randomly reduces the chance of bias selecting results
ANOVA hypotheses statements
One-way ANOVA
One-Way ANOVA
Hypotheses statements
Test statistic
=
𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
𝑊𝑖𝑡ℎ𝑖𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Under the null hypothesis both the between and within group variances estimate the
variance of the random error so the ratio is assumed to be close to 1.
Null Hypothesis
Alternate Hypothesis
One-Way ANOVA
One-Way ANOVA
One-Way ANOVA Excel Output
Treatme
Analysis of Variance (ANOVA)
This document provides an overview of analysis of variance (ANOVA) techniques, including one-way and two-way ANOVA. It defines key terms like factors, interactions, F distribution, and multiple comparison tests. For one-way ANOVA, it explains how to test if three or more population means are equal. For two-way ANOVA, it notes you must first test for interactions between two factors before testing their individual effects. The Tukey test is introduced for identifying specifically which group means differ following rejection of a one-way ANOVA null hypothesis.
INFERENTIAL TECHNIQUES. Inferential Stat. pt 3
Inferential statistics are techniques that allow us to use these samples to make generalizations about the populations from which the samples were drawn. ... The methods of inferential statistics are (1) the estimation of parameter(s) and (2) testing of statistical hypotheses.
mean comparison.pptx
The document provides information on statistical techniques for comparing means between groups, including t-tests, analysis of variance (ANOVA), and their assumptions and applications. T-tests are used to compare two groups, while ANOVA allows comparison of three or more groups and controls for increased Type I error rates. Steps for conducting t-tests, ANOVA, and post-hoc tests using SPSS are outlined along with examples and interpretations.
mean comparison.pptx
The document discusses statistical methods for comparing means between groups, including t-tests and analysis of variance (ANOVA). It provides information on different types of t-tests (one sample, independent samples, and paired samples t-tests), assumptions of t-tests, and how to perform t-tests in SPSS. It also covers one-way ANOVA, including its assumptions, components of variation, properties of the F-test, and how to run a one-way ANOVA in SPSS. Examples are provided for each statistical test.
Data Processing and Statistical Treatment.pptx
This document discusses various topics in data processing and statistical treatment. It begins by explaining how data is categorized, coded, and tabulated. It then discusses the importance of statistical treatment and describes descriptive and inferential problems. Specific statistical tests and analyses are defined, including parametric vs non-parametric tests, measures of central tendency, variability, correlation, t-tests, and methods for comparing means. Examples of outputs like frequency tables are provided.
Research 101: Inferential Quantitative Analysis
This document provides an overview of quantitative inferential analysis techniques, including:
- Inferential statistics are used to test hypotheses and draw conclusions about populations based on sample data, using specialized software.
- Basic concepts include null and alternative hypotheses, significance levels, p-values, and that statistical inferences are probabilistic rather than deterministic.
- Common analysis techniques described are the general linear model, structural equation modeling, ANOVA for comparing groups, factorial designs, and other techniques like factor analysis, discriminant analysis, logistic regression, and path analysis.
anova.ppt
This document provides an overview of analysis of variance (ANOVA) and experimental design. It discusses parametric and nonparametric test procedures, with parametric procedures making stronger assumptions about the data like normal distributions. One-way and two-way ANOVA are introduced for analyzing experiments with one or two factors. Key concepts in ANOVA are partitioning total variation into variation due to treatments and random error, and using F-ratios to test if treatment means are significantly different. Examples show how to set up ANOVA in StatGraphics and evaluate the significance of main effects and interactions.
anova.ppt
This document provides an overview of analysis of variance (ANOVA) and experimental design. It discusses parametric and nonparametric test procedures, with parametric procedures making stronger assumptions about the data like normal distributions. One-way and two-way ANOVA are introduced for analyzing experiments with one or two factors. Key concepts in ANOVA are partitioning total variation into variation due to treatments and random error, and using F-ratios to test if treatment means are significantly different. Examples are provided to illustrate hypothesis testing and interpreting ANOVA results from StatGraphics software.
anova.ppt
This document provides an overview of analysis of variance (ANOVA) and experimental design. It discusses parametric and nonparametric test procedures, with parametric procedures making stronger assumptions about the data like normal distributions. One-way and two-way ANOVA are introduced for analyzing experiments with one or two factors. Key concepts in ANOVA are partitioning total variation into variation due to treatments and random error, and using F-ratios to test if treatment means are significantly different. Examples show how to set up ANOVA in StatGraphics and evaluate the results.
One_-ANOVA.ppt
Here are the steps to analyze this data using one-way ANOVA:
1. State the null and alternative hypotheses:
H0: There is no difference in perception between the three bank sectors
H1: There is a difference in perception between at least one of the bank sectors
2. Conduct a one-way ANOVA with bank sector (public, private, foreign) as the independent variable and perception score as the dependent variable.
3. Check that the assumptions of ANOVA are met (normality, homogeneity of variances)
4. Report the ANOVA table results, including F-value and p-value.
5. Make a conclusion based on the p-value:
- If p
Basic of Statistical Inference Part-V: Types of Hypothesis Test (Parametric)
The fifth part of the basic of statistical inference series is concerned with explaining different kinds of parametric tests.
Analysis of Variance
This document provides an introduction to analysis of variance (ANOVA). It discusses previous statistical tests learned, the logic and calculations of ANOVA, and examples of hypothesis testing using ANOVA. ANOVA allows comparison of three or more population means using their sample means. It partitions total variability into two components - variability between groups and variability within groups. The F-ratio compares the two and is evaluated to determine if there are statistically significant differences between population means. Post hoc tests are used after a significant F-ratio to determine exactly which group means differ.
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Welability of Carbon and alloy steel -2.pptx
This document discusses the weldability of carbon and alloy steels. It covers two main topics: 1) Cleanliness of the weld is important to prevent inclusions and porosity, and protection of the base metal and proper groove geometry and solidification conditions are also important. 2) The properties of the heat-affected zone (HAZ) depend on factors like hardenability, toughness, and structural hardness, which can be affected by the carbon equivalent and alloying element concentrations. Carbon equivalent values above 0.7 generally indicate materials that are not suitable for welding without preheat. Common alloying elements like manganese, silicon, nickel, chromium, molybdenum, and vanadium are discussed in
Welability of Carbon and alloy steel.pptx
This document discusses the weldability of carbon and alloy steels. It states that weldability is affected by various parameters related to the steels, including the process used, the purpose of the welding, and fabrication conditions. It lists some key parameters that influence weldability, such as oxidation tendency, chemical composition, mechanical properties, and physical properties. The document also notes that carbon steels are an alloy of iron and carbon, with carbon occupying interstitial spaces in the alloy. It indicates that high carbon steels typically contain 0.5-0.8% carbon.
Weldability of Specific Material.pptx
The document discusses the weldability of stainless steel. Weldability refers to the ability of materials, usually metals and alloys, to be welded together. Stainless steel contains at least 12% chromium which forms a thin protective oxide layer, preventing corrosion. There are different types of stainless steel including ferritic, austenitic, and martensitic. Austenitic stainless steel is widely used due to its ductility, toughness, and corrosion resistance but requires 5-10% ferrite content to prevent solidification cracking during welding. The Schaeffler diagram is used to evaluate stainless steel compositions. Slow cooling in the heat affected zone can cause chromium carbide formation and weld decay.
Gas and other Welding Shailesh Dewangan.pptx
This document discusses various welding and metal joining processes. It describes gas welding (oxy-acetylene welding) where a flame is used to heat and join metal parts. Resistance welding uses electric current to heat and join metals under pressure. Solid state welding techniques like friction and ultrasonic welding join materials without melting them. The document also covers soldering and brazing which use filler metals to join parts through capillary action without melting the base metals.
Arc Welding Shailesh.pptx
This document discusses various welding processes, including arc welding. It describes arc welding as the most common metal joining method that uses an electric arc between an electrode and base metal to generate heat and join metals. The document categorizes common welding processes and provides examples, such as gas metal arc welding, tungsten inert gas welding, submerged arc welding, electro-slag welding, and atomic hydrogen welding. It provides brief descriptions and applications for some of these processes.
Hot Working and Cold working Operation.pptx
The document discusses the differences between hot working and cold working of metals. Hot working involves deforming metals above their recrystallization temperature but below their melting point, allowing deformation and recovery to occur simultaneously. Cold working deforms metals below the recrystallization temperature, resulting in work hardening rather than recovery. Hot working reduces required stresses and improves properties, while cold working increases strength and hardness but reduces ductility. Both methods have advantages and disadvantages depending on the desired final properties and shape of the metal.
Cutting Fluids.pptx
Cutting fluids, also known as coolants or lubricants, are liquids applied to cutting tools and workpieces during machining operations to provide cooling and lubrication. There are three main types of cutting fluids: cutting oils based on mineral or vegetable oils, emulsifiable oils which are oil-water emulsions, and chemical or synthetic fluids which are water-based solutions containing dissolved chemicals. Cutting fluids lower machining temperatures, reduce friction and tool wear, improve surface finish, and increase tool life. They are applied through nozzles, flooding, misting, or through the tool itself depending on the machining application and facilities.
Numerical Problem.pptx
- An orthogonal cutting operation is performed on aluminum with a cutting speed of 5.6 m/s, feed of 0.25 mm/rev, and depth of cut of 2.0 mm.
- The lathe has a mechanical efficiency of 0.85.
- The specific energy values for aluminum need to be determined to calculate the power required and power drawn from the lathe.
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Centrigugal casting Shailesh.pptx
Centrifugal casting is a casting process that uses centrifugal force to distribute molten metal evenly throughout a rotating mold. There are three main types: true centrifugal casting produces hollow cylindrical parts like pipes; semi-centrifugal casting fills the entire mold; and centrifuge casting produces multiple parts in a single mold fed by a central sprue. Centrifugal casting provides benefits like reduced need for coring, easier impurity removal, lighter equipment, and near 100% yield. However, it is limited to axially symmetrical shapes and requires machining of inner diameters.
Bias variance dichotomy
This document discusses bias-variance dichotomy in machine learning. It defines bias as the difference between predicted and actual values, and variance as how much predictions scatter relative to each other. High bias means underfitting data while high variance means overfitting. The goal is to minimize both for the best model. Examples show that low bias and low variance leads to the best accuracy on both training and testing data, while high bias or variance for either leads to worse accuracy on testing data even if training accuracy is high.
Engineering Economy Costing and Accounting
This document provides an overview of the course Engineering Economy taught by Dr. Shailesh Dewangan at BIT Mesra. The course covers topics related to time value of money, including simple and compound interest, cash flows, interest rates, and economic equivalence. It defines key terms and concepts and provides examples to illustrate time value of money calculations. The document also discusses how inflation impacts interest rates and economic decisions. References for further reading on engineering economy and time value of money are listed at the end.
Safety audit and risk assessment
This document discusses safety audits and risk assessments. It defines a safety audit as a systematic examination of an organization's health and safety systems to identify issues and recommend corrective actions. There are two types of audits: internal and external. The objectives of a safety audit are to evaluate safety performance, verify regulatory compliance, and advise on improvements. Risk assessment involves evaluating risks to determine if they are tolerable or not. It may lead to policies to reduce intolerable risks through elimination, substitution, or procedural controls. The document provides examples of using a risk matrix to rate risks based on likelihood and consequence.
Hazard and Risk Assessment
1. The document discusses techniques for hazard identification and risk assessment, including plant safety inspections, job safety analysis, accident investigations, safety audits, and hazard operability studies.
2. Plant safety inspections aim to identify unsafe acts or conditions for both workers and machinery by examining housekeeping, material handling, guarding of machines, lighting, and use of personal protective equipment.
3. Job safety analysis breaks jobs down into steps to identify hazards at each step and develop safety measures to eliminate hazards.
Characteristics of Maintenance
Dr. Shailesh Kumar Dewangan discusses maintenance engineering concepts including maintenance times, frequency, and costs. There are different types of maintenance times such as corrective, preventive, and administrative/logistic delay times. Maintenance frequency is characterized by metrics like mean time between maintenance (MTBM) and mean time between repair (MTBR), with the main objective being to maximize MTBR. MTBM is calculated based on scheduled (preventive) and unscheduled (corrective) maintenance times. Maintainability is represented by a function relating the probability of repair time. Maintenance costs include labor and materials to keep equipment operational.
Maintenance engineering Introduction
The document discusses maintenance engineering and its objectives. Maintenance engineering aims to optimize equipment, procedures, and budgets to improve maintainability, reliability, and availability. Its key objectives are to maximize equipment performance, prevent failures and breakdowns, increase system reliability, minimize interruptions, and prevent safety hazards. Maintainability, reliability, availability, and supportability are important factors that maintenance engineering considers and aims to improve through preventive and corrective maintenance practices. The product life cycle involves four phases - concept development, validation, production, and operation - that maintenance engineering and maintainability influence.
Fault Tree analysis
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TIME DIVISION MULTIPLEXING TECHNIQUE FOR COMMUNICATION SYSTEM
Time Division Multiplexing (TDM) is a method of transmitting multiple signals over a single communication channel by dividing the signal into many segments, each having a very short duration of time. These time slots are then allocated to different data streams, allowing multiple signals to share the same transmission medium efficiently. TDM is widely used in telecommunications and data communication systems.
### How TDM Works
1. **Time Slots Allocation**: The core principle of TDM is to assign distinct time slots to each signal. During each time slot, the respective signal is transmitted, and then the process repeats cyclically. For example, if there are four signals to be transmitted, the TDM cycle will divide time into four slots, each assigned to one signal.
2. **Synchronization**: Synchronization is crucial in TDM systems to ensure that the signals are correctly aligned with their respective time slots. Both the transmitter and receiver must be synchronized to avoid any overlap or loss of data. This synchronization is typically maintained by a clock signal that ensures time slots are accurately aligned.
3. **Frame Structure**: TDM data is organized into frames, where each frame consists of a set of time slots. Each frame is repeated at regular intervals, ensuring continuous transmission of data streams. The frame structure helps in managing the data streams and maintaining the synchronization between the transmitter and receiver.
4. **Multiplexer and Demultiplexer**: At the transmitting end, a multiplexer combines multiple input signals into a single composite signal by assigning each signal to a specific time slot. At the receiving end, a demultiplexer separates the composite signal back into individual signals based on their respective time slots.
### Types of TDM
1. **Synchronous TDM**: In synchronous TDM, time slots are pre-assigned to each signal, regardless of whether the signal has data to transmit or not. This can lead to inefficiencies if some time slots remain empty due to the absence of data.
2. **Asynchronous TDM (or Statistical TDM)**: Asynchronous TDM addresses the inefficiencies of synchronous TDM by allocating time slots dynamically based on the presence of data. Time slots are assigned only when there is data to transmit, which optimizes the use of the communication channel.
### Applications of TDM
- **Telecommunications**: TDM is extensively used in telecommunication systems, such as in T1 and E1 lines, where multiple telephone calls are transmitted over a single line by assigning each call to a specific time slot.
- **Digital Audio and Video Broadcasting**: TDM is used in broadcasting systems to transmit multiple audio or video streams over a single channel, ensuring efficient use of bandwidth.
- **Computer Networks**: TDM is used in network protocols and systems to manage the transmission of data from multiple sources over a single network medium.
### Advantages of TDM
- **Efficient Use of Bandwidth**: TDM all
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The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
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the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
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The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
A review on techniques and modelling methodologies used for checking electrom...
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
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Analysis of Variance
- 1. Dr. Shailesh Kumar Dewangan Assistant Professor Department of Production Engineering, BIT Mesra Ranchi Analysis of Variance (ANOVA) ANOVA
- 2. 2 Analysis of Variance (ANOVA) ANOVA is a fundamental step in the Design of Experiment, which is a dominant statistical tool aimed at statistically quantifying interactions between independent variables through their methodical modifications to determine their impact on the predicted variables. The ANOVA pre requires the following assumptions: • The treatment data must be normally distributed, • The variance must be the same for all treatments, • All samples are randomly selected. SS Regression = σ ො 𝑦 − 𝑦 2 SS Error = σ 𝑦 − ො 𝑦 2 SS Total = SS Regression + SS Error y = observed response, ˆy = fitted response, and y = mean response. ANOVA, is a statistical method that separates observed variance data into different components to use for additional tests.
- 3. 3 Types of ANOVAAnalysis • Dependent Variable – Analysis of variance must have a dependent variable that is continuous. ... • Independent Variable – ANOVA must have one or more categorical independent variable like Sales promotion. ... • Null hypothesis – All means are equal. What is ANOVA formula? The Anova test is performed by comparing two types of variation, the variation between the sample means, as well as the variation within each of the samples. Formula represents one-way Anova test statistics: Alternatively, F = MST/MSE MST = SST/ p-1
- 4. 4 Degree of Freedom DOF: Indicates the number of independent pieces of information involving the response data needed to calculate the sum of squares. The degrees of freedom for each component of the model are: 𝑫𝑭 𝑹𝒆𝒈𝒓𝒆𝒔𝒔𝒊𝒐𝒏 = 𝒑 − 𝟏 𝑫𝑭 𝑬𝒓𝒓𝒐𝒓 = 𝒏 − 𝒑 𝑻𝒐𝒕𝒂𝒍 = 𝒏 − 𝟏 • where n = number of observations and p = number of terms in the model Mean Square: In an ANOVA, the term Mean Square refers to an estimate of the population variance based on the variability among a given set of measures. The calculation for the mean square for the model terms is: 𝑴𝑺 𝑻𝒆𝒓𝒎 = 𝑨𝒅𝒋 𝑺𝑺 𝑻𝒆𝒓𝒎 𝑫𝑭 𝑻𝒆𝒓𝒎
- 5. 5 F-value and P value of ANOVA F-value is the measurement of distance between individual distributions. As the F- value goes up, the P-value goes down. F is a test to determine whether the interaction and main effects are significant. The formula for the model terms is: 𝑭 = 𝑴𝑺 𝑻𝒆𝒓𝒎 𝑴𝑺 𝑬𝒓𝒓𝒐𝒓 The degrees of freedom for the test are: Numerator = degrees of freedom for term, Denominator = degrees of freedom for error Larger values of F support rejecting the null hypothesis that there is not a significant effect P-value is used in hypothesis tests to help you decide whether to reject or fail to reject a null hypothesis. The p-value is the probability of obtaining a test statistic that is at least as extreme as the actual calculated value, if the null hypothesis is true. A commonly used cut-off value for the p-value is 0.05. For example, if the calculated p-value of a test statistic is less than 0.05, you reject the null hypothesis.
- 6. 6 Model Adequacy Check Before the conclusions from the analysis of variance are adopted, the adequacy of the underlying model should be checked it is always necessary to • Examine the fitted model to ensure that it provides an adequate approximation to the true system; • Verify that none of the least squares regression assumptions are violated. Now we consider several techniques for checking model adequacy. Before the full model ANOVA, several R2 are presented. The ordinary R2 is 𝑹𝟐 = 𝑺𝑺𝒓𝒆𝒈𝒓𝒆𝒔𝒔𝒊𝒐𝒏 𝑺𝑺𝒕𝒐𝒕𝒂𝒍 • R2 (R-sq) Coefficient of determination; indicates how much variation in the response is explained by the model. The higher the R2, the better the model fits your data. The formula is: 𝑹𝟐 = 𝟏 − 𝑺𝑺 𝑬𝒓𝒓𝒐𝒓 𝑺𝑺 𝑻𝒐𝒕𝒂𝒍 Adjusted R2 accounts for the number of factors in your model. The formula is: 𝑹𝟐 = 𝟏 − 𝑴𝑺 (𝑬𝒓𝒓𝒐𝒓) 𝑺𝑺 𝑻𝒐𝒕𝒂𝒍/𝑫𝑭 𝑻𝒐𝒕𝒂𝒍
- 7. Thank You