Shalabh
shalab@iitk.ac.in
shalabh1@yahoo.com
Department of Mathematics & Statistics
Indian Institute of Technology Kanpur, Kanpur - 208016 (India)

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MTH 513A : Analysis of Variance

Classes will begin on 5 January 2022 and continue in ONLINE MODE unless announced for offline or hybrid mode.

Course Contents: Analysis of completely randomized design, randomized block design, Latin squares design; Split plot, 2ⁿ and 3ⁿ factorials with total and partial confounding, two-way non-orthogonal experiment, BIBD, PBIBD; Analysis of covariance, missing plot techniques; First and second order response surface designs.

Online Course Platform: Course can be accessed through online platform MooKit available at https://hello.iitk.ac.in .

Expected topics to be covered: Likelihood ratio test for general linear hypothesis; Test of hypothesis for one and more than one linear parametric functions; Likelihood ratio test in in one way model; analysis of variance in one way model; multiple comparison tests; Analysis of completely randomized, randomized block and Latin squares designs; missing plot techniques; General intrablock and interblock analysis of variance in Incomplete block designs; Balanced incomplete block design (BIBD); Intrablock analysis of variance in BIBD; Interblock analysis of variance in BIBD; Recovery of information in BIBD; Intrablock analysis of variance in partial balanced incomplete block design (PBIBD); 2ⁿ factorial experiments with total confounding, partial confounding and fractional replications; Analysis of covariance; Introduction to 3ⁿ factorials.

Books:

H. Scheffe: The Analysis of Variance, Wiley, 1961.

H. Toutenburg and Shalabh: Statistical Analysis of Designed Experiments, Springer 2009.

D. C. Montagomery: Design & Analysis of Experiments, 5th Edition, Wiley 2001(Low price edition is available).

Who can explain better than himself who invented - Enjoy the book by Sir RA Fisher - The Design of Experiments

Reference Books:

D. D. Joshi: Linear Estimation and Design of Experiments, Wiley Eastern, 1987.

George Casella: Statistical Design, Springer, 2008.

Max D. Morris: Design of Experiments- An Introduction Based on Linear Models, CRC Press, 2011.

N. Giri: Analysis of Variance, South Asian Publishers, New Delhi 1986.

H. Sahai and M.I. Ageel: The Analysis of Variance-Fixed, Random and Mixed Models, Springer, 2001.

Aloke Dey: Incomplete Block Design, Hindustan Book Agency 2010.

Grading scheme: Quiz: 30%, Assignments: 30% Mid semester examination: 20%, End semester examination: 20%.

Quiz conduct: Quizzes will be conducted through the MooKIT platform. If anyone has any internet issue, please inform. The pdf file of the quiz will be sent using appropriate mode. Preferable mode is the discussion forum on MooKit platform.

Class Schedule: Time table: Tuesday: 8:00-8:50, Wednesday: 10-10:50, Thursday: 8:00-8:50, Friday: 8:00-8:50,

We will meet on zoom every Wednesday from 10:00-10:50 AM (as of now) over Zoom to discuss and tutorial.

Another session to meet will be announced after the class depending upon the contents.

The zoom link will be forwarded through email. Better you install Zoom.

Announcements:

Classes will begin on 5 January 2022 and continue to hold in ONLINE MODE unless announced for offline or hybrid mode. First class will be on 5 January 22 at 10 AM.

1. The lecture notes and videos are prepared in detail assuming some students may not have studied statistics in B.Sc.

2. Details about quiz will be shared through email.

3. All the students need to access the video lectures only from the MooKIT platform.

4. Relevant notes and slides are provided inside the MooKIT platform.

5. A Google upload link for the submission of all the assignments will be shared over email.

6. Assignments are to be done using only R software, wherever required.

7. The lecture slides in pdf format, lecture notes in pdf format and videos are available in MooKit.

8. The recordings of online classes will be uploaded on youtube link which will be shared after the class.

Note: The grading scheme may change due to COVID-19 pandemic situation. It will be changed in consultation with the students.

Contact hours: 24 X 7, by email, phone, what's app. (If possible and not so urgent, avoid calling between 12-9 AM.) Please raise the course related queries inside the MooKit platform under "Forums" only.

Assignments

Note: Please submit the codes, commands, screenshot of output and interpretations along with the text output in a single file. The link for online submission of the assignments has been sent to the students through email.

Feel in Class

Lecture notes for your help (If you find any typo, please let me know)

For the course MTH513 A, Lecture Notes 1 are only for the quick revision.

In other notes, try to follow only the content that is covered in the video lectures. The notes may have more contents for completeness. The required notes are uploaded in the MooKit platform.

Lecture Notes 1 : Results on Linear Algebra, Matrix Theory and Distributions

Lecture Notes 2 : General Linear Hypothesis and Analysis of Variance

Lecture Notes 3 : Experimental Design Models

Lecture Notes 4 : Experimental Designs and Their Analysis

Lecture Notes 5 : Incomplete Block Designs

Lecture Notes 6 : Balanced Incomplete Block Design (BIBD)

Lecture Notes 7 : Partially Balanced Incomplete Block Design (PBIBD)

Lecture Notes 8 : Factorial Experiment

Lecture Notes 9 : Confounding

Lecture Notes 10 : Partial confounding

Lecture Notes 11 : Fractional Replications

Lecture Notes 12 : Analysis of Covariance

The video lectures are also available at Swayam Prabha DTH Channel 16 YouTube (Click here).

Slides and Videos used in the lectures:

Video Lecture links	Lecture Slides download links	Brief Description	Lecture Title	Lecture No.
Click here Lecture 1	Click here Lecture 1	Results from Matrix Theory and Random Variables	Vectors and Matrices	1
Click here Lecture 2	Click here Lecture 2	Results from Matrix Theory and Random Variables	Random Vectors and Linear Estimation	2
Click here Lecture 3	Click here Lecture 3	General Linear Hypothesis and Analysis of Variance	Regression and Analysis of Variance Models	3
Click here Lecture 4	Click here Lecture 4	General Linear Hypothesis and Analysis of Variance	ANOVA models and Least Squares Estimation of Parameters	4
Click here Lecture 5	Click here Lecture 5	General Linear Hypothesis and Analysis of Variance	Least squares and Maximum Likelihood Estimation of Parameters	5
Click here Lecture 6	Click here Lecture 6	General Linear Hypothesis and Analysis of Variance	Test of Hypothesis for Equality of Parameters	6
Click here Lecture 7	Click here Lecture 7	Multiple comparison test	Test of Hypothesis for Linear Parametric Functions	7
Click here Lecture 8	Click here Lecture 8	Multiple comparison test and confidence intervals	Analysis of Variance in One Way Fixed Effect Model	8
Click here Lecture 9	Click here Lecture 9	One way analysis of variance	CCD and Multiple Comparison Tests	9
Click here Lecture 10	Click here Lecture 10	Two way analysis of variance	Multiple Comparison Tests	10
Click here Lecture 11	Click here Lecture 11	Two way analysis of variance with interaction	Multiple Comparison Tests Based on Confidence Intervals and Test of Hypothesis for Variance	11
Click here Lecture 12	Click here Lecture 12	Experimental Design Models	Basics for ANOVA in Experimental Design Models	12
Click here Lecture 13	Click here Lecture 13	Experimental Design Models	One-Way Classification in Experimental Design Models	13
Click here Lecture 14	Click here Lecture 14	Experimental Design Models	Two-way classification without interaction in Experimental Design Models	14
Click here Lecture 15	Click here Lecture 15	Experimental Design Models	Two-way classification with interaction in Experimental Design Models	15
Click here Lecture 16	Click here Lecture 16	Experimental Design Models	Tukey's Test for Non-additivity	16
Click here Lecture 17	Click here Lecture 17	Experimental Designs and Their Analysis	Basics of Design of Experiments	17
Click here Lecture 18	Click here Lecture 18	Experimental Designs and Their Analysis	Completely Randomized Design	18
Click here Lecture 19	Click here Lecture 19	Experimental Designs and Their Analysis	Randomized Block Design	19
Click here Lecture 20	Click here Lecture 20	Experimental Designs and Their Analysis	Basics in Latin Square Design	20
Click here Lecture 21	Click here Lecture 21	Experimental Designs and Their Analysis	Analysis in Latin Square Design and Missing Plot Technique	21
Click here Lecture 22	Click here Lecture 22	Incomplete Block Designs and Their Analysis	Basics of Incomplete Block Designs	22
Click here Lecture 23	Click here Lecture 23	Incomplete Block Designs and Their Analysis	Basics and Estimation of Parameters	23
Click here Lecture 24	Click here Lecture 24	Incomplete Block Designs and Their Analysis	Estimation of Parameters in IBD	24
Click here Lecture 25	Click here Lecture 25	Incomplete Block Designs and Their Analysis	Analysis of Variance in IBD	25
Click here Lecture 26	Click here Lecture 26	Incomplete Block Designs and Their Analysis	Properties of Treatment and Block Totals	26
Click here Lecture 27	Click here Lecture 27	Incomplete Block Designs and Their Analysis	More Properties of Treatment and Block Totals	27
Click here Lecture 28	Click here Lecture 28	Incomplete Block Designs and Their Analysis	Interblock Analysis of Variance	28
Click here Lecture 29	Click here Lecture 29	Incomplete Block Designs and Their Analysis	Recovery of Interblock Information in IBD	29
Click here Lecture 30	Click here Lecture 30	Balanced Incomplete Block Design	Basic Definitions in BIBD	30
Click here Lecture 31	Click here Lecture 31	Balanced Incomplete Block Design	Basic Definitions and Intrablock Analysis of Variance in BIBD	31
Click here Lecture 32	Click here Lecture 32	Balanced Incomplete Block Design	Intrablock Analysis of Variance and Other Tests in BIBD	32
Click here Lecture 33	Click here Lecture 33	Balanced Incomplete Block Design	Recovery of Interblock Information	33
Click here Lecture 34	Click here Lecture 34	2ⁿ Factorial Experiments	Terminologies and Notations	34
Click here Lecture 35	Click here Lecture 35	2ⁿ Factorial Experiments	ANOVA in 2² Factorial Experiment	35
Click here Lecture 36	Click here Lecture 36	2ⁿ Factorial Experiments	ANOVA in 2³ and 2ⁿ Factorial Experiment	36
Click here Lecture 37	Click here Lecture 37	2ⁿ Factorial Experiments	Understanding Confounding in 2² Factorial Experiment	37
Click here Lecture 38	Click here Lecture 38	Analysis with partial confounding	Definitions and Confounding Arrangement	38
Click here Lecture 39	Click here Lecture 39	Partial Confounding	Partial Confounding in 2²Factorial Experiment	39
Click here Lecture 40	Click here Lecture 40	Partial Confounding	Partial Confounding in 2² and 2³ Factorial Experiments	40

Video Lectures:

The following video lectures were created to help the students during COVID-19 pandemic 2020. It was not possible to do editing after the first recording due to the lockdown. The videos may have some minor slips as they were recorded in a single shot without much preparation and editing. I request you to kindly ignore the slips. The videos have been uploaded on www.youtube.com.

Enormous thanks to Prof. Satyaki Roy, Media Technology Centre and Wonderful Persons over there for their support in creating the videos.

These videos were prepared for the 2020 MSc Statistics students at IIT Kanpur. The 2021 MSc Statistics students at IIT Kanpur are requested to follow from MooKit.

Lecture 1: YouTube Link : Slides of Video Lecture 1 : Factorial Experiment

Lecture 2: YouTube Link : Slides of Video Lecture 2 : Factorial Experiment

Lecture 3: YouTube Link : Slides of Video Lecture 3 : Factorial Experiment

Lecture 4: YouTube Link : Slides of Video Lecture 4 : Factorial Experiment

Lecture 5: YouTube Link : Slides of Video Lecture 5 : Factorial Experiment

Lecture 6: YouTube Link : Slides of Video Lecture 6 : Confounding (Video recording has been done at home, without any editing etc. So please excuse for minor slips of the tongue)

Lecture 7: YouTube Link : Slides of Video Lecture 7-8 : Confounding (Video recording has been done at home, without any editing etc. So please excuse for minor slips of the tongue)

Lecture 8: YouTube Link : Slides of Video Lecture 7-8 : Confounding (Video recording has been done at home, without any editing etc. So please excuse for minor slips of the tongue)

About the Students :

How Design of Experiment evolved - An interesting article from https://www.sciencehistory.org/distillations/ronald-fisher-a-bad-cup-of-tea-and-the-birth-of-modern-statistics

Ronald Fisher, a Bad Cup of Tea, and the Birth of Modern Statistics

A lesson in humility begets a scientific revolution.

By Sam Kean | August 6, 2019

Detail of an advertisement for Dutch tea company Van Nelle, ca. 1929.

Wikimedia Commons

In offering his colleague a cup of tea, Ronald Fisher was just being polite. He had no intention of kicking up a dispute much less remaking modern science.

At the time, the early 1920s, Fisher worked at an agricultural research station north of London. A short, slight mathematician with rounded spectacles, he'd been hired to help scientists there design better experiments, but he wasn't making much headway. The station's four o'clock tea breaks were a nice distraction.

One afternoon Fisher fixed a cup for an algae biologist named Muriel Bristol. He knew she took milk with tea, so he poured some milk into a cup and added the tea to it.

That's when the trouble started. Bristol refused the cup. "I won't drink that," she declared.

Fisher was taken aback. "Why?"

"Because you poured the milk into the cup first," she said. She explained that she never drank tea unless the milk went in second.

The milk-first/tea-first debate has been a bone of contention in England ever since tea arrived there in the mid-1600s. It might sound like the ultimate petty butter battle, but each side has its partisans, who get boiling mad if someone makes a cup the "wrong" way. One newspaper in London declared not long ago, "If anything is going to kick off another civil war in the U.K., it is probably going to be this."

As a man of science Fisher thought the debate was nonsense. Thermodynamically, mixing A with B was the same as mixing B with A, since the final temperature and relative proportions would be identical. "Surely," Fisher reasoned with Bristol, "the order doesn't matter."

"It does," she insisted. She even claimed she could taste the difference between tea brewed each way.

Fisher scoffed. "That's impossible."

Ronald Fisher in his youth, undated.

Barr Smith Library, University of Adelaide

This might have gone on for some time if a third person, chemist William Roach, hadn't piped up. Roach was actually in love with Bristol (he eventually married her) and no doubt wanted to defend her from Fisher. But as a scientist himself, Roach couldn't just declare she was right. He'd need evidence. So he came up with a plan.

"Let's run a test," he said. "We'll make some tea each way and see if she can taste which cup is which."

Bristol declared she was game. Fisher was also enthusiastic. But given his background designing experiments he wanted the test to be precise. He proposed making eight cups of tea, four milk-first and four tea-first. They'd present them to Bristol in random order and let her guess.

Bristol agreed to this, so Roach and Fisher disappeared to make the tea. A few minutes later they returned, by which point a small audience had gathered to watch.

The order in which the cups were presented is lost to history. But no one would ever forget the outcome of the experiment. Bristol sipped the first cup and smacked her lips. Then she made her judgment. Perhaps she said, "Tea first."

They handed her a second cup. She sipped again. "Milk first."

This happened six more times. Tea first, milk first, milk first again. By the eighth cup Fisher was goggle-eyed behind his spectacles. Bristol had gotten every single one correct.

It turns out adding tea to milk is not the same as adding milk to tea, for chemical reasons. No one knew it at the time, but the fats and proteins in milk-which are hydrophobic, or water hating-can curl up and form little globules when milk mixes with water. In particular, when you pour milk into boiling hot tea, the first drops of milk that splash down get divided and isolated.

Surrounded by hot liquid, these isolated globules get scalded, and the whey proteins inside them-which unravel at around 160 degree Faherniet -change shape and acquire a burnt-caramel flavor. (Ultra-high-temperature pasteurized milk, which is common in Europe, tastes funny to many Americans for a similar reason.) In contrast, pouring tea into milk prevents the isolation of globules, which minimizes scalding and the production of off-flavors.

As for whether milk-first or tea-first tastes better, that depends on your palate. But Bristol's perception was correct. The chemistry of whey dictates that each one tastes distinct.

Bristol's triumph was a bit humiliating for Fisher-who had been proven wrong in the most public way possible. But the important part of the experiment is what happened next. Perhaps a little petulant, Fisher wondered whether Bristol had simply gotten lucky and guessed correctly all eight times. He worked out the math for this possibility and realized the odds were 1 in 70. So she probably could taste the difference.

Muriel Bristol Roach, undated.

Lawes Agricultural Trust

But even then he couldn't stop thinking about the experiment. What if she'd gotten just one cup wrong out of eight? He reran the numbers and found the odds of her guessing "only" seven cups correctly dropped from 1 in 70 to around 1 in 4. In other words, accurately identifying seven of eight cups meant she could probably taste the difference, but he'd be much less confident in her ability-and he could quantify exactly how much less confident.

Furthermore, that lack of confidence told Fisher something: the sample size was too small. So he began running more numbers and found that 12 cups of tea, with 6 poured each way, would have been a better trial. An individual cup would carry less weight, so one data point wouldn't skew things so much. Other variations of the experiment occurred to him as well (for example, using random numbers of tea-first and milk-first cups), and he explored these possibilities over the next few months.

Now this might all sound like a waste of time. After all, Fisher's boss wasn't paying him to dink around in the tearoom. But the more Fisher thought about it, the more the tea test seemed pertinent. In the early 1920s there was no standard way to conduct scientific experiments: controls were rare, and most scientists analyzed data crudely. Fisher had been hired to design better experiments, and he realized the tea test pointed the way. However frivolous it seemed, its simplicity clarified his thinking and allowed him to isolate the key points of good experimental design and good statistical analysis. He could then apply what he'd learned in this simple case to messy real-world examples-say, isolating the effects of fertilizer on crop production.

Fisher published the fruit of his research in two seminal books, Statistical Methods for Research Workers and The Design of Experiments. The latter introduced several fundamental ideas, including the null hypothesis and statistical significance, that scientists worldwide still use today. And the first example Fisher used in his book-to set the tone for everything that followed-was Muriel Bristol's tea test.

His intellectual acumen, however, did not insulate Fisher from the prejudices of his time when it came to class, race, and colonialism. Fisher was a well-known eugenicist and was steadfast in those beliefs throughout his life. When, in the aftermath of World War II, UNESCO formed a coalition of scientists to wrestle with Nazi science and provide the scientific backbone for the universal condemnation of racism, Fisher was among those who officially objected to what he saw as the project's "well-intentioned" but misguided mission, affirming his belief that groups differed "in their innate capacity for intellectual and emotional development."

But such convictions have done little to tarnish Fisher's legacy. He became a legend in biology for helping to unite the gene theory of Gregor Mendel with the evolutionary theory of Charles Darwin. But his biggest contribution to science remains his work on experimental design. The reforms he introduced are so ubiquitous that they're all but invisible nowadays-the sign of a true revolution.