Shalabh
shalab@iitk.ac.in
shalabh1@yahoo.com
Department of Mathematics & Statistics
Indian
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, 2^{n} and 3^{n} 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); 2^{n} factorial experiments with total confounding, partial confounding and fractional replications; Analysis of covariance; Introduction to 3^{n} 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.
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
Lecture Notes 7
Lecture Notes
8
Lecture Notes
9
Lecture Notes 10
Lecture Notes
11
Lecture Notes
12
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 |
2^{n} Factorial Experiments |
Terminologies and Notations |
34 |
Click here Lecture 35 |
Click here Lecture 35 |
2^{n} Factorial Experiments |
ANOVA in 2^{2} Factorial Experiment |
35 |
Click here Lecture 36 |
Click here Lecture 36 |
2^{n} Factorial Experiments |
ANOVA in 2^{3} and 2^{n} Factorial Experiment |
36 |
Click here Lecture 37 |
Click here Lecture 37 |
2^{n} Factorial Experiments |
Understanding Confounding in 2^{2} 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 2^{2 }Factorial Experiment |
39 |
Click here Lecture 40 |
Click here Lecture 40 |
Partial Confounding |
Partial Confounding in 2^{2} and 2^{3} 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.
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
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."
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.
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.