This lecture defines the outcome and the covariate, reviews summaries and distributions for regression, and gives the course plan (Lecture 1 · p.1–38).
Unit 01Nomenclature and data example
Plan · Slides p.1–6
p.1 title · p.2 outcome \(y\) · p.3 covariate \(x\) · p.4 brainhead data · p.5 variables and first rows · p.6 scatterplot.
How Can we Describe The Data? · Lecture 1 · p.9
You have by now seen a few ways of describing the data:
• Univariate summaries:
• Mean
• Variance/Standard Deviation
• Bivariate summaries:
• Covariance
• Correlation
Let’s review them briefly
A summary is one number that describes one property of a data column.
A univariate summary uses one variable. This unit covers it.
A bivariate summary shows how two variables change together. Unit 03 covers it.
Slides p.7–8 build the same list in parts.
02.2Mean (expectation) and linearity Slides p.10–11
WhatSlides p.10–11
Review: Mean · Lecture 1 · p.10–11
Mean/Expectation:
\[E[Y]=\int y f(y)\,dy\]
Linearity of expectation: For random variables \(Y_1,\dots,Y_m\) and constants \(a_i,b_i\) for \(i=1,\dots,m\):
\[E\left[\sum_{i=1}^m (a_iY_i+b_i)\right]=\sum_{i=1}^m a_iE[Y_i]+\sum_{i=1}^m b_i\]
A random variable \(Y\) is a quantity whose value chance sets. Small \(y\) is one possible value.
The probability density function (PDF) \(f(y)\) tells how likely \(Y\) is near \(y\). \(f(y)\ge 0\) and \(\int f(y)\,dy=1\).
The expectation \(E[Y]\) is the average of \(Y\) over many repeats. The PDF sets it, not data.
Constants \(a_i\), \(b_i\) are fixed, not random.
Linearity: a constant factor moves outside \(E\). \(E\) of a sum is the sum of the \(E\) values. \(E\) of a constant is the constant.
HowAdded
Use linearity to simplify \(E[\cdot]\)
Write the expression as \(\sum_i (a_iY_i+b_i)\).
Read each \(a_i\) and \(b_i\).
Replace each \(a_iY_i\) with \(a_iE[Y_i]\).
Keep each \(b_i\).
Add the terms.
Self-check: The result contains only \(E[Y_i]\) values and constants.
ExampleAdded
\(E[Z]=2\). Calculate \(E[3Z+1]\).
1
\(E[3Z+1]=3E[Z]+1\)
linearity, \(a_1=3,\ b_1=1\)
2
\(=3(2)+1\)
\(E[Z]=2\)
3
\(=6+1\)
4
\(=7\)
02.3Sample mean Slides p.12
WhatSlides p.12
Review: Mean · Lecture 1 · p.12
For observations \(y_1,\dots,y_n\), the sample mean is:
\[\bar y=\frac{1}{n}\sum_i^n y_i\]
\(n\) is the sample size, the number of observations.
An estimate is a number from data that replaces an unknown population value. \(\bar y\) estimates \(E[Y]\).
\(\sum_i^n\) adds the terms for \(i=1,\dots,n\). The slides also write \(\sum_i\).
HowAdded
Calculate the sample mean \(\bar y\)
Count the observations to get \(n\).
Add the observations.
Divide the sum by \(n\).
Self-check: \(\bar y\) is between the smallest and largest observation, in the unit of \(y_i\).
Example 1 · First four peopleAdded
Brain weights (g): 1530, 1297, 1335, 1282.
1
\(\bar y=\frac14(1530+1297+1335+1282)\)
\(n=4\)
2
\(=\frac14(5444)\)
3
\(=1361\) g
Self-check: \(1282\le 1361\le 1530\).
Example 2 · All 236 peopleAdded
\(n=236\) (Slides p.4). The sum over the course data file is 303086 g (not on the slides). Min. 1012 g, Max. 1635 g (Slides p.23).
1
\(\bar y=\frac{1}{236}(303086)\)
\(n=236\)
2
\(=1284.263\) g
Self-check: \(1012\le 1284.263\le 1635\). Slides p.23 prints 1284.
Example 3 · Linearity on dataAdded
Grams to kilograms: \(y_i'=0.001\,y_i\).
1
\(\bar y'=0.001\,\bar y\)
linearity, \(a=0.001,\ b=0\)
2
\(=0.001(1284.263)\)
3
\(=1.284263\) kg
02.4Variance and its two properties Slides p.13–14
WhatSlides p.13–14
Review: Variance · Lecture 1 · p.13–14
Variance
\[\mathrm{Var}[Y]=E\left[(Y-E[Y])^2\right]=E[Y^2]-E[Y]^2\]
Properties:
• \(\mathrm{Var}(aY+b)=a^2\mathrm{Var}(Y)\)
• For independent random variables \(X\) and \(Y\): \(\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{var}(Y)\)
The deviation \(Y-E[Y]\) is the difference between \(Y\) and its mean.
The variance is the expectation of the squared deviation. It is zero or more. More spread gives a larger variance.
The standard deviation is \(\sqrt{\mathrm{Var}[Y]}\). It has the unit of \(Y\). The variance has that unit squared.
Section 02.6 shows that the two forms are equal.
\(X\) and \(Y\) are independent when the value of one does not change the distribution of the other.
\(\mathrm{Var}[Y]\), \(\mathrm{Var}(Y)\), and \(\mathrm{var}(Y)\) are the same quantity.
Property 1: a shift \(b\) does not change the spread. A factor \(a\) multiplies the variance by \(a^2\).
HowAdded
Get the variance of a transformation
Write the target as \(aY+b\) or \(X+Y\).
For \(aY+b\): remove \(b\).
For \(aY+b\): write \(a^2\mathrm{Var}(Y)\).
For \(X+Y\): make sure that \(X\) and \(Y\) are independent.
For \(X+Y\): write \(\mathrm{Var}(X)+\mathrm{var}(Y)\).
Self-check: The result is zero or more. The sign of \(a\) has no effect.
Example 1 · \(\mathrm{Var}(aY+b)\)Added
\(Z\) has standard deviation 0.25. Calculate \(\mathrm{Var}(3Z+1)\).
1
\(\mathrm{Var}(Z)=0.25^2\)
2
\(=0.0625\)
3
\(\mathrm{Var}(3Z+1)=3^2\,\mathrm{Var}(Z)\)
Property 1, \(a=3,\ b=1\)
4
\(=9(0.0625)\)
5
\(=0.5625\)
Example 2 · \(\mathrm{Var}(X+Y)\)Added
\(X\), \(Y\) independent, with standard deviations 0.25 and 1. Calculate \(\mathrm{Var}(X+Y)\).
Q1 (Slides p.18 Exercise). Write \(\mathrm{Var}(X+Y)\) with \(\mathrm{Var}(X)\), \(\mathrm{Var}(Y)\), and \(\mathrm{cov}(X,Y)\). Check it on the 5-person population.
All 236 people (sample values): \(130413.76+14084.06+2\times34014.61=144497.82+68029.22=212527.04\).
Q2. Find \(\mathrm{Var}(X-Y)\). With \(s_x^2=130413.76\), \(s_y^2=14084.06\), and sample covariance 34014.61, calculate the sample variance of \(x-y\).
The difference from 7993433.4 comes from rounding.
Unit 04Toward linear regression: simple and multiple models
This unit gives only the form of each model. Lecture 2 calculates the line.
Plan · Slides p.21–24
p.21 (footer 11/27) three questions, simple model · p.22 (12/27) line on the scatterplot · p.23 (13/27) more covariates · p.24 (14/27) multiple model, course aims.
04.1Three questions and the simple model Slides p.21
WhatSlides p.21
Toward Linear Regression · Lecture 1 · p.21
How do we characterize the relationship between \(x\) and \(y\)?
How do we predict \(y\) given \(x\)?
How does the mean of \(y\) change when \(x\) increases by \(a\).
We can answer questions like these with simple linear regression:
\[y_i=\beta_0+\beta_1x_i+\epsilon_i\]
Relationship: how \(y\) changes when \(x\) increases.
\(a\) is any given change in \(x\), for example \(a=100\) cm³.
A parameter is a fixed, unknown number in the model. \(\beta_0\) and \(\beta_1\) are parameters, the same for all \(i\).
Intercept \(\beta_0\): the height of the line \(\beta_0+\beta_1x\) at \(x=0\).
Slope \(\beta_1\): the increase in line height when \(x\) increases by 1.
Error \(\epsilon_i\): \(y_i\) minus the line height \(\beta_0+\beta_1x_i\). Each observation has its own error.
Simple linear regression: \(y\) equals a straight line in one covariate plus an error.
HowAdded
Write one data row in the model form
Find the \(y\) and \(x\) columns.
Select row \(i\). Read \(x_i\) and \(y_i\).
Put them into \(y_i=\beta_0+\beta_1x_i+\epsilon_i\).
Keep \(\beta_0\), \(\beta_1\), and \(\epsilon_i\) as letters.
Self-check: Only \(\beta_0\), \(\beta_1\), \(\epsilon_i\) are unknown. The index of \(\epsilon\) equals the row number.
Example · Row 1 of the brainhead dataAdded
Row 1 (Slides p.23): M, 20-46, head size 4512, brain weight 1530.
1
\(y=\) brain weight, \(x=\) head size
How step 1
2
\(i=1:\ x_1=4512,\ y_1=1530\)
How step 2
3
\(y_1=\beta_0+\beta_1x_1+\epsilon_1\)
4
\(1530=\beta_0+\beta_1\cdot 4512+\epsilon_1\)
How step 3
The model splits 1530 g into the line height at 4512 cm³ and the error \(\epsilon_1\).
Why · The error is a vertical distance, and \(\beta_1\) is the slopeAdded
1
\(1530=\beta_0+\beta_1\cdot 4512+\epsilon_1\)
Example, line 4
2
\(\epsilon_1=1530-(\beta_0+\beta_1\cdot 4512)\)
3
\(\epsilon_1=y_1-(\beta_0+\beta_1x_1)\)
Change in line height when \(x\) increases by \(a\):
1
\([\beta_0+\beta_1(x+a)]-[\beta_0+\beta_1x]\)
2
\(=\beta_0+\beta_1x+\beta_1a-\beta_0-\beta_1x\)
3
\(=\beta_1a\)
4
\(a=1:\ =\beta_1\)
the slope
Lecture 2 adds assumptions that connect the line height to the mean of \(y\).
04.2A line on the scatterplot Slides p.22
WhatSlides p.22
Simple Linear Regression · Lecture 1 · p.22
Simple Linear Regression: \(y_i=\beta_0+\beta_1x_i+\epsilon_i\)
[Scatterplot: Brain Weight (grams) versus Head Size (cm^3), with a straight line through the points.]
The line is \(\beta_0+\beta_1x\) for one set of values that Lecture 2 calculates.
HowAdded
Read a scatterplot with a line
Find the direction of the line.
Up from left to right: \(\beta_1\) is positive.
Select one point. Find if it is above or below the line.
Above: \(\epsilon_i>0\). Below: \(\epsilon_i<0\).
Examine the spread. A point far from the line has a large \(|\epsilon_i|\).
Self-check: Measure the distance vertically, at the same \(x_i\). \(\epsilon_i\) compares only heights.
Example · The line on Slides p.22Added
Figure 4-1. Brainhead points and the line of Slides p.22 (green). Orange: row 1 and its vertical distance to the line.
Steps 1–2: the line goes up. The slope is positive.
Steps 3–4: row 1, \((4512,1530)\), is above the line. Its vertical distance is positive.
Step 5: points are above and below the line. Each observation needs its own \(\epsilon_i\).
Why · The picture shows a straight lineAdded
1
\(y_i=\beta_0+\beta_1x_i+\epsilon_i\)
Slides p.21
2
\(y_i-\epsilon_i=\beta_0+\beta_1x_i\)
3
\(\beta_0+\beta_1x\) is a first-degree function of \(x\)
its graph is a straight line
04.3More than one covariate Slides p.23
WhatSlides p.23
What if we have multiple covariates? · Lecture 1 · p.23
The slide prints the first 6 rows and a summary of each column. The values are below.
First 6 rows (236 rows in total): the table in Section 01.5 shows these rows.
Summary of the two number columns:
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
head.size
2773
3390
3614
3638
3876
4747
brain.wgt
1012
1208
1280
1284
1351
1635
The columns gender and age contain text, not numbers. Each has length 236.
A categorical column contains group labels, not numbers. gender and age are categorical.
Median: the middle value of the sorted column.
1st Qu. and 3rd Qu. (quartiles): approximately 25% and 75% of the values are not larger than them.
HowAdded
Find the possible covariates in a data table
Count the columns.
Find the \(y\) column.
Each other column is a possible covariate.
Find if each one contains numbers or text.
Put a number column into the equation directly.
Change a text column into numbers first (week 6, categorical covariates, Lecture 1 p.37).
Read the range (Min. to Max.) and center (Median, Mean) of each number column.
Self-check: Possible covariates = columns − 1. Min. ≤ 1st Qu. ≤ Median ≤ 3rd Qu. ≤ Max.
Example · The brainhead data on Slides p.23Added
1
number of columns \(=4\): gender, age, head.size, brain.wgt
How step 1
2
\(y=\) brain.wgt
How step 2
3
possible covariates: head.size, gender, age
How step 3
4
head.size: numbers; gender, age: text
How steps 4–6
5
head.size: range 2773 to 4747 cm³, median 3614, mean 3638
How step 7
6
brain.wgt: range 1012 to 1635 g, median 1280, mean 1284
possible covariates \(=\) columns \(-\) outcome columns
2
\(=4-1\)
3
\(=3\)
Simple linear regression uses one of the three. A model with more than one \(x\) can use all.
04.4Simple and multiple linear regression Slides p.24
WhatSlides p.24
Toward Multiple Linear Regression · Lecture 1 · p.24
Simple Linear Regression:
\[y_i=\beta_0+\beta_1x_i+\epsilon_i\]
Multiple Linear Regression:
\[y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i3}+\epsilon_i\]
This course will focus on developing multiple linear regression
Theoretically/mathematically (derive estimators)
Practically (how to fit these models [with software])
How to choose and compare a model (which \(x_{ij}\) to include)
How to evaluate the appropriateness of the model/assumptions (model diagnostics)
\(x_{ij}\): covariate \(j\) of observation \(i\).
A coefficient is the parameter that multiplies a covariate, for example \(\beta_2\) for \(x_{i2}\).
An estimator is a formula that calculates an unknown parameter from data.
Fit: calculate the parameter values from the data.
Assumptions: conditions that the model sets on the errors and other parts.
Model diagnostics: methods that check the model and its assumptions with data.
HowAdded
Write the multiple linear regression equation
Find the \(y\) column.
List the covariates for the model.
Number them \(j=1,2,\dots\).
For each \(j\), write \(\beta_jx_{ij}\).
Write \(y_i=\beta_0+\) (sum of the terms) \(+\epsilon_i\).
Self-check: Parameters = covariates + 1. Each \(x\) has first index \(i\). No \(\beta\) has index \(i\).
Example · Three covariates for the brainhead dataAdded
1
\(y_i=\) brain.wgt of person \(i\)
How step 1
2
\(x_{i1}=\) head.size, \(x_{i2}=\) gender, \(x_{i3}=\) age group
Q2. In \(y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i3}+\epsilon_i\), \(x_{i2}\) is gender. What is \(x_{52}\)? How many parameters \(\beta\) are there?
Answer
\(x_{52}\) is the gender of person 5: M (Slides p.23).
\(3+1=4\): \(\beta_0,\beta_1,\beta_2,\beta_3\).
Q3. From the summary on Slides p.23, give the median head size. Between which two values are the middle 50% of brain weights?
Answer
Median head size: 3614 cm³.
Between 1st Qu. and 3rd Qu.: 1208 g and 1351 g.
Unit 05Review: normal, chi-square, t, and F distributions
Later lectures use these four distributions for inference about regression.
Plan · Slides p.25–28
p.25 normal, linear combinations · p.26 chi-square = sum of squared standard normals · p.27 t = standard normal / \(\sqrt{\text{chi-square}/\nu}\) · p.28 F = ratio of two chi-squares, each over its degrees of freedom.
A distribution describes how likely each value of a random variable is. "\(Z\sim N(\mu,\sigma^2)\)": \(Z\) has the normal distribution with mean \(\mu\), variance \(\sigma^2\).
Distribution parameters (\(\mu\), \(\sigma^2\), \(\nu\)) are fixed numbers that set its shape.
Moments: the slides use this name for \(E[Z]\) and \(\mathrm{Var}(Z)\).
05.1Normal distribution Slides p.25
Lecture 3 shows that \(\hat\beta_1\) is a linear combination of independent normals.
WhatSlides p.25
Review: Normal Distribution · Lecture 1 · p.25
\[Z\sim N(\mu,\sigma^2)\quad\text{with parameters } \mu \text{ and } \sigma^2\]
PDF:
\[f(z)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(z-\mu)^2}{2\sigma^2}\right]\]
Moments:
\[E[Z]=\mu,\qquad \mathrm{Var}(Z)=\sigma^2\]
For independent \(Z_i\overset{ind}{\sim}N(\mu_i,\sigma^2)\), \(U=\sum_i^n(a_iZ_i+b_i)\) is normally distributed:
\[U\sim N\left(\sum_i^n(a_i\mu_i+b_i),\ \sum_i^n a_i^2\sigma_i^2\right)\]
The normal PDF is a bell curve, symmetric about \(\mu\) ("mu"). \(\sigma^2\) ("sigma squared") sets its width.
Standard normal: \(N(0,1)\).
\(\exp[t]=e^t\).
\(\overset{ind}{\sim}\): "are independent and have the distribution".
A linear combination multiplies each \(Z_i\) by \(a_i\), adds \(b_i\), and sums.
The slide writes \(\sigma^2\) in the assumption and \(\sigma_i^2\) in \(\mathrm{Var}(U)\). With all \(\sigma_i^2=\sigma^2\), \(\mathrm{Var}(U)=\sum_i^n a_i^2\sigma^2\).
Figure 5-1 · PDF of \(N(0,1)\). Highest point \(f(0)=0.3989\) (Example).
HowAdded
Find the distribution of \(U=\sum_i^n(a_iZ_i+b_i)\)
Make sure that each \(Z_i\) is normal.
Make sure that the \(Z_i\) are independent.
Read \(a_i\) and \(b_i\).
Read \(\mu_i\) and \(\sigma^2\).
Mean: add \(a_i\mu_i+b_i\) over all terms.
Variance: add \(a_i^2\sigma^2\) over all terms. Do not use \(b_i\).
Write \(U\sim N(\text{mean},\ \text{variance})\).
Self-check: The variance is positive.
Calculate the normal PDF \(f(z)\)
Put \(z\), \(\mu\), \(\sigma^2\) into the formula.
Calculate the exponent \(-(z-\mu)^2/(2\sigma^2)\).
The slide states that \(U\) is normal. The mean and variance follow from linearity, \(\mathrm{Var}(aZ+b)=a^2\mathrm{Var}(Z)\), and Property 2 (Section 02.4).
Lecture 3 uses it for tests of the regression coefficients.
WhatSlides p.27
Review: t-Distribution · Lecture 1 · p.27
\[Y\sim t_\nu\quad\text{with } \nu \text{ degrees of freedom}\]
PDF:
\[f(y)=\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi\nu}\,\Gamma(\frac{\nu}{2})}\left(1+\frac{y^2}{\nu}\right)^{-\frac{\nu+1}{2}}\]
Moments:
\[E[Y]=0 \text{ if } \nu>1,\ \text{NA otherwise}\]
\[\mathrm{Var}(Y)=\frac{\nu}{\nu-2} \text{ if } \nu>2,\ \infty \text{ otherwise}\]
For independent \(Z\sim N(0,1)\) and \(X\sim\chi^2_\nu\)
\[\frac{Z}{\sqrt{X/\nu}}\sim t_\nu\]
The slide prints the PDF denominator as \(\sqrt{\pi\nu}\,\frac{\nu}{2}\), with no \(\Gamma\). The total area is 1 only with \(\Gamma(\frac{\nu}{2})\). This page uses \(\Gamma(\frac{\nu}{2})\).
\(t_\nu\) is a bell curve, symmetric about 0. Its center is lower than \(N(0,1)\), and its tails are thicker.
A tail is the part of the curve far from the center. Thicker tails make extreme values more likely.
\(\nu\) is the degrees of freedom of the chi-square in the denominator.
NA (not available): for \(\nu=1\), \(\int y f(y)\,dy\) does not converge. The mean does not exist.
\(\infty\): for \(\nu\le 2\), \(\int y^2 f(y)\,dy\) does not converge.
Figure 5-3 · PDFs of \(t_1\) (orange), \(t_5\) (blue), \(N(0,1)\) (black, dashed). As \(\nu\) increases, \(t_\nu\) comes near \(N(0,1)\).
Q2. \(Z_1,Z_2,Z_3,Z_4\overset{iid}{\sim}N(0,1)\). Name the distribution of
(a) \(Z_2^2+Z_3^2+Z_4^2\); (b) \(\dfrac{Z_1}{\sqrt{(Z_2^2+Z_3^2+Z_4^2)/3}}\); (c) \(\dfrac{Z_1^2/1}{(Z_2^2+Z_3^2+Z_4^2)/3}\).
For (a) and (b), give the mean and variance.
Answer
(a) \(\chi^2_3\). Mean 3, variance \(2\cdot3=6\).
(b) \(Z_1\) is \(N(0,1)\), independent of \(X\sim\chi^2_3\): \(t_3\). Mean 0 (\(\nu=3>1\)). Variance \(3/(3-2)=3\).
(c) \(\chi^2_1/1\) over \(\chi^2_3/3\), independent: \(F(1,3)\).
Q3. With the PDF on Slides p.27, calculate \(f(0)\) for \(Y\sim t_1\). What is \(E[Y]\)?
Footer numbers 19/27 to 27/27 are on PDF pages 30 to 38. This unit uses PDF page numbers.
06.2Course content and reference books Slides p.30
WhatSlides p.30
Course Overview · Lecture 1 · p.30
Review & Simple Linear Regression (2 weeks)
Multiple Linear Regression (4 weeks)
Model Diagnostics (3 weeks)
Model Building (3 weeks)
Extensions (time permitting)
Textbook: None required. If you want a different perspective:
Introduction to Regression Modeling by Abraham and Ledolter, 2006 (available online, see course outline for link)
Gelman, A., Hill, J., & Vehtari, A. (2020). Regression and other stories. Cambridge University Press.
Any other introductory book on linear regression
SLR: simple linear regression. MLR: multiple linear regression.
Model building: selecting the variables of a model and their form.
Extensions: topics past linear regression, only if time is available.
HowAdded
Find the block for a topic
Count the covariates that the topic uses.
One covariate: block 1.
Two or more: block 2.
Checking a fitted model: block 3.
Selecting variables: block 4.
Self-check: Blocks 1–4 add to 12 weeks.
Example · Check the number of weeksAdded
1
\(2+4+3+3\)
blocks 1–4, Slides p.30
2
\(=6+3+3\)
3
\(=12\) weeks
Block 5 has no fixed length.
Why · The order of the blocksAdded
1
Start with one covariate (SLR).
2
Extend to two or more (MLR).
3
Fit a model before you check it (diagnostics).
4
Check models before you select one (model building).
06.3Grading Slides p.31
WhatSlides p.31
Grading · Lecture 1 · p.31
Assignments: 15%
Midterm 1: 15%
Midterm 2: 15%
Final: 50%
Participation: 5%
Weight: the percentage of the final grade from one component.
Midterm: an exam in the middle of the term.
Participation: a grade from your Piazza posts (Slides p.34).
Length is proportional to weight. A = Assignments, M1 = Midterm 1, M2 = Midterm 2, P = Participation.
HowAdded
Calculate a final grade
Write the percentage score of each component.
Multiply each score by its weight.
Add the five products.
Self-check: The weights add to 100%.
Example · A final gradeAdded
Scores: Assignments 80%, Midterm 1 70%, Midterm 2 75%, Final 72%, Participation 4 of 5 (80%).
1
\(0.15(80)+0.15(70)+0.15(75)+0.50(72)+0.05(80)\)
Slides p.31
2
\(=12+10.5+11.25+36+4\)
3
\(=73.75\)
Why · The weights add to 100%Added
1
\(15+15+15+50+5\)
Slides p.31
2
\(=45+50+5\)
3
\(=100\)
06.4Assignments Slides p.32
WhatSlides p.32
Assignments · Lecture 1 · p.32
Due:
Assignment 1: 5pm Friday, September 25, 2026
Assignment 2: 5pm Friday, October 30, 2026
Assignment 3: 5pm Friday, November 27, 2026
Each worth 5 % of final grade
Goal: Practice what you've learned in class, extend your knowledge independently
Typically 4 questions with multiple parts
Usually include a theoretical question, an applied analysis question, and a simulation-based question
You may discuss with other students, but you must write up your own solutions
Assignments up to 24 hours late will receive a penalty of 50%. If you are unable to complete an assignment due to severe illness or extenuating circumstances, you must email the instructor before the deadline.
The slide item about software is not quoted.
Theoretical question: you answer with a mathematical derivation.
Applied analysis question: you fit a model to real data and explain the result.
Simulation-based question: you generate random data and study how a method behaves.
HowAdded
Complete an assignment
Record the deadline: 5pm Friday.
Discuss with other students (optional).
Write your own solutions.
If severe illness stops you, email the instructor before the deadline.
Self-check: \(3\times 5\%\) equals the 15% on Slides p.31.
Example · Check the assignment weightAdded
1
\(3\times 5\%\)
Slides p.32
2
\(=15\%\)
Slides p.31
Why · The due dates spread over the termAdded
1
A1 to A2: Sept 25 to Oct 30 = 35 days
2
A2 to A3: Oct 30 to Nov 27 = 28 days
3
A1 to A3: Sept 25 to Nov 27 = 63 days
06.5Midterms and final exam Slides p.33
WhatSlides p.33
Midterms & Final Exam · Lecture 1 · p.33
Two in-class midterms (1hr20min)
October 8th & November 12th
You must attend the correct section!
Each worth 15% of final grade
Final exam (2hrs30min)
Scheduled by the registrar during final exam period
Worth 50% of final grade
Coverage is cumulative
Will test theory & application
Bring a non-programmable calculator
If you are unable to attend a midterm due to severe illness or extenuating circumstances, you must email the instructor before the deadline/midterm start time. Grades for a missed midterm will be shifted to the final exam.
The slide also says that exams ask you to read the printed output of a fitted model.
In-class: during the usual class time and room. October 8 and November 12 are Thursdays.
Registrar: the university office for exams.
Cumulative: each exam can test all material from the start of the course.
HowAdded
New weights after a missed midterm
Set the missed midterm to 0%.
Add its 15% to the final.
Keep the other weights.
Self-check: The weights add to 100%.
Example · A student misses Midterm 1Added
1
Final weight \(=50\%+15\%\)
Slides p.33
2
Final weight \(=65\%\)
Why · The sum stays at 100%Added
1
\(15+0+15+(50+15)+5\)
2
\(=15+15+65+5\)
3
\(=100\)
06.6Participation Slides p.34
WhatSlides p.34
Participation · Lecture 1 · p.34
Worth 5% of final grade
Based on (public) discussions on Piazza
0 pts: You did not post on Piazza this term
1 pts: You posted on Piazza once
2 pts: You posted on Piazza twice
3 pts: You posted on Piazza several times
4 pts: You asked and answered several questions on Piazza
5 pts: You regularly posted high quality questions and gave high quality answers to student questions
Piazza: the online question-and-answer forum of the course.
Only public posts (visible to all students) count.
HowAdded
Get the participation grade
Post publicly on Piazza.
Ask questions.
Answer other students.
Do this for the full term.
Self-check: Find your current level on Slides p.34.
06.7Communication Slides p.35
WhatSlides p.35
Communication · Lecture 1 · p.35
Slides, assignments, etc. will all be posted on LEARN
Announcements will be made in class and on LEARN
All questions about course content should be posted on Piazza (not email)
If you have personal concerns, you can email the TAs/instructor directly.
LEARN: the university course website (D2L Brightspace).
TA: teaching assistant.
Where to send a question
Course content: Piazza, not email.
Personal concern: email a TA or the instructor.
06.8Tutorials Slides p.36
WhatSlides p.36
Tutorials · Lecture 1 · p.36
Scheduled for Fridays at 2:30 in this room
There will be 4 formal tutorials: Sept 18th, Oct 2nd, Oct 23rd, Nov 6th
Work through practice problems with TA
In other weeks TAs will use the tutorial time slot for office hours in this room
Tutorial: a class where a TA helps you do practice problems.
Office hour: a time when a TA answers your questions.
06.9Schedule Slides p.37
WhatSlides p.37
Simple Linear Regression · Slides p.37
Week 1: Introduction. TA Office hour.
Week 2: Simple linear regression, least squares estimation, inference. Tutorial 1 (Sept 18th).
Week 3: Prediction & Review of matrix algebra. TA Office hour. Assignment 1 due 5pm (Sept 25th).