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R代写-STA 304/1003
时间：2021-03-31
STA 304/1003 Winter 2021: Week 7
A1 regrades
Engagement Activity II
Missed Test guideline
Today: Ratio and regression estimation
Readings: Chapter 6; exclude §6.5, 6.9
Upcoming:
I March 8 (new date!): Assignment 2
I March 15: Drop day
I March 22: Test 2
Shivon Sue-Chee Ratio and Regression Estimation 1
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 2
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 3
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 4
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 5
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 6
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 7
Examples
Parameters of interest:
Ratio: R =
µy
µx
Population Mean: µy
Population Total: τy
Examples:
Y -
X -
Y -
X -
Y -
X -
Shivon Sue-Chee Ratio and Regression Estimation 8
Motivating Examples
1 Estimate population size N (Laplace):
use total # of births × birth rate
2 Estimate total sugar content (Eg. 6.2):
use total weight × av. sugar content / av. weight
3 Estimate population ratio (Ex. 6.3):
amount spent on food
household income
Shivon Sue-Chee Ratio and Regression Estimation 9
Example (Lohr, Ch.4): Estimate N
Laplace wanted to estimate population of France (1802)
first sampled 30 communes (districts)
y1, . . . , y30 – number of persons in commune i
x1, . . . , x30 – number of registered births in commune i
additional information: 1 million births in the whole country
n∑
i=1
yi = 2, 037, 615
n∑
i=1
xi = 71, 866∑
yi∑
xi
= 28.3
estimate of population 28.3× 1million = 28.3 million
with x and y correlated, less variability in y¯/x¯ that Ny¯
Shivon Sue-Chee Ratio and Regression Estimation 10
Example (§6.2): Estimate total
Estimate the sugar content of a truck-load of oranges
sample n oranges: measure sugar content yi and weight xi
τy
τx
=
τy =
τˆy =
How do we get τx?
What is the Ch.4 solution?
Shivon Sue-Chee Ratio and Regression Estimation 11
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 12
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 13
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 14
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 15
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 16
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 17
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 18
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 19
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 20
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 21
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 22
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 23
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 24
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 25
Estimating a population ratio: Exercise 6.3
Ratio: Y - Money spent on food/yr vs
X - total yearly household income
26000 28000 30000 32000 34000
30
00
40
00
50
00
60
00
x(income)
y(
fo
od
e
xp
en
di
tu
re
)
plot the data! any influential points?
Shivon Sue-Chee Ratio and Regression Estimation 26
Example: Exercise 6.3- estimate R
> exer63x = scan()
1: 25100 32200 29600 35000 34400 26500 28700
8: 28200 34600 32700 31500 30600 27700 28500
15:
Read 14 items
> exer63y = scan()
1: 3800 5100 4200 6200 5800 4100 2900
8: 3600 3800 4100 4500 5100 4200 4000
15:
Read 14 items
> plot(exer63x,exer63y, xlab="x(income)",ylab="y(food expenditure)")
> mean(exer63y)/mean(exer63x)
[1] 0.1443687
> mean(exer63x)
[1] 30378.57
> mean(exer63y)
[1] 4385.714
=⇒ r = R̂ = 0.144 V̂ (r) =
Shivon Sue-Chee Ratio and Regression Estimation 27
Stats In the News: Meat causes Cancer?!
The New York Times
Shivon Sue-Chee Ratio and Regression Estimation 28
Stats In the News: WHO report on meat and cancer
Cancer deaths worldwide, on a yearly basis:
I Tobacco - about a million
I Alcohol -600,000
I Diets high in processed meat -about 34,000
22 scientists from 10 countries reviewed more than 800 studies
‘linking what people ate with cancers they developed later’
‘Often such studies can’t prove a causal link’
Conclusion: people should follow diets “lower in red and processed
meat.”
Shivon Sue-Chee Ratio and Regression Estimation 29
Ratio estimation formulas
Toolboxes (6.1-6.7)
Estimator Estimated Variance
r =
∑n
i=1 yi∑n
i=1 xi
V̂ (r) =
(
1− nN
)(
1
µ2x
s2r
n
)
τˆy = V̂ (τˆy ) =
µˆy = V̂ (µˆy ) =
where s2r =
∑n
i=1(yi−rxi )2
n−1
Shivon Sue-Chee Ratio and Regression Estimation 30
Ratio estimation formulas
Toolboxes (6.1-6.7)
Parameter Estimator Estimated Variance
R =
µy
µx
r = R̂ =
∑n
i=1 yi∑n
i=1 xi
= y¯x¯ V̂(r) =
(
1− nN
)(
1
µ2x
s2r
n
)
τy τˆy =
y¯
x¯ τx = rτx V̂(τˆy ) = τ
2
x V̂(r)
µy µˆy =
y¯
x¯ µx = rµx V̂(µˆy ) = µ
2
x V̂(r)
where
s2r =
∑n
i=1(yi − rxi )2
n − 1
Shivon Sue-Chee Ratio and Regression Estimation 31
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 32
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 33
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 34
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 35
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 36
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 37
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 38
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 39
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 40
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 41
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 42
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 43
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 44
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 45
Regression estimation formulas (§6.6)
Regression estimation: used instead of ratio estimation if there is a
linear relationship between y and x but not necessarily through the
origin
Regression line: yˆi = a + bxi
By least squares method:
a = y¯ − bx¯ and b =
∑n
i=1(yi − y¯)(xi − x¯)∑n
i=1(xi − x¯)2
Alternatively, yˆi = y¯ + b(xi − x¯)
To estimate µy by linear(L) regression, using µx at xi ,we get
µˆyL = y¯ + b(µx − x¯)
V̂(µˆyL) =
Shivon Sue-Chee Ratio and Regression Estimation 46
Regression estimation formulas (§6.6)
Regression estimation: used instead of ratio estimation if there is a
linear relationship between y and x but not necessarily through the
origin
Regression line: yˆi = a + bxi
By least squares method:
a = y¯ − bx¯ and b =
∑n
i=1(yi − y¯)(xi − x¯)∑n
i=1(xi − x¯)2
Alternatively, yˆi = y¯ + b(xi − x¯)
To estimate µy by linear(L) regression, using µx at xi ,we get
µˆyL = y¯ + b(µx − x¯)
V̂(µˆyL) =
Shivon Sue-Chee Ratio and Regression Estimation 47
Regression estimation formulas (Toolbox 6.24-6.26)
µˆyL = y¯ + b(µx − x¯)
V̂(µˆyL) =
(
1− n
N
) 1
n
∑n
i=1(yi − a− bxi )2
n − 2 =
(
1− n
N
) MSE
n
note on notation:
MSE =
∑n
i=1(yi − a− bxi )2
n − 2 =
SSE
n − 2
many books call this MSR = SSR/(n − 2) for “mean square
residuals” and “sum of squared residuals”
because MSE usually means “variance + bias-squared”= the mean
squared error of a biased estimator
Shivon Sue-Chee Ratio and Regression Estimation 48
Example 6.9
Simple Least Squares Regression
> achieve=scan()
1: 39 43 21 64 57 47 28 75 34 52
11:
Read 10 items
> calc = scan()
1: 65 78 52 82 92 89 73 98 56 75
11:
Read 10 items
> plot(achieve, calc, main="Figure 6.6")
> # ordinary least squares regression
> fit=lm(calc~achieve)
> summary(fit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 40.7842 8.5069 4.794 0.00137 **
achieve 0.7656 0.1750 4.375 0.00236 **
Residual standard error: 8.704 on 8 degrees of freedom
Multiple R-squared: 0.7052,Adjusted R-squared: 0.6684
F-statistic: 19.14 on 1 and 8 DF, p-value: 0.002365
20 30 40 50 60 70
60
70
80
90
Figure 6.6
achieve
ca
lc
Shivon Sue-Chee Ratio and Regression Estimation 49
Example 6.9
‘Deviations from sample average’ Regression
> # least squares regression on mean centred data
> fittedmodel = lm(calc~ I(achieve - mean(achieve)))
> # I( ... ) treats the arguments numerically
> summary(fittedmodel)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 76.0000 2.7523 27.613 3.19e-09 ***
I(achieve - mean(achieve)) 0.7656 0.1750 4.375 0.00236 **
Residual standard error: 8.704 on 8 degrees of freedom
Multiple R-squared: 0.7052,Adjusted R-squared: 0.6684
F-statistic: 19.14 on 1 and 8 DF, p-value: 0.002365
> 2*sqrt((1-10/486)*(sum(residuals(fittedmodel)^2)/8)/10)
[1] 5.447734
> #Knowing mu_x=52, estimate mu_y
> mean(calc)+ 0.7656*(52-mean(achieve))
[1] 80.5936
Shivon Sue-Chee Ratio and Regression Estimation 50
Regression Example 6.9 Summary
Figure 6.6: least squares line from sample has intercept a = 40.7842
and slope b = 0.7656
y¯=76, x¯=46
µx=52, µˆyL=80.6
V̂(µˆyL)=7.42
Margin of error: 2
√
V̂(µˆyL)=5.45
Check residual plot to determine whether the simple linear model is
an appropriate fit.
See Figure 6.7: the linear model seems appropriate; no obvious
pattern or outlier values.
Shivon Sue-Chee Ratio and Regression Estimation 51
Summary
Ratio estimation is most appropriate when the relationship between y
and x is linear through the origin.
Regression estimator performs better than ratio estimator when the
population relationship moves away from a straight line through the
origin (i.o.w., intercept close to zero).
As the population relationship exhibits more curvature, the regression
estimator becomes more biased.
Shivon Sue-Chee Ratio and Regression Estimation 52
Summary
Ratio estimation is most appropriate when the relationship between y
and x is linear through the origin.
Regression estimator performs better than ratio estimator when the
population relationship moves away from a straight line through the
origin (i.o.w., intercept close to zero).
As the population relationship exhibits more curvature, the regression
estimator becomes more biased.
Shivon Sue-Chee Ratio and Regression Estimation 53
Summary
Ratio estimation is most appropriate when the relationship between y
and x is linear through the origin.
Regression estimator performs better than ratio estimator when the
population relationship moves away from a straight line through the
origin (i.o.w., intercept close to zero).
As the population relationship exhibits more curvature, the regression
estimator becomes more biased.
Shivon Sue-Chee Ratio and Regression Estimation 54
Homework
Read §6.1 – §6.8
EX: 6.1, 6.2, 6.6, 6.9, 6.12, 6.13, 6.14, 6.16
Sampling from real populations: EX 6.4
Shivon Sue-Chee Ratio and Regression Estimation 55
学霸联盟
A1 regrades
Engagement Activity II
Missed Test guideline
Today: Ratio and regression estimation
Readings: Chapter 6; exclude §6.5, 6.9
Upcoming:
I March 8 (new date!): Assignment 2
I March 15: Drop day
I March 22: Test 2
Shivon Sue-Chee Ratio and Regression Estimation 1
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 2
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 3
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 4
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 5
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 6
Ratio and regression estimation (Ch. 6)
most surveys measure population values on more than one variable
even if yi , i = 1, . . . ,N is the variable of most interest
consider one additional auxiliary or subsidiary variable xi , i = 1, . . . ,N
auxiliary: x is correlated with y , y ∝ x
obtain a random sample of paired measurements:
y1, . . . , yn
x1, . . . , xn
use information in x to improve estimation of population parameters
related to y , such as µy , τy , or
µy
µx
Shivon Sue-Chee Ratio and Regression Estimation 7
Examples
Parameters of interest:
Ratio: R =
µy
µx
Population Mean: µy
Population Total: τy
Examples:
Y -
X -
Y -
X -
Y -
X -
Shivon Sue-Chee Ratio and Regression Estimation 8
Motivating Examples
1 Estimate population size N (Laplace):
use total # of births × birth rate
2 Estimate total sugar content (Eg. 6.2):
use total weight × av. sugar content / av. weight
3 Estimate population ratio (Ex. 6.3):
amount spent on food
household income
Shivon Sue-Chee Ratio and Regression Estimation 9
Example (Lohr, Ch.4): Estimate N
Laplace wanted to estimate population of France (1802)
first sampled 30 communes (districts)
y1, . . . , y30 – number of persons in commune i
x1, . . . , x30 – number of registered births in commune i
additional information: 1 million births in the whole country
n∑
i=1
yi = 2, 037, 615
n∑
i=1
xi = 71, 866∑
yi∑
xi
= 28.3
estimate of population 28.3× 1million = 28.3 million
with x and y correlated, less variability in y¯/x¯ that Ny¯
Shivon Sue-Chee Ratio and Regression Estimation 10
Example (§6.2): Estimate total
Estimate the sugar content of a truck-load of oranges
sample n oranges: measure sugar content yi and weight xi
τy
τx
=
τy =
τˆy =
How do we get τx?
What is the Ch.4 solution?
Shivon Sue-Chee Ratio and Regression Estimation 11
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 12
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 13
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 14
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 15
Example: Estimate mean and post-stratification
Suppose population is 50% male, 50% female
We select a sample of size 100, and record yi = weight of ith person
Goal: estimate the average weight in the population, µy
The sample turns out to have just 20 mean, and 80 women
Men Women
n1 = 20 n2 = 80
y¯1 = 180 pounds y¯2 = 110 pounds
y¯ = 124 pounds
Adjusted to a more realistic value:
y¯st = 0.5(180) + 0.5(110) = 145
Shivon Sue-Chee Ratio and Regression Estimation 16
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 17
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 18
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 19
Examples: Estimating a ratio
Example 6.1:
R = mean monthly cost in 2002/ mean monthly cost in 1994
See Table 6.1: y¯ = 901.5, x¯ = 695.8,
r = Rˆ = y¯/x¯ = 1.296
Example: estimate the average number of fish caught per hour by
anglers visiting a lake
x – number of hours fished, y – number of fish caught
Example: estimate the average amount that undergraduate students
spent on textbooks
x – number of textbooks bought, y – total cost
Example (see §6.0 and §6.9): estimate the mean number of students
per section in elementary courses
x – numbers of sections, y – enrollments
Shivon Sue-Chee Ratio and Regression Estimation 20
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 21
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 22
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 23
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 24
Why ratio estimation?
1 To estimate a ratio
E.g. average yield per acre, percentage of magazine pages devoted to
advertising, mean enrollment per section (§6.0), Consumer Price
Index (§6.2)
2 To estimate a population total, but N is unknown
E.g. oranges, Laplace’s
3 If x and y are correlated, the ratio estimator of µy or τy could have
smaller variance than the variance of the simpler estimators
4 To adjust estimates from a sample to reflect demographics
E.g. Current Population Survey (§6.2)
This is called “post-stratification”
5 To adjust estimates for non-response (§11.6)
Shivon Sue-Chee Ratio and Regression Estimation 25
Estimating a population ratio: Exercise 6.3
Ratio: Y - Money spent on food/yr vs
X - total yearly household income
26000 28000 30000 32000 34000
30
00
40
00
50
00
60
00
x(income)
y(
fo
od
e
xp
en
di
tu
re
)
plot the data! any influential points?
Shivon Sue-Chee Ratio and Regression Estimation 26
Example: Exercise 6.3- estimate R
> exer63x = scan()
1: 25100 32200 29600 35000 34400 26500 28700
8: 28200 34600 32700 31500 30600 27700 28500
15:
Read 14 items
> exer63y = scan()
1: 3800 5100 4200 6200 5800 4100 2900
8: 3600 3800 4100 4500 5100 4200 4000
15:
Read 14 items
> plot(exer63x,exer63y, xlab="x(income)",ylab="y(food expenditure)")
> mean(exer63y)/mean(exer63x)
[1] 0.1443687
> mean(exer63x)
[1] 30378.57
> mean(exer63y)
[1] 4385.714
=⇒ r = R̂ = 0.144 V̂ (r) =
Shivon Sue-Chee Ratio and Regression Estimation 27
Stats In the News: Meat causes Cancer?!
The New York Times
Shivon Sue-Chee Ratio and Regression Estimation 28
Stats In the News: WHO report on meat and cancer
Cancer deaths worldwide, on a yearly basis:
I Tobacco - about a million
I Alcohol -600,000
I Diets high in processed meat -about 34,000
22 scientists from 10 countries reviewed more than 800 studies
‘linking what people ate with cancers they developed later’
‘Often such studies can’t prove a causal link’
Conclusion: people should follow diets “lower in red and processed
meat.”
Shivon Sue-Chee Ratio and Regression Estimation 29
Ratio estimation formulas
Toolboxes (6.1-6.7)
Estimator Estimated Variance
r =
∑n
i=1 yi∑n
i=1 xi
V̂ (r) =
(
1− nN
)(
1
µ2x
s2r
n
)
τˆy = V̂ (τˆy ) =
µˆy = V̂ (µˆy ) =
where s2r =
∑n
i=1(yi−rxi )2
n−1
Shivon Sue-Chee Ratio and Regression Estimation 30
Ratio estimation formulas
Toolboxes (6.1-6.7)
Parameter Estimator Estimated Variance
R =
µy
µx
r = R̂ =
∑n
i=1 yi∑n
i=1 xi
= y¯x¯ V̂(r) =
(
1− nN
)(
1
µ2x
s2r
n
)
τy τˆy =
y¯
x¯ τx = rτx V̂(τˆy ) = τ
2
x V̂(r)
µy µˆy =
y¯
x¯ µx = rµx V̂(µˆy ) = µ
2
x V̂(r)
where
s2r =
∑n
i=1(yi − rxi )2
n − 1
Shivon Sue-Chee Ratio and Regression Estimation 31
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 32
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 33
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 34
Example 6.2: Estimating a population total
To estimate total sugar content of truckload of oranges
y is sugar content; x is weight; τx = 1800lbs is weight of the
truckload (easily obtained)
10∑
i=1
yi = 0.246,
10∑
i=1
xi = 4.35, r =
0.246
4.35
τˆy =
V̂ (τˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 35
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 36
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 37
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 38
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 39
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 40
Example 6.3: Estimating a population mean
µy = mean acreage in sugarcane, in 1999, across N = 32 counties
sample 6 counties and record: yi = mean acreage in sugarcane, 1999
and xi = mean acreage in sugarcane, 1997
plus we know µx = mean acreage across all 32 counties
µˆy =
V̂ (µˆy ) =
Shivon Sue-Chee Ratio and Regression Estimation 41
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 42
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 43
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 44
Improved estimation using regression (§6.6)
Ratio estimation:
- uses τx/
∑n
i=1 xi to improve estimation of µy or τy
- works well when y ∝ x
Regression estimation:
- can be used if y − a ∝ x
-there is a linear relationship between y and x but not necessarily
through the origin
- get estimate of µy
Example 6.9: x- score on SAT math, y - final grade in calculus course
Class example: x- handspan, y -height
Shivon Sue-Chee Ratio and Regression Estimation 45
Regression estimation formulas (§6.6)
Regression estimation: used instead of ratio estimation if there is a
linear relationship between y and x but not necessarily through the
origin
Regression line: yˆi = a + bxi
By least squares method:
a = y¯ − bx¯ and b =
∑n
i=1(yi − y¯)(xi − x¯)∑n
i=1(xi − x¯)2
Alternatively, yˆi = y¯ + b(xi − x¯)
To estimate µy by linear(L) regression, using µx at xi ,we get
µˆyL = y¯ + b(µx − x¯)
V̂(µˆyL) =
Shivon Sue-Chee Ratio and Regression Estimation 46
Regression estimation formulas (§6.6)
Regression estimation: used instead of ratio estimation if there is a
linear relationship between y and x but not necessarily through the
origin
Regression line: yˆi = a + bxi
By least squares method:
a = y¯ − bx¯ and b =
∑n
i=1(yi − y¯)(xi − x¯)∑n
i=1(xi − x¯)2
Alternatively, yˆi = y¯ + b(xi − x¯)
To estimate µy by linear(L) regression, using µx at xi ,we get
µˆyL = y¯ + b(µx − x¯)
V̂(µˆyL) =
Shivon Sue-Chee Ratio and Regression Estimation 47
Regression estimation formulas (Toolbox 6.24-6.26)
µˆyL = y¯ + b(µx − x¯)
V̂(µˆyL) =
(
1− n
N
) 1
n
∑n
i=1(yi − a− bxi )2
n − 2 =
(
1− n
N
) MSE
n
note on notation:
MSE =
∑n
i=1(yi − a− bxi )2
n − 2 =
SSE
n − 2
many books call this MSR = SSR/(n − 2) for “mean square
residuals” and “sum of squared residuals”
because MSE usually means “variance + bias-squared”= the mean
squared error of a biased estimator
Shivon Sue-Chee Ratio and Regression Estimation 48
Example 6.9
Simple Least Squares Regression
> achieve=scan()
1: 39 43 21 64 57 47 28 75 34 52
11:
Read 10 items
> calc = scan()
1: 65 78 52 82 92 89 73 98 56 75
11:
Read 10 items
> plot(achieve, calc, main="Figure 6.6")
> # ordinary least squares regression
> fit=lm(calc~achieve)
> summary(fit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 40.7842 8.5069 4.794 0.00137 **
achieve 0.7656 0.1750 4.375 0.00236 **
Residual standard error: 8.704 on 8 degrees of freedom
Multiple R-squared: 0.7052,Adjusted R-squared: 0.6684
F-statistic: 19.14 on 1 and 8 DF, p-value: 0.002365
20 30 40 50 60 70
60
70
80
90
Figure 6.6
achieve
ca
lc
Shivon Sue-Chee Ratio and Regression Estimation 49
Example 6.9
‘Deviations from sample average’ Regression
> # least squares regression on mean centred data
> fittedmodel = lm(calc~ I(achieve - mean(achieve)))
> # I( ... ) treats the arguments numerically
> summary(fittedmodel)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 76.0000 2.7523 27.613 3.19e-09 ***
I(achieve - mean(achieve)) 0.7656 0.1750 4.375 0.00236 **
Residual standard error: 8.704 on 8 degrees of freedom
Multiple R-squared: 0.7052,Adjusted R-squared: 0.6684
F-statistic: 19.14 on 1 and 8 DF, p-value: 0.002365
> 2*sqrt((1-10/486)*(sum(residuals(fittedmodel)^2)/8)/10)
[1] 5.447734
> #Knowing mu_x=52, estimate mu_y
> mean(calc)+ 0.7656*(52-mean(achieve))
[1] 80.5936
Shivon Sue-Chee Ratio and Regression Estimation 50
Regression Example 6.9 Summary
Figure 6.6: least squares line from sample has intercept a = 40.7842
and slope b = 0.7656
y¯=76, x¯=46
µx=52, µˆyL=80.6
V̂(µˆyL)=7.42
Margin of error: 2
√
V̂(µˆyL)=5.45
Check residual plot to determine whether the simple linear model is
an appropriate fit.
See Figure 6.7: the linear model seems appropriate; no obvious
pattern or outlier values.
Shivon Sue-Chee Ratio and Regression Estimation 51
Summary
Ratio estimation is most appropriate when the relationship between y
and x is linear through the origin.
Regression estimator performs better than ratio estimator when the
population relationship moves away from a straight line through the
origin (i.o.w., intercept close to zero).
As the population relationship exhibits more curvature, the regression
estimator becomes more biased.
Shivon Sue-Chee Ratio and Regression Estimation 52
Summary
Ratio estimation is most appropriate when the relationship between y
and x is linear through the origin.
Regression estimator performs better than ratio estimator when the
population relationship moves away from a straight line through the
origin (i.o.w., intercept close to zero).
As the population relationship exhibits more curvature, the regression
estimator becomes more biased.
Shivon Sue-Chee Ratio and Regression Estimation 53
Summary
Ratio estimation is most appropriate when the relationship between y
and x is linear through the origin.
Regression estimator performs better than ratio estimator when the
population relationship moves away from a straight line through the
origin (i.o.w., intercept close to zero).
As the population relationship exhibits more curvature, the regression
estimator becomes more biased.
Shivon Sue-Chee Ratio and Regression Estimation 54
Homework
Read §6.1 – §6.8
EX: 6.1, 6.2, 6.6, 6.9, 6.12, 6.13, 6.14, 6.16
Sampling from real populations: EX 6.4
Shivon Sue-Chee Ratio and Regression Estimation 55
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