R语言实战：回归与方差分析详解

语言回归分析

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1.3. HISTORY

We have added the y



x (solid) line to the plot. Now a student scoring, say one standard deviation

above average on the midterm might reasonably expect to do equally well on the ﬁnal. We compute the

least squares regression ﬁt and plot the regression line (more on the details later). We also compute the

correlations.

> g <- lm(final ˜ midterm,stat500)

> abline(g$coef,lty=5)

> cor(stat500)

midterm final hw total

midterm 1.00000 0.545228 0.272058 0.84446

final 0.54523 1.000000 0.087338 0.77886

hw 0.27206 0.087338 1.000000 0.56443

total 0.84446 0.778863 0.564429 1.00000

We see that the the student scoring 1 SD above average on the midterm is predicted to score somewhat

less above average on the ﬁnal (see the dotted regression line) - 0.54523 SD’s above average to be exact.

Correspondingly, a student scoring below average on the midterm might expect to do relatively better in the

ﬁnal although still below average.

If exams managed to measure the ability of students perfectly, then provided that ability remained un-

changed from midterm to ﬁnal, we would expect to see a perfect correlation. Of course, it’s too much to

expect such a perfect exam and some variation is inevitably present. Furthermore, individual effort is not

constant. Getting a high score on the midterm can partly be attributed to skill but also a certain amount of

luck. One cannot rely on this luck to be maintained in the ﬁnal. Hence we see the “regression to mediocrity”.

Of course this applies to any







situation like this — an example is the so-called sophomore jinx

in sports when a rookie star has a so-so second season after a great ﬁrst year. Although in the father-son

example, it does predict that successive descendants will come closer to the mean, it does not imply the

same of the population in general since random ﬂuctuations will maintain the variation. In many other

applications of regression, the regression effect is not of interest so it is unfortunate that we are now stuck

with this rather misleading name.

Regression methodology developed rapidly with the advent of high-speed computing. Just ﬁtting a

regression model used to require extensive hand calculation. As computing hardware has improved, then

the scope for analysis has widened.

2.3. MATRIX REPRESENTATION

is linear but





is not. Some relationships can be transformed to linearity — for example y



ε can be linearized by

taking logs. Linear models seem rather restrictive but because the predictors can transformed and combined

in any way, they are actually very ﬂexible. Truly non-linear models are rarely absolutely necessary and most

often arise from a theory about the relationships between the variables rather than an empirical investigation.

2.3 Matrix Representation

Given the actual data, we may write









but the use of subscripts becomes inconvenient and conceptually obscure. We will ﬁnd it simpler both

notationally and theoretically to use a matrix/vector representation. The regression equation is written as



Xβ



where y









, ε









, β









and









1 x

 

1 x







The column of ones incorporates the intercept term. A couple of examples of using this notation are the

simple no predictor, mean only model y



































We can assume that Eε



0 since if this were not so, we could simply absorb the non-zero expectation for

the error into the mean µ to get a zero expectation. For the two sample problem with a treatment group

having the response y



with mean µ

and control group having response z

 

with mean µ

have























1 0



1 0

0 1

 

0 1





























2.4 Estimating β

We have the regression equation y



Xβ



ε - what estimate of β would best separate the systematic com-

ponent Xβ from the random component ε. Geometrically speaking, y



while β



where p is the

number of parameters (if we include the intercept then p is the number of predictors plus one).

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