![6 INTRODUCTION
random variables Y1, Y2, . . . , Yn. Greek letters are used to denote parameters
and the corresponding lower case roman letters are used to denote estimators
and estimates; occasionally the symbol b is used for estimators or estimates.
For example, the parameter β is estimated by b
β or b. Sometimes these con-
ventions are not strictly adhered to, either to avoid excessive notation in cases
where the meaning should be apparent from the context, or when there is a
strong tradition of alternative notation (e.g., e or ε for random error terms).
Vectors and matrices, whether random or not, are denoted by bold face lower
and upper case letters, respectively. Thus, y represents a vector of observations
y1
.
.
.
yn
or a vector of random variables
Y1
.
.
.
Yn
,
β denotes a vector of parameters and X is a matrix. The superscript T
is
used for a matrix transpose or when a column vector is written as a row, e.g.,
y = [Y1, . . . , Yn]
T
.
The probability density function of a continuous random variable Y (or the
probability mass function if Y is discrete) is referred to simply as a proba-
bility distribution and denoted by
f(y; θ)
where θ represents the parameters of the distribution.
We use dot (·) subscripts for summation and bars (−
) for means; thus,
y =
1
N
N
X
i=1
yi =
1
N
y · .
The expected value and variance of a random variable Y are denoted by
E(Y ) and var(Y ), respectively. Suppose random variables Y1, . . . , YN are in-
dependent with E(Yi) = µi and var(Yi) = σ2
i for i = 1, . . . , n. Let the random
variable W be a linear combination of the Yi’s
W = a1Y1 + a2Y2 + . . . + anYn, (1.1)
where the ai’s are constants. Then the expected value of W is
E(W) = a1µ1 + a2µ2 + . . . + anµn (1.2)
and its variance is
var(W) = a2
1σ2
1 + a2
2σ2
2 + . . . + a2
nσ2
n. (1.3)](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-24-320.jpg)
![DISTRIBUTIONS RELATED TO THE NORMAL DISTRIBUTION 7
1.4 Distributions related to the Normal distribution
The sampling distributions of many of the estimators and test statistics used
in this book depend on the Normal distribution. They do so either directly be-
cause they are derived from Normally distributed random variables or asymp-
totically, via the Central Limit Theorem for large samples. In this section
we give definitions and notation for these distributions and summarize the
relationships between them. The exercises at the end of the chapter provide
practice in using these results which are employed extensively in subsequent
chapters.
1.4.1 Normal distributions
1. If the random variable Y has the Normal distribution with mean µ and
variance σ2
, its probability density function is
f(y; µ, σ2
) =
1
√
2πσ2
exp
"
−
1
2
y − µ
σ
2
#
.
We denote this by Y ∼ N(µ, σ2
).
2. The Normal distribution with µ = 0 and σ2
= 1, Y ∼ N(0, 1), is called the
standard Normal distribution.
3. Let Y1, . . . , Yn denote Normally distributed random variables with Yi ∼
N(µi, σ2
i ) for i = 1, . . . , n and let the covariance of Yi and Yj be denoted by
cov(Yi, Yj) = ρijσiσj,
where ρij is the correlation coefficient for Yi and Yj. Then the joint distri-
bution of the Yi’s is the multivariate Normal distribution with mean
vector µ = [µ1, . . . , µn]T
and variance-covariance matrix V with diagonal
elements σ2
i and non-diagonal elements ρijσiσj for i 6= j. We write this as
y ∼ MVN(µ, V), where y = [Y1, . . . , Yn]
T
.
4. Suppose the random variables Y1, . . . , Yn are independent and Normally
distributed with the distributions Yi ∼ N(µi, σ2
i ) for i = 1, . . . , n. If
W = a1Y1 + a2Y2 + . . . + anYn,
where the ai’s are constants, then W is also Normally distributed, so that
W =
n
X
i=1
aiYi ∼ N
n
X
i=1
aiµi,
n
X
i=1
a2
i σ2
i
!
by equations (1.2) and (1.3).
1.4.2 Chi-squared distribution
1. The central chi-squared distribution with n degrees of freedom is de-
fined as the sum of squares of n independent random variables Z1, . . . , Zn](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-25-320.jpg)
![Plate VIII
The fringe of the robe from the Tomb of Charlemagne. To face p. 291.
Plate VII shows one medallion of a piece of silk found on the body
of Charlemagne when the grave was opened in the time of the
present German Emperor. It is certainly not of Charlemagne’s time.
But it seems a fairly safe guess to suppose that his well-known
regard for his favourite beast Abulabaz, who died only four years
before him, caused his son to have the body wrapped in one of the
robes decorated with elephants which we know that he possessed;
and that either in the year 1000, when Otho III opened the tomb, or
in 1166, under Barbarossa, when Charlemagne was canonized, this
piece of silk replaced the decayed robe originally buried there. We
know of the two elephant-robes referred to from Anastasius[270], who
gives an enormous list of the art works in gold and silver and silk and
cloth of gold which were wrought for Leo III, Charlemagne’s
contemporary. One item is “two robes of Syrian purple, with borders
of cloth of gold wrought with elephants”. These robes Leo gave to
Charlemagne.
We can all but give the exact date of this remarkable Byzantine
beast. The inscription breaks off exactly where the date came. The
Greek inscription worked in the stuff (Plate VIII) sets forth that it was
made “under Michael the great chamberlain and controller of the
privy purse of the emperor, when Peter was the manager of](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-27-320.jpg)
![Zeuxippos”, i. e. the Byzantine court factory in Negropont. Then
comes the tantalizing Indictionos (? B), and the date is lost.
Dreger, in his Europäische Weberei und Stickerei,[271] gives some
early examples of elephants in art. His Figure 37b shows an archaic
silver relief of an elephant with a castle containing armed men. His
Figure 37a shows a silk stuff of the seventh or eighth century, of
Asiatic manufacture, with circular medallions containing elephants,
griffins and winged horses, hippogryffs; and he remarks that “the
elephant is one of the most holy beasts of Buddhism”. This silk stuff
is shown in our Plate IX from a photograph of the original. A
comparison of these elephants with the elephant shown in Plate VIII
makes it fairly clear that the Charlemagne stuff is later than the other,
while in all of the details of the beast itself, ears, three toes, eye,
trunk, they are exactly the same. Each has a tree behind the
elephant; but while the Charlemagne tree is a piece of stiff
conventional work, the other is a natural tree with leaves and fruit,
much resembling the vegetable ornamentation of some early
Egyptian stuffs. Another feature pointing in the same direction is the
thirty-two conventional patterns on the circular enclosing border.
These in the earlier piece are twenty-eight plain disks.
There is an example of sculptured elephants something like this
one, but much more like the real beast, especially about the feet.
The elephants are the legs of the ivory chair[272] of Urso, at
Canossa; he was Bishop of Bari and Canossa 1078-89.
Something should be said about the language spoken by the
people of France and Germany in the times with which we are
dealing, the reference to a rustic tongue being not infrequent.
In the Council convened by Charlemagne at Tours in the year 813,
equally representing Eastern France and Western France, Austrasia
and Neustria, Germany and the Galliae, the bishops in the
Transalpine Empire were enjoined to be diligent in preaching, and to
take care that their discourses should be rendered either into
Romana Rustica or into Theotisc or Deutsch, that all might
understand. It may be of interest to give the earliest specimens we](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-28-320.jpg)
![In 841, after the dreadful battle of Fontenai near Vézelay in
Burgundy, where Charles-le-Chauve and Louis-le-Germanique
combined against their brother Lothar and their nephew Pepin and
defeated them, they held a Congress at Strassburg to confirm their
alliance.
Louis and Charles each made announcement in Latin of the
purpose of their agreement, and of their intention to take in public an
oath each to other. That done, Louis, as the elder, first took the oath.
Being the ruler of the German portion of the empire, he took the oath
in the language of the Franks, the Romance tongue, Rustica
Romana, in order that the adherents of Charles might hear and
understand his undertaking. These were the words of his oath,
probably read by a chancellor, for the Latin account[273] says haec
se servaturum testatus est:—
“Pro Deo amur et pro christian poblo et nostro commun salvament,
dist di[274] in avant, in quant Deus savir et podir me dunat[275], si
salvarai eo cist meon fradre Karlo, et in aiudha et in cadhuna[276]
cosa, si cum om[277] per dreit[278] son fradra salvar dist, in o quid il
mi altresi fazet[279]; et ab Ludher nul plaid numquam prindrai, qui
meon vol[280] cist meon fradre Karle in damno sit.”
Then Charles said the same in the language of the Germans, the
Teudisc or Deutsch tongue. The Latin account uses a different
phrase here, haec eadem verba testatus est.
“In Godes minna ind in thes christianes folches ind unser bedhero
gehaltnissi, fon thesemo dage frammordes, so fram so mir Got
geuuizci indi mahd furgibit, so haldi thesan minan bruodher, soso
man mit rehtu sinan bruodher scal, in thiu, thaz er mig so sama duo;
indi mit Ludheren in nohheiniu thing ne gegango, the minan uuillon
imo ce scadhen uuerdhen.”
The peoples then swore an oath, each in their own, not the
other’s, tongue. The Frank people swore in the Romance language:
—](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-30-320.jpg)
![“Si Lodhuvigs sagrament, que son fradre Karlo iurat, conservat, et
Karlus meos sendra de suo part non los tanit, si io returnar non l’int
pois: ne io ne neuls, cui eo returnar int pois, in nulla aiudha contra
Lodhuuuig nun li iv er.”
The others then swore in the Teudisc language:—
“Oba Karl then eid, then er sinemo bruodher Ludhuuuige gesuor,
geleistit, indi Ludhuuuige min herro then er imo gesuor forbrihchit, ob
ih inan es iruuenden ne mag: noh ih no thero nohhein, then ih es
iruuenden mag, uuidhar Karle imo ce follusti ne uuirdhit.”
Plate X
The abrenuntiatio diaboli of Archbishop Boniface. To face p. 295.
An example of language nearly a hundred years earlier than this is
found in the renunciation of the devil and the declaration of belief in
God which our own Boniface required of his converts from
paganism. The form is found attached to the decrees of a
Council[281] held by Boniface, probably in the year 743. It exists in a](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-31-320.jpg)
![The suggested date of the letter is a.d. 757-786.
Mention was made on page 57 of the inscriptions which exist on
the great shaft of a cross in the churchyard of Bewcastle in
Cumberland. These inscriptions are the earliest extant pieces of
English prose. They give the names of the King of Mercia, Wulfhere,
his queen and her sister, with the date “first year of Ecgfrith King of
this realm”, that is, a.d. 670. We have another inscription dated in
Ecgfrith’s reign, that, namely, on the dedication stone of the basilica
of St. Paul at Jarrow, “in the 15th year of King Ecgfrith and the fourth
year of Abbat Ceolfrid”, so that the manner of dating the Bewcastle
cross was that usual at the time; the Jarrow inscription is in
Latin[282]. Plate XI shows a facsimile of all except the two top lines
(which were beyond my reach) of the main inscription on the
Bewcastle cross, a copy of which is given in a note on page 57. The
runes on Plate XI begin with the gar of Wothgar, the second of the
three persons who “set up this slender token of victory in memory of
Alchfrith once King and son of Oswy”, the half-brother of King
Ecgfrith; mention has been made of him on page 9.](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-33-320.jpg)
![probably about fourteen years later. King Ecgfrith was slain by the
Picts in 685, and the Angles were never dominant in the south-west
of Scotland after his death. Plate XII shows a portion of the many
runes on this great monument, which is described at pages 235-254
of my little book on Theodore and Wilfrith. Reading across the top
and down the right side the runes are as follows:—
Krist wæs on rodi hwethræ ther fusæ fearran kwomu æththilæ til
anum ic thæt al bih[eald]. Christ was on the cross, and there
hastening from far came they to the noble prince. I that all beh[eld].
Beginning at the top again and reading down the left side, we
have:—
Mith strelum giwundad alegdun hiæ hinæ limwoerignæ gistoddun
him (æt his licæs heafdum). With missiles wounded, they laid Him
down limb-weary, they stood at His body’s head.](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-36-320.jpg)
![Ep. 147. June 26,
800.
CHAPTER XVIII
Alcuin’s latest days.—His letters mention his ill health.—His appeals for the
prayers of friends, and of strangers.—An affectionate letter to Charlemagne.—The
death scene.
Alcuin’s health began to break in the later part of
the year 800, or early in 801. In June, 800, he
wrote a letter to Arno of Salzburg which shows that
he had in the first days of the month travelled with Karl from Tours to
Aix by way of Orleans and Paris, and after a debate with Felix the
Adoptionist had returned to Tours. We do not find in this long letter
any mention of failing health. Indeed, he overflows with
affectionateness, a feeling always displayed in his letters to Arno. “I
am sending to your dearness three little gifts; a tent to protect your
venerated head from the rain[283], a bed-cover to keep warm your
sacred breast, and a glass in which your bread may be dipped at
table, that whenever they are used they may bring to your sanctity a
recollection of my name.”
In this letter he describes his debate with Felix.
“I have had a great dispute with the heretic Felix in presence of the
lord king and holy fathers. He was obdurate; would recognize the
authority of no one who took an opposite view; held himself to be
wiser than all in this, that he was more foolish than all. But the divine
clemency touched his heart; he confessed that he had of late been
carried away by a false opinion; he professed that he held firmly the
Catholic faith. We could not see into his mind, and we left the cause
to the Judge of secret things. We handed him over to Laidrad [the
Bishop of Lyon (798-814)] our dearest son, who is to keep him and
see whether it is true that he believes, and whether he will write
letters condemning the heresy which he has preached. The king had
intended to send him to Archbishop Riculf [of Maintz] to be kept and
chastised; and his presbyter, who is worse than his master, was to](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-37-320.jpg)
![Ep. 189. May,
802.
Ep. 189. May,
802.
Ep. 193.
be sent to you and your providence. But now that they say they are
converted to the Catholic faith, they have been handed over to
Laidrad, who is to test their sincerity.”
On May 24, 801, Alcuin received a letter from
Arno, Archbishop of Salzburg, at nine o’clock in the
morning, and the messenger told him he must
leave again at three in the afternoon. In the course of the six hours
he dictated fourteen paragraphs in reply. One of these concerned his
health. “My Candidus has been able to tell you all about my
weakness. It is therefore superfluous to write on the subject, except
to say that all bodily fitness has left me, and pleasures of the world
have fled far away.”
The interesting remark in this letter that the messenger from
Salzburg to Tours, a distance of some six hundred miles, must go
back in six hours is not the only interesting detail. We learn, also,
that many letters were lost in the difficulties of the journey. The eyes
of the Sassenach of to-day, who rides some forty miles in a Scottish
mail-cart in Sutherland, and sees the letters shied out into kailyards
and steadings, are opened to the possibilities of loss in primitive
methods of letter-carrying. The admirable arrangements of the early
Roman empire, for conveyance of men and things, had been thrown
into chaos long before Alcuin’s time, and special messengers, or
“runners”, were used by important people for the transmission of
letters.
“To Arno. My devotedness is greatly grieved by
the unfaithfulness of those whom I have trusted
with letters to you. Last year I sent to you on your
return from Italy two letters, and I also sent to you other two to meet
you on your arrival at the palace [Aachen]. I do not know that any of
them reached your presence.”
Alcuin wrote to the Emperor Charles in 802, or
possibly in 803, begging that he might be allowed
to stay quietly at St. Martin’s, Tours. “I am so very weak in body that I
am unequal to any more travelling or labour. To speak truth, all the
fitness and strength of my body has left me; it has gone, and day by](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-38-320.jpg)
![Ep. 194.
Ep. 196. a.d. 802-
3.
Ep. 198. a.d. 802-
3.
Ep. 230.
Ep. 271. a.d. 804.
day will be further away; I fear it will never come back to me in this
world.”
Again, writing to Arno in 802 or 803, he tells him
how he longs to see him at St. Martin’s, “not for the
sake of your black hair, but for your most sweet eyes and lovable
talk.” Though bidden to the palace, where he would have met him,
his poor little body was too weak for the journey: he could not go.
In another letter to Arno he writes: “I have been
summoned to my lord David [Charlemagne], but
my bodily weakness prevented my going: the will of
God detained me.” We have his letter of excuse to
the emperor. He begins with the simile of the aged
soldier, unable not only to bear the weight of armour, but even to
support his own body. Then he proceeds: “To speak simply, let not
the mind of my lord be inflamed against me for my delay; I am not
strong enough to come. A more favourable opportunity may occur.”
He became more than ever pressing in his
entreaties that his friends would pray for him. In
seeking for the prayers of others we find him turning to a part of
England of which we do not appear to have any other mention in his
letters, namely, Norfolk and Suffolk, called then the dioceses of
Elmham and Dunwich (see page 159). In like manner, and for a like
purpose, he wrote to the brethren of Candida Casa, i. e. Whithorn in
Galloway, the following letter:—
“I pray the unanimity of your piety to have my
name in memory. Deign to intercede for my
littleness in the church of your most holy father Nynia the bishop.
“He shone bright with many virtues, as has recently been related
to me by a skilful poem which our faithful disciples the scholars of
the church of York have sent. In that poem I have discerned in that
which I have read there both the skill of the writer and the holiness of
him who wrought the miracles. Wherefore, I pray you, by your holy
intercessions to commend me to his prayers, that by the most holy
prayers of the same your father, and by the assiduous intercessions
of your love, I may receive pardon for my sins, by the mercy of the](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-39-320.jpg)
![Ep. 170. a.d. 801,
early autumn.
God Christ, and may come to the communion of saints who have
bravely conquered the labours of the world, and have received the
crown of perpetual praise.
“I send to the body of our holy father Nyniga (sic) a robe of whole
silk, that my name may be remembered, and that I may merit to have
always the pious intercession both of him and of you.
“May Christ’s right hand protect and rule you, brothers.”
Here is a very touching letter, which sets clearly before our eyes
the dear affectionate old man—old as men then counted age—
beaten at last by bodily weakness, while his heart was as loving as
ever. It is addressed to “the most longed for lord David, most worthy
of all honour.”
“Day by day, with hungry intentness of heart, my
ears hanging on the words of messengers, I
wondered anxiously what they could tell me of my
most sweet lord David: when he would come home; when he would
return to his own land. At last, though late, the wished-for voice
sounded in the ears of my desire: ‘He will soon come. He has
already crossed the Alps, he whose presence thou hast desired, O
Albinus, with such fervour of mind.’ And then I cried over and over
again with tearful voice: ‘O Lord Jesu, why dost thou not give me the
wings of an eagle? Why dost thou not grant me the translation of the
prophet Abacuc[284] for one day, or even one hour, that I might
embrace and kiss the steps of him my dearest one, and—above all
that can be loved in this world—see the most clear eyes of my
sweetest one, and hear his most joyous words. And why dost thou,
mine enemy of fever, oppress me at this inopportune time, and not
permit me to have my wonted alacrity of body, so that, though tardily,
that might be accomplished which promptly it cannot do.’”
The dates and the story of his final illness and his death are found,
as we have seen (Ch. II), in the life written about twenty years later
by a pupil of Alcuin’s favourite priest Sigulf, and more concisely in
the Annals of Pettau, a monastery not far from Salzburg, and
therefore likely to be well-informed. Some of the touching facts
should be repeated here.](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-40-320.jpg)
![Ep. 186. a.d. 801-
2.
APPENDIX A
(Page 26)
It would appear that when Alcuin was not allowed by Charlemagne
to retire to Fulda, as he had wished to do, an impulse of affectionate
responsibility brought him to pour himself out in advice and help to
those with whom he had hoped to spend his last days. This is his
letter to the monks of Fulda.
“To the most holy, and by us with all love to be
cherished, the brethren of the holy Boniface[285],
our father and protector, the humble levite Alchuin
wishes eternal beatitude in Christ.
“I am mindful of your most sweet love, with which you most
benignantly received me long ago with all joy. Greatly as I then was
glad in your presence, so greatly is my mind now tortured in your
absence, desiring to see you whom it loves, to have present you
whom it esteems. Since this is denied to the eyes of the flesh, let
love be made perpetual by spiritual presence; love which can come
to an end has never been true love.
“Let us therefore aim at that which is never to have an end, where
is blessed eternity and eternal blessedness. That ye may deserve to
attain to this, let no labour affright you, no blandishments of this life
keep you back. Let there always burn in your hearts the love of Him
that appeared as their companion on the way to the two apostles,
who, when He was removed from their carnal eyes, said ‘Did not our
hearts burn within us, while He talked with us by the way, and while
He opened to us the Scriptures?’ In the writings of the holy fathers
let us seek Him whom they, not yet learned in the Scriptures,
understood. Now all is open; now He has opened the meaning of
Whom it was said ‘Then opened He their understanding, that they
might understand the Scriptures.’ Now the gospel truth shines forth](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-42-320.jpg)
![in all the world; now the enigmas of the prophets are clearer than the
sun in the churches of Christ. This light of truth follow ye with your
whole soul and understand Christ; in it love Christ, follow Christ; that
cleaving to His most sacred footsteps ye may merit to have in His
most holy presence life eternal.
“Be mindful of the apostolic mandate,[286] ‘My brethren, be ye
stedfast, unmovable; always in the work of the Lord; forasmuch as
ye know that your labour is not in vain in the Lord.’ Be stedfast in
your own place and in the devotion of your purpose. Leave not your
most holy father. Stand about his sepulchre, that he may offer your
prayers to Almighty God. Desire not the vanities of the world, but
love celestial blessings. ‘And,’ as the teacher of the Gentiles says,
[287] ‘be not conformed to this world, but be ye transformed by the
renewing of your mind.’ It is a base thing for a monk to lose the
spiritual warfare and to immerse himself in the affairs of the world.
“Let there be no murmurings among you, no hatreds, no envyings,
no evil speakings.[288] Judge not one another. Let everything be
done in humility and concord, in obedience to those set over you, not
to the eye only but from the heart, as in the presence of God. Let
your obedience, your love, your humility be known to all, that very
many may be taught by your good examples and may advance in
the salvation of their souls.
“If the venerable father Bouulf,[289] my most loved friend, is unable
by reason of his weakness to observe the full hardness of the life by
rule, judge ye not him, but obey him from your heart and love him as
a father, for he will have to give account of your souls. He labours for
you in wanderings and journeyings, that you may live quiet and keep
the life by rule and have what is necessary for your bodies. Do you
act as very dear sons. Fear God, love God, and have care of your
most holy father in your prayers, that he may live in long prosperity
with you, and that he with you and you with him may merit to have
everlasting life.
“Warn, instruct, teach your young men in all holy discipline and
Catholic doctrine; that they may be held worthy to stand in the place
of you and to send up prayers for you wherever you may remain.](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-43-320.jpg)
![Warn them about chastity of body, about confession of their sins,
about study and manual labour without murmur, and about all things
which seem necessary at their age. And let them become subject to
their elders and masters in good humility, in most pious religion. And
do you who are older afford to them good examples, so that they
may learn not from your words only but by the religion of your life.
Let them not be given to luxury, not slaves to drink, not despisers,
not following empty games; but let them learn to be good servants in
the house of God, that by the intercession of holy Boniface their
father they may deserve to receive from the God Christ blessing and
favour.
“And as to myself, I pray you have me in perpetual recollection
with yourselves in your holy prayers. For the time is at hand which
no man can escape. Let each one prepare himself, that he may
appear in the presence of his God not naked but clothed with good
practices.
“I have sent a pall for the body of the holy Boniface our father, on
whose holy intercession for my sins I place great reliance; that I, a
sinner, may even merit pardon in that day, when your holiness shall
receive the crown of eternal blessedness.
“To you, O most holy presbyters, I have sent a little collection of
words for the Mass, for use on various days on which any one
desires to offer prayers to God, whether in honour of the Holy
Trinity[290], or in love of wisdom, or in tears of penitence, or in perfect
love, or asking for angelic support, or in address to any one of all the
saints; or if any one wish to offer prayers for his own sins, or for any
living friend, or for many friends, or for brothers departing this life; or,
especially, when one wishes to invoke the intercessions of blessed
Mary, mother of God, ever virgin; or when any desires to chant and
invoke by his prayers the most pious presence of the most holy
Boniface your father. All these things we have been at the pains to
send to you by the intuition of love, praying your humility to receive
benignantly that which with the fullest love we send you. Let each
make of it such use as each pleases; and blame me not in this office
of love. Let each be fully persuaded in his own mind[291] and do](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-44-320.jpg)
![Ep. 10. a.d. 786.
APPENDIX B
(Page 91)
We have the report which the legates George and Theophylact
sent to Pope Hadrian on their mission. No reference is made in it to
the matter of the Archbishopric of Lichfield. Iaenbricht is still the sole
southern archbishop, and Higbert of Lichfield is only bishop.
“Your holy prayers favouring us, we set sail with
joyous countenance obeying your commands. But
the tempter hindered us with a contrary wind. He who stills the
waves hearkened unto your deprecatory entreaty, calmed the blue
strait, led us across to a safe haven, and brought us to the shore of
the English unharmed, but afflicted with many dangers.
“We were received first by Iaenberht, Archbishop of the holy
church of Dorovernia,[292] whose other name is Cantia, where the
holy Augustine rests in the body; dwelling there we gave him the
necessary information.
“Going on thence, we arrived at the dwelling of Offa, King of the
Mercians. With great joy, for reverence of the blessed Peter and
honour of your apostolate, he received both us and the messages
sent from the highest see. Then Offa the King of the Mercians, and
Cynewulf the King of the West Saxons, came together in a council to
which we delivered your holy writings; and they forthwith promised
that they would correct the vices named.[293] Then, after counsel
held with the said kings, pontiffs, and elders of the land, considering
that that corner of the world stretches far and wide, we gave
permission to Theophylact, the venerable bishop, to go to the King of
the Mercians[294] and the parts of Britain.
“I for my part, taking with me the companion whom your most
excellent King Karl sent with us out of reverence to your apostolate,](https://image.slidesharecdn.com/9402-250316212005-0c9587fd/85/An-Introduction-to-Generalized-Linear-Models-3rd-Edition-Annette-J-Dobson-46-320.jpg)
An Introduction to Generalized Linear Models 3rd Edition Annette J. Dobson
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- 5. An Introduction to Generalized Linear Models 3rd Edition Annette J. Dobson Digital Instant Download Author(s): Annette J. Dobson, Adrian Barnett ISBN(s): 9781584889502, 1584889500 Edition: 3 File Details: PDF, 4.21 MB Year: 2008 Language: english
- 7. An Introduction to Generalized Linear Models Third Edition C9500_FM.indd 1 4/8/08 9:54:27 AM
- 8. Analysis of Failure and Survival Data P. J. Smith The Analysis of Time Series — An Introduction, Sixth Edition C. Chatfield Applied Bayesian Forecasting and Time Series Analysis A. Pole, M. West and J. Harrison Applied Nonparametric Statistical Methods, Fourth Edition P. Sprent and N.C. Smeeton AppliedStatistics—HandbookofGENSTAT Analysis E.J. Snell and H. Simpson Applied Statistics — Principles and Examples D.R. Cox and E.J. Snell Bayesian Data Analysis, Second Edition A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin Bayesian Methods for Data Analysis, Third Edition B.P. Carlin and T.A. Louis Beyond ANOVA — Basics of Applied Statistics R.G. Miller, Jr. Computer-Aided Multivariate Analysis, Fourth Edition A.A. Afifi and V.A. Clark A Course in Categorical Data Analysis T. Leonard A Course in Large Sample Theory T.S. Ferguson Data Driven Statistical Methods P. Sprent Decision Analysis — A Bayesian Approach J.Q. Smith Elementary Applications of Probability Theory, Second Edition H.C. Tuckwell Elements of Simulation B.J.T. Morgan Epidemiology — Study Design and Data Analysis, Second Edition M. Woodward Essential Statistics, Fourth Edition D.A.G. Rees Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models J.J. Faraway A First Course in Linear Model Theory N. Ravishanker and D.K. Dey Generalized Additive Models: An Introduction with R S. Wood Interpreting Data — A First Course in Statistics A.J.B. Anderson An Introduction to Generalized Linear Models, Third Edition A.J. Dobson and A.G. Barnett Introduction to Multivariate Analysis C. Chatfield and A.J. Collins Introduction to Optimization Methods and Their Applications in Statistics B.S. Everitt Introduction to Probability with R K. Baclawski Introduction to Randomized Controlled Clinical Trials, Second Edition J.N.S. Matthews Introduction to Statistical Methods for Clinical Trials Thomas D. Cook and David L. DeMets Large Sample Methods in Statistics P.K. Sen and J. da Motta Singer Linear Models with R J.J. Faraway Markov Chain Monte Carlo — Stochastic Simulation for Bayesian Inference, Second Edition D. Gamerman and H.F. Lopes Mathematical Statistics K. Knight CHAPMAN & HALL/CRC Texts in Statistical Science Series Series Editors Bradley P. Carlin, University of Minnesota, USA Julian J. Faraway, University of Bath, UK Martin Tanner, Northwestern University, USA Jim Zidek, University of British Columbia, Canada C9500_FM.indd 2 4/8/08 9:54:27 AM
- 9. Modeling and Analysis of Stochastic Systems V. Kulkarni Modelling Binary Data, Second Edition D. Collett Modelling Survival Data in Medical Research, Second Edition D. Collett MultivariateAnalysisof VarianceandRepeated Measures — A Practical Approach for Behavioural Scientists D.J. Hand and C.C. Taylor Multivariate Statistics — A Practical Approach B. Flury and H. Riedwyl Practical Data Analysis for Designed Experiments B.S. Yandell Practical Longitudinal Data Analysis D.J. Hand and M. Crowder Practical Statistics for Medical Research D.G. Altman A Primer on Linear Models J.F. Monahan Probability — Methods and Measurement A. O’Hagan Problem Solving — A Statistician’s Guide, Second Edition C. Chatfield Randomization, Bootstrap and Monte Carlo Methods in Biology, Third Edition B.F.J. Manly Readings in Decision Analysis S. French Sampling Methodologies with Applications P.S.R.S. Rao Statistical Analysis of Reliability Data M.J. Crowder, A.C. Kimber, T.J. Sweeting, and R.L. Smith Statistical Methods for Spatial Data Analysis O. Schabenberger and C.A. Gotway Statistical Methods for SPC and TQM D. Bissell Statistical Methods in Agriculture and Experimental Biology, Second Edition R. Mead, R.N. Curnow, and A.M. Hasted Statistical Process Control — Theory and Practice, Third Edition G.B. Wetherill and D.W. Brown Statistical Theory, Fourth Edition B.W. Lindgren Statistics for Accountants S. Letchford Statistics for Epidemiology N.P. Jewell Statistics for Technology — A Course in Applied Statistics, Third Edition C. Chatfield Statistics in Engineering — A Practical Approach A.V. Metcalfe Statistics in Research and Development, Second Edition R. Caulcutt Survival Analysis Using S—Analysis of Time-to-Event Data M. Tableman and J.S. Kim The Theory of Linear Models B. Jørgensen Time Series Analysis H. Madsen C9500_FM.indd 3 4/8/08 9:54:27 AM
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- 11. Texts in Statistical Science Annette J. Dobson University of Queensland Herston, Australia Adrian G. Barnett Queensland University of Technology Kelvin Grove, Australia An Introduction to Generalized Linear Models Third Edition C9500_FM.indd 5 4/8/08 9:54:28 AM
- 12. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-950-2 (Softcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Dobson, Annette J., 1945- An introduction to generalized linear models / Annette J. Dobson and Adrian G. Barnett. -- 3rd ed. p. cm. -- (Chapman & Hall/CRC texts in statistical science series ; 77) Includes bibliographical references and index. ISBN 978-1-58488-950-2 (alk. paper) 1. Linear models (Statistics) I. Barnett, Adrian G. II. Title. III. Series. QA276.D589 2008 519.5--dc22 2008013034 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C9500_FM.indd 6 4/8/08 9:54:28 AM
- 13. Contents Preface 1 Introduction 1 1.1 Background 1 1.2 Scope 1 1.3 Notation 5 1.4 Distributions related to the Normal distribution 7 1.5 Quadratic forms 11 1.6 Estimation 12 1.7 Exercises 15 2 Model Fitting 19 2.1 Introduction 19 2.2 Examples 19 2.3 Some principles of statistical modelling 32 2.4 Notation and coding for explanatory variables 37 2.5 Exercises 40 3 Exponential Family and Generalized Linear Models 45 3.1 Introduction 45 3.2 Exponential family of distributions 46 3.3 Properties of distributions in the exponential family 48 3.4 Generalized linear models 51 3.5 Examples 52 3.6 Exercises 55 4 Estimation 59 4.1 Introduction 59 4.2 Example: Failure times for pressure vessels 59 4.3 Maximum likelihood estimation 64 4.4 Poisson regression example 66 4.5 Exercises 69 5 Inference 73 5.1 Introduction 73 5.2 Sampling distribution for score statistics 74
- 14. 5.3 Taylor series approximations 76 5.4 Sampling distribution for MLEs 77 5.5 Log-likelihood ratio statistic 79 5.6 Sampling distribution for the deviance 80 5.7 Hypothesis testing 85 5.8 Exercises 87 6 Normal Linear Models 89 6.1 Introduction 89 6.2 Basic results 89 6.3 Multiple linear regression 95 6.4 Analysis of variance 102 6.5 Analysis of covariance 114 6.6 General linear models 117 6.7 Exercises 118 7 Binary Variables and Logistic Regression 123 7.1 Probability distributions 123 7.2 Generalized linear models 124 7.3 Dose response models 124 7.4 General logistic regression model 131 7.5 Goodness of fit statistics 135 7.6 Residuals 138 7.7 Other diagnostics 139 7.8 Example: Senility and WAIS 140 7.9 Exercises 143 8 Nominal and Ordinal Logistic Regression 149 8.1 Introduction 149 8.2 Multinomial distribution 149 8.3 Nominal logistic regression 151 8.4 Ordinal logistic regression 157 8.5 General comments 162 8.6 Exercises 163 9 Poisson Regression and Log-Linear Models 165 9.1 Introduction 165 9.2 Poisson regression 166 9.3 Examples of contingency tables 171 9.4 Probability models for contingency tables 175 9.5 Log-linear models 177 9.6 Inference for log-linear models 178 9.7 Numerical examples 179 9.8 Remarks 183 9.9 Exercises 183
- 15. 10 Survival Analysis 187 10.1 Introduction 187 10.2 Survivor functions and hazard functions 189 10.3 Empirical survivor function 193 10.4 Estimation 195 10.5 Inference 198 10.6 Model checking 199 10.7 Example: Remission times 201 10.8 Exercises 202 11 Clustered and Longitudinal Data 207 11.1 Introduction 207 11.2 Example: Recovery from stroke 209 11.3 Repeated measures models for Normal data 213 11.4 Repeated measures models for non-Normal data 218 11.5 Multilevel models 219 11.6 Stroke example continued 222 11.7 Comments 224 11.8 Exercises 225 12 Bayesian Analysis 229 12.1 Frequentist and Bayesian paradigms 229 12.2 Priors 233 12.3 Distributions and hierarchies in Bayesian analysis 238 12.4 WinBUGS software for Bayesian analysis 238 12.5 Exercises 241 13 Markov Chain Monte Carlo Methods 243 13.1 Why standard inference fails 243 13.2 Monte Carlo integration 243 13.3 Markov chains 245 13.4 Bayesian inference 255 13.5 Diagnostics of chain convergence 256 13.6 Bayesian model fit: the DIC 260 13.7 Exercises 262 14 Example Bayesian Analyses 267 14.1 Introduction 267 14.2 Binary variables and logistic regression 267 14.3 Nominal logistic regression 271 14.4 Latent variable model 272 14.5 Survival analysis 275 14.6 Random effects 277 14.7 Longitudinal data analysis 279 14.8 Some practical tips for WinBUGS 286
- 16. 14.9 Exercises 288 Appendix 291 Software 293 References 295 Index 303
- 17. Preface The original purpose of the book was to present a unified theoretical and conceptual framework for statistical modelling in a way that was accessible to undergraduate students and researchers in other fields. The second edition was expanded to include nominal and ordinal logistic regression, survival analysis and analysis of longitudinal and clustered data. It relied more on numerical methods, visualizing numerical optimization and graphical methods for exploratory data analysis and checking model fit. These features have been extended further in this new edition. The third edition contains three new chapters on Bayesian analysis. The fundamentals of Bayesian theory were written long before the development of classical theory but practical Bayesian analysis has only recently become avail- able. This availability is mostly thanks to Markov chain Monte Carlo methods which are introduced in Chapter 13. The increased availability of Bayesian analysis means that more people with a classical knowledge of statistics are trying Bayesian methods for generalized linear models. Bayesian analysis offers significant advantages over classical methods because of the ability formally to incorporate prior information, greater flexibility and an ability to solve complex problems. This edition has also been updated with Stata and R code, which should help the practical application of generalized linear models. The chapters on Bayesian analyses contain R and WinBUGS code. The data sets and outline solutions of the exercises are available on the publisher’s website: http://www.crcpress.com/e products/downloads/. We are grateful to colleagues and students at the Universities of Queens- land and Newcastle, Australia, and those taking postgraduate courses through the Biostatistics Collaboration of Australia for their helpful suggestions and comments about the material. Annette Dobson and Adrian Barnett Brisbane
- 19. CHAPTER 1 Introduction 1.1 Background This book is designed to introduce the reader to generalized linear models; these provide a unifying framework for many commonly used statistical tech- niques. They also illustrate the ideas of statistical modelling. The reader is assumed to have some familiarity with classical statistical principles and methods. In particular, understanding the concepts of estima- tion, sampling distributions and hypothesis testing is necessary. Experience in the use of t-tests, analysis of variance, simple linear regression and chi-squared tests of independence for two-dimensional contingency tables is assumed. In addition, some knowledge of matrix algebra and calculus is required. The reader will find it necessary to have access to statistical computing facilities. Many statistical programs, languages or packages can now perform the analyses discussed in this book. Often, however, they do so with a different program or procedure for each type of analysis so that the unifying structure is not apparent. Some programs or languages which have procedures consistent with the approach used in this book are Stata, R, S-PLUS, SAS and Genstat. For Chapters 13 to 14 programs to conduct Markov chain Monte Carlo methods are needed and WinBUGS has been used here. This list is not comprehensive as appropriate modules are continually being added to other programs. In addition, anyone working through this book may find it helpful to be able to use mathematical software that can perform matrix algebra, differentiation and iterative calculations. 1.2 Scope The statistical methods considered in this book all involve the analysis of re- lationships between measurements made on groups of subjects or objects. For example, the measurements might be the heights or weights and the ages of boys and girls, or the yield of plants under various growing conditions. We use the terms response, outcome or dependent variable for measurements that are free to vary in response to other variables called explanatory vari- ables or predictor variables or independent variables—although this last term can sometimes be misleading. Responses are regarded as random variables. Explanatory variables are usually treated as though they are non-
- 20. 2 INTRODUCTION random measurements or observations; for example, they may be fixed by the experimental design. Responses and explanatory variables are measured on one of the following scales. 1. Nominal classifications: e.g., red, green, blue; yes, no, do not know, not applicable. In particular, for binary, dichotomous or binomial variables there are only two categories: male, female; dead, alive; smooth leaves, serrated leaves. If there are more than two categories the variable is called polychotomous, polytomous or multinomial. 2. Ordinal classifications in which there is some natural order or ranking be- tween the categories: e.g., young, middle aged, old; diastolic blood pressures grouped as ≤ 70, 71–90, 91–110, 111–130, ≥ 131 mmHg. 3. Continuous measurements where observations may, at least in theory, fall anywhere on a continuum: e.g., weight, length or time. This scale includes both interval scale and ratio scale measurements—the latter have a well- defined zero. A particular example of a continuous measurement is the time until a specific event occurs, such as the failure of an electronic component; the length of time from a known starting point is called the failure time. Nominal and ordinal data are sometimes called categorical or discrete variables and the numbers of observations, counts or frequencies in each category are usually recorded. For continuous data the individual measure- ments are recorded. The term quantitative is often used for a variable mea- sured on a continuous scale and the term qualitative for nominal and some- times for ordinal measurements. A qualitative, explanatory variable is called a factor and its categories are called the levels for the factor. A quantitative explanatory variable is sometimes called a covariate. Methods of statistical analysis depend on the measurement scales of the response and explanatory variables. This book is mainly concerned with those statistical methods which are relevant when there is just one response variable although there will usually be several explanatory variables. The responses measured on different subjects are usually assumed to be statistically independent random variables although this requirement is dropped in Chapter 11, which is about correlated data, and in subsequent chapters. Table 1.1 shows the main methods of statistical analysis for various combinations of response and explanatory variables and the chapters in which these are described.The last three chapters are devoted to Bayesian methods which substantially extend these analyses. The present chapter summarizes some of the statistical theory used through- out the book. Chapters 2 through 5 cover the theoretical framework that is common to the subsequent chapters. Later chapters focus on methods for analyzing particular kinds of data. Chapter 2 develops the main ideas of classical or frequentist statistical mod- elling. The modelling process involves four steps:
- 21. SCOPE 3 Table 1.1 Major methods of statistical analysis for response and explanatory vari- ables measured on various scales and chapter references for this book. Extensions of these methods from a Bayesian perspective are illustrated in Chapters 12–14. Response (chapter) Explanatory variables Methods Continuous Binary t-test (Chapter 6) Nominal, >2 categories Analysis of variance Ordinal Analysis of variance Continuous Multiple regression Nominal & some Analysis of continuous covariance Categorical & continuous Multiple regression Binary Categorical Contingency tables (Chapter 7) Logistic regression Continuous Logistic, probit & other dose-response models Categorical & continuous Logistic regression Nominal with Nominal Contingency tables >2 categories (Chapters 8 & 9) Categorical & continuous Nominal logistic regression Ordinal Categorical & continuous Ordinal logistic (Chapter 8) regression Counts Categorical Log-linear models (Chapter 9) Categorical & continuous Poisson regression Failure times Categorical & continuous Survival analysis (Chapter 10) (parametric) Correlated Categorical & continuous Generalized responses estimating equations (Chapter 11) Multilevel models
- 22. 4 INTRODUCTION 1. Specifying models in two parts: equations linking the response and explana- tory variables, and the probability distribution of the response variable. 2. Estimating fixed but unknown parameters used in the models. 3. Checking how well the models fit the actual data. 4. Making inferences; for example, calculating confidence intervals and testing hypotheses about the parameters. The next three chapters provide the theoretical background. Chapter 3 is about the exponential family of distributions, which includes the Nor- mal, Poisson and Binomial distributions. It also covers generalized linear models (as defined by Nelder and Wedderburn 1972). Linear regression and many other models are special cases of generalized linear models. In Chapter 4 methods of classical estimation and model fitting are described. Chapter 5 outlines frequentist methods of statistical inference for general- ized linear models. Most of these methods are based on how well a model describes the set of data. For example, hypothesis testing is carried out by first specifying alternative models (one corresponding to the null hypothesis and the other to a more general hypothesis). Then test statistics are calculated which measure the “goodness of fit” of each model and these are compared. Typically the model corresponding to the null hypothesis is simpler, so if it fits the data about as well as a more complex model it is usually preferred on the grounds of parsimony (i.e., we retain the null hypothesis). Chapter 6 is about multiple linear regression and analysis of vari- ance (ANOVA). Regression is the standard method for relating a continuous response variable to several continuous explanatory (or predictor) variables. ANOVA is used for a continuous response variable and categorical or qualita- tive explanatory variables (factors). Analysis of covariance (ANCOVA) is used when at least one of the explanatory variables is continuous. Nowadays it is common to use the same computational tools for all such situations. The terms multiple regression or general linear model are used to cover the range of methods for analyzing one continuous response variable and multiple explanatory variables. Chapter 7 is about methods for analyzing binary response data. The most common one is logistic regression which is used to model relationships be- tween the response variable and several explanatory variables which may be categorical or continuous. Methods for relating the response to a single con- tinuous variable, the dose, are also considered; these include probit anal- ysis which was originally developed for analyzing dose-response data from bioassays. Logistic regression has been generalized to include responses with more than two nominal categories (nominal, multinomial, polytomous or polychotomous logistic regression) or ordinal categories (ordinal logis- tic regression). These methods are discussed in Chapter 8. Chapter 9 concerns count data. The counts may be frequencies displayed in a contingency table or numbers of events, such as traffic accidents, which need to be analyzed in relation to some “exposure” variable such as the num-
- 23. NOTATION 5 ber of motor vehicles registered or the distances travelled by the drivers. Mod- elling methods are based on assuming that the distribution of counts can be described by the Poisson distribution, at least approximately. These methods include Poisson regression and log-linear models. Survival analysis is the usual term for methods of analyzing failure time data. The parametric methods described in Chapter 10 fit into the framework of generalized linear models although the probability distribution assumed for the failure times may not belong to the exponential family. Generalized linear models have been extended to situations where the re- sponses are correlated rather than independent random variables. This may occur, for instance, if they are repeated measurements on the same sub- ject or measurements on a group of related subjects obtained, for example, from clustered sampling. The method of generalized estimating equa- tions (GEEs) has been developed for analyzing such data using techniques analogous to those for generalized linear models. This method is outlined in Chapter 11 together with a different approach to correlated data, namely multilevel modelling in which some parameters are treated as random vari- ables rather than fixed but unknown constants. Multilevel modelling involves both fixed and random effects (mixed models) and relates more closely to the Bayesian approach to statistical analysis. The main concepts and methods of Bayesian analysis are introduced in Chapter 12. In this chapter the relationships between classical or frequentist methods and Bayesian methods are outlined. In addition the software Win- BUGS which is used to fit Bayesian models is introduced. Bayesian models are usually fitted using computer-intensive methods based on Markov chains simulated using techniques based on random numbers. These methods are described in Chapter 13. This chapter uses some examples from earlier chapters to illustrate the mechanics of Markov chain Monte Carlo (MCMC) calculations and to demonstrate how the results allow much richer statistical inferences than are possible using classical methods. Chapter 14 comprises several examples, introduced in earlier chapters, which are reworked using Bayesian analysis. These examples are used to illustrate both conceptual issues and practical approaches to estimation, model fitting and model comparisons using WinBUGS. Further examples of generalized linear models are discussed in the books by McCullagh and Nelder (1989), Aitkin et al. (2005) and Myers et al. (2001). Also there are many books about specific generalized linear models such as Hosmer and Lemeshow (2000), Agresti (1990, 1996), Collett (2003a, 2003b), Diggle et al. (2002) and Goldstein (2003). 1.3 Notation Generally we follow the convention of denoting random variables by upper case italic letters and observed values by the corresponding lower case letters. For example, the observations y1, y2, ..., yn are regarded as realizations of the
- 24. 6 INTRODUCTION random variables Y1, Y2, . . . , Yn. Greek letters are used to denote parameters and the corresponding lower case roman letters are used to denote estimators and estimates; occasionally the symbol b is used for estimators or estimates. For example, the parameter β is estimated by b β or b. Sometimes these con- ventions are not strictly adhered to, either to avoid excessive notation in cases where the meaning should be apparent from the context, or when there is a strong tradition of alternative notation (e.g., e or ε for random error terms). Vectors and matrices, whether random or not, are denoted by bold face lower and upper case letters, respectively. Thus, y represents a vector of observations y1 . . . yn or a vector of random variables Y1 . . . Yn , β denotes a vector of parameters and X is a matrix. The superscript T is used for a matrix transpose or when a column vector is written as a row, e.g., y = [Y1, . . . , Yn] T . The probability density function of a continuous random variable Y (or the probability mass function if Y is discrete) is referred to simply as a proba- bility distribution and denoted by f(y; θ) where θ represents the parameters of the distribution. We use dot (·) subscripts for summation and bars (− ) for means; thus, y = 1 N N X i=1 yi = 1 N y · . The expected value and variance of a random variable Y are denoted by E(Y ) and var(Y ), respectively. Suppose random variables Y1, . . . , YN are in- dependent with E(Yi) = µi and var(Yi) = σ2 i for i = 1, . . . , n. Let the random variable W be a linear combination of the Yi’s W = a1Y1 + a2Y2 + . . . + anYn, (1.1) where the ai’s are constants. Then the expected value of W is E(W) = a1µ1 + a2µ2 + . . . + anµn (1.2) and its variance is var(W) = a2 1σ2 1 + a2 2σ2 2 + . . . + a2 nσ2 n. (1.3)
- 25. DISTRIBUTIONS RELATED TO THE NORMAL DISTRIBUTION 7 1.4 Distributions related to the Normal distribution The sampling distributions of many of the estimators and test statistics used in this book depend on the Normal distribution. They do so either directly be- cause they are derived from Normally distributed random variables or asymp- totically, via the Central Limit Theorem for large samples. In this section we give definitions and notation for these distributions and summarize the relationships between them. The exercises at the end of the chapter provide practice in using these results which are employed extensively in subsequent chapters. 1.4.1 Normal distributions 1. If the random variable Y has the Normal distribution with mean µ and variance σ2 , its probability density function is f(y; µ, σ2 ) = 1 √ 2πσ2 exp " − 1 2 y − µ σ 2 # . We denote this by Y ∼ N(µ, σ2 ). 2. The Normal distribution with µ = 0 and σ2 = 1, Y ∼ N(0, 1), is called the standard Normal distribution. 3. Let Y1, . . . , Yn denote Normally distributed random variables with Yi ∼ N(µi, σ2 i ) for i = 1, . . . , n and let the covariance of Yi and Yj be denoted by cov(Yi, Yj) = ρijσiσj, where ρij is the correlation coefficient for Yi and Yj. Then the joint distri- bution of the Yi’s is the multivariate Normal distribution with mean vector µ = [µ1, . . . , µn]T and variance-covariance matrix V with diagonal elements σ2 i and non-diagonal elements ρijσiσj for i 6= j. We write this as y ∼ MVN(µ, V), where y = [Y1, . . . , Yn] T . 4. Suppose the random variables Y1, . . . , Yn are independent and Normally distributed with the distributions Yi ∼ N(µi, σ2 i ) for i = 1, . . . , n. If W = a1Y1 + a2Y2 + . . . + anYn, where the ai’s are constants, then W is also Normally distributed, so that W = n X i=1 aiYi ∼ N n X i=1 aiµi, n X i=1 a2 i σ2 i ! by equations (1.2) and (1.3). 1.4.2 Chi-squared distribution 1. The central chi-squared distribution with n degrees of freedom is de- fined as the sum of squares of n independent random variables Z1, . . . , Zn
- 26. Discovering Diverse Content Through Random Scribd Documents
- 27. Plate VIII The fringe of the robe from the Tomb of Charlemagne. To face p. 291. Plate VII shows one medallion of a piece of silk found on the body of Charlemagne when the grave was opened in the time of the present German Emperor. It is certainly not of Charlemagne’s time. But it seems a fairly safe guess to suppose that his well-known regard for his favourite beast Abulabaz, who died only four years before him, caused his son to have the body wrapped in one of the robes decorated with elephants which we know that he possessed; and that either in the year 1000, when Otho III opened the tomb, or in 1166, under Barbarossa, when Charlemagne was canonized, this piece of silk replaced the decayed robe originally buried there. We know of the two elephant-robes referred to from Anastasius[270], who gives an enormous list of the art works in gold and silver and silk and cloth of gold which were wrought for Leo III, Charlemagne’s contemporary. One item is “two robes of Syrian purple, with borders of cloth of gold wrought with elephants”. These robes Leo gave to Charlemagne. We can all but give the exact date of this remarkable Byzantine beast. The inscription breaks off exactly where the date came. The Greek inscription worked in the stuff (Plate VIII) sets forth that it was made “under Michael the great chamberlain and controller of the privy purse of the emperor, when Peter was the manager of
- 28. Zeuxippos”, i. e. the Byzantine court factory in Negropont. Then comes the tantalizing Indictionos (? B), and the date is lost. Dreger, in his Europäische Weberei und Stickerei,[271] gives some early examples of elephants in art. His Figure 37b shows an archaic silver relief of an elephant with a castle containing armed men. His Figure 37a shows a silk stuff of the seventh or eighth century, of Asiatic manufacture, with circular medallions containing elephants, griffins and winged horses, hippogryffs; and he remarks that “the elephant is one of the most holy beasts of Buddhism”. This silk stuff is shown in our Plate IX from a photograph of the original. A comparison of these elephants with the elephant shown in Plate VIII makes it fairly clear that the Charlemagne stuff is later than the other, while in all of the details of the beast itself, ears, three toes, eye, trunk, they are exactly the same. Each has a tree behind the elephant; but while the Charlemagne tree is a piece of stiff conventional work, the other is a natural tree with leaves and fruit, much resembling the vegetable ornamentation of some early Egyptian stuffs. Another feature pointing in the same direction is the thirty-two conventional patterns on the circular enclosing border. These in the earlier piece are twenty-eight plain disks. There is an example of sculptured elephants something like this one, but much more like the real beast, especially about the feet. The elephants are the legs of the ivory chair[272] of Urso, at Canossa; he was Bishop of Bari and Canossa 1078-89. Something should be said about the language spoken by the people of France and Germany in the times with which we are dealing, the reference to a rustic tongue being not infrequent. In the Council convened by Charlemagne at Tours in the year 813, equally representing Eastern France and Western France, Austrasia and Neustria, Germany and the Galliae, the bishops in the Transalpine Empire were enjoined to be diligent in preaching, and to take care that their discourses should be rendered either into Romana Rustica or into Theotisc or Deutsch, that all might understand. It may be of interest to give the earliest specimens we
- 29. have of these native languages. Philologically, these examples are of the very highest importance. Plate IX Silk stuff of the seventh or eighth century. To face p. 292.
- 30. In 841, after the dreadful battle of Fontenai near Vézelay in Burgundy, where Charles-le-Chauve and Louis-le-Germanique combined against their brother Lothar and their nephew Pepin and defeated them, they held a Congress at Strassburg to confirm their alliance. Louis and Charles each made announcement in Latin of the purpose of their agreement, and of their intention to take in public an oath each to other. That done, Louis, as the elder, first took the oath. Being the ruler of the German portion of the empire, he took the oath in the language of the Franks, the Romance tongue, Rustica Romana, in order that the adherents of Charles might hear and understand his undertaking. These were the words of his oath, probably read by a chancellor, for the Latin account[273] says haec se servaturum testatus est:— “Pro Deo amur et pro christian poblo et nostro commun salvament, dist di[274] in avant, in quant Deus savir et podir me dunat[275], si salvarai eo cist meon fradre Karlo, et in aiudha et in cadhuna[276] cosa, si cum om[277] per dreit[278] son fradra salvar dist, in o quid il mi altresi fazet[279]; et ab Ludher nul plaid numquam prindrai, qui meon vol[280] cist meon fradre Karle in damno sit.” Then Charles said the same in the language of the Germans, the Teudisc or Deutsch tongue. The Latin account uses a different phrase here, haec eadem verba testatus est. “In Godes minna ind in thes christianes folches ind unser bedhero gehaltnissi, fon thesemo dage frammordes, so fram so mir Got geuuizci indi mahd furgibit, so haldi thesan minan bruodher, soso man mit rehtu sinan bruodher scal, in thiu, thaz er mig so sama duo; indi mit Ludheren in nohheiniu thing ne gegango, the minan uuillon imo ce scadhen uuerdhen.” The peoples then swore an oath, each in their own, not the other’s, tongue. The Frank people swore in the Romance language: —
- 31. “Si Lodhuvigs sagrament, que son fradre Karlo iurat, conservat, et Karlus meos sendra de suo part non los tanit, si io returnar non l’int pois: ne io ne neuls, cui eo returnar int pois, in nulla aiudha contra Lodhuuuig nun li iv er.” The others then swore in the Teudisc language:— “Oba Karl then eid, then er sinemo bruodher Ludhuuuige gesuor, geleistit, indi Ludhuuuige min herro then er imo gesuor forbrihchit, ob ih inan es iruuenden ne mag: noh ih no thero nohhein, then ih es iruuenden mag, uuidhar Karle imo ce follusti ne uuirdhit.” Plate X The abrenuntiatio diaboli of Archbishop Boniface. To face p. 295. An example of language nearly a hundred years earlier than this is found in the renunciation of the devil and the declaration of belief in God which our own Boniface required of his converts from paganism. The form is found attached to the decrees of a Council[281] held by Boniface, probably in the year 743. It exists in a
- 32. Vatican manuscript (Vat. Palat, nro. 577, fol. 6, 7), which Pertz and other scholars believe to be of contemporary date. The form is of such extreme interest that I have had that part of it which is at the foot of folio 6 photographed, by the kind help of a friend in the Vatican Library; see figure 10, the four lowest lines. This is the form:— “Forsachistu diobolae? Ec forsacho diabolae. End allum diobolgelde? End ec forsacho allum diobolgeldae. End allum dioboles uuercum? End ec forsacho allum dioboles uuercum and uuordum thunaer ende uuoden ende saxnote ende allum them unholdum the hira genotas sint. Gelobistu in Got alamehtigen fadaer? Ec gelobo in Got alamechtigen fadaer. Gelobistu in Crist Godes suno? Ec gelobo in Crist Godes suno. Gelobistu in halogen Gast? Ec gelobo in halogen Gast.” An isolated piece of early “Saxon” is found in one of the letters contained in vol. iii of the Epistolae of the Monumenta Germaniae Historica, the volume containing Epistolae Meròwingici et Karolini Aevi (Berlin, 1892). The letter is No. 146 of the “letters of Boniface and Lull”. It is written by a poor and humble monk to a personage described as reverentissimus atque sanctissimus, who would appear to have had the reputation of not carrying out his purposes. The proverb looks like the eighth century; Brandl thinks that it is pre- Christian. The dialect is probably Northumbrian, varied by a West- Saxon or a German scribe. “I hear of thee that thou proposest to make a journey: I exhort thee not to fail. Do what thou hast begun. Remember the Saxon saying Oft daedlata dôme foreldit Sigisitha gahuem suuyltit thi âna”. That is, Often the tardy man (deed-late) loses glory, some victory; thus he dies solitary.
- 33. The suggested date of the letter is a.d. 757-786. Mention was made on page 57 of the inscriptions which exist on the great shaft of a cross in the churchyard of Bewcastle in Cumberland. These inscriptions are the earliest extant pieces of English prose. They give the names of the King of Mercia, Wulfhere, his queen and her sister, with the date “first year of Ecgfrith King of this realm”, that is, a.d. 670. We have another inscription dated in Ecgfrith’s reign, that, namely, on the dedication stone of the basilica of St. Paul at Jarrow, “in the 15th year of King Ecgfrith and the fourth year of Abbat Ceolfrid”, so that the manner of dating the Bewcastle cross was that usual at the time; the Jarrow inscription is in Latin[282]. Plate XI shows a facsimile of all except the two top lines (which were beyond my reach) of the main inscription on the Bewcastle cross, a copy of which is given in a note on page 57. The runes on Plate XI begin with the gar of Wothgar, the second of the three persons who “set up this slender token of victory in memory of Alchfrith once King and son of Oswy”, the half-brother of King Ecgfrith; mention has been made of him on page 9.
- 34. Plate XI Runes incised on the Bewcastle Cross. To face p. 296.
- 35. Plate XII Runes incised on the Ruthwell Cross. To face p. 297. The earliest pieces of English verse in existence in their original form are found on the Cross at Ruthwell in Dumfriesshire, a monument of equal magnificence with the Bewcastle Cross, and
- 36. probably about fourteen years later. King Ecgfrith was slain by the Picts in 685, and the Angles were never dominant in the south-west of Scotland after his death. Plate XII shows a portion of the many runes on this great monument, which is described at pages 235-254 of my little book on Theodore and Wilfrith. Reading across the top and down the right side the runes are as follows:— Krist wæs on rodi hwethræ ther fusæ fearran kwomu æththilæ til anum ic thæt al bih[eald]. Christ was on the cross, and there hastening from far came they to the noble prince. I that all beh[eld]. Beginning at the top again and reading down the left side, we have:— Mith strelum giwundad alegdun hiæ hinæ limwoerignæ gistoddun him (æt his licæs heafdum). With missiles wounded, they laid Him down limb-weary, they stood at His body’s head.
- 37. Ep. 147. June 26, 800. CHAPTER XVIII Alcuin’s latest days.—His letters mention his ill health.—His appeals for the prayers of friends, and of strangers.—An affectionate letter to Charlemagne.—The death scene. Alcuin’s health began to break in the later part of the year 800, or early in 801. In June, 800, he wrote a letter to Arno of Salzburg which shows that he had in the first days of the month travelled with Karl from Tours to Aix by way of Orleans and Paris, and after a debate with Felix the Adoptionist had returned to Tours. We do not find in this long letter any mention of failing health. Indeed, he overflows with affectionateness, a feeling always displayed in his letters to Arno. “I am sending to your dearness three little gifts; a tent to protect your venerated head from the rain[283], a bed-cover to keep warm your sacred breast, and a glass in which your bread may be dipped at table, that whenever they are used they may bring to your sanctity a recollection of my name.” In this letter he describes his debate with Felix. “I have had a great dispute with the heretic Felix in presence of the lord king and holy fathers. He was obdurate; would recognize the authority of no one who took an opposite view; held himself to be wiser than all in this, that he was more foolish than all. But the divine clemency touched his heart; he confessed that he had of late been carried away by a false opinion; he professed that he held firmly the Catholic faith. We could not see into his mind, and we left the cause to the Judge of secret things. We handed him over to Laidrad [the Bishop of Lyon (798-814)] our dearest son, who is to keep him and see whether it is true that he believes, and whether he will write letters condemning the heresy which he has preached. The king had intended to send him to Archbishop Riculf [of Maintz] to be kept and chastised; and his presbyter, who is worse than his master, was to
- 38. Ep. 189. May, 802. Ep. 189. May, 802. Ep. 193. be sent to you and your providence. But now that they say they are converted to the Catholic faith, they have been handed over to Laidrad, who is to test their sincerity.” On May 24, 801, Alcuin received a letter from Arno, Archbishop of Salzburg, at nine o’clock in the morning, and the messenger told him he must leave again at three in the afternoon. In the course of the six hours he dictated fourteen paragraphs in reply. One of these concerned his health. “My Candidus has been able to tell you all about my weakness. It is therefore superfluous to write on the subject, except to say that all bodily fitness has left me, and pleasures of the world have fled far away.” The interesting remark in this letter that the messenger from Salzburg to Tours, a distance of some six hundred miles, must go back in six hours is not the only interesting detail. We learn, also, that many letters were lost in the difficulties of the journey. The eyes of the Sassenach of to-day, who rides some forty miles in a Scottish mail-cart in Sutherland, and sees the letters shied out into kailyards and steadings, are opened to the possibilities of loss in primitive methods of letter-carrying. The admirable arrangements of the early Roman empire, for conveyance of men and things, had been thrown into chaos long before Alcuin’s time, and special messengers, or “runners”, were used by important people for the transmission of letters. “To Arno. My devotedness is greatly grieved by the unfaithfulness of those whom I have trusted with letters to you. Last year I sent to you on your return from Italy two letters, and I also sent to you other two to meet you on your arrival at the palace [Aachen]. I do not know that any of them reached your presence.” Alcuin wrote to the Emperor Charles in 802, or possibly in 803, begging that he might be allowed to stay quietly at St. Martin’s, Tours. “I am so very weak in body that I am unequal to any more travelling or labour. To speak truth, all the fitness and strength of my body has left me; it has gone, and day by
- 39. Ep. 194. Ep. 196. a.d. 802- 3. Ep. 198. a.d. 802- 3. Ep. 230. Ep. 271. a.d. 804. day will be further away; I fear it will never come back to me in this world.” Again, writing to Arno in 802 or 803, he tells him how he longs to see him at St. Martin’s, “not for the sake of your black hair, but for your most sweet eyes and lovable talk.” Though bidden to the palace, where he would have met him, his poor little body was too weak for the journey: he could not go. In another letter to Arno he writes: “I have been summoned to my lord David [Charlemagne], but my bodily weakness prevented my going: the will of God detained me.” We have his letter of excuse to the emperor. He begins with the simile of the aged soldier, unable not only to bear the weight of armour, but even to support his own body. Then he proceeds: “To speak simply, let not the mind of my lord be inflamed against me for my delay; I am not strong enough to come. A more favourable opportunity may occur.” He became more than ever pressing in his entreaties that his friends would pray for him. In seeking for the prayers of others we find him turning to a part of England of which we do not appear to have any other mention in his letters, namely, Norfolk and Suffolk, called then the dioceses of Elmham and Dunwich (see page 159). In like manner, and for a like purpose, he wrote to the brethren of Candida Casa, i. e. Whithorn in Galloway, the following letter:— “I pray the unanimity of your piety to have my name in memory. Deign to intercede for my littleness in the church of your most holy father Nynia the bishop. “He shone bright with many virtues, as has recently been related to me by a skilful poem which our faithful disciples the scholars of the church of York have sent. In that poem I have discerned in that which I have read there both the skill of the writer and the holiness of him who wrought the miracles. Wherefore, I pray you, by your holy intercessions to commend me to his prayers, that by the most holy prayers of the same your father, and by the assiduous intercessions of your love, I may receive pardon for my sins, by the mercy of the
- 40. Ep. 170. a.d. 801, early autumn. God Christ, and may come to the communion of saints who have bravely conquered the labours of the world, and have received the crown of perpetual praise. “I send to the body of our holy father Nyniga (sic) a robe of whole silk, that my name may be remembered, and that I may merit to have always the pious intercession both of him and of you. “May Christ’s right hand protect and rule you, brothers.” Here is a very touching letter, which sets clearly before our eyes the dear affectionate old man—old as men then counted age— beaten at last by bodily weakness, while his heart was as loving as ever. It is addressed to “the most longed for lord David, most worthy of all honour.” “Day by day, with hungry intentness of heart, my ears hanging on the words of messengers, I wondered anxiously what they could tell me of my most sweet lord David: when he would come home; when he would return to his own land. At last, though late, the wished-for voice sounded in the ears of my desire: ‘He will soon come. He has already crossed the Alps, he whose presence thou hast desired, O Albinus, with such fervour of mind.’ And then I cried over and over again with tearful voice: ‘O Lord Jesu, why dost thou not give me the wings of an eagle? Why dost thou not grant me the translation of the prophet Abacuc[284] for one day, or even one hour, that I might embrace and kiss the steps of him my dearest one, and—above all that can be loved in this world—see the most clear eyes of my sweetest one, and hear his most joyous words. And why dost thou, mine enemy of fever, oppress me at this inopportune time, and not permit me to have my wonted alacrity of body, so that, though tardily, that might be accomplished which promptly it cannot do.’” The dates and the story of his final illness and his death are found, as we have seen (Ch. II), in the life written about twenty years later by a pupil of Alcuin’s favourite priest Sigulf, and more concisely in the Annals of Pettau, a monastery not far from Salzburg, and therefore likely to be well-informed. Some of the touching facts should be repeated here.
- 41. Early in 804 he was evidently failing. He prayed earnestly that he might die on the day on which the Holy Spirit came upon the Apostles in tongues of fire. All through Lent he was able to move about, night after night, to the several basilicas of saints which were included in the monastery of St. Martin, cleansing himself from his sins with much groaning. He kept the solemnity of the Lord’s Resurrection; but on the night of the Ascension he fell upon his couch, oppressed by languor even unto death, and unable to speak. The Annals of Pettau tell us that this was a paralytic stroke, and that it fell on Thursday, May 8, in the evening, after sunset. On the third day before his death he recovered the power of speech, and with a voice of exultation sang through his favourite antiphon, O clavis David, based upon Isaiah xxii. 22: “The key of the house of David I will lay upon his shoulder; so he shall open and none shall shut, he shall shut and none shall open.” Then he repeated a number of verses from several psalms: “Like as the hart desireth the waterbrooks.” “O how amiable are Thy dwellings, Thou Lord of hosts; blessed are they that dwell in Thy house.” “Unto Thee do I lift up mine eyes.” “One thing I have desired of the Lord.” “Unto Thee, O Lord, will I lift up my soul.” And others of like kind. On the day of Pentecost, matins having been said, at full dawn, just at the hour at which he was wont to enter the church for Mass, the holy soul of Alcuin was released from the flesh. He had prayed months before that he might die on Whit Sunday; on Whit Sunday he died.
- 42. Ep. 186. a.d. 801- 2. APPENDIX A (Page 26) It would appear that when Alcuin was not allowed by Charlemagne to retire to Fulda, as he had wished to do, an impulse of affectionate responsibility brought him to pour himself out in advice and help to those with whom he had hoped to spend his last days. This is his letter to the monks of Fulda. “To the most holy, and by us with all love to be cherished, the brethren of the holy Boniface[285], our father and protector, the humble levite Alchuin wishes eternal beatitude in Christ. “I am mindful of your most sweet love, with which you most benignantly received me long ago with all joy. Greatly as I then was glad in your presence, so greatly is my mind now tortured in your absence, desiring to see you whom it loves, to have present you whom it esteems. Since this is denied to the eyes of the flesh, let love be made perpetual by spiritual presence; love which can come to an end has never been true love. “Let us therefore aim at that which is never to have an end, where is blessed eternity and eternal blessedness. That ye may deserve to attain to this, let no labour affright you, no blandishments of this life keep you back. Let there always burn in your hearts the love of Him that appeared as their companion on the way to the two apostles, who, when He was removed from their carnal eyes, said ‘Did not our hearts burn within us, while He talked with us by the way, and while He opened to us the Scriptures?’ In the writings of the holy fathers let us seek Him whom they, not yet learned in the Scriptures, understood. Now all is open; now He has opened the meaning of Whom it was said ‘Then opened He their understanding, that they might understand the Scriptures.’ Now the gospel truth shines forth
- 43. in all the world; now the enigmas of the prophets are clearer than the sun in the churches of Christ. This light of truth follow ye with your whole soul and understand Christ; in it love Christ, follow Christ; that cleaving to His most sacred footsteps ye may merit to have in His most holy presence life eternal. “Be mindful of the apostolic mandate,[286] ‘My brethren, be ye stedfast, unmovable; always in the work of the Lord; forasmuch as ye know that your labour is not in vain in the Lord.’ Be stedfast in your own place and in the devotion of your purpose. Leave not your most holy father. Stand about his sepulchre, that he may offer your prayers to Almighty God. Desire not the vanities of the world, but love celestial blessings. ‘And,’ as the teacher of the Gentiles says, [287] ‘be not conformed to this world, but be ye transformed by the renewing of your mind.’ It is a base thing for a monk to lose the spiritual warfare and to immerse himself in the affairs of the world. “Let there be no murmurings among you, no hatreds, no envyings, no evil speakings.[288] Judge not one another. Let everything be done in humility and concord, in obedience to those set over you, not to the eye only but from the heart, as in the presence of God. Let your obedience, your love, your humility be known to all, that very many may be taught by your good examples and may advance in the salvation of their souls. “If the venerable father Bouulf,[289] my most loved friend, is unable by reason of his weakness to observe the full hardness of the life by rule, judge ye not him, but obey him from your heart and love him as a father, for he will have to give account of your souls. He labours for you in wanderings and journeyings, that you may live quiet and keep the life by rule and have what is necessary for your bodies. Do you act as very dear sons. Fear God, love God, and have care of your most holy father in your prayers, that he may live in long prosperity with you, and that he with you and you with him may merit to have everlasting life. “Warn, instruct, teach your young men in all holy discipline and Catholic doctrine; that they may be held worthy to stand in the place of you and to send up prayers for you wherever you may remain.
- 44. Warn them about chastity of body, about confession of their sins, about study and manual labour without murmur, and about all things which seem necessary at their age. And let them become subject to their elders and masters in good humility, in most pious religion. And do you who are older afford to them good examples, so that they may learn not from your words only but by the religion of your life. Let them not be given to luxury, not slaves to drink, not despisers, not following empty games; but let them learn to be good servants in the house of God, that by the intercession of holy Boniface their father they may deserve to receive from the God Christ blessing and favour. “And as to myself, I pray you have me in perpetual recollection with yourselves in your holy prayers. For the time is at hand which no man can escape. Let each one prepare himself, that he may appear in the presence of his God not naked but clothed with good practices. “I have sent a pall for the body of the holy Boniface our father, on whose holy intercession for my sins I place great reliance; that I, a sinner, may even merit pardon in that day, when your holiness shall receive the crown of eternal blessedness. “To you, O most holy presbyters, I have sent a little collection of words for the Mass, for use on various days on which any one desires to offer prayers to God, whether in honour of the Holy Trinity[290], or in love of wisdom, or in tears of penitence, or in perfect love, or asking for angelic support, or in address to any one of all the saints; or if any one wish to offer prayers for his own sins, or for any living friend, or for many friends, or for brothers departing this life; or, especially, when one wishes to invoke the intercessions of blessed Mary, mother of God, ever virgin; or when any desires to chant and invoke by his prayers the most pious presence of the most holy Boniface your father. All these things we have been at the pains to send to you by the intuition of love, praying your humility to receive benignantly that which with the fullest love we send you. Let each make of it such use as each pleases; and blame me not in this office of love. Let each be fully persuaded in his own mind[291] and do
- 45. always such things as are pleasing to God and all saints, that with them they may be found worthy to enjoy the perpetual vision of our Lord Jesus Christ. “May the Lord God hearken to your holy blessedness mindful of me in all holy supplication, and deign to grant unto you present felicity and future beatitude, my most loved brothers. “I beg that you make known to me by letter from your blessedness, if this letter reaches you, and what it pleases your prudence to do. What it is mine to do I have done, fulfilling the office of affection in the love and honour of our Lord Jesus Christ.”
- 46. Ep. 10. a.d. 786. APPENDIX B (Page 91) We have the report which the legates George and Theophylact sent to Pope Hadrian on their mission. No reference is made in it to the matter of the Archbishopric of Lichfield. Iaenbricht is still the sole southern archbishop, and Higbert of Lichfield is only bishop. “Your holy prayers favouring us, we set sail with joyous countenance obeying your commands. But the tempter hindered us with a contrary wind. He who stills the waves hearkened unto your deprecatory entreaty, calmed the blue strait, led us across to a safe haven, and brought us to the shore of the English unharmed, but afflicted with many dangers. “We were received first by Iaenberht, Archbishop of the holy church of Dorovernia,[292] whose other name is Cantia, where the holy Augustine rests in the body; dwelling there we gave him the necessary information. “Going on thence, we arrived at the dwelling of Offa, King of the Mercians. With great joy, for reverence of the blessed Peter and honour of your apostolate, he received both us and the messages sent from the highest see. Then Offa the King of the Mercians, and Cynewulf the King of the West Saxons, came together in a council to which we delivered your holy writings; and they forthwith promised that they would correct the vices named.[293] Then, after counsel held with the said kings, pontiffs, and elders of the land, considering that that corner of the world stretches far and wide, we gave permission to Theophylact, the venerable bishop, to go to the King of the Mercians[294] and the parts of Britain. “I for my part, taking with me the companion whom your most excellent King Karl sent with us out of reverence to your apostolate,
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