Lexis Diagrams

Griffith Feeney

June 1996

Lexis Diagrams

• represent relationships between sets ofpersons and events

• are a specialized, highly effective visuallanguage

• like any other language, require study andpractice for effective use

Life Lines

• The history of any individual’s membershipin a population is represented graphically oncoordinate axes by a straight line or brokenstraight line segment which begins at theage and time of this individual’s first entryto the population and ends at the time andage of last exit from the population. Thisline is called a life line.

Life Lines

Time

Age

t u

a

b

Repeated Entry and Exit

• The life line of an individual with a singlespell of membership in a population is asingle straight line segment.

• The life line of an individual who enters andexits the population more than once is abroken straight line segment, each segmentrepresenting a spell of membership.

Life Lines Not (Usually) Drawn

• Except occassionally for illustrativepurposes, life lines for actual persons arenot drawn, only imagined.

• Imagined life lines form the basis forvisualizing sets of persons as describedbelow.

The Age-Time Plane

• This half plane is called the age-time plane.

• The age axis is drawn down because tablesshowing statistics by age show the youngestage in the first row with older ages insubsequent rows.

• Other orientations (age axis going up, ageand time axes reversed) are sometimesfound.

Referencing Points and Lines onthe Age-Time Plane

• Points are referenced by their time and agecoordinates, e.g., (t,a) for the point at time tand age a

• Lines are referenced by their endpoints, e.g.,the line connecting the points (t,a) and (u,b)

• Broken life lines are referenced by theirconstituting segments

Exact Age andAge in Completed Years

• A persons exact age at any given time is thetime elapsed since this person’s birth.

• A person’s age in completed years at anygiven time is the greatest integer less thanthis person’s exact age.

• Age in completed years is also referred to asage at last birthday.

Time and Time Periods

• Time refers to a point in time and isspecified by writing, e.g., 1984.754, withthe appropriate number of places after thedecimal point being determined by context.

• Time period refers to an interval beginningand ending at specified times.

• Time and time period are analogous to exactage and age group.

Dates and Times

• Dates, expressed as day, month, and year,represent time periods, not times.

• For many purposes, time specified within a24 hour period is sufficient precision.

• When translating dates to times, someconvention, e.g., mid-day, is needed.

Decimal Translation of Dates

• To calculate the fraction of a year (365days) represented by a given date, add thedays in months preceding the given monthto the given day minus one half and divideby the number of days in the year.

• Three decimal places (e.g., 1980.123) givesprecision to the day and allows inversion.

Calendar Years and Census Years

• Calendar years begin at 0000 hours onJanuary 1 and end at 2400 hours onDecember 31.

• Census years begin and end at thereference time of a specified census.

Visualizing Sets of Individuals

• Any straight line segment in the age-timeplane represents the set of individualswhose life lines intersect this line.

• This is the Lexis diagram representationprinciple for sets of individuals.

• The principle is general, but practical usesinvolve only a few special cases.

Individuals in population at time t

• The half line perpendicular to the time axisand intersecting it at time t represents theset of all individuals in the population attime t.

• Because this is the set of individuals whoshould be enumerated by a census at time t,it is sometimes called the census set of thepopulation at time t

Individuals in population at time t

Age

Timet

Individuals aged x to x+n at time t

• The straight line connecting the points (t,a)and (t,a+n) represents the set of allindividuals in the population at time t whoare aged a to a+n at time t.

• Partitioning the half line representing totalpopulation at time t into segmentscorresponds to distributing the population attime t by age group.

Individuals aged x to x+n at time t

Age

Time

x

x+n

t

Individuals attaining age xbetween times t and t+n

• The line connecting the points (t,a) and(t+n,a) represents the set of all individualsin a population who reach exact age abetween time t and time t+n.

• In the special case a = 0, this line representsbirths to the population during time t andtime t+n.

Individuals attaining age xbetween times t and t+n

Age

Time

a

t t+m

EXERCISE 1Draw a Lexis diagram indicating:• all persons in a population at (time) 1980;

• persons aged 0-4 at time t;

• persons aged 5-9 at time t+5;

• persons reaching exact age 5 between time tand time t+5;

• how are the last three sets related? (Hint:draw some illustrative life lines.)

Representing Deaths

• A death is represented on the age-time planeby a point whose coordinates (t,a) are thetime t at which the event occurred and theage a of the individual to whom it occurred.

• The point representing a death lies at theend of an individual’s life line.

• Points representing actual deaths, like lifelines, are (usually) imagined, not drawn.

Visualizing Sets of Deaths

• Any two dimensional set in the age-timeplane represents the set of deaths repre-sented by points within it.

• This is the Lexis diagram representationprinciple for deaths.

• The principle applies to any twodimensional set, but most applicationsinvolve a few special cases.

Deaths between times t and t+n

• Deaths to a population between time t andtime t+n are represented by the open-endedrectangle bounded by the time axis and theperpendiculars to the time axis at times tand t+n.

• The same set in the age-time plane mayrepresent events of any other type, e.g.,births, migrations, emigrations.

Deaths between times t and t+n

t t+n

Age

Time

Deaths to individuals aged x to x+nbetween times t and t+m

• The set of deaths occuring in a populationbetween times t and t+m to persons aged xto x+n at the time of death is represented bythe rectangle formed by the points (t,x),(t+m,x), (t,x+n), and (t+m,x+n).

• The disaggregation of deaths by age isrepresented by partitioning the vertical stripat (t,t+m) with lines at age group breaks.

Deaths to individuals aged x to x+nbetween times t and t+m

Age

Time

x

x+n

t t+n

EXERCISE 2Draw a Lexis digram indicating:

• deaths to a population during 1980-1984;

• deaths by 5 year age group through 20-24;

• deaths in the open-ended age group 25+;

• deaths during 1980-1984 to individuals bornduring 1980-1984;

• deaths during 1980-84 to persons 25 andover in 1980.

Deaths of individuals bornbetween times t and t+m

• The set of all deaths to members of the birthcohort born between times t and t+m isrepresented by the open-ended diagonalstrip bounded by the line connecting thepoints (t,0) and (t+m,0) and the diagonallines extending down and to the right fromthe endpoints of this line.

Deaths of individuals bornbetween times t and t+mt t+n

Age

Time

Deaths of individuals aged x tox+n at the time of death

• The set of all deaths of individuals aged x tox+n at the time of death is represented bythe doubly open-ended horizontal stripbounded by the lines that are perpendicularto the age axis and pass through the points xand x+n

Deaths of individuals aged x tox+n at the time of death

Age

Time

x

x+n

Lexis Diagram Translation

• Working with Lexis diagrams requires twoway translation.

• We need to translate verbal descriptions intoLexis diagram representations.

• We need to translate Lexis diagramrepresentations into verbal descriptions.

• There are two general translation methods.

The Method of Extremes

• Applies to sets of persons and deaths.

• Identify the extreme possibilities consistentwith the description; each extreme will be apoint in the age-time plane.

• Draw lines connecting these points to forma line segment (set of individuals) or planefigure such as a rectangle, parallelogram, ortriangle (set of events).

Individuals reaching exact age xbetween times t and t+m

• There are two extremes, reaching age x attime t exactly and reaching age x at timet+m exactly.

• The corresponding points are (t,x) and(t+m,x).

• The line connecting these points is the Lexisdiagram representation.

Individuals reaching exact age xbetween times t and t+m

Age

Time

x

t t+m

Infant deaths during year Y tochildren born during year Y

• There are three extreme possiblitiescorresponding to birth and death at thebeginning and end of the year.

• Born at beginning of year, die immediately

• Born at beginning of year, die at end of year

• Born at end of year, die immediately

• Question: Why not four possibilities?

(Continued)

• The points corresponding to these threeextreme cases are (y denotes beginning ofyear Y):

• point (y,0), point (y,1), and point (y+1,0)

• The desired representation is the triangleformed by the lines connecting these threepoints (draw it!)

Infant deaths during year Y tochildren born during year Y

Age

TimeYear Y

Age 1

EXERCISE 3: Draw a Lexisdiagram representing:

• Deaths during 1980-1984 to persons aged 0-4 at the beginning of 1980.

• Deaths during 1980-1984 to persons aged 0-4 at the beginning of 1980 who were aged0-4 at the time of death.

• Deaths during 1980-1984 to persons aged 0-4 at the beginning of 1980 who were aged5-9 at the time of death.

(continued)

• What is the relationship between these threesets of deaths?

• What is the relationship between these threesets of deaths and the sets of personsrepresented in Exercise 1?

The Method of Intersections

• Does not apply to sets of individuals.

• Identify references to time period, agegroup, and/or birth cohort.

• Draw the corresponding vertical, horizontal,or diagonal strips.

• The Lexis diagram representation is theintersection of these strips.

Method of IntersectionsExample 1

• Draw the representation of the set of deathsoccuring at ages x to x+n to individualsborn between times t and t+m.

• For ease of reference, label points ondiagram and refer to figures formed bygiven points, e.g., parallelogram abcd.

(continued)

x

x+n

a b

c d

t t+m

EXERCISE 4Draw Lexis diagram representations of

• Deaths between times t and t+m to personsaged x to x+n at time t .

• Deaths during year Y to individuals aged xin completed years at the beginning of yearY that occur to individuals aged x+1 incompleted years at the time of death.

Generalization 1:Other Classes of Events

• Everything said thus far about representingsets of deaths applies to any other class ofevents, births, migrations, marriages,divorces, IUD insertions, and so on.

• The point representing any event must lieon the life line of the individual whoexperiences the event.

• The class of events must be identified.

Generalization 2:Other Duration Variables

• Age is defined as time elapsed since birth.

• Duration variables measure time elapsedsince occurence of a specified event, such asfirst marriage, birth of a child, or insertionof an IUD.

• The age axis in the Lexis diagram may bereplaced by an axis representing anyduration variable.

Example 1Ever Married Women

• Consider the subpopulation of a givenpopulation that consists of ever marriedwomen.

• Women enter this population by marryingfor the first time and leave it by dying.

• Lexis diagrams for this population mayshow either age or duration of marriage asthe non-time axis.

Example 2Parity One Women

• Consider the subpopulation of a givenpopulation that consists of women withexactly one child.

• Women enter this population by having afirst birth and leave it either by having asecond birth or by dying.

• Lexis diagrams may show either age oropen birth interval as the non-time axis.

The End